Computers as embodied mathematics and logic:

Implications for computer applications, HCI and AI


 by Kristo Ivanov, prof.em., UmeŒ University

Dec 2015, rev. 2017, Research review v200103-1605



Introduction and literature
Intuitionism and formalism
Rejected parts of Brouwer's dissertation
Life, art, and mysticism
Commentary on "Life, art, and mysticism"
Symptoms: ontology of logic
Symptoms: syntax, semantics, pragmatics
Buddhist models of the mind
Analytical psychological approach





In the search for an understanding of the "essence" of computation in order to better grasp the intellectual impact of information technology I decided to look into controversies about the foundations of mathematics and logic as bases of formalisms that are embodied in computers, such as boolean algebra and binary arithmetic for digital electronics, algorithm theory and numerical analysis, analytic geometry for graphics, and symbolic logic for software development. In an age of computer graphics, acoustics, and haptic technology it is easy to get a perception that knowledge of meaning of computation and of computers, not to mention its ethical implications if any, is less relevant for design of new equipment or use of what is already available. The meaning of "use" itself is obfuscated by the seeing computers as tools instead of theory-laden instruments, as well as by the availability of techniques of human-computer interaction - HCI - and of so called artificial intelligence - AI, mediated through HCI. The matter is further complicated by the submersion of computation into continuously arising neologisms and terms such as virtual reality or digital materiality without a clarification, for instance, of whether reality is material and digitality is mathematical.


Resuming: the main message of this article is that only if we understand what we are doing with the basic formalisms of mathematics and related logic, geometry and such, we will understand what we are doing with computers. My writing is also motivated by my purpose of facilitating the further investigation of systemically related issues by potential readers. In order to save space and simplify the editing of this work, this is done by means of rich bibliographical references indicated in the links associated to the underlines of key words, available only for those who need them while reading the text on a computer that is connected to the Internet. For further details about the style of the style of my writing this text I wish to refer at once to the last paragraph of the conclusion of the whole essay.


Before going further: as a preliminary didactic advice on how to read the text that follows it is to be noted that it is not written in a narrow linear logical sequence with the purpose of supporting one single thesis, motivating extremely logical readers to stop reading this text exclamating: "The text is too long - what do you want to say?" Logic itself together with mathematics is here called into question. Nobody would dare to ask the same question of, say, a criticism of quantum physics unless the critic criticizes a detail in a logical-mathematical edifice that for the rest is hoped to be left intact supporting the old established mainstream thesis.


On several occasions I witnessed the "mystical" fascination experienced in interaction with computers: it appears in "religious" wars about which is the best computer brand or programming language, but most clearly in computer addiction including mobile phones and and in games, and unfortunately less clearly in misjudging benefits if not even costs of computers such as in business, government and private life. The latter is partly portrayed in the "productivity paradox", not to mention the recurring myth of AI, i.e. "artificial intelligence" taking over humanity and the whole world, notwithstanding opportunely ignored or misunderstood research like the reported in The Design of Inquiring Systems (more about it below.) The only courageous direct hint of critique I met in my searches had been Clifford Truesdell's in his An Idiot's Fugitive Essays on Science (1984), especially its essays on The scholar, a species threatened by the professions and the apparently controversial The computer: ruin of science and threat to mankind. I had also read about a famous controversy on the foundations of mathematics and its relation to logic. It attracted my attention since mathematics and logic as embodied in computers have been often if not always considered as the most reliable fields of scientific method, human knowledge and their philosophical guides. I also was puzzled by the fact that such foundational controversies were mostly ignored in today's computing and information science, as if the controversies had been solved. It turns out that probably they were eventually ignored because of "more important" military and industrial-commercial perceptions of a trivial and opportune "good enough".


My earlier, naively ambitious attempts in this vague research direction recall the "remember not the sins of my youth" but can have some general interest for their references. They may be surveyed in old papers like Presuppositions of formal methods for development of computer systems (pdf, 1983), Logic and psycho-logic: A Logical-psychological perspective of computer support (pdf, 1990), and more generally in a draft of a research program on Logic and psycho-logic: A logical-psychological perspective of computer support (pdf, 1990). They are also mentioned marginally but significatively in related context such as in my essay Belief and Reason (1993).


In my latest efforts I did study the recurrent name of a key-mathematician involved in the controversies mentioned above: L.E. Jan Brouwer. For this purpose I concentrated upon his "ideological manifesto" in

Jan Brouwer, Intuitionism and Formalism (pdf-format) in the Bulletin of the American Mathematical Society (Vol. 37, No.1, 1999, pp. 55-64 reprinted from Vol 20, No. 2, 1913, pp. 81-96.)

The second and third texts consist of an introduction to Brouwer original book in Dutch and the book itself:

Walter P. Van Stigt, Introduction to Life, Art and Mysticism (1996, pdf)

That is, the book got a for our present secular epoch disconcerting, unfortunate if not outright repelling title that in its English translation appears as

Jan Brouwer, Life, Art and Mysticism (1905, pdf)

Both the last mentioned works were published as articles in the Notre Dame Journal of Formal Logic, (vol. 37, No. 3, Summer 1996.) Because of my need for a better understanding of van Stigt's English translation from the Dutch I checked my readings of this Brouwer's book against its Italian translation, that is,

Jan Brouwer, Vita, Arte e Mistica, (2015), the first published book in a language other than Dutch, with an introduction by Lorenzo Perilli and a very pertinent commenting essay by numerical analyst Paolo Zellini.

The last two main papers in by Brouwer himself, translated into English that I will consider are presented in

Jan Brouwer: "Mathematics, science, and language", (original lecture in German, year 1928) in Paolo Mancosu From Brouwer to Hilbert: The debate on the Foundations of Mathematics in the 1920s. Oxford University Press, (1998), pp. 45-53. Also, more complete, in William Bragg Ewald, ed. From Kant to Hilbert: A source book in the foundations of mathematics. Vol. 2. (1996). Oxford Univ. Press (pp. 1170-1185). Commentaries by Vladimir Tasic in Mathematics and the roots of postmodern thought, pp. 44ff. and notes p. 163 as well as by Mark van Atten in the Stanford Encyclopedia of Philosophy.


Walter P. van Stigt's translation of The rejected parts of Brouwer's dissertation on the foundations of mathematics, in Historia Mathematica, Vol 6, No. 4 (1979).

This historically and theoretically meaningful rejection of parts of what became a famous mathematical dissertation bears meaningful resemblances with the case of another doctoral dissertation of Erik Persson on realistic computing (cf. virtual reality) that I was appointed to be faculty opponent for at Lund university in 2003.


The issue is extremely complex and the more so for a non-mathematician like me who studied mainly engineering maths such as required for electronic engineering. Such a complexity is obvious the simple context of the misuse of mathematics such as in Ralph Abraham's contribution to Mathknow: Mathematics, Applied Science and Real Life (2009), but much more in writings that deal with the controversy in which Brouwer was involved. See especially in the kind of summary offered (in Italian) by Giovanni Sambin, Per una dinamica nei fondamenti (pdf) i.e. "For a dynamics in foundations", 2005 or the list of research subjects in the list of research publications by Richard Tieszen, or in earlier historical times, in Imre Lakatos (ed.) Problems in the Philosophy of Mathematics: Studies in Logic and the Foundations of Mathematics (1967). In the last mentioned work one can see, for example, contributions by Y. Bar-Hillel who was an acknowledged inspiration for defining the concept of data and information in the original school of Swedish Theoretical Analysis of Information Systems by the grand old man of Swedish computer information systems, prof. Bšrje Langefors. The latter's conception of what data, information and systems are or how they should be defined was later questioned by me in The Systems Approach to Design and Inquiring Information Systems, presaging my need of the present ongoing study.


The complexity of the involved issues that, however, also hides some main questions, is also evidenced by whoever seeks orientation in further comments or explanations in the following complementary sources 

Luitzen Egbertus Jan Brouwer in Stanford Encyclopedia of Philosophy


Dirk van Dalen, Mystic, Geometer, and Intuitionist: The Life of L.E.J.Brouwer, (volumes 1 and 2, 2013), or volume 1 (1999) and volume , (2015) as in secondary sources such as in the review by Bonnie Shulman of the volume 1, by Richard Tieszen of its chapter 2 "Mathematics and Mysticism" (pdf) in Philosophia Mathematica (3, Vol. 8, 2000, pp. 217-224, and by I. Grattan-Guinness in Bulletin of The American Mathematical Society (Vol. 36, No. 4, 1999, pp. 529-532). Refs. to more reviews here.

Paolo Zellini, La Ribellione del numero [The rebellion of the number]. Adelphi (1985), esp. pp. 20-30 and 133ff. Summary of Spanish translation here.

Other parallel cross-checking literature that has been consulted is

L.E.J. Brouwer, in Wikipedia


Raymond L. Wilder, "Relativity of standards of mathematical rigor", in Dictionary of the history of ideas, vol. III, pp. 170-177 (on Brouwer esp. p. 176f.), the author having also written the relevant book Mathematics as a cultural system (1981).


Stephen Orr, Brouwer's Mysticism – The Hegelian in all of us

L.E.J. Brouwer, a Mathematician on Self,  Hermitary


Henk Barendregt, Buddhist models of the mind and the common core thesis on mysticism (pdf), an article that is relevant for our studies of the relation of Brouwer's thought to later psychology


Mark van Atten & Robert Tragesser, Mysticism and Mathematics: Brouwer, Gšdel and the common core thesis (pdf, orig. published in W.Deppert and M. Rahnfeld Klarheit in Religionsdingen, Leipziger UniversitŠtsverlag, 2003, pp. 145-160)


Brouwer's Philosophy of Mathematics (pdf). A review article of L. J. Brouwer, Collected Works, North-Holland/American Elsevier. Vol.1 and Vol. 2, 1975-1976) in Erkenntnis 15 (1980), pp. 105-126


J.J. O'Connor & E.F. Robertson, Luitzen Egbertus Jan Brouwer (biography)


Philosophy of Mathematics: Intuitionism. In


Ernst Snapper, The Three Crises in Mathematics: Logicism, Intuitionism and Formalism. (pdf)


L.E.J. Brouwer, Consciousness, philosophy, and mathematics. In Paul Benacerraf & Hilary Putnam (eds.) Philosophy of Mathematics: Selected Readings. New York: Cambridge University Press, 1983 pp 90-96 (Excerpted from 10th International Congress of Philosophy, Amsterdam, 1948. Proceedings I, Fascicule II (Amsterdam: North Holland, 1949), pp. 1243-1249.

L.E.J. Brouwer, Lectures on Intuitionism. Historical Introduction and Fundamental Notions. Source: Brouwer's Cambridge Lectures on Intuitionism (1951). Publ. Cambridge Univ. Press, 1981. (Most of first lecture plus appendix of fragments.)


Miriam Franchella, Brouwer and Nietzsche: Views about Life, Views about Logic. History and Philosophy of Logic, 36:4, 367-391. <>


Robert J. Leonard, Ethics and the Excluded Middle: Karl Menger and Social Science in Interwar Vienna. Isis 89, No. 1 (March 1998), 1-26. (Published by the University of Chicago Press on behalf of the History of Science Society.) The introductory abstract sets the stage: "This account of Menger shows how, on the eve of Hitler's arrival in Vienna, social scientific, mathematical, and political debates there were deeply intertwined."

There are indeed many secondary sources as the above, which attempt to survey and resume both Brouwer's heritage and the controversy at the turn of the past century as represented mainly by two protagonists, Brouwer himself and the famous mathematician David Hilbert. In mentioning these sources I refrain from considering others which to different degrees are more "technical" in the sense of specialized and of difficult understanding by the educated general reader including myself. Some examples are

Edward Nelson, Understanding Intuitionism (pdf), at Princeton University


Alan Weir, Formalism in the Philosophy of Mathematics, in the Stanford Encyclopedia of Philosophy


Mark van Atten, The Development of Intuitionistic Logic in the Stanford Encyclopedia of Philosophy


Wikipedia's Intuitionism


Rosalie Iemhoff, Intuitionism in the Philosophy of Mathematics in the Stanford Encyclopedia of Philosophy


W.W. Tait, Gšdel's interpretation of intuitionism (pdf)

I only checked such secondary sources to the extent that I could ascertain that nothing in them seemed to contradict my reflections in the present article. It will be essentially grounded in the text found in Brouwer's own aforementioned late (1999) article "Intuitionism and Formalism" and in his aforementioned early (1905) book Life, Art, and Mysticism including its introductions written by Van Stigt and Perilli, plus van Stigt's editing of The rejected parts of Brouwer's dissertation on the foundations of mathematics (1979.)







Brouwer starts his article Intuitionism and Formalism with a concise definition of science as consisting of the isolation of causal sequences supplemented by human activity in view of bringing about desired phenomena, and leading to natural laws that often treat only the mutual relations between the results of counting and measuring with certain degrees of mathematical approximation in experimental situations. The latter qualification of "approximation" was not supposed to hold for the so called exact sciences mathematics and geometry, they were "exact". By the beginning of the 20th century two different schools could be distinguished: intuitionism (largely French) claimed that mathematical exactness exists in the human intellect, while formalism (largely German) claimed that such exactness exist on paper.


In this present essay I intend later to deepen the question of what is meant and what is to be meant by the terms exactness beyond what is to be inferred from my PhD dissertation on the concept of accuracy of information, and further by the terms in intellect vs. on paper. All this considering that ultimately we need to understand what happens today, and why we apparently seek ultimate salvation in the exactness of computers and related communication technology.


To understand the clash between these two currents of intuitionism and formalism we leave for the moment aside the third current of logicism as suggested in the mentioned article by Ernst Snapper, The Three Crises in Mathematics: Logicism, Intuitionism and Formalism. Brouwer starts dwelling in what he calls an old form of Kantian intuitionism that had already been completely abandoned, in which time and space were taken as forms of conception inherent in human reason. The axioms of arithmetic and geometry were "synthetic a priori" judgments, i.e. independent of experience and not capable of analytic demonstration, maintaining their exactness in the world of experience as in the abstract, because of their proof or disproof was simply unthinkable.


Brouwer goes on illustrating the opposed view of formalism that negates the human mind's capacity of containing inherent judgments or exact images of geometric or number axioms. From simply assumed axioms we can deduce other relations between mathematical entities by means of logical reasoning, they having no significance except for the possible efficacy of their projection into nature. Modern philosophers of science often addressed this latter question by studying Galileo Galilei, as Alexandre KoyrŽ does e.g. in ƒtudes d'histoire de la pensŽe philosophique (1961/1971, pp. 348ff.) and in his ƒtudes d'histoire de la pensŽe scientifique (1966/1973, pp. 186ff.). Brouwer recalls that the common attitude to that question is exemplified by what appears to me as irresponsible:

"To the philosopher or to the anthropologist but not to the mathematician, belongs the task of investigating why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well the positive of certain propositions are valid",

which is an intuitionistic tenet of the rejection of the excluded middle. The article continues explaining why the latest developments of mathematics at the turn of the century as well as the discovery of non-euclidean geometry seemed to contradict the Kantian conception of the "a priori" on which relied intuitionism, and was an apparent confirmation of formalism. The apriori question is didactically explained also in certain system philosophies as expounded in West Churchman's The Design of Inquiring Systems (chap. 6). Brouwer, however, does not yet intuit the systems problem that, as we will see below, he touched upon in the context of his pioneer attack on the environmental-ecological problem at the beginning of his earlier book Life, Art and Mysticism. Brouwer concentrates instead upon what is most important for us here: the basis for mathematical and logical thinking that is embodied in computers.


The contradiction of the early Kantian apriori fundaments of intuitionism, did not convince Brouwer about the rightness of formalism but rather about the need of revising intuitionism by abandoning Kant's apriority of space while concentrating on the apriority of time in the human intellect. And this is the intuitionistic heart of the matter that involves the conception of the human intellect in its relation to mathematics and logic. Later on shall we dwell on the question of the structure of human intellect in the light of Carl Jung's psychological conception of intellect, mind or psyche. Here I will refer Brouwer's "neo-intuitionistic" – as he calls it - elaboration of the workings of an intellect that is taken as given:

"This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness." [My italics.]

Brouwer continues with calling this two-oneness (cf. bi-unicity) as an individual's primordial or basal intuition of mathematics in which the connected and the separate, the continuous and the discrete are united. In this present paper of mine I wish to expose how, in the context of analytical psychology, this very same basal intuition of mathematics can be connected to basal conceptions of the human mind: Marie-Louise von Franz dedicates the chapter 5 of her book Number and Time (Northwestern Univ. Press, 1970/1974) to the "The Number Two", including several explicit references to Brouwer (pp. 70n, 71. 87, 96n). It represents psychological research that borrows some additional scientific respectability from the involvement of the renowned physicist Wolfgang Pauli. It indicates the potentiality for future research to establish a bridge to both computers and psychic processes beyond "small unimportant parts of the brain" or of the "head" (cf. later, below), along the lines of the present paper of mine. And this with due regards to the historical roots of false equation Psyche=Mind=Head=Ego, alternatively Skull=Brain=Mind=Psyche, as recalled by James Hillman in his The Myth of Analysis (Northwester Univ. Press, 1972, pp. 153-154, 245.) These consideration may ultimately explain the reasons for the ultimate failure of Brouwer's attempts to keep mathematics ethical and "clean".

The article considered here, on intuitionism and formalism, goes observing that the so called primordial or basal intuition mentioned above is not only of mathematics but also of geometry because since Descartes we can reduce all geometries to arithmetic by means of the calculus of coordinates. Brouwer continues noting that all mathematical sets can be developed out of the basal intuition, and that in the construction of these sets neither the ordinary language nor any symbolic language can have any other role than that of serving as a non-mathematical auxiliary, to assist the mathematical memory or to enable different individuals to build up the same set.


Formalism is instead compelled to presuppose the existence of a world of mathematical objects, and independent of the thinking individual, obeying the laws of classical logic and whose objects may possess with respect to each other the "relation of a set to its elements".


By applying technically the above to the concepts of finite and infinite sets, Brouwer sees that the intuitionist can never feel assured of the exactness of a mathematical theory by such guarantees as the proof of its being non-contradictory, the possibility of defining its concepts by a finite number of words, or the practical certainty that it will never lead to a misunderstanding in human relations.


For our purposes in this present paper, however, it is enough to note that there is the use of the term intellect, intuition, abstraction from emotional content, unification of the continuous and the discrete, individual, independence of thinking, understanding, and human relations. In doing so Brouwer calls into question problems that also belong to philosophy, psychology and sociology. In particular, the remarkable need of abstraction from emotional content is also mentioned in the context of the history of mathematical notations, as represented e.g. in the classical work by Florian Cajori. At Brouwer's time (1913) such areas were not so sharply distinguished and there was no depth psychology that I relate to the analytical psychology of Carl Jung, or of systems thinking that I relate in particular to aforementioned work of West Churchman. They would have stood in contrast to the formalist idea still alive today of mathematical systems and computer systems as sets and relations between elements or components of sets. In particular, Carl Jung started at the time immediately after the turn of the century to elaborate his own psychology that to a great extent incorporated the above mentioned terms used by Brouwer, something that obviously is necessary for evaluating the need and the consequences of trying to "abstract from emotions". Brouwer had implicitly already questioned formalism in science and techno-systems by his early perception of systemic ecological problems, implicitly suggesting the need for a systems theory . In view of the poverty of his contemporaneous psychology he had based his more psychological stands in Life, Art, and Mysticism (published 1905 but with positions maintained until the end of his life) upon Christian, Buddhist and Hindu thought as perceived at that time in the German cultural sphere.


As an important curiosity regarding to abstract from emotional content, as well regarding the difference between intelligence as smartness or wisdom, we may recall what Plato famously writes in his Republic VI (§491e):

"[...] shall we not similarly affirm that the best endowed souls become worse that the others under a bad education? Or do you suppose that great crimes and unmixed wickedness spring from a slight nature and not from a vigorous one corrupted by its nurture, while a weak nature will never be the cause of anything great, either for good or evil?".

The continuation of the present paper will then consist in considering details of the above as they can be found or deduced in Brouwer's Life, Art and Mysticism. This will show that he was a precursor of the elaboration of the concept of systems, and it will allow us to revert from his references to Eastern conceptions of mind or intellect to their Christian and Jungian correspondents. This will also explain the paradox of the success of intuitionistic ideas in algorithm theory, and the fateful hegemony of formalism (David Hilbert, John von Neumann, Alan Turing), allowing for the intrusion of military, commercial and industrial interests in the development of, and applications to computation.

That is, many issues that today are masked by a-philosophical and a-ethical talks and neologisms as virtual reality without understanding reality, digital materiality without understanding neither materia nor digits or number, compositional design without realizing (and therefore ignoring) that it deals with systems thinking, and such. Among other things, as a "bonus" or byproduct, it should also become clear that Brouwer's apparently misogynistic, problematic, and paradoxical attitude to womanhood in the above mentioned work has serious historical antecedents in the Christian tradition (cf. "habet mulier animam"), and at the sources of modern psychology as exemplified by Juan Luis Vives and their further treatment by Carl Jung. It can also be the result of a misunderstanding based on a sort of Jungian introverted thinking-intuition attitude, and an identification of his psyche with elements of the unconscious in the form of the concept of anima (anima obsession, or possession, popularly and roughly illustrated on the Internet). Earlier examples of applications of Jung's psychology in my work can be found in my texts The illusion of communicative information, Political Correctness, and Ethics in Technology.


In particular it will be necessary to deal with the meaning of the intuitionistic hypothesis of "the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness" (my italics.) It could mean that despite its apparent "acrobatic" complexity, mathematical and logical thinking is a sort of minimal thinking. It engages the most simple, primitive apriori level of the so called intellect, sparing psychic energies for acrobatic performance and consequently aggressive, attractive "success" in all fields and for all simpler purposes that do not require other higher functions of the so called intellect. Obviously, however, this does not prevent that attitudes and functions of the psyche in contact with the unconscious, such as introversion, extraversion, thinking, feeling, sensation, and intuition will later interact with the psychologically, intellectually flawed results of applying computer operations or communications. This will in turn influence, for instance, the dynamics of language, the apprehension of graphic images, and aesthetic or ethic dimensions. These issues are observed, for instance, by Richard Stivers in Technology and Magic: The Triumph of the Irrational (e.g. pp. 26 and 63) and accords with the philosophy of technology that was addressed in my essay on Trends in philosophy of technology.


Ultimately the application of "Jungian" analytical psychology may uncover also the reasons for the failure of Brouwer's intentions with intuitionism, non only in replacing formalism but mainly in guaranteeing the connection between mathematics and ethics. On the contrary, the aforementioned work by Zellini suggests how some ideas of intuitionism could be kidnapped for the development and consequent misuse of later algorithm theory.






The meaningful resemblances with the case of another doctoral dissertation on "realistic computing" (cf. virtual reality) that I mentioned above have to do with what is mentioned in The rejected parts of Brouwer's dissertation on the foundations of mathematics. They are about Brouwer pleading passionately with his doctoral (thesis) advisor prof. Korteweg about the latter's suggestion that he remove the "philosophical" first part of his draft.


The Rejected Parts published year 1907 have a very tight connection with the controversial book Life, Art and Mysticism, published a couple of years earlier, in 1905. Its controversiality is evidenced by the comments that are bestowed by van Stigt and partly borrowed from others, as follows: a generally solipsistic philosophy (as I understand it, nearly epistemological solipsism) and a rather pessimistic and misanthropic outlook on and attitude to life, an extreme view on life science and his fellowmen, a passionate involvement in a romantic revolt against intellectualism and industrialization, Brouwer fulminates against all things human and singles out the human intellect as the cause of all evil, pessimistic views, appreciation of mysticism and Eastern philosophies, and low regard for women, pessimism and a mystical attitude to life, bizarre flavour, etc.


I purport to show in the course of this article that most of this pessimism is motivated as a contrast to the extreme optimism in the viewing of science in general and formal sciences in particular by the turn of the century. Most if not all the other judgments above can be clarified by enframing Brouwer's psyche in Carl Jung's typology (his Collected Works, vol. 6) as an "introverted thinking-intuitive type whose attitude to women is indeed an attitude to, or influence from unconscious femininity as represented by the Jungian image of anima, which, by the way, may have been the positive source of his uncommon insights. Brouwer had not recourse to what today is depth psychology since Jung, following Freud, was starting working at it at about the same time. Furthermore, the perceived bizarre mysticism or mystical attitude to life would be explained and incorporated in Jung's psychology and its integration with Eastern philosophies. Most objections to Brouwer's "mysticism" are, for instance, based on Western misunderstandings of mysticism that are unraveled in Harold Coward's chapter on the issue in his book Jung and Eastern Thought (1985, chap. vii, reprinted from Philosophy East and West, vol. 29, 1979.) All this complication was necessary for Brouwer in order for him to affirm, on the basis of his time's philosophical and psychological knowledge, the relevance of a moral value judgment for the practice (rather than "application") of mathematics and what we nowadays might call the use of computers.


The apparently successful projection of mathematical structures into the natural world since Galilei, depending upon what it to be meant by success (cf. earlier reference to KoyrŽ) may be the result of forcing nature into preconceived ways of exploitative use of natural resources, and of prior knowledge. The "success" in form of forced consistency of various areas of modern physics may portray the commonality of interests in its applications. Its maintained consistency by means of a plethora of esoteric "mathematical tools" and "thought experiments" (cf. the mind-blowing context of so called EPR-paradox) may be analog to the forced consistency obtained in the past in Ptolemaic astronomy by means of gradual patchwork-adjustments of computation of orbits of celestian bodies, until the serendipitous advent and posterior rationalization of the Copernican revolution.


In presenting The rejected parts of Brouwer's dissertation, Van Stigt refers to Brouwer's interpretation of causality as essentially mathematical: the ability to link events in the mind, to see sequences and repetition of sequences in time, to link sensations as the immediate source of awareness of time and discreteness. It is the source of man's power to predict the future and interfere in the course of events. This "intellectual or mathematical" way of looking at the world is not only a one-sided concentration and interpretation of reality: by ignoring and wilfully removing aspects which deviate from the expected course of events, man supplements and creates more regularity than exists in nature, he makes the world linear or "one-sided". The regularity observed in nature is due to the nature of the measuring instruments and physical science has value only as weapon, not concerning life. It is clearly inferior and has nothing to do with religion or wisdom. More in detail, in Brouwer's own words:

"Man has the faculty, accompanying all his interactions with nature, of objectifying the world, of seeing in the world causal systems in time. The primordial phenomenon is simply the intuition of time in which repetition of "thing in time and again thing" is possible, but in which (and this is a phenomenon outside mathematics) a sensation can fall apart in component qualities, so that a single moment can be lived through a sequence of qualitatively different things. One can, however, restrict oneself to the mere sensation of theses sequences as such, independent of the various degrees to which objects are perceived in the world outside are to be feared or desired. (The attention is reduced to an intellectual observation.) The human tactics of "acting purposively" then consists in replacing the end by the means (a later occurrence in the intellectually observed sequence by an earlier occurrence) when the human instinct feels that chance favours the means."

This description by Brouwer recalls the famous problem of technological "creep" or unforeseen applications or consequences, good and bad, of particular technologies. It characterizes most advances of technology and also falsely legitimizes the eventual benefits (possibly rewarded by Nobel prizes) of basic research that is seen as motivated (but not financed!) by "curiosity". Its "unforeseen" also recalls modern decision and systems theory in that it suggests how "probability" that he calls chance, in practice, is misunderstood and misused in the minimal thinking with the most primitive apriori level of the so called intellect (e.g. in evaluations of risks involved by nuclear power plants). This in contrast to the recommendations and deeper problems framed by modern systems theory (Churchman's Design of Inquiring Systems, chap. 3 on "the anatomy of goal seeking", and chap. 10 on "basic models of inquiring systems".)


From all this follows Brouwer's disapproval of applied mathematics that today we see in computers: it promotes man's rule (i.e. not God's rule) and lack of wisdom that will spoil environment making it intolerable. And this was written at a time when there was no relevant environment consciousness except for the intuitions of a beginning consciousness of environmental systemic aspects of "nature" in the work of Alexander von Humboldt, George Perkins Marsh, and John Muir that Swedish readers can appreciate in the summary by Tomas Öberg in Svenska Dagbladet (18 October 2016.) Van Stigt continues mentioning a glimmer of hope and optimism in that Brouwer believes that pure "mathematics practiced for its own sake can achieve all the harmonyÉsuch as found in architecture and music. ..."


It is my experience that this recourse to architecture and consequently "design" as a main modern expression of the hope of being able to integrate art with science has appeared in an unfortunate destructive way in Western society in general and university research in particular, because of intuitions much less sophisticated than Brouwer's. It appeared through the bizarre sudden shift of emphasis from the sixties' logical positivism (in Sweden B.Langefors inspired by Y.Bar-Hillel) and the following marxist systems development, over to the concept of design while forgetting the eighties- and nineties's pragmatist dialectical systems. This motivated me to write down several reflections, some of them in "telegraphic-powerpoint" style, such as Ethics and Politics in Design and Systems Cultures, and in a more readable structured way in my The Systems Approach to Design and Inquiring Information Systems. At the time I could get some inspiration from Christoper Norris' What's Wrong with Postmodernism (1990) but I could not yet avail myself of the insights of works relating more clearly socialism to posmodernism, such as Stephen Hicks' Explaining Postmodernism: Skepticism and Socialism from Rousseau to Foucault (2004, expanded edition 2011).


It is symptomatic that the USA domination of Western university research ignores such thoughts that are egregiously common in e.g. the French cultural sphere. One example is Alain Gras' writings (in French) that include references to Galilei and others, such as about Abatement as spiritual fact (in Entropia, No. 11, Autumn 2011, pp. 30-44.) Heidegger's bombastic "standing-reserve" in his The Question Concerning Technology echoes Brouwer's early clear and simple insights. This may also be gradually obvious in the field of HCI (human-computer interaction) when considering that ongoing progressive computerization of society and of social interaction violently forces humans to (attempt to) adapt their more or less unconscious mental models and behavior to preconceived computerized structures. Such phenomena can be exemplified and perceived as problems in the use of computers, particularly in problems of human-computer interactions. Swedish readers can appreciate one best description of how computer technology "rapes" the minds of citizens in Anna-Lena Laurén's article "App, app, app - det börjar likna Bolsjevism!" [App, app, app - it begins to resemble Bolshevism!" (Dagens Nyheter 14 October 2018) and the follow-up by Ylva Hasselberg "Hejda digitalbolsjevismen" [Stop the digital Bolshevism] (Dagens Nyheter 23 December 2018). See also the description of bugs such as in the ticket system of Stockholm's public transportation, (En bugg i SL:s biljettsystem, i.e. A bug in the ticket system of Stockholm's Transportation, in Swedish language, Metro-Stockholm, 22 Dec. 2015.) It becomes more a question of forcing human thought, trust and behavior, relinquishing self-reliance, into anonymous structures that are required for the functioning of computer software and hardware as much as supposedly for humans' purposes. This is the kernel for understanding why the issues of this essay are relevant for the field of HCI and consequently also for AI (artificial intelligence).


In the long run such forcing of human thought, trust and behavior for the functioning of computer software may gradually extend in the form of police surveillance of the whole society seen as dependent upon integrated computer networks. As such, the phenomenon will be analog to what happens with basic infrastructure of electric and associated nuclear power plants. In a French book Grandeur et Dépendance: Sociologie des macro-systèmes techniques (PUF, 1993, pp. 248ff.), the author Alain Gras, also a student of man-machine interaction, has indeed a section dedicated to generalized surveillance seen as remedy to human danger in technical systems. Ultimately all this stuff, in the absence of a valid religious "infrastructure", overflows politically in bizarrely advanced anarchist analyses such as Tiqqun's The Cybernetic Hypothesis.


Phenomenologically, in the sense of dealing with the description and classification of phenomena, this kind of forced thinking and behavior, at its extremes in industrial techno-science including computer applications, may be related to the savant syndrome, i.e. a "success" obtained in narrowly defined tasks, paradoxically enframed in complex computer systems, adapted to very narrowly focused brain-psychic activity, and having to be mitigated by an in-depth study of reports of broader mathematical minds such as portrayed in the book Fascinating Mathematical People , or exemplified by physicists like Isaac Newton, Wolfgang Pauli and Werner Heisenberg. At the same time this might explain the parallel aversion in certain quarters against systems thinking in general and social-humanistic thinking such as in West Churchman's systems approach and Carl Jung's psychology. Time-sequential thinking, and in particular logic linear thought cannot by definition be envisaged to take into account events outside the immediate time sequence that are then perceived as irrelevant (to the linear logical chain.)


I will continue here excerpting and occasionally commenting words taken from the text of The Rejected Parts (of Brouwer's dissertation) emphasizing Van Stigt's observation that the final version of Brouwer's dissertation on the Foundations contains much of his negative appreciation of applied mathematics and logic and remains of his moral disapproval. The Rejected Parts, van Stigt writes "are important for the clear statement they give of Brouwer's motivation for his fundamental theories, for the expressed positive valuation of pure mathematics, and for providing a further context of what, in cryptic form, still exists in his Foundations." Now over to the text of the Rejected Parts that I edit together with some of its notes (in parentheses), and reproduced in italics while my own interspersed comments are written in normal font.

All human life originated in a one-sided constriction of nature [literally: "making one-sided -- human concentration on one single aspect of nature and adapting nature accordingly] and has protracted its existence in an "externalization," man impregnating nature with the human self and repressing other one-sided developments. This externalization by man, making his environment subservient to the full development of his humanity, appears to us (if we new the world intellectually, i.e. with a mathematical causal eye) as a process whereby nature itself becomes linear and regular and all other life repressed or adapted to mankind. (Since the adaptation of the environment leads human life further and further away from the natural state which originally supported man, this conquered and adapted environment will ultimately become intolerable to mankind.)

The above observation by Brouwer can be seen as a premonition of the environmental crisis of our time, including pollution, global warming and the rest.

We find linearity and regularity, for example in bees; there it does not result in any sort of special power. But man has the faculty, accompanying all his interactions with nature, of objectifying the world, of seeing in the world causal systems in time. (This "seeing", however, is a human act of externalization: there is not real  e x i s t e n c e  of objective natural phenomena as can be ascribed to nature itself: the seeing originates in man, is an expression of man's will alone, independent of nature which itself exists independent of man's will.)

The process that Brouwer calls objectifying and externalization recalls what Carl Jung considers when discussing the process of extraversion in his conceptualization of extraversion vs. introversion. This text shows how Brouwer could be accused of solipsism when focusing on his claim that there is no real existence (his own italics) of objective natural phenomena as can be ascribed to nature itself, if one ignores his following qualification that nature itself exists independent of man's will. Such an accusation seems to me to follow from lack of understanding for what he calls an act of externalization, which turns my attention to what much later Jung would come to name projection, pending the question of whether it is wilful or not which in turn requires an understanding of the act of willing. The unravelling of this psychology under the keywords extraversion and projection is scattered around in Jung's Collected Works, CW, volume 6, Psychological Types. An analysis that recalls Brouwer's reasoning above is to be found in CW vol. 8 The Structure and Dynamics of the Psyche, (p. 451 , § 864), as in West Churchman's The Systems Approach and its Enemies, (the whole chap. 3 on "Logic: General, and p. 147), but our and Brouwer's focus is now upon the role of mathematics. Brouwer continues:

The primordial phenomenon is simply the intuition of time in which repetition of "thing in time and again thing" is possible, but in which (and this is a phenomenon outside mathematics) a sensation can fall apart in component qualities, so that a single moment can be lived through as a sequence of qualitatively different things. One can, however, restrict oneself to the mere sensation of these sequences as such, independent of the emotional content, i.e. independent of the various degrees to which objects perceived in the world outside are to be feared or desired. (The attention is reduced to an intellectual observation.)

This last observation required that the reader is attentive to Brouwer's use of the adjective intellectual as related to intellect, which in turn requires a recourse to psychology where intellect would probably correspond to Jung's thinking on the basis of the perception or rather sensation raised by the object-thing.

The human tactics of "acting purposively" then consists in replacing the end by the means (a later occurrence in the intellectually observed sequence by an earlier occurrence) when the human instinct feels that chance favours the means. However, since the link between end and means is observed in the intellect without the control of more central instincts (a restriction which will make the process even more intensive and more generally applicable), the reliability of the human conviction that the parts of the sequence belong together in reality is far from absolute and can constantly be disproved; this is experienced by the intellect as a discovery "that the rule no longer applies".

Brouwer also observes that the "tactics" in general is the source of human power, to discover regularity in a limited domain of phenomena independently of other moments and other phenomena, which therefore can remain completely concealed from the intellectual observation: to maintain the certainty of observed regularity as much and as long as possible, one tries to isolate systems, i.e. exclude observations which disturb this regularity. When one fails or has the courage to not exclude these observations may appear the so called serendipity that characterizes breakthroughs. One example I suggest with some reticence is the Nobel prize in physiology or medicine awarded in year 2018 for the "discovery of cancer therapy by inhibition of negative immune regulation". The serendipity was expressed at the award ceremony as being that instead of concentrating on the direct causal chain of destruction of cancer cells, the wider immune system was taken into account. My point is that a still wider approach may be required, without necessarily entering into the controversies of so called integrative medicine as it is understood today, as it relates to Asklepian medicine and cultural studies of the archetypal image of physician's existence by Károly Kerényi.


Brouwer's observation above is consistent with the principles of modern decision making where the mentioned "chance that favours the means" corresponds to probability as conceptualized e.g. in West Churchman et al. Thinking for Decisions: Deductive Quantitative Methods (1975), where, furthermore, the ends are supplemented with the individual worth or value of the ends. To counter or compensate for the above mentioned attempts to "isolate systems" was in turn what motivated the later development of systems theory. But, we repeat that our focus here is the role of mathematics as it is embodied in computers where such mathematics is necessarily disembodied from its applications, raising the question of the meaning of its success or "successful application":

The process of objectifying the world through the primordial intuition of "repetition in time" and "following in time" gains in generality by the construction of mathematics from the same primordial intuition, without reference to direct applicability. In this way man has a ready-made supply of unreal causal sequences at his disposal, just waiting for an opportunity to be projected into reality. One should bear in mind that in mathematical systems with no time coordinate, all relations in practical applications clearly become causal relations in time; e.g. Euclidean geometry when applied to reality shows a causal connection between the results of different measurements made by means of the group of rigid bodies. Needless to say, in the application of a mathematical system, in general, only a fraction of the elements and substructures finds their correspondence in reality; the remainder plays the role of and unreal "physical hypothesis." Similarly, even with a limited development of method, the observed sequences no longer consist exclusively of phenomena evoked by man himself (acts without any direct instinctive aim, but carried out solely to complete the causal system into a more manageable one). The simples example is the sound image (or written symbol) of number as a result of counting, or the sound image (or written symbol) of number as a result of measuring (this example shows how infinitely many causal sequences can be brought together under the viewpoint of one single law of causality on the basis of a mapping the numbers through mathematical induction.) [My boldface.]

I hope to not exaggerate the importance of this section that for the first time gives me the impression of understanding the comments about Galilei's famous mathematization of science. My intuition is that this contributes to an explanation of the "success" of the mathematization of modern science that, to take only one example from the mathematized field of quantum mechanics, is the explanation of its (mathematical) success as suggested in the story of the famous physicist Paul Dirac as related to the earlier mentioned Wolfgang Pauli. If his atheism did not approach him to God then it approached him to perceiving that he became gradually godlike by means of mathematics as described in Wikipedia's excerpt from an article he wrote in the May 1963 edition of Scientific American:

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe."

Then, Dirac claims, instead of asking why, that "we simply have to accept it" and in the bargain that we have to accept that man (beginning with Dirac himself) becomes more godlike the more he mathematizes his conception of the universe. Dirac's colleague Wolfgang Pauli summarized all this by his famous remark (also in Wikipedia, later applied by others to Richard Dawkins in his evolutionism) that "Well, our friend Dirac has got a religion and its guiding principle is 'There is no God and Paul Dirac is His prophet.'" This is part of the great but limited discussion of Eugene Wigner's article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" in Communications in Pure and Applied Mathematics, vol. 13, No. 1 (February 1960), as summarized in context by Krister Renard in an essay written in Swedish with the analog title of "Matematikens osannolika användbarhet" (lecture in English for sale here). A more superficial summary with further references is found in Wikipedia. Brouwer's approach suggests instead man's distancing himself from God and it seems consistent with later detailed discussions of the matter such as in the historical context of Galileo Galilei, in Alexandre Koyré's Études d'Histoire de la Pensée Scientifique (1966, especially p. 189f., 211f., 258.) Contrary to Koyré, however, Brouwer does not stop at affirmations such as considering mathematics (and geometry) as apriori bases for modern experimental science. He goes further and questions the apriori base of mathematics itself (in the apriority of time, and its relation to the so far undefined intellect). What is remarkable, however, is Brouwer's early observation, (on the basis of what Koyré inadvertently acknowledges in the ultimate necessity of "experience"), of the environmental impact of mathematized industrial science at a time when environmental issues had not yet made a societal impact. And it is now a so far undefined "environment" that must include people who get affected by the computerization of the world.

The strategy of objectifying the world forces man even more to eliminate the "deviating" influences and thereby to abnormalize his environment. The nature of the phenomena within a certain domain changes not only through the elimination of the influences which deviate in this domain itself, but also through the degeneration of the environment of these phenomena because of the removal of influences which deviate with respect to another quite different group of phenomena. [...] In this way science, in a process of increasing self-perfection, will strengthen its power to obtain results but debase the  v a l u e  of these results.

We observe in this context that "mathematical viewing" is only instinctive, i.e. justified insofar as it is directed to a world which is considered to be external; to try and direct it to inner perception is a serious error (moreover, there would never be any agreement between the results of mathematical viewing from different viewpoints). What Kant describes as "Transcendental analytic" can only be described as idle play.

Brouwer combines this process of degeneration and devaluation with a consequent observation that the passionless language (also reason for creation of mathematical notations, cf. the earlier reference to Cajori) of objectification of the work in mathematical systems allows the emotional content to be completely different (and ignored) for different individuals. It this way the resulting agreement between mathematical systems of reality in different individuals allows the enforcement of a particular will over all others "out of fear or desire associated with certain elements in the system." Fear and desire are basic categories in Brouwer's analysis in several other contexts.


In my attempts to understand the reference above to "idle play" I also tried to understand something of the intricate field of philosophy of mathematics, and in particular the philosophically "canonized" Immanuel Kant's philosophy of mathematics where, Philip Kitcher notes (in Kant and the Foundations of Mathematics), that pure mathematics includes geometry, arithmetic, algebra, kinematics, "pure mechanics" and, he thinks, analysis. Encouraged by the example of colleagues who had studied Kant's Critique of Pure Reason (F. Max Müller's translation, Anchor Books, 1966) as a step for understanding information systems I dwelled in its treatment of mathematics in its Transcendental analytic, such as its mentioning of "counting" (p. 61, A76f., B102f.) or "number" (p. 65, A:83f., B:112f.) in the section III of Book I, about "Pure concepts of understanding, or of the categories." Not finding anything that recalls Brouwer's problem (and this may be the reason for his description of "idle play",) I went over to what is considered as the explicit basis for his famous philosophy of mathematics, in the Critique's Transcendental Dialectic, "The discipline of pure reason in its dogmatical use" (pp. 465-479, A710ff., B742ff.). I found there something that may be consistent with Brouwer's message, namely the following (p. 471f., A725f, B:749f.):

"The great success which attends reason in its mathematical use produces naturally the expectation that it, or rather its method, would have the same success outside the field of quantities also by reducing all concepts to intuitions which may be given a priori, and by which the whole of nature might be conquered [...] Nor there seem to be any lack of confidence on the part of those who are masters in the art of mathematics, or of high expectations on the part of the public at large, as to their ability of achieving success, if they only would try it."

Further summarizing comments about all this can be found in the Stanford Encyclopedia of Philosophy but for me it is still an open question whether Kant's reasoning implies or is consistent with Brouwer's conceptions and his condemnation of "idle play" and consequences for the application of "mathematical machines" to human activities rather than to phenomena of natural science.


Brouwer continues exposing the difference between the mentioned "passionless language" and the language of mysticism which may convey contemplative thoughts that are independent of superficial human collusion and animal emotions of fear and desire, appearing as meaningless to those who seek communication of mathematical systems. In this way he tries to foster the understanding of the remarkable book he had written some years earlier, Life, Art and Mysticism, already mentioned above.

Is it surprising that not only do we succeed in observing sequences which repeat themselves again and again, but that so many groups of phenomena affecting our naive senses in totally different ways can be brought together under a few general aspects which are covered by simple constructible mathematical systems? This really would be a miracle, were it not for the simple fact that the physicist concerns himself with the projection of the phenomena on his measuring instruments, all constructed by a similar process from rather similar  s o l i d  bodies. It is therefore not surprising that the phenomena are forced to record in this similar medium either similar laws or no laws. For example the laws of astronomy are no more than the laws of our measuring instruments when used to follow the course of heavenly bodies

This last sentence is indicated by van Stigt as having been crossed out by Brouwer himself, and this may be an indication that Brouwer had overstepped his own intentions in an improper expression that would give the impression of sheer solipsism. On the other hand, however, it may be question of interpretations of the meaning of "solipsism" since for all practical purposes the important thing is the relation between human observers and so called reality, rather than philosophical reality per se that can be cheerfully neglected as it already is in computer science in its dealings with "virtual reality". Observations of nature and environment are important in that they point to the mysterious and seldom addressed question of the strange, not to say mysterious capacity of mathematics to act "humanly", i.e. to foster human understanding or rather prediction and manipulation of nature as if it, mathematics, were the "language" of nature. What both Brouwer and we ourselves are trying to do here is to understand what happens to our conception not only of nature, but also of anything partly falling outside nature such as humans when we apply that sort of eyeglasses constituted by the mathematical machines, in the broad meaning of mathematics. And this, besides the already mentioned Design of Inquiring Systems, may be of utmost importance for understanding the import of late developments of so called artificial intelligence (AI), such as "Deep learning" or applications of the type "Xiaoice" or "AlphaGo", not to mention "Emotional AI" as well as some of the criticism directed against it, such as "What emotional AI fails to grasp about emotion".

A basic distinguishing faculty of humans with respect to nature is free will, despite many people including scientists are trying today to "deconstruct" it in terms of determinism and atheism. Our discussion can be seen as touching the question of free will as related to nature, while free will is coerced and cast into predetermined forms by imposed structures of human-computer interaction, and thereby also mediated human-human interaction such as so called social media. I am not sure to which extent one has to deepen a proper understanding of the conundrum of free will in order to understand what is happening and its consequences, including computer addiction as a result of playing havoc or gambling with the degrees of freedom of the will. I see, however, that such an understanding is absent in dealings with so called artificial intelligence - AI - that eschew teleology in favor of mechanism, as in mathematical speculations on "infinity and the mind" exemplified by Rudy Rucker. They have not yet incorporated the lessons from e.g. West Churchman's The Design of Inquiring Systems. Humility in face of this matter may in due time be fostered by an in-depth study of Kant's discussion of free will in his Critique of Pure Reason, second division: Transcendental Dialectic, Book II, chap. II, section 9 on "Explanation of the cosmological idea of freedom in connection with the general necessity of nature" (pp. 371ff., A:538ff., B:566ff.). Nevertheless such a discussion will probably not take place because of the inaccessible complexity of this type of literature and of Kant in particular. Many modern philosophically and technically sophisticated researchers still dwell in the sixties' analytic philosophy or cultural marxism, and many have swiftly moved over to phenomenology, post-structuralism, post-phenomenology and such, and some of them to anthroposophical criticism, all of them apparently avoiding evaluation of variants of pragmatism that otherwise is praised in the Western USA-influenced culture.

Science, therefore, makes sense only when man in his struggle against nature and his fellow men, uses the calculations of counting and measuring; in other words, physical science has value only as a  w e a p o n, it does not concern life - indeed it is a disturbing and distracting factor like  e v e r y t h i n g  in any way connected with struggle.

In this way Brouwer continues touching upon controversial issues which require a deepening of the philosophy and theology of technology that I have otherwise considered in my earlier mentioned Trends in Philosophy of Technology. Theologically it is a matter of the relation between man's and God's will that to my knowledge is programatically considered in Carl Mitcham and Jim Grote's (eds.) Theology and Technology: Essays in Christian Analysis and Exegesis. Theological discussions seem to be centered on the meaning of Genesis 3:19 "By the sweat of your face You will eat bread" as related to human reason as a gift of God in Genesis 1:27 "God created man in His own image, in the image of God He created him", supposing the legitimacy of using reason for at least partly redeeming himself. Besides common knowledge that much if not most technology derives from military needs Brouwer is also known to have been since early youth interested in theology, which he took very seriously, as shown in his profession of faith (see below).

But mathematics practised for its own sake can achieve all the harmony (i.e., an overwhelming multiplicity of different visible, simple structures which one and the same all-embracing edifice) such as can be found in architecture and music, and also yield all the illicit pleasures which ensue from the free and full development of one's faculties without external force.

Brouwer suggests here the source of all speculations associated in later years to the treatment of computer applications in terms of aesthetics as in architecture, which I tried to problematize in my The search for a theory of hypermedia, and music, whose "mystical" relation to mathematics is known since antiquity and Brouwer implicitly suggests by his reference to the "primordial intuition" of time that is common to both mathematics and music. In the computer age all seems to have started with Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid, which does not seem to eschew secular "mysticism" when properly described in Wikipedia as "not about the relationships of mathematics, art, and music, but rather about how cognition emerges from hidden neurological mechanisms. In that book Hofstadter presents an analogy about how the individual neurons of the brain coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants." Brouwer forestalled the misuse of the ant-analogy by pointing out the "illicit pleasures" in the last quotation of his above, something that would come to be the object of justified controversy about Herbert Simon's ant-analogy in his book The Sciences of the Artificial (reviewed by West Churchman in Contemporary Psychology, vol. 15, 6 June 1970, pp. 385f.) In pursuing the question of illegitimacy Brouwer continues referring to the mathematician Henri Poincaré in his La Valeur de la Science [The value of science]:

Poincaré [...] is inclined to reduce all aesthetic affections to such an affection of harmony. Perhaps his notion of aesthetic affection is simply an affection of harmony; but even according to him it is more: he says "outside science and aesthetic there is nothing but "le pur néant" [The pure nothingness]. He therefore seems to believe that it may be this aesthetic affection which is referred to as the highest good for mankind, preserved with such great difficulty. This shows the blinding effect which the immoral, free and full development of human faculties has also on him.

This is what I recalled in other computer-contexts, often associated with the turn towards "Design", as being the well known but seldom recognized popular version of aestheticism, as paragraphs with this keyword are found e.g. in my


East and West of Information Systems,

The Systems Approach to Design,

Whither Computers and Systems?,

Information as Debate,

Trends in Philosophy of Technology, and

Ethics and Politics in Design and System Cultures (seminar notes)


This represents a particular thread in European culture during at least the last 200 years and whose immense potential for mind-blowing confusion is illustrated in the discussion of moral aestheticism and of the early mentioned books on postmodernism. Van Stigt's edited Rejected Parts of Brouwer's dissertation continues with a discussion of the creation of mathematical systems in the exterior world as a moment in the process of externalization, i.e. "of holding out against the exterior world." It is followed by considerations about Kant's Transcendental Aesthetics, about objectivation, language in philosophy, causality and free morality. The discussion, however, becomes gradually more intricate also because rejected paragraphs require for their understanding to be related to other paragraphs that were not rejected and do appear only in Brouwer's dissertation. Such discussion falls therefore outside the limits of space and purpose of my present text. It is enough to acknowledge that externalization is coupled to processes considered in analytical psychology.


A presentation and commented summary of Brouwer's book on Life, Art and Mysticism is not necessary since, supplemented with a description of its historical-cultural context, it is available on the net, in the previously mentioned Dirk van Dalen's Mystic, Geometer, and Intuitionist: The Life of L.E.J.Brouwer, (pp. 63-74.) An appreciation is also found in Walter P. Van Stigt, Introduction to Life, Art and Mysticism. My purpose here in to highlight some particular thoughts that may show to be relevant for the conception of computers as embodiment of mathematics, and of logic as understood by Brouwer.

Anticipating many later and modern issues in philosophy of technology, Brouwer starts noting problems of environmental destruction while apparently influenced by the Rousseau-like "theory of the natural human", meaning, as Wikipedia expresses it, that Brouwer "looked to a hypothetical State of Nature as a normative guide". This does obviously mean that there are important theological presuppositions for the bases and the consequent understanding of the purposes of his work. It does not require that Brouwer should have borrowed the rather confusingly convolute theological presuppositions of Rousseau. Dirk van Dalen in chapter 1, "Child and student" of the first volume of his book Mystic, Geometer, and Intuitionist, concludes (p. 20), possibly on the basis of Brouwer's early profession of faith in the early 1900 in the Dutch Remonstrant Church (summary here) that in his philosophical views he "had adopted a rigorous, Schopenhauer-like, view of the world, religion, and his fellow human beings." His profession of faith in a rather "liberal" brand of Protestantism explains, by the way, his critical stance on certain forms of established religion as expressed in chapter 3, "Man's Downfall, Caused by the Intellect") such as his statement that "Art and religion in this world are only grand morphine industries" (p. 398.) A theological analysis of Brouwer's thought would certainly require a deeper understanding of this issue.

In the book's chapter 2 on Turning into Oneself, Brouwer talks about the need of withdrawing from the attitude that causes environmental destruction by "turning into oneself". Van Dalen explains how this must be interpreted theologically and psychologically in that Brouwer had very early focused his attention of the essence of the ego as related to "given" perceptions, rather independent upon the objective world out there, and therefore he appeared rather "solipsistic". Here we have, again, a connection to Jung's psychology in the sense of introverted ego-thinking supplemented by intuition, i.e. (in Jung's conception) unconscious perception.


The reach of Brouwer's psychological ambitions would be reiterated so late as in 1948 when he is reported (by van Dalen, pp. 832f.) to have reiterated his views about the inverse part of the "turning into oneself", i.e. the original fatidic "exodus of consciousness from its deepest home", which would consist of three phases: the naive phase (creation of the world of sensation), the isolated causal phase (where causal acts take place), and the social phase (in which cooperation with the individuals finds its place.) In the causal phase I understand that also mathematical projection takes place, regardless whether mathematical patterns are developed in the social phase.


In this way Brouwer was also touching the problem of consciousness, which eludes the most modern discussions of artificial intelligence. In order to understand the immense problem that Brouwer was trying to address, starting in his youth and at a time when most mathematicians (and psychologists) had neither the interest nor the knowledge necessary to grapple with it, we can point out Erich Neumann's The Origins and History of Consciousness, (1954, German orig. 1949.) In Neumann's terms Brouwer seems to be concerned with a spirituality that had lost touch with reality and the instincts: "The form which this kind of degeneration usually takes in the West is not spiritual inflation, but sclerosis of consciousness where the ego identifies with consciousness as a form of spirit. In most cases this means identifying spirit with intellect, and consciousness with thinking" (p. 386.)


In the rest of chapter 2, in two pages (393f.), Brouwer repeatedly mentions the term "self" at least 15 times, including the "you" of "yourself" corresponding to the "ego", without attempting to define it. As far as I can see this self may correspond, in Jung's terms to both his own concept of self and, especially, to what is as yet unconscious. This is so because the self as an empirical concept related to the psyche designates the "whole range of psychic phenomena in man". As expressed in Jung's Collected Works volume 6 or CW6 (Psychological Types, § 789 and 797), this is almost identical to the definition of the psyche as "the totality of all psychic processes, conscious as well as unconscious". The editors of CW6 add, however, that the inference from this comparison would seem to be that every individual, by virtue of having a psyche, is potentially the self. It is only a question of "realizing" it. But the realization, if ever achieved, is the work of a lifetime, possibly under the guidance of a spiritual adviser that once would have been a priest or, in later times, a "therapeut". That is, under the guidelines for a sort of yoga master or, in our Christian tradition, according Loyola's spiritual exercises and Clorivière's guidelines for prayer (also in Swedish translation) or, in our modern times, a "gnostic" analytic psychotherapeut. In this case Brouwer is only ignoring the lifelong work that is necessary for the "return to the self" and seems to be assuming the it can be done, albeit requiring an "effort" to "overcome an inertia", as an act of free will for "self-reflection" that "can withdraw from the world of causality". This optimistic attitude of Brouwer may be attributed to his feeling of identification with the experiences of the mystic Jakob Boehme, his main inspiration in these thoughts: Brouwer must have needed to use the words of Boehme for describing his own early experiences.


For our purposes in this essay it must be finally mentioned that Brouwer offers in this essay an indirect clue about the reason for the fascination exerted by computers. Among the difficulties met for the "return to the self" he mentions the downfall caused by "fear and an obsession with saving, born from the illusion of time", and "the desire and lust for power, born from the illusion of space." Considering that on several occasions he refers to the need to free oneself from the shackles of worldly fear and desire, it follows that the computer, especially in the form of mobile smartphones with their instant spatial communications paradoxically offers an illusion of freedom from the illusions of both time and space.

In chapter 3 about Man's Downfall, Caused by the Intellect Brouwer can be seen as presenting a criticism of science. He dwells upon the downfall caused by the intellect chained to fear and desire. Man is chained to its "intellect", and he is incapable of lifting itself in self-reflection. Unfortunately, in the whole argument, Brouwer leaves "intellect" as undefined as the earlier self, and one of our purposes here is to relate this intellect to what especially later psychology would come to name as psyche, with due regard to former Greek philosophy and Christian conceptions of spirit, soul and body, such as e.g. described in a CBN (Christian Broadcasting Network) entry. Anyway, in Brouwer's own words:

This highly valued intellect has enabled man and forced him to go on living in desire and fear, rather than from a salutary sense of bewilderment take refuge in self-reflection. Intellect has made him forfeit the amazing independence and directness of his rambling images by connecting them with each other rather than with the self. In this way the intellect made him persist with apparent security in the conviction of a "reality", which man in his arrogance has made himself and had tied to causality, but in which in the end must feel totally powerless.

In this life of lust and desire the intellect renders man the devilish service of linking two images of the imagination as means and end. Once in the grip of desire for one thing he is made to strive after another as a means to that end [...]

Brouwer goes on explaining that the act aimed at the means, however, always overshoots the mark to some extent - the means has a direction of its own, diverted at an angle however small from that of the end. It acts, therefore not only in the direction of the end, but also in other dimensions that we could call unforeseen consequences while many perceive as an end what was originally [and for others, we may add] only a means. Brouwer notes that if this "deceptive jump" from end to means is repeated several times, it may happen that a direction is pursued which not only deviates but opposes and counteracts the original one. Brouwer applied this insight to the development of industry and the consumption of natural resources including the balance of nature, expanding his reasoning to the production of industrial tools and therefore to the essence of technology. Unforeseen consequences lead repeatedly to the uncovering of hidden assumptions and consequent anxiety about the future, craving for the power to predict its course: that is, requiring a science to support technology. Or:

Science, which in its original form was wholly subservient to industry, has made up all kinds of general assertions in and about the world of perception. These come true as long as it pleases God; but one day they will suddenly be contradicted by facts and then our scientists will claim, "O yes, of course, we always made this or that tacit assumption." In their incompetence they then set about complicating the issue even further and making so-called corrections and improvements.


And, continuing:


But science does not confine itself to serving industry: again the means becomes and end in itself, and science is practised for its own sake. Bodily awareness has strayed so far away that it is all concentrated in the human head, ignoring and excluding the rest of the body. At the same time man becomes convinced of his own existence as an individual and that of a separate and independent world of perception. At this stage there are radical changes in the direction of man's attention, and these constitute scientific thinking. For scientific thinking is nothing but a fixation of the direction of will within the confines of the head, and a scientific truth no more than an infatuation of desire living exclusively in the human head.(My emphasis.)

We can see here a possible source of modern phenomenological approaches with emphasis upon the body, such as those that were considered in my essay Trends in philosophy of technology with special focus on information technology. The interest for the "body" as a attempted contrast to the "head" can also appear in queer forms of "feminist techno-science", partly based on feminist interpretations of (pdf) "quantum theory". A problematic feature of the analysis above is that in face of the lack of an appropriate psychology the ego of the psyche is equated with the "head", paradoxically further contrasted to the "body" as if the head including its implicitly considered and often emphasized "brain" did not belong to the body. In some late phenomenological approaches the body is further differentiated from the "skin" in the context of "touching", as if it were a sort of metaphysical differentiation between the ("its") body and the environment or "the other", apart from or based on the logical Aristotelian considerations of sense of touch. In view of analytical psychology, however, the basic problem that Brouwer encounters in his analysis is the inability to analyze the situation of the ego itself with its consciousness and will. This is the source of innumerable analytic difficulties including lately such as in debates about the philosophy of free will among analytic philosophers such as Harry Frankfurt and David Widerker, which has wide implications for informational decision theory. We find there a systematic and unreflective use of the apparently unproblematic concept of (a unique) "agent" corresponding to one "head", after it having been widely problematized in Churchman's systems approach (The Design of Inquiring Systems, chap. 3) by means of the split between researcher, manager and client, not to mention the problem about their "head" or "intellect", mentioned above. Back to Brouwer:

Every branch of science will therefore run into deeper trouble; when it climbs too high it is almost completely shrouded in even greater isolation, where the remembered results of that science take on an independent existence. The "foundations" of this branch of science are investigated, and that soon becomes a new branch of science. One then begins to search for the foundations of science in general and knocks up some "theory of knowledge". As they climb higher and higher confusion grows until they are all completely deranged. Some in the end quietly give up; having thought for a long time about the elusive link between the intuiting consciousness (which develops from the perceptional world) and the perceptional world itself (which in turn only exists through and in the forms of the intuiting consciousness) - a confusion which arose from their own sin of constructing a perceptional world - they then plug the hole with the concept of the  e g o, which was self-created with and at the same time as their perceptional world; and they say, "Yes, of course, something must remain incomprehensible, and that something is the ego that comprehends."

The above has direct implications for the whole of computer science, especially the quagmire of AI, its neologisms such as "neuromorphism", and the phraseology associated to its already mentioned "deep learning", machine learning and data mining, which are built upon the concept of mathematical computing machines and presupposes the view of (theory of) information as a theory of knowledge. Brouwer explains further how this in turn is a temptation for the assumption of an incomprehensible and therefore unchallenged monolithic concept of ego. In doing so, however, Brouwer ignores both the need and the possibility of an analysis (and synthesis) of the ego as related to consciousness and to the unconscious as proposed in analytical psychology or exemplified in the earlier reference to Erich Neumann. From these shortcomings originate also affirmations that have raised the suspicion of solipsism in the context of a construction of the perceptional world.

Brouwer continues describing and foreseeing results of this state of things as reflected in environmental destruction, menaces to human physical and mental health with consequent rise of the medical and entertainment industry (that today culminates in computer-supported "edutainement"). They imply behaviors that, at Brouwer's time, can be seen as presaging the new age hype, accompanied by the "myth of drugs" and followed by its repeated disappointments. Sometimes it is perceived (e.g. by B. G. Rosenqvist in Svenska Dagbladet, in "The drug that formed the ideal society", in Swedish) how the concept of "drug" salvation appears in Aldous Huxley's late utopic book The Island (1962). It seems to be the result of his reflections upon Huxley's (related to George Orwell) earlier famous dystopic book Brave New World (1932), recalling the European cultural environment of the early Brouwer who, however, had the insight of preferring Christian salvation.


In a chapter 4 dedicated to "atonement" Brouwer, in a preaching prophetic tone that characterizes the whole book, recommends a sort of mystical attitude to life that borrows even words from Jacob Boehme and resembles both Buddhism and Christianity in the spirit of Thomas à Kempis' Imitation of Christ. It is easy to guess that such an attitude in the ambit of Western science could only be possible in dedication to mathematics for its own sake. The step from there to mysticism is natural and is portrayed in the chapter that follows:

Chapter 5 is dedicated to "language" and I have already used some of its argument in my earlier essay on Information and Debate. In this chapter Brouwer criticizes logic for being "life in the human brain" using here the term brain instead of the earlier reference to "head" indicating the he uses them as synonyms. Among other things he states that:

[R]idiculous is the use of language when one tries to express subtle nuances of will which are not part of the living reality of those concerned, when for example so-called philosophers or metaphysicians discuss among themselves morality, God, consciousness, immortality, or the free will. These people do not even love each other, let alone share the same subtle movements of the soul; sometimes they even do not know each other personally. They either talk at cross-purposes or each builds his own little logical system which lacks any connection with reality. For logic is life in the human brain; it may accompany life outside the brain but it can never guide it by virtue of its own power. Indeed, if there is a harmony of will, logic may well fall by the wayside ...


Ridiculous [...] is the use of language when there is an argument and people try to come to an agreement by means of reasoning. [...] In everyday life language only makes sense as a means of holding the already harmonious will of two people together on one path. [...] Language can accompany man's will to dominate the will of others or his will to keep the movement of wills together; for example, the war cry of Red Indians accompanies the will to break the will of others.

Yes, language as the source of logic can be used, in the absence of mutual love or friendship, for "man's will to dominate the will of others", the more so with a science embedded in abstruse "mathematical tools" that disregard not only the meaning of mathematics but also the difference between tools and theory-based instruments. Such tools insulate the scientific communities from a shared democratic understanding and criticism by the general public and other narrow scientific specialties who do not share knowledge about the particular mathematical tools. Let us also remind that this may be illustrated by what happens in the Internet's social media's logically structured communications when clusters of people, including children in schools, get to bully other clusters or, especially, individuals, or start revolutionary gatherings, today always with supposedly democratic aims. The question about language was part or the polemics of that time regarding the relation between logic and mathematics where Brouwer saw mathematics fundamental and originally individual, and independent from logic seen as based upon the need for (necessarily imperfect) language communication. The implications for computer communications beyond logic and along language are then clear. For example, Facebook, Twitter, group-blogs and so called search motors have already been noted for facilitating the pooling together of people in sub-cultures with similar convictions and for achieving what Brouwer mentions as "language, which presumes a harmony of will, may well be used to accompany strife and combat." It is possible to figure out the import of it all for the filtering implied in the process of peer review and publication in established influential scientific journals.


This touches upon cultural aspects of "rhetoric" that I have previously (1997-1998) considered in a paper on The East and West of Information Systems. In the context of a Swedish blog debate about climate and global warming I have seen a member (pseudonym "Nils G") of one dominant sub-culture irrupting into the community of another deviant sub-culture and requesting that they turn more "objective" and hoping to be allowed to start including further opposing arguments. This attitude reveals a misunderstanding of the "Hegelian inquiring systems", in West Churchman's conception of The Design of Inquiring Systems, since they presuppose a recognition of the fact that the most important debates are indeed "objectively" driven by the participant parties' passioned conviction that they are right in very important matters. This implies a recognition that all people are in different degrees legitimately influenced by feelings, intuitions, unknowns or unconscious factors. One can claim either to be "objective" or to require necessary additional arguments, but others can legitimately refuse to agree about such a requirement, as it is indeed the case regarding the legitimacy of e.g. the Holocaust, paedophilia or, earlier, atheism itself. In the best case, a synthesis of opposing views is supposed to be obtained by a an "external" observer of the opposition (having another related parallel passion.) This may include the case of the creation of a special, "theory-laden" communicative language such as in pastoral counseling or psychoanalysis. Hegelian approaches to language not to mention logic, however, were not within the horizon of Brouwer at the time, the less so for mathematics based on a "primordial intuition", despite his acquaintance with Hegelian philosophy through his early contact with Gerard Bolland.


In the context of "ridiculous use of language" which in my mind recalls the modern Western phenomenon of "political correctness" Brouwer refers in two occasions in this chapter 5 to the "brain" in the context of language as

Only in the very narrowly restricted domains of the imagination such as in the exclusively intellectual sciences—which are completely separated from the world of perception and therefore touch the least upon the essentially human—only there can mutual understanding be maintained for some time. There is little scope for misunderstanding notions such as “equal” and “triangle,” but even then two different people will never feel them in exactly the same way. Even in the case of the most restricted sciences, logic and mathematics - a sharp distinction between these two is hardly possible - no two different people will have the same conception of the fundamental concepts on which these two sciences are constructed; and yet their wills are parallel, and in both there is a small, unimportant part of the brain which forces their attention in the same way.

Once someone is imprisoned in the belief of a logically coherent (i.e. conceived without pain in a certain region of the brain) complex of externalities, which he calls "reality", it becomes rather difficult to follow him in his folly, and even more difficult to try to evoke in him a particular emotion or state of mind be means of words which he can only interpret in accordance with his reality. [My bold face]

These latest considerations indicate that the diffidence of people towards mathematics needs not be a result of their identification with the fox in what Aesop means in the fable of the fox and the sour grapes, as rarely gifted mathematicians and rationalist sympathizers may be tempted to have it. It may be even the other way round, regarding diffidence towards whatever recalls religion, mindfully cherished and respected by a majority of world's population. Mathematics and logic can even be a preferred opportunity for young gifted individuals to excel in them since it is natural that their figuratively "small unimportant part of the brain" is the main part of the brain that is available to them: other "parts of the brain" and their interactions have not yet become active in "important" adult matters such as work-life, love, suffering, death and religion. This is something to think about when praising children and youngsters for their "brainy" superiority to elders in intuitive proficiency (Brouwer's "primal intuition in time sequences") in handling new computer gadgets that are "fingered" in the spirit of Richard Stiver's Technology as Magic that is "it works but have no idea of why - what happens" and do not know what "works" means. Similarly, individuals gifted with a mathematical mind may be perceived as being polymaths or universal geniuses in numerous intellectual fields thanks to their mental reduction of the fields to mathematical or logical form. This seems to be common in the area of computer science as exemplified by computer scientists such as Herbert Simon, or lately in the case of luminaries in climate research and global warming, such as climate research such as Hans Joachim Schellnhuber. It is gift that also enhances the capability to win debates cheaply by reducing complex systemic issues to oversimplified linear logic sequences of computable apparent "arguments" (most often using undefined concepts) as I illustrate in two other texts on "debates" and on "climate change and global warming". Nevertheless, depending upon the particular personality of the thinker, it can also enhance the capability to unravel complex issues to the point of legitimately bedazzling the reader or the listener. It may be the case with a remarkable essay by Olivier Rey on Que faire des différences? [What to make of the differences?] where differences concern also those between citizen and foreigner, or man and woman, a sore point in the issue of equality or of radical feminism.


After further considerations that include the drift from defective language to a consequent philosophy based on a supposed meaning of language Brouwer foresees a rude awakening in front of perceived deficiencies and contradictions or a flight from the "world of intellectual perception with its man-made laws" or to the highly acclaimed "science" [sic, within quotation marks.] His diffidence concerning language is echoed by my own considerations about the difficulty if not impossibility of debate in Information as Debate, touching matters of major importance concerning ethics and politics of science. This type of insights motivates Brouwer to move further into the four last extensive chapters 6-9 of his book with the titles of Immanent Truth, Transcendent Truth, The Freed Life, and Economics. In doing so he can be interpreted as entering gradually into the domains of "mysticism" where the term, however, must not be understood as in the popular daily secularized verbiage of modern Western languages.


Chapter 6 is dedicated to cultural criticism in those forms of human enlightenment and communication beyond the aforementioned limitations of language, as evidenced by worldly arts and literature, which arein my view erroneously taken by "Nietzscheans" like Heidegger and Alain de Botton as aestheticis tools for a "religion for atheists": art as religion. The subsequent chapters 7 to 9 take the necessary further step into the forms represented by religion or mysticism, a mysticism understood critically as, for instance, in Harold Coward's book Jung and Eastern Thought (1985, chap. vii) and echoing some of the most serious currents of Western philosophy such as in the treatment of "Quietism" and the contrast between Leibniz and Schopenhauer as depicted in Émilienne Naert's Leibniz et la Querelle du Pur Amour [Leibniz and the Dispute of Pure Love] (1959.) In the chapter on Le Quiétisme vu par Leibniz et Schopenhauer [Quietism seen by Leibniz and Schopenhauer], pp. 232ff., the descriptions in the text confirm the proximity of Brouwer to Schopenhauer's thought. The relevance of the latter for the philosophy of psychology is indicated by Carl Jung's references to the latter, which take almost as much space in his Collected Works' General Index (volume 20) as the number of references to Plato. It must be noted that the complexity of the background of the question requires, for a proper interpretation, the kind of theological knowledge that is advanced, for instance, in Fr. Raymond Gawronski, S.J. Word and Silence (1995) as noted also in my related article on the hopelessness of communication in debates.


There is no need for me to go deep into the mentioned last chapters 7 to 9 that need no further explanations beyond the meaning of mysticism, and are found in the original book in question with the text available on the net in pdf-format at the link already specified earlier, above. They are already summarized in the likewise mentioned introduction to and overviews of Brouwer's book that are the best and most detailed source for understanding his thought, such as Walter P. Van Stigt, Introduction to Life, Art and Mysticism (1996, pdf).

The main necessary observation is a reference to a part of chapter 6 that deals with the repeatedly criticized and misunderstood treatment of womanhood, or Brouwer's supposed "misogyny". I think that the a key paragraph for an in-depth understanding of what Brouwer is expressing is the following:

There is a balance between man's burden of guilt and the burden of labor and toil imposed on him. A similar balance is found between woman's wantonness, her inborn capacity for karma burdening, and the measure of femininity which this world offers in temptation. In a world of humble acceptance of given karma there would be no women.

I sense that these thoughts are the author's probably unconscious attempt to capture the theological meaning of the biblical "original sin", made by a man with superior intelligence who was struggling, in terms which later would be conceived by analytical psychology, to defend himself from a dangerously close contact with his anima. The latter, being unconscious, was then projected into womanhood, i.e. "women considered collectively". It is remarkable that Brouwer reached so far and deep in his analysis at the turn of the century, without detailed theological knowledge and at a time when analytical psychology was still being conceived. This was probably made possible by his understanding and application of Christian mysticism and Eastern thought. His shortcomings in this respect probably caused the ultimate failure of Brouwer's program for a supposed ethical pure mathematics. The failure of the program led among other things to the purely instrumental (and commercial-industrial) mathematics of algorithm theory for computers (cf. the reference to Paolo Zellini above). Nevertheless Brouwer's shortcomings appear to me as incomparably milder than the misandry or present ongoing and tragic misunderstandings of masculinity corresponding to the animus. This goes on in branches of modern feminism, as I suggest in a writing about the case study of a renewed glorification of SCUM - "Society for Cutting Up Men", and it is a feminism that is closely related to the spread in Western societies of a sort of feminization that critics denominate as cultural marxism and political correctness. An unraveling of the socio-psychological implication of these problems will probably necessitate a study and development of the insights of the previously mentioned book The Myth of Analysis by James Hillman.


We should not forget, however, that for our purposes of connecting Brouwer's criticism of science to the embodiment of mathematics and logic (in computers) the most relevant aspect seems to be the phenomenon of projection in science, mentioned above in the context of his supposed misogyny. While it can be said that modern science consists in differentiating the observer's own projection into nature (cf. astrology and alchemy) from what it is "actually there", the reader will have observed in the present text up to here that the term projection used by Brouwer appears in several forms such as projection into nature of mathematical structures, as (synonymously) act of externalization, projection into reality, projection of phenomena on measuring instruments, or mathematical projection. This is also related to the question of mathematical-logical isomorphism with "reality", that in my view is the object of complex misunderstandings as exemplified in a dissertation by Marius Cohen on Reality as a Mathematical Structure, in contrast to ambitious attempts as by Krister Renard in his essay on Predicate Logic and on (in Swedish) mathematics with a corresponding video of a lecture (in English). This will be the object of a future section.


Continuing our concern with chapter 6, should be noted that it is dedicated to a rather detailed analysis of various artful, often false, forms of communication beyond science and ordinary language, as in literature of epics, comedy and tragedy and music addressing the illusion of time, and visual arts (that at the time, without multimedia) addressing the illusion of space, in order to convey what the author calls immanent truth. It is often not perceived that such an idea has already made its presence felt in the ramifications of computer science. It has happened, e.g., in the fashionable recourse to musical "improvisation", or "architecture" and "design" as a source of inspiration in the study of applications of late computer technology or multimedia, prompting my own attention in papers dealing with the Kantian classification of arts as well as his critique of judgment.


Chapter 7 on "transcendent truth" follows the failures of the search for immanent truth in chapter 6 and is said to represent the Kingdom of God. It also represents Brouwer's courageous confession of the possibility of moving from the daily world to his realm of faith and his distancing from the world. This world today, one hundred years after his writing, is explicitly and visibly dominated by the dogmas of religious secularization. It purports to show the apparent impossibility to live a pious and genuine, intransigent Christian life in a modern society where "God is dead" and all transcendent truths such as in music, visual arts, literature and poetry are submerged in contrafactions or fake products. The only solution would be a commitment to sheer mysticism that relinquishes the hope of language communication. As Brouwer writes, adding later quotations by the mystic Boehme and the Bhagavad Gita (that, together, with Hindu and Chinese philosophy, also is symptomatically referred to by West Churchman in his The Systems Approach and its Enemies) such a commitment offers the link to our original question of computation and systems:

It treats the questions posed by metaphysics, such as immortality, freewill, the meaning of art and religion, and the foundations of morality, as riddles hatched by the intellect; in doing so it removes all mystery and yet shows the impossibility of solving such questions by reasoning.

Chapter 8 on "the freed life" seems to depict the final result of above commitment to mysticism: refraining from debate or the preaching the truth, in terms that recall the commitment to quietism mentioned above, apparently implying also care in contact with society's current political correctness:

For a man therefore life will move toward absolute solitude [...] he listens carefully and waits patiently for the revelation of inner contradictions of that intellect [...] begins to resent all his links with society; he is forced to exercise extreme care in human company [...] disturbing influences will only help his patient move away from human society...

Brouwer's style of writing shifts thereafter towards poetry or description of a dream that is sheer mysticism and, for modern understanding, requires the help of depth or analytical psychology, especially in terms of the meaning of "introversion" or, more specifically, thinking-introverted attitude (as popularly described on the net) as related to the process of individuation which required a plunge into Carl Jung's Collected Works, mainly vol. 6 on Psychological Types.

Ultimately, one could say that the coupling of the critique of language in chapter 5 to the references to mysticism in the following chapters 6 to 8 may resume the essence of the "Word" and its connection to philosophical and theological "Logos", as famously outlined by Thomas Aquinas in the first lectures of chapter 1 of the Commentary on the Prologue of the Gospel of St. John (recently translated into Swedish as Kommentar till Johannes-Prologen, (Svenska Katolska Akademien, 2015.)


Chapter 9, the last one, on "Economics" refers to "one more evil", describing the "free life" in detachment to a fallen world, in terms that most readers will most probably label as pessimistic and misanthropic. It can also be understood, however, as a extremely blunt interpretation of the Bible's "fall of man", as the biblical interpretation of Genesis chapter 3, combined with a questioning and mocking of modern man's ambition to rescue this world by means of economic (profitable) technology and own concepts of justice. The text is hard to grasp against our background of modernity but, for instance, the concept of a ridiculous justice pending towards the "socialism" of Brouwer's time, has its counterpart in Blaise Pascal's Pensées (examples: pages of the original manuscript 69, 165, 169, 453, or, in general, the section V in the French edition, Nelson Éditeurs, 1955. English translation with introduction by T.S. Eliot, at project Gutenberg, here, see Index, "justice" and "injustice".)


Special importance for the purposes of this present paper of mine about computers is the example of Pascal's scientific interest as mathematician and physicist who defended the scientific method and designed a mechanical calculator as precursor of computers. As he explains in the preface of the Pensées, however, he relinquished these interests in favour of the perceived premises for his scientific and engineering work, acquiring the posthumous status of "Christian philosopher". Throughout Étienne Périers preface to the Pensées' French edition of year 1955, appear considerations that recall Brouwer's extremely quietist thoughts, which permeate this last chapter 9, also recalling the fundamental Christian message of Thomas à Kempis' The Imitation of Christ.


The Italian translation of Brouwer's book surveyed above includes an erudite commentary (not yet translated from the Italian) that borrows particular interest from the fact that it is written by the previously mentioned numerical analyst, and professor of mathematics Paolo Zellini. He is widely known in Italian cultural world for books beyond the narrow understanding of his discipline, relating to the history, philosophy and essence of mathematics, some of them having been translated into other languages including English and German.


Zellini starts surveying Brouwer's initial motivations for the study of mathematics, as they were related to his religiosity and early commitment to the Remonstrantse Kerk. He sees that Brouwer perceives that faith requires the historically famous and controversial sacrifice of the intellect. Brouwer perceives that the so called intellect (that Jung would criticize for being equated to the rationality of the ego, disregards the whole psyche) deserves suspicion because of its failure in this human world, a failure that we today would relate to cultural impact of technology and environmental issues such as climate change. This perceived failure, apparently disregarding the "success" of science, is exemplified by the corruption of instrumental science put at the service of commerce and industry, and, in general, of petty human desires residing in the "head" as site of this undefined intellect. Such insights would have led Brouwer to a basic ascetic and solipsistic, mysogynous vision of life that was to be reflected in his lifelong way to conceptualize and do mathematics.


For my purposes here it is important to note once again the absence of a discussion of what is to be meant by intellect and its seat in the head that today would be most probably equated further to the brain, in the spirit of what has been critically called for brain mythology. It is a term that was duly considered by Jung himself on several occasions (especially in his Psychological Types, CW6, §516) and it returns nowadays in other purely scientific contexts such as in the journal Brain Structure and Function. It reminds that neuropsychology presupposes informatics in terms of understanding of what data and information is as well as understanding of the relation between (space?) structure and (time?) function such as presented in West Churchman's The Design of Inquiring Systems (chap. 3, pp.43ff.), a work that indirectly introduces also the relevance of Carl Jung. Occasionally some few of these aspects appear in critical articles such as in Jorge Ibáñez, Is the brain a computer?. All these considerations seem to be ignored in "strategic neuro-research" or "computational brain science", as suggested in modern or postmodern mathematical dreams and speculations recalling brain mythology. It is a worldwide network also exemplified by key names such as Florian Markowetz and it can be seen as sort of trendy computerization of old basic ideas of mathematical biology into "computational biology" that merges with the powerful rhetoric of "artificial intelligence". The press reports that computers will be developed with en "inbuilt own will", while I have met engineers who with support of certain philosophers deny that humans themselves have a free will. This is a field originally expressed by Nicolas Rashevsky, and developed later by his followers within a mathematical-logical concept of systems diverging from Churchman's ideas and represented by Robert Rosen and Herbert Simon. Simon, in particular, is an excellent example of how a mathematical mind can appear as a polymath or type of renaissance genius by mathematizing basic knowledge in several different fields of knowledge. It is one example of the dazzling effect of mathematics (and logic) that today is experienced in its embodiment in computer technology.


In Zellini's interpretation Brouwer's perspective is cast in a sort or apocalyptic dress that makes it natural to accept the basic impossibility of changing or improving the world. It opens only the ascetic option of silencing desires and sensations that imprison man in the logic of an alienated world. Zellini remarks vividly how these thoughts of Brouwer embarrassed his entourage who tried to silence them, starting from the publication of his dissertation and up to the posthumous publication of his collected works. This reminds me of society's attitude to analogous peculiarities such as Isaac Newton's or Carl Jung's misunderstood interests for alchemy and, in general, for what is nowadays seen as occult studies. In my own experience I met the same attitude in my function as faculty opponent according to the canons of a Swedish doctoral dissertation. The event, concerning the relations between computer science and gnosticism has been well documented and available for the readers' understanding the dynamics of such attitudes.


Brouwer's quotations from intellectual signposts such as Meister Eckhart, Jacob Böhme and Bhagavad Gita were considered embarrassing, despite of their portrayal of what Zellini denotes as the climate of philosophical introspection that mathematics was going through at the transition between the 19th and 20th centuries. It was a matter of contrasting a specifically mathematical thought and the nature of the intellect and human will, as it was being done for decades in Europe because of uncertainties and obscurities raised by research on the logical foundations of number and mathematical infinity. Brouwer's theses of Life, Art and Mysticism were kept unchanged in his more mature and creative thought during the following decades. They influenced Brouwer's choice of theoretical presuppositions and helped to clarify the concept of algorithm that supports all informatics and scientific calculus since the design of the first big digital computers around the fifties. Zellini sees as equally important his so called Fixed Point Theorem that today is a necessary presupposition of all computational and applied sciences.


Significantly, Zellini concludes a subchapter of his essay in the Italian book translating Life, Art and Mysticism musing over that Brouwer would not have welcomed the forthcoming applications of his achievements that paradoxically both implemented and at the same time betrayed his deepest intentions. In Zellini's words: "But this surely was not the only occasion in which the unforeseeable course of history ended in overthrowing the meaning and purpose of a project." My own reflection upon that is that this overthrowing or at least the illusion and delusion are a hint about the defective psychology if not also theology and consequently politics underlying Brouwer's thought. A symptom may also be the paradox implied by his obvious use of his own intellect outside mysticism in order to discuss and counter an undefined or misunderstood intellect in the practice of mathematics.


Zellini continues, however, showing how Brouwer can be understood as working out his own intellect in order to free it from those dimensions that chain it to the external world that is governed by the causal links of necessity and evil. In this Zellini performs a tour de force by adducing Plato (e.g. Phaedo, 83 a6-b1), Jakob Böhme, Meister Eckhart, Simone Weil, H. de Balzac, Bhagavad Gita, Orphic tradition (on the relation between Time and Necessity), Émile Boutroux, Luther's thought, Augustin of Hippo, and coupling to Hebrew mysticism as well as to Hegel's philosophy. In this respect Zellini treats Brouwer's process of developing mathematics in a analogous way to how Carl Jung treated the development of psychology, i.e. taking into account the historical records of West, without explaining away philosophy, apparent "occultism", and religion as background of the obscure modern, supposedly scientific "intellect". That is: it is a question of taking into account the concerns and efforts of past generations as much as today's environmental thought about e.g. pollution and climate change in supporting claims to take into account the lives of future generations.


Zellini recalls that it is the effort of interiorizing basic human thought that motivates Brouwer to ground mathematics upon the intuition of time, suggested by Immanuel Kant in his Critique of Pure Reason (p.91, B: 153-156) to be the main condition for the intuition of ourselves, of our internal state. I suggest that those who want to muse upon Kantian bases of Brouwer's mathematical thought direct themselves to the mentioned Critique in the section on the Method of Transcendentalism, chap. 1, section 1 on The discipline of pure reason in its dogmatical use (p. 465ff., esp. 471, A:721-725; B:749-753). It is the intuition of time that when leaves the interiorizing effort and relates to the intuition of space in the observation of objects in the real world raises sets of sensations related by an intuition of (chains of) causality. The extension of the causal nexus into various sets of sensations supplies a mathematical theoretical repertory of sequences of data that only need the opportunity of being projected into the real world. It is this projection that justifies the apparent success of applied mathematics, which, however, programmatically ignores (until further notice of new problems) whatever is extraneous to the chains that happen to have been observed and considered. This requires (my note) the so called control of the so called "environment" if it is going to "work", and where such environment includes forced human behavior including its operation of machines and computers where uncontrolled or uncontrollable aspects (of behavior) are called "errors". They are then said to be due to the "human factor" as seen and defined by the implicit occasional observer who happens to have the interpretative prerogative. All this includes also the phenomenon of elders having to buy and with the help of children to learn to use "survival kits" in the form of mobile phones in order to manage details in their daily lives.


Following the post-Kantian developments with new non-Euclidean geometries (Bolyai, Gauss, Lobachevsky), Zellini remarks that the human intellect is organized for the elaboration of various geometries, independently from our daily empirical reality. Brouwer removed from the empirical space all connotations of mathematical necessity. Mathematics exists independently from any external experience and the latter is completely independent of mathematics. Consequently mathematics as well as geometry with its Cartesian reduction to arithmetic is independent of externally perceived Euclidean reality. Mathematics is best developed on the basis of a space-independent inner reality and on a willful action deployed in time, displaying succession as the form of the principle of reason in time. Numeration and calculus would be brought back to successions (cf. algorithms, below) that could be maintained free from externally raised necessity or fear , free from ambition and desire and from the ballast of real object. Compare the symptomatic paradoxical attraction exerted in computer addiction and especially by computer gaming [my note]. In all this sort of play and its mysterious oscillation between internal and external, Zellini emphasizes Brouwer's attribution of it to the role of the whimsical free will, and its mysterious oscillation between the knowing and acting Ego, internal and external, self-conscious or seduced by power. In the context of will, Brouwer's conceptions seem to be influenced by the philosophy of Schopenhauer and I find it interesting for my purposes here to note that Carl Jung's psychology in its unraveling of the concept of will gives due consideration to Schopenhauer: in Jung's collected works' General Index (CW20) there are more than 80 entries for references to his work.


Zellini repeatedly emphasizes Brouwer's distinction between internal and external by means of the original intuition of "bi-uniqueness". It consists in retaining a first sensation together with a successive one in order to set up a first distinction between present and past. Zellini also reminds that the repetition of this mechanism stands at the basis of the concept of iteration and, consequently of algorithm, adducing a relevant reference to a classical paper by A.N. Komolgorov and V.A. Uspensky On the definition of an algorithm (American Mathematical Society Transactions, 2nd series, XXIX, 1963.) For our purposes here, and for an understanding of internal-external with respect to the interiority of uncorrupted thought, it is also important to recall that the iterative application of an algorithmic operator presupposes the operator as well as the applicator, i.e. a stability or permanence that is a sort of "selfhood". Or, as Zellini expresses it, there is an affinity between Brouwer and idealistic philosophy. It was evidenced through the mediation of Hermann Weyl (cf., inter alia, his Philosophy of Mathematics and Natural Science) and expressed by Novalis inasmuch "Selfhood [Selbstheit] is the fundament of all knowledge since it is the fundament of permanence in what is changing."


According to Zellini, Brouwer seems to have equated selfhood with the constructional activity of the mathematical (mathematician's) mind while proving the "existence" of mathematical entities, and thereby invalidating the logical principle of the excluded middle. In general Brouwer was proving that the logical principles in current use cannot be the fundament of truth or of the discovery of truth. We cannot apply logic in a world that is too complex to be dominated by logical laws. In this way Brouwer's criticism of the intellect became a circumstantial criticism of logic. In that context Zellini also shows a criticism of the mathematical continuum (such as in set theory or in topology) that becomes more "digital" and ultimately betrays Brouwer's own intentions. Ultimately it leads, under the pressure of the military and industrial establishment, to the abandon the personal constructional element and allowing itself to be automatized. I wish to complete one of Ludwig Wittgenstein's sentences inspired by Brouwer and quoted by Zellini: "Mathematical logic has completely perverted mathematicians' thought, letting go for structure of facts what in reality is a superficial interpretation of the forms of our everyday language." I wish to add that it is this fake "analysis of the structure of facts" that we nowadays gradually embody into the software of computers in our computerized society.


In following his own thought further, Zellini (p. 176) adduces Weyl's interest for the subjectivism of G. Fichte's philosophy that problematizes what is external vs. internal to the observer, recalling in me as reader what in Jung becomes the relation between the Ego and the Self. Fichte in his The Vocation of Man exposes the principle that exteriority is only the mode in which things get articulated in our conscience.


All these considerations and Zellini's further discussion of the concepts of mathematical continuum and infinity indicate their import in set theory and the attempts to order reality by means of enumeration. I see this as a reminder of the concept of mathematical systems and, further, the elaboration of difference between them and the concept of system itself as exposed by Churchman in his book The Design of Inquiring Systems. It includes a discussion between what is internal and external. This is done in terms of a problematization of the concept of input (p. 107) and, ultimately, by means of references to Carl Jung (p. 244f., 261-263, 272, 277) and his conceptualization of the psyche and the Ego vs. the Self. Some clarification of the references to Jung can be found in Churchman's later work on The Systems Approach and its Enemies (p. 130ff.).


My conclusions from Zellini's valuable commentary of Brouwer's work is that it indirectly uncovers details of the reasons for its paradoxical failure with regard to his deep personal commitment - and this despite its important intuitions about the apparent success of mathematics in science in general, and computer science in particular. This failure reveals that the flight from a misleading causality cannot be resolved by a flight into misunderstood internality. According to developments in psychology conceived after Brouwer's formative years, the so called internality can also suffer from irruptions of "external" forces that pervert the quest for the Self.


If we are to complain for the complexity of the issue, it must be understood that it is complex and the fault is ours, in our wishful thinking that it must and can be philosophically simplified mirroring the military-industrial, social-engineering, techno-science, and technostructure frames of mind that often unconsciously dominate our secularized big science culture. To give a taste of what this all is about I will finally take the following excerpt from one comment in Amazon's presentation of Zellini's translated book A Brief History of Infinity, an infinity that today is perceived as such in the computerized world-wide-web but is supposed to be solved by means of an unknown algorithmic power.

"Paolo Zellini's sources are wide ranging, almost intimidatingly so. We readers encounter the philosophical thoughts of the Platonists, Aristotle, the Pythagoreans, Anaximander, the Chaldeans, Duns Scotus, St. Thomas Aquinas, Giordano Bruno, Nicholas of Cusa, Raymond Lull, Descartes, Leibniz, Goethe, Kant, Hegel, Russell, Simone Weil, Quine, Popper, Wittgenstein, and many others. Similarly, on the literary front we meet Cervantes, Kafka, Borges, Musil, and others. Mathematicians are prominent also. Zellini discusses the provocative ideas of Descartes, Newton, Leibniz, Dedekind, Poincare, Cauchy, Weierstrass, Bolzano, Frege, Du Bois-Raymond, Cantor, Russell, Whitehead, Gödel, Von Neumann, Zermelo, Skolem, Brouwer, and many others."

To appreciate the difference between Zellini's and other's range of interests and corresponding possible insights, we may consider the alternative approach represented by Rudy Rucker in (on the net) Infinity and the Mind: The Science and Philosophy of the Infinite (1982-2005), and published by Princeton Science Library (2004).






This section is based on Miriam Franchella's article "On Brouwer and Nietzsche: Views about Life, Views about Logic". History and Philosophy of Logic, 36:4, 367-391 (2015). The article consists of a comparison between Brouwer and Nietzsche regarding in their views in five respects: attitudes to life, science, mathematics, and logic. It relies heavily on original works including several other studies of Brouwer and Nietzsche. I will not dwell more deeptly into the matter except for stating my conviction that the whole complex issue is related to the historical division of logic between formal and material logic.


Nietzsche is seen by Franchella as well aware, as Brouwer is, of the suffering in human life. Nevertheless he declaimed a joyful positive conception of it "with an aim of enhancing and defending the disruptive nature of life in its originality and creativity, from those who wanted to downgrade it to mere survival, scheme, rule." He would observe mathematics and science including physics within this life-promoting and survival framework. Because of this, and differently from Brouwer, Nietzsche sees it important to dominate the world. For him it is important that the "intellect" sets a "subject" before "objects" that are generated or named by its metaphors, connecting them to one another with causal links. Metaphors mediate between physiological perception and imagination, from image to sound to word. Language maintains a useful yet fictitious "reality" in daily life, (that for Brouwer consisted in maintaining two or more wills cooperating in a single direction.) What we perceive is only some part of the continuous flowing of reality. "Causality is appearance. Our belief in causality is due to our faith in strength that we experience when we do something, yet force moves nothing; the force we feel does not 'put our muscles in action'. Causality is only created by thought, which introduces constraint in the process of succession."


So, Nietzsche is seen as criticizing the alleged truth of science in a way that has similarities but also differences with respect to Brouwer.: the scientific world is made up of "fictitious entities interrelated in a fictional way for the sake of survival", while truth (the thing-in-itself) would be the absolute, the non-perspective, the non-anthropocentric. For Nietzsche the content of science is an epistemological lie regulated by society, and such knowledge or lie, like mathematics and logic, is finalized, the result of perspectives. In its claim to truth science is a faith, and as such, it can be likened to religion. From Franchella's relate it seems obvious that Nietzsche abhors religion mediated by priests. I see that he rather adopts science itself as his "religion" where survival (until further notice?) is his sort of god. I am tempted to regard postmodern or smartly denominated "non modern" (with the purpose to escape critics of postmodernism) scientific intellectualism as being influenced by these Nietzschean thoughts, as represented, for instance in France, by studies in the tradition around Bruno Latour, about the social construction of science. It would explain the puzzling de-emphasis on the question of truth in science and on normativity, on why and how to do or not to do science, in favour of a dazzling brilliance in aestheticizing descriptivity, celebrating creativity and survival in the form of career success. It is here we find the basic difference between Nietzsche and Brouwer.


Nietzsche is seen as not devoting much attention to mathematics. What is interesting is that for him it was based on the concept of the "Ego" which is configured as a unity, and on the existence of identical objects. This is so because Nietzsche as quoted from this Posthumous Fragments notes that we have borrowed the concept of unity from our "I" concept, our oldest article of faith which, as seen earlier in this text, has been definitively questioned by later analytical psychology. And in his Human, All too Human: "The invention of the laws of numbers is made on the basis of the original error that there are several similar things (but there is in fact nothing equal), at least that there are things (but there is no 'thing'). The presumption of multiplicity always presupposes that there is something which manifests itself in many instances, but the error here is already being felt, already there are beings or units which do not exist." Nietzsche is quote again stating the our feelings of space and time are false for they lead to logical contradictions, despite of the results getting perfect certainty in their relationships: "One can build on them - until the last end, when erroneous basic assumptions get in contradiction with the results, e.g. in the theory of atoms." [My translations from the German quotations.]


Not having read Nietzsche's whole original text and not fully understanding the reasoning behind all this, I happen to think about quantum physics. Several physicists colleagues of mine have confessed that it "works" but that they do not understand it and believe that nobody really understands, whatever understanding really means. A popular review that is appropriate for the general reader of this paper of mine is found in "Relativity and quantum mechanics: the battle for the universe" in The Guardian, 4 November 2015. An overview is also to be found in sections of Wikipedia's entry on "Physical paradox" where there are references about piecemeal ad-hoc mathematical manipulations "conveniently sidestepping the philosophical issue of what actually occurs". So, truth has been relegated to philosophers or, rather, to conflicts between philosophical schools in the name of the idea that "it works", possibly meaning that e.g. nuclear bombs really explode, i.e. far from the scientific, pragmatist meaning of "implementation" that depends upon mutual understanding.


Franchella goes on noting that Nietzsche extends the earlier mentioned insights to geometry that "is also based on an untruthful leveling of sensory experiences: we introduce them into reality to make it static, predictable and controllable so that it becomes reassuring" or, in Nietzsche's own words (still in Human, All Too Human): "The same is true of mathematics, which would certainly not have arisen if one had known from the beginning that there is no exact straight line, no real circle, no absolute quantity measure in nature."


So the difference from Brouwer can be inferred by what was shown above about his reliance on Kant's view of foundations of mathematics, but like Brouwer Nietzsche is seen concerned about the morality of mathematics as part of human knowledge. Nevertheless it is not judged so negatively since, like science and logic, it requires honesty and discipline in introspection in its practice, and is a useful lie which is regulated by (a trusted!? - cf. marxism) society, becoming acceptable to be transmitted as truth. What I do not see discussed is the problem of the endorsement of utilitarianism that is implicit in the appeal to usefulness.


Regarding logic Nietzsche notes that since its Aristotelian conception it assumes the identity of (self-identical) objects, that things dealt in logical sentences are or remain equal in the course of reasoning and application of its conclusions or, as I understand it, that they are stable even if we perceive that they are in continuous change. This means that the will to logical truth can only be carried out after a fundamental falsification of all happenings. Nietzsche sees that from this follows that there is an "instinct" which is capable of both means, first the forgery and then the implementation of a point of view: logic does not come from the will to truth.


I see the mention of "instinct" as an implicit reference to concepts that would later be developed in analytical psychology, justifying that references to Nietzsche occupy more than a whole page in the Index volume (CW20) of Jung's collected works. This is emphasized further in Franchella's quoted German text: "The course of logical thoughts and conclusions in our present brain corresponds to a process and struggle of instincts, which in themselves are all very illogical and unjust: we usually only experience the result of the struggle, so quickly and so hiddenly is this ancient mechanism now playing within us." My translation of Nietzsche's quoted German text goes on considering the origin of the logical, suggesting that it had a survival value in the course of evolution, in the sense (as I understand it) that for survival it is better to shoot first and think later, i.e. to assume likeness and oversee details, or practice trial and error rather than to hesitate in doubts and subtleties. Franchella quotes and expands on a sentence in German: logic soothes and gives confidence. "It does not emerge from objective data in our possession but it is a pattern that we sneak into the continuous flow of perceptions in order to control it. It is, like science and mathematics, a useful simplification to avoid drowning in uncertainty or anxiety but it is not 'true' in the profound sense of the term." In the course of academic debates about which I did write a particular debate-essay, I have sensed this all in terms of an unwillingness of people with logical minds to question definitions of fundamental terms they happen to use, which often amounts to an unwillingness to question the basic assumptions of their logical trains of thought.




Perhaps this is the place for making a pause in order to see how presuppositions of logic, like "instincts" or "struggle of instincts", also appear in the now academically popular field of phenomenology such as in the discussion of the "Origins of Geometry" found in Edmund Husserl's famous The Crisis of European Sciences (1970). Let us disregard for the moment that geometry after Descartes belongs to mathematics rather than logic. Husserl writes (p. 377) about intersubjective meaning and "the same merely factual presuppositions of understanding":

It is a general conviction that geometry, with all its truths, is valid with unconditioned generality for all men, all times, all peoples [...] The presuppositions of principle for this conviction have never been explored because they have never been seriously made a problem. But is has also become clear to us that every establishment of a historical fact which lays claim to unconditioned objectivity likewise presupposes this invariant or absolute a priori. Only [through the disclosure of this a priori] [interpolation by phenomenologist Walter Biemel] can there be an a priori science extending beyond all historical facticities, all historical surrounding worlds, peoples, times, civilizations; only in this way can a science as aeterna veritas appear. Only on this fundament is based the secured capacity of inquiring back from the temporarily depleted self-evidence of a science to the primal self-evidences. - Do we not stands here before the great and profound problem-horizon of reason [...] a root in the essential structure of what is generally human, through which a teleological reason running throughout all historicity announces itself [...] grounded upon the foundations of the universal historical a priori [...] leads further to the indicated highest question of a universal teleology of reason. [My emphasis.]

For purposes of honesty I must account for the fact that after the appendix on origins of geometry it is Eugen Fink writes in Husserl's book a critical appendix on the Problem of the "Unconscious" that, however, I do not know whether Husserl himself would have approved despite his original mathematical mind. Fink himself considers initially an objection to his text that follows, that is tries to interpret the "un-conscious" according to the methodical means for understanding consciousness. But he inverts the objection observing the ever growing tendency of "depth-psychology" (a term used for gathering the theories and therapies associated with the names of among others Sigmund Freud, William James and Carl Jung) to conceive of consciousness as a mere stratum of the concrete man and to oppose to it other dimensions of life not traceable to consciousness.


Fink does not comment the philosophy of the unconscious but states that the above study of the unconscious is philosophical naïveté consisting of an omission, because one thinks that one is already acquainted with what the "conscious" or consciousness is and dismisses the task of first making into a prior subject matter what consciousness is. This would show our illusion that consciousness is something immediately given, while the "intentional analysis" of phenomenology destroys this illusion and leads one into a science of a new sort "where one gradually learns to see and grasp for the first time what consciousness is". As an account of intentional analysis puts it: "The researcher mentally examines the object or state-of-affairs by taking different imagined perspectives and making modifications, like adding or subtracting or changing different features, to determine what is essential to the meaning and what is not." For a "naïve" layman this sounds like a mixture of ideas in introspection and meditation or, for me, this looks as a paradoxical naïve attempt at an individualization of the "Hegelian inquiring systems" as expounded and finally published in the earlier mentioned The Design of Inquiring Systems. The main difference from Nietzsche's talk about struggle of instincts is that the phenomenological approach is a search for the universal historical a priori and a universal teleology of reason by means of a so called intentional analysis through the imagining of perspectives and such. Further comments on phenomenology impersonated by Husserl's main heir, Martin Heidegger, are to be found in an earlier essay of mine on Ethics in Technology.




Returning to Franchella, another quotation from Nietzsche's German text (translated by me) that illustrates his conception of the essence of logic, which in turn has had such a large influence on modern western thought, including views on religion:

"As far as the logicians are concerned, I will never tire of underlining a little brief fact which is disinclined to be accepted by these superstitious people, namely, that a thought comes when it will, and not when 'I' want. So that it is a falsification of the fact that the subject 'I' is the condition of the predicate 'think'. It thinks: that that this 'it' is precisely that old celebrated 'I' is, mildly speaking, only an assumption, a claim, above all no 'immediate certainty'".

I see this sentence as a clue to Nietzsche's cultural contribution to the deconstruction of the Ego and its reconstruction" into the concept of psyche or mind that is illustrated further along Franchella's quotations. The purpose was to put the concept of Ego in its proper context, as it is the purpose of so called depth psychology and, in particular, in analytical psychology. Unfortunately Nietzsche's approach and limitations, including his treatment of religion as it appears in Franchella's chosen quotations of Nietzsche, opened the way for its derailment in the direction of postmodernism and nihilism.


Franchella goes further in considering inadequate Nietzsche's criticism of logic as presupposing the existence both of objects and subject as in ordinary language, since "from Frege [considered 'father' of analytic philosophy] onwards, the way in which logic expresses the sentences of ordinary language is in terms of function-argument, borrowed from mathematics and certainly neutral from an ontological point of view." I think that here Franchella, despite of her familiarity with Brouwer's work, misses his main point in that mathematics also distorts reality in its application to the same. Nietzsche himself thought of solving in Greek terms the problem of the shortcomings of "rational Socratic-Apollonian-Sophoclean" logic-mathematics as tools of survival, by balancing them with the help of the pre-Socratic and pre-Euripides Dionysian kind of creative art-aesthetics (that today is represented by the trend towards "design"): "Perhaps there is a kingdom of wisdom from which the logician is exiled? Perhaps art is even a necessary correlative and supplement to science?"


Here is not the place to expand on explanations of these references to Nietzsche's thought that can be overviewed in e.g. Wikipedia or in the Stanford Encyclopedia of Philosophy including its entry on Nietzsche's Life and Works. As already stated, my choice is to refer to analytical psychology and its process of individuation, which has incorporated such thoughts and those of the aforementioned Schopenhauer's who also had influenced Brouwer.


Franchella concludes the essay with a very informative final overview and comparison between Nietzsche and Brouwer who partly agree about science, including applied mathematics and logic, seeing it as a biased human product with problematic "sinful" morality. This is so because for both of them it consists in projecting a "useful" pattern of causality onto the world or in fostering survival and human control of other humans and environment. Nietzsche, however does not consider it sinful as Brouwer does because of its luring us into the sinful world. And yet can be said that also Nietzsche sees it as "sinful" in its one-sidedness, in its wanting, for purposes of survival, to "keep everything under control" at the cost of suffocating creativity as the booming of the vital, the individual, the unique, the special. Brouwer solves the sinfulness by means of abstention or retreat from the world and from any "useful" application of mathematics. He sees doing mathematics as a sort of "introverted" intuitive and in this sense artistic activity. Nietzsche solves his own brand of sinfulness by advocating an "extroverted" balancing of scientific with artistic activity, which recalls a vague resemblance to the idea of "bicameralism", which I prefer to see as a balancing of psychological functions in the process of individuation.







In texts about language it is typically stated that the study of language is often divided into syntax, semantics and pragmatics. A similar division except for pragmatics appears in computer science where the construction and application of computer programs are investigated. The distinction between syntax (sentence form) and semantics (word and sentence meaning) is said to be fundamental to the study of language. Syntax is seen as the collection of rules that govern how words are assembled into meaningful sentences.


Despite of the frequent "dogmatic" use of these terms it is seldom mentioned what is the source of this classification and the symptomatically confusing interrelations it implies are exposed in the Stanford Encyclopedia of Philosophy under the ambitious label of "The Structure of Scientific Theories". An understanding of the source is found in the study of semiotics and its history, relating the whole matter to Charles Sanders Peirce as exponent of a branch of the philosophical pragmatism that through William James and Edgar Singer Jr. influenced the formative years of the previously mentioned system theorist West Churchman. It is interesting to note how syntax, syntax, semantics and pragmatics are more systematically defined by the "father" of semiotics, Charles Morris (1901-1979) as rendered in Wikipedia:


(1) Syntactics/Syntax: relations among or between signs in formal structures.


(2) Semantics: relation between signs and the things to which they refer; their signified denotata, or meaning.

(3) Pragmatics: relation between signs and sign-using agents or interpreters.


What I wish to show here is how the vain search for truth in these contexts evidences what Brouwer objects to regarding the merging of mathematics with logic in syntax, which is the most common concept mentioned in computer science, and consequently the shortcomings of the "secondary" problematic talk about semantics that is the subject treated below, while pragmatics is left completely vague. Consequently the whole division and construction of the tripartite syntax-semantics-pragmatics can be suspected for anonymously doing away with or relativize the concept of truth under the label of "semantic theory of truth".


The Polish-American logician and mathematician Alfred Tarski who dedicated to mathematical logic felt the necessity to try to clarify the concept of truth that evades the typical syntax studies of the computer field. For this purpose he wrote a noteworthy and symptomatic paper that in the English translation has the title The Semantic Conception of Truth: and the Foundations of Semantics(in Philosophy and Phenomenological Research, Vol.4, No. 3, March 1944, pp. 341-376, review in the J. of Symbolic Logic, vol.9, nr. 3, 1944). Tarski initially defines semantics as a discipline which, "speaking loosely, deals with certain relations between expressions of a language and the objects (or 'states of affairs') 'referred to' by those expressions". Without defining language except for presupposing only formal languages, he then goes on in the following 30 pages circumambulating around the concept of truth in examples of series of logical sentences. This includes the use of "recursive procedure" in the definition of "satisfaction" for the definition of truth, but without some sort (of Brouwerian?) reflection upon the concept of recursiveness. Before this the author states that "the problem of definition of truth obtains a precise meaning and can be solved in a rigorous way only for those languages whose structure has been exactly specified. For other languages - thus, for all "natural, "spoken" languages - the meaning of the problem is more or less vague, and its solution can have only an approximate character." The conceptual bewilderment of this formal approach to semantics can be easily overviewed in what is usually considered as semantics in computer science, where pragmatics appears under the name of "implementation" of the function of the programs - i.e. that they "work". This prevents from seeing that not only programming languages but also particular software products or programs should be considered as (axiomatic) theories about what (activity) is being computerized.


We recognize in all this Brouwer's reservations against the "truth" or validity of logic applied to this world, which should include computers and computerization. It is therefore symptomatic that it is difficult to see a conclusion in Tarski's essay, except for exercises of logical interchanges between syntax and formal semantics. There are only some final remarks: he declares having doubts in connection with the evaluation of scientific achievements in terms of their applicability. He believes "that it is inimical to the progress of science to measure the importance of any research exclusively or chiefly in terms of its usefulness and applicability." Furthermore: "It seems to me that there is a special domain of very profound and strong human needs related to scientific research, which are similar in many ways to aesthetic and perhaps religious needs. And it seems to me that the satisfaction of these needs should be considered an important task of research." Paradoxically, the author'js final words after these final remarks are a statement that (nevertheless) nothing of this is relevant to the content of the article itself, showing that the honorific link between science, aesthetics and religion is just only honorific or, rather, only nominally ennobling the type of science he represents.


These early developments that we find in Tarski show up in plenitude how seldom we come to more elaborate discussions of the ontology of logic. The problem already appears in what was mentioned above in the beginning to this essay when we mentioned contributions by Y. Bar-Hillel who was an acknowledged inspiration for defining the concept of data and information in the original school of Swedish Theoretical Analysis of Information Systems by the grand old man of computer information systems, prof. Börje Langefors. This affected the latter's conception of what data, information and systems are or how they should be defined, showing up in the basic phenomena of how the use of computers is perceived as obliterating the concepts of time and space in terms of simultaneity and ubiquity. In fact, despite of mathematics and logic being able to manipulate, as in physics, these concepts of time and space - they are absent in their theory. In this respects they recall the psychic phenomenon of dreams, and suggest a reason for the suspension of consciousness as well for the phenomenon of dependency upon computer gaming as related to virtual reality and augmented reality. In fact, the analytic psychology of Carl Jung was considering such related phenomena in an essay named "The dreamlike world of India" (Abstracts of the Collected Works of C.G. Jung, ed. C.L. Rothgeb et. al., orig. in CW10, Civilization in Transition). From the professional point of view of computer and information science, however, these matters are reflected in the Tarski/Bar-Hillel/Langefors attempts to work with concepts of information other than those in mathematics and logic. This was attempted through the introduction of syntax-semantics (and rather neglected pragmatics), and the consequent introduction of so-to-say "molecules" of information (informational elementary "messages") composed of "atomic" units or terms of an elementary message of information were the object or entity (identifier) and therefore space, the characteristicum (property part composed of variable type and variable value), and the time of measurement (or time during which the object is affirmed or predicted to hold the characteristic). In the spirit of the Churchman-Singer teleological theory of measurement (cf. The Design of Inquiring Systems) I supplemented or complemented them (in my doctoral thesis om Quality-Control of information) with the error term. (Details in my papers "A subsystem in the design of informatics", and "The systems approach to design", the latter in Information Systems Frontiers, Vol. 3, No. 1, 2001).


Leaving this aside but subsumed, a review of the ontology of logic will now be considered in particular in the form of a review of Nietzsche's approach that we already considered above in Franchella. This is the case of an article by Steven D. Hales with the title "Nietzsche on Logic", in Philosophy and Phenomenological Research, Vol. 56, No. 4 (Dec. 1996), pp. 819-835.


Hales starts with a valuable summary of Nietzsche's criticism of logic that is similar to the summary at the end of Franchella's text surveyed above. His particular contribution is that he purports to show the paucity of secondary literature criticizing Nietzsche's criticism and to show that Nietzsche's original criticism does not detract from the value and necessity of logic. He does so starting from an unstated position that seems to be well represented by Tarski, above. This means that in contrast to Franchella he does not consider Brouwer and his relation between logic and mathematics.


Hales notes that Nietzsche's own knowledge of logic "seems a bit quirky" since he was certainly aware of only traditional Aristotelian logic and had read Kant and Hegel and their followers, criticizing them as "philosophical laborers" for showing (in "Beyond Good and Evil) the data of the past into rigid logical formulas. For me this is particularly revealing since it was also my strong impression in reading Kant, especially his Critique of Pure Reason.


But then Hales reveals indirectly his standpoint by noting that Nietzsche (who lived in the second part of the nineteenth century) seems wholly ignorant of the "stars of the nineteenth-century logic". All of them were at the time his not yet famous contemporaries, but are today considered as fathers of modern logic, Augustus de Morgan, George Boole, Gottlob Frege. Hales mentions also John Venn and C.S. Peirce without noting, however, that Peirce's most famous contribution as pragmatics (cf. above) and pragmatism is symptomatically totally absent in what he calls modern logic.


Hales notes that Nietzsche in Human, All Too Human declares that logic rests on assumptions that do not correspond to anything in the real world, and in The Twilight of the Idols finds " of formulae, sign-systems: such as logic and applied logic, mathematics. In these reality does not appear at all, not even as a problem; just as little as does the question what value a system of conventional signs such as constitutes logic can possibly possess." And in The Will to Power: "Logic (like geometry and arithmetic) applies only to fictitious entities that we have created. Logic is the attempt to comprehend the actual world by means of a scheme of being posited by ourselves; more correctly, to make it formulable and calculable for us." And "The world seems logical to us because we have made it logical". Or, further, "Logic is merely slavery within the fetters of language." This latter remark recalls Brouwer's observation that logic is deduced from the structure of ordinary language, as the previous ones recall his noting the mathematical reality of the world is a projection of our own causal perceptions.


With a sense of relief Hales observes that despite Nietzsche's obvious reservations about logic he is aware of its strengths as well, e.g. for drawing correct conclusions (and convincing others, the problem being their premises). Hales, however, sees a contradiction between Nietzsche's view that logic opposes life while at the same time affirming that logic is useful for life. Hales apparently misses that the one "life" refers to creativity while the second life refers to sheer biological survival. Hales also sees that Nietzsche weakens his claim that logic ("and rationality") is necessary for life when he otherwise claims that it is merely necessary for thinking, meaning that one can indeed live without logic and rationality. In my view all this follows from Hales' lack of stringency in defining thinking as well as life. He seems to sense this when he subsequently needs to refer to Nietzsche writing (in On Truth and Lies in a Non-moral Sense) that "everything which distinguishes man from the animals depends upon his ability to volatilize perceptual metaphors in a schema, and thus dissolve an image into a concept." Hales then reformulates it in his own words that so, humans engage in a process of abstraction from sensory impressions to form concepts and demarcate objects, ultimately giving raise to the "great edifice of concepts" which "exhales...logic". So, despite of trying to define thinking, without relating it to sensory impressions and images (even "primordial" ones, as done in analytical psychology), Hales does it in such a manner to be able to conclude that Nietzsche admits that giving up logic means ceasing to think. And this thinking is done in opposition to "mythically inspired people" or "the man of intuition" who live in a dream where "the web of concepts is torn by art" or intuition shatters the existing conceptual edifice with new metaphors, myths, and art. So, to refuse logic would be to refuse (to think under the constraints of) language.


Hales considers on one occasion that it is possible to see Nietzsche, as I do, as attacking formal logic, but endorsing clear argumentation, rationality and thinking unpolluted by superstition. Unfortunately Hales does not pursue what clarity, argument, rationality and superstition ultimately are. He rather goes on giving two reasons against the formal-logic/rational-thought dichotomy. The first one is that Nietzsche does not "clearly" separate the issues of formal logic and rational thought. But, let alone "clearly", at his time there was no express formal logic, and still today there is not consensus about what rationality is. If formal logic had already been available to Nietzsche in his lifetime he would probably have been eager to raise objections against it and about the concept of "form". What happens here is that Hales unperceived takes stand for formal logic. And this becomes more evident in his giving the second reason against the dichotomy: Nietzsche is said to consider the constraint of [natural!] language to be essential for thinking and hence for rational thinking (my square brackets and italics). He also is said to regard logic as the infrastructure of language (while I interpret Brouwer as seeing language as the "infrastructure" of logic in the sense that logic is extracted from ordinary language). And here comes the remarkable conclusion of Hales: "Thus logic as the formal semantics of natural language and of thinking is inextricably tied together for Nietzsche". (My italics.)


So, in trying to pin down "what exactly is Nietzsche's complaint against logic, then?" Hales goes on questioning whether it is true that logic presupposes the existence of things that Nietzsche considers fictitious in their use in logic. Hales recurs to a sort of rhetorical argument in observing that Nietzsche with his "imprecise and rudimentary understanding of logic" used an inappropriate methodology and he would have needed some of the tools acquired in the past century of logical development. "Ignoring what has been learned about logic since Nietzsche's time is simply a Luddite approach to a technical issue." (My italics, is it a technical issue?). Secondly Hales purports to illuminate problems that Nietzsche "could indicate only dimly" by using "the concepts and clarity of modern vocabulary". He claims that Nietzsche in his claims that logic falsifies reality or makes no claims about reality ignores that modern logic is divided into syntax and semantics. Hales notes then that it is the syntactical aspect of logic that is formally aloof from the world and that it is not the business of syntax to worry about reality. Its business is formally aloof from the world: "It provides the rules for the manipulation of operators, connectives, quantifiers, predicate letters, variables, and constants of the formal system, how symbols can be moved around, and how theorems are to be proven from the axioms."


In doing so he recalls in a reader like me some of the observations we made above about Tarski, that are to be renewed in what follows.


Hales reminds that in modern logic the "interpretation" or "meaning" of the formulas of logic is the business of semantics, specifying "domains of entities of universes of discourse, along with an interpretation function that leads us from the symbols supplied by the syntax to the entities in the domain. The function assigns a unique object in the domain to each constant, tells us which things the variables can stand for, and provides an extension in the domain for each predicate letter.


But: what are these mentioned entities, things or objects if not Nietzsche's criticized things? And here comes Hales explanation of how the problem can be explained away in the theory of modern logic. Yes, thingness is a requirement of logic, we need things but the nature of these things is a further question, one that is strictly speaking beyond the purview of logic and more properly the subject of metaphysics or ontology. (My emphasis.)


In this way I think that Hales indeed explains away the whole problem together with the third aspect of language mentioned at the beginning of this chapter, namely the pragmatics of language as the infrastructure of logic. This is so because semantic functions assigned to work out syntactic symbols are assumed to be relating to non-controversial authoritative and objective consensual observers, reality being "out there" in a "Gods Eye" "externalist". This is what Churchman's The Design of Inquiring Systems covers under Lockean inquiring systems, or naive empiricism, raising also the need of differentiating and relating concepts of intuition to concepts of postulation in the philosophy of F.S.C. Northrop (cf. his The Logic of the Sciences and the Humanities, and its use in my PhD dissertation on Quality-Control of Information). And Hales feels uncomfortable in discussing this, observing that "realism and antirealism are the focus of much current debate, and are notoriously slippery terms." The "current" debate refers to the advent of used and abused postmodernism in academic America around the eighties-nineties as considered in an essay adduced in Hale's article, on "Nietzsche's Prefiguration of Postmodern American Philosophy".


Consequently Hales goes on into the need of "an investigation into the semantics of natural language" observing, however, the failure of Rudolf Carnap to consider Nietzsche's concerns. I direct those interested in such investigation to acknowledge my attempts to comment the problems of the influence of Rudolf Carnap and Yehoshua Bar-Hillel upon the field of informatics in Sweden, as presented in my paper on A subsystem in the design of informatics (1995). It is symptomatic that in order to make sense of the above talk on semantics and about the claimed difference between meaninglessness and falsity relies on acrobatic differentiation between so called grammatical syntax and logical syntax. And so on, after a series of manipulations it seems that the ultimate purpose of the juggling with syntax and semantics, and the ignoring of pragmatics is the attempt to make logic into a tool for discovery or construction of truth, where the concept of truth itself is not called into question but, rather, definitionally explained away.


I think that something meaningful arises in these reasonings when (p. 829) Hales writes that Nietzsche contention is that "we categorize our sensory phenomena in a way that suits our ends and purposes." Or (cf. The Will to Power) "humans have simply chosen those interpretations that allow them to live and promote their interests. This is how we have made the world logical, and formulatable and calculable for us."


This is particularly meaningful because it indicates the need to move and enrich the domains of logic and mathematics with a knowledge of their relation to teleology. This is what is done in the already mentioned work of Churchman in The Design of Inquiring Systems where initially, partly in the tradition of C. S. Peirce also mentioned above, the second chapter on Leibnizian inquiring systems does away with the primacy of badly understood logic and is subsequently expanded into teleology, naive empiricism, Kant, Hegel, and the rest. What Hales seems to be doing, however, is to miss the point by quoting the expressions "our ends and purposes", "their interests" and "calculable for us". The core of the problem is instead the question of our, their, and us, that is, who is we, they, and us, who is the (competent according to whom) observer using the logic (and mathematics) and whose empirical findings for whose and what purposes. Despite of not following Hegel, Nietzsche is said to have been well acquainted with this work and this explains his particular sensitivity to the issue as expressed in the emphasis on the perspectival question. And we have not yet considered the hypothesis of the observer as one more "thing" among other things. What about the unity and stability of the observer - considered, for example, in analytical psychology? There was a time in which the "atom" also was conceived as a indivisible and stable unit, prior to its disintegration.


Hales, however, we have seen seems to solve the problem by relegating it to "metaphysics or ontology". Even if things are fictitious, then logic does indeed work with these fictitious things, and (p. 830) " this does not the slightest damage to logic". He finds that, as Nietzsche is quoted to say, logicians rely on faith and are superstition (positing as realities all those hypotheses such as substance, attribute, object, subject, action, etc.), this is a fact about logicians, and not one about logic itself. It would be a curious experiment to compare this declaration with another most frequent nowadays that the problems caused by faith in technology are a fact about the technicians and scientists and not about technology or science itself. It is interesting to realize that in the latter case the escape is guaranteed by claiming that indeed the problems are caused by "the users" of technology, such as politicians deflagrating atomic bombs or buyers and abusers of computers, not by the technicians and scientists. But this escape is obviously not available in the case of logic because of: who are the users of logic?


This is, again, a core idea in Hales' article. What he is indeed revolting against is what I myself also refuse in the interpretation of Nietzsche's provocative declaration quoted from Beyond Good and Evil, that "it is no more than a moral prejudice that truth is worth more than mere appearance". I mean that such an interpretation must be done questioning the meaning of truth, and criticizing Nietzsche himself including his denigration of theology, instead of only criticizing his criticism of logic. Nietzsche's denigration of theology in terms of his stand against Christianity and its relation to Buddhism is a complex matter to be detailed here. But I think that it is especially symptomatic in giving an opportunity for understanding a basic shortcoming of his philosophy, which in turn influenced so many others including Heidegger and "postmodern existentialism". Hales devotes several lines to expose Nietzsche's illustration of non-identity, or non-persistence of individuals or objects, by means of an indirect reference to the Torah commandment "Eye for an eye" (Exodus 21: 23-25), to be hereby differentiated from New Testament's "Love your neighbour as yourself" (Mark 12:31). As I understand it the message is that every action is uniquely tied to one (changing, in flux) individual and cannot, with "the instinct of the herd" which is related to the "herd morality" (see also Nietzsche and Morality) or what today recalls political correctness, be logically attributable to any one in the future, the less so in taking for granted that an injury will be repaid with a injury. Franchella (p. 383) had even illustrated this very same matter with a German quotation from Nietzsche's Posthumous Fragments that I painstakingly translated for myself but because of its complexity I reproduce only in the original German for the benefit of readers who understand it:

Der Calcul ‘thue nichts, was dir selber nicht angethan werden soll’ verbietet Handlungen um ihrer schŠdlichen Folgen willen: der Hintergedanke ist, daß eine Handlung immer vergolten wird. [ . . . ] Dagegen ist der Spruch werthvoll, weil er einen Typus Mensch verrŠth: es ist der Instinkt der Heerde, der sich mit ihm formulirt—man ist gleich, man nimmt sich gleich: wie ich dir, so du mir—Hier wird wirklich an eine €quivalenz der Handlungen geglaubt, die, in allen realen VerhŠltnissen, einfach nicht vorkommt. Es kann nicht jede Handlung zurŸckgegeben werden: zwischen wirklichen ‘Individuen’ giebt es keine gleiche Handlung, folglich auch keine ‘Vergeltung’ . . . Wenn ich etwas thue, so liegt mir der Gedanke  vollkommen fern, daß Ÿberhaupt dergleichen irgend einem Menschen mšglich sei: es gehšrt mir . . . Man kann mir Nichts zurŸckzahlen, man wŸrde immer eine ‘andere’ Handlung gegen mich begehen. (NF=Nachgelassen Fragmente, Posthumous Fragments, 1888, 22[1])

The problem seems to be that Nietzsche extends this mechanism to the Christian message of love that, indeed, in order to be genuine love must not presuppose a retribution with an identical action of love, and consequently it is not a herd mentality. Hales picks up this question in its implications for the explanation of Nietzsche's interest in prudential reasoning and claim that it does depend upon transtemporal persistence.


In her article, Franchella mentions Nietzsche's considering in essay II of the Genealogy of Morals the relation debtor/creditor as responsible for "instilling memory" in humanity, since the former must remember the debt and the latter that he is owed the repayment. The debtor must also remember the he himself is a thing that persists through time or that he is the same person as the one who acquired the debt. Wish to avoid punishment encourages his belief in persistent identity and repayment. "Thus this kind of reasoning leads to a belief in a continuing ego or self, a belief that gets displaced onto other objects and so creates the concept 'thing'." It impresses me is that in a personal debate with a friend of mine who does not believe in free will, he still accepts the need of crime punishment because it encourages the abstention from crime due to fear of punishment. This seems to imply a paradox in that punishment is supposed to both encourage persistence of an ego and the disbelief in the existence of a free will, at the same time as Nietzsche makes an issue of (whose) Will to Power.


It also impresses me in Nietzsche's reasoning his paradoxical use of "he" or "his" belief, which is postulating the persistence of a not only a "he" but especially an "ego or self". The latter are important distinct concepts in analytical psychology, not being a question of ego or self but, rather ego and self. I expect to emphasize towards the end of this essay their distinction as the key to resolving the apparent paradoxes in the discussions up to now.


Disregarding all this, Hales, however, goes on in a defence of his own position on formal logic instead of deepening himself into Nietzsche's criticism. Hales goes so far as to defend truth by means of reference to a need for "plenty of wholesome multivalent logics" and by claiming that interest for the psychology of logicians is pure ad hominem, to be comparable with what would be dismissing Nietzsche's later work on account of his insanity. And his criticism of identity would be another manifestation of his "anti-realism" about "things". After a divagation about the concept of thing and change, in particular about the bundle theory of objects prefigured by Berkeley and Hume, Hales concludes that Nietzsche's critique is really about the applicability of logic than it is about logic per se. He sees it as a critique of semantics and about Nietzsche's own misunderstanding of semantics as necessarily referring to objects that Hales resumes under the concept of "realistic metaphysics". Hales reiterates that ''a realist semantics is not the only one possible, and the universes of discourse can just as well be populated with Nietzschean fictions as they can be with things-in-themselves [whatever they are]." (My italics and square brackets.)


Consequently Hales concludes in the last paragraphs of his article that "The charge that logic or language is misleading is ultimately a criticism of those who are thereby misled and is not an objection that undermines logic as a science of thought or as a formal representation of natural language." Questions about logic are relegated by him to metaphysics and "faith" of logicians, i.e. to their sort of "religion". While metaphysic(s) is a word that today is used for depicting something imaginary or fanciful to be kept out of science and serious discourse, the Webster's Third New International Dictionary also offers the definition as a system of first principles or philosophy underlying a particular study or subject of inquiry, and a division of philosophy that includes ontology and cosmology, treating of the relations obtaining between the underlying reality and its manifestations.


All the above may look rather "theoretical-abstract" it it were not for its consequences in late problematic trends in The Logic of Information (2019), a book with the subtitle A Theory of Philosophy as Conceptual Design, authored by the widely promoted Luciano Floridi (also visible on YouTube). In the preface he introduces the book as being a (third) middle ground between the first two books in a planned tetralogy labelled initially, as a pun among colleagues, Principia Philosophiae Informationis (in analogy to Principia Mathematica?) and up to now consisting of The Philosophy of Information and The Ethics of Information plus a planned one on The Politics of Information. In the preface the book is alternately presented as (1) A book neither in the epistemological tradition nor about ontology, rather a book on the logic of design and hence of making, transforming, refining, and improving the objects of our knowledge, (2) Asking what is the conceptual logic of information modeling, i.e. generating a description of some structural properties of a system, (3) A study in the conceptual logic of semantic information, (4) A constructionist study in the conceptual logic of semantic information both as a model (mimesis) and as a blueprint (poiesis). As Wikipedia puts it the goal to develop a constructionist philosophy, where design, modelling and implementation replace analysis and dissection.


In my own words in the present text I would say that Floridi tries to leave his early background in analytic philosophy and logical empiricism (in Wikipedia on Floridi: "analysis and dissection") towards a teleological systems theory (in Wikipedia on Floridi: "design, modelling and implementation"), despite of grounding his philosophy of technology in the logical positivist Herbert Simon and his pragmatism in C.S. Peirce instead of William James. But by shuffling around with undefined terms such as concept and conceptual), system, structure and structural (without its relation to function-functional), design, as well as epistemology with related ontology and semantics which I touched above in this section, he creates an esoteric quagmire, or as his writes "this third volume too is not a page-turner, to put it mildly". Such esoteric quagmire prevents criticism by cultured lay non-specialist readers, and prevents the understanding of constructivist design as being related the ethically laden teleology with the consequent ignoring of ethical-theological issues considered in Churchman's The Design of Inquiring Systems, as I expound in my essay on Information and Theology. I think that the attraction exerced by Floridi's work relies on his extreme formal skill in building logical networks out of disparate esoteric terms, or what Churchman calls "Leibnizian inquiring systems", which academically seem to legitimize the troubling chaos of the computerized world. As Floridi himself acknowledges: "I may be moving out of the shadows of my three philosophical heroes [Plato, Descartes and Kant?]. Not a plan, but this is what happens when you follow your reasoning wherever it leads you."


I add: "apparently regardless of what is that leads and should lead the reasoning". In terms of analytic psychology, for an extremely logically thinking and empirical psyche, this means to be "artistically" and unconsciously led by feelings and intuition. It is an interesting case of "enantiodromia", which also may characterize the mind of those who drive the process of computerization of society. The end was paradoxically postmodernism, and soon post- and transhumanism, trends that are already being studied.





We have now seen various approaches to the relation between mathematics and logic that are embodied in computer, and the meaning of their "application" as it is represented by their relation with reality. I suspect that this may be way of also getting at the "core" of the issue of philosophy of technology for which I refer eventually interested readers to another earlier essay of mine titled Trends in Philosophy of Technology.


We must now pause on what ultimately was referred to above as the non-controversial authoritative and objective consensual observer of the reality "out there" in a "Gods Eye" "externalist" perspective. The reader may have noted that Nietzsche, for all his insights, claims that all is flux or change, and there is no truth but only appearances in perspective. But who and how has paradoxically guaranteed that this claim of Nietzsche is true? And if things are changing bundles of changing properties, who can observe change from a changing platform of observation? And if logicians or the "users" of logical-mathematical computers claim that their guaranteed consensual competence assures that there is persistence of identity in their own personality, their own judgment, and in the elements of their logical formulas or computer-software, then how do they know that this is true?


It seems obvious to me that the first requirement is the identity and stability of the observer, logician or not. In the course of this article we have several times already referred to analytical psychology but there are readers or "observers" that this has already been superseded by later modern neuropsychiatry or neuropsychology. Others feel that psychology and psychiatry is subsumed under religion or dharma, including Christianity, Hinduism and Buddhism.


Therefore I could have started by considering an article by Henk Barendregt on [pdf] "Buddhist models of the mind and the common core thesis on mysticism" (in One Hundred Years of Intuitionism 1907-2007: The Cerisy Conference. Eds. Mark van Atten, Pascal Boldini, Michel Bourdeau, Gerhard Heinzmann. Springer, 2008). When compared with the apparently more detailed and deepgoing Buddhist Epistemology by S.R. Bhatt and A. Mehrotra, it has one great merit of trying to reach quite far, until comparing Brouwer's and Kurt Gödel's attitudes to mysticism, reminding of the "mystical" experience of people getting involved with computers and, in particular, in computer games as mentioned at the beginning of this article of mine. It is a mysticism that reveals the price our culture pays in form of computer abuse and dependence, and meaningfully contrasts with the wholesale disposal of "metaphysics-ontology" together with real semantics and pragmatics in formal logic as we saw in Hales, above.


The problem with Barendregt's article, however, is that in explaining Brouwer experiencing of the essence of consciousness it refers to an extremely complex body of Eastern knowledge that easily overpowers us. Western readers who live in a barely understood and often negated Judeo-Christian culture cannot really expect to understand and intellectually assimilate a foreign one as represented by what Barendregt, in a rather eclectic exposition, calls "AM". It is done without any particular reference to what the letters AM stand for, introducing it as a "model of conscious cognition inspired by the Abhidhamma and [or] Abhidharma" (see also in The Stanford Encyclopedia of Philosophy). The words themselves are said to mean higher-teaching, and they are said (in Wikipedia) to have been considered in the West alternatively as "Buddhist psychology", "Buddhist phenomenology", "Process philosophy", "Philosophy, psychology and ethics", "extending into ontology, epistemology and metaphysics", or simply "among the major achievements of the classical period of Indian philosophy." Among other things it analyzes six types of causes (compared with Aristotle's four), reminding Brouwer's emphasis of causality we have seen earlier in this text. To give an idea, Barendregt reminds that the work consists of seven volumes comprising more than 5000 pages, leading its students to study commentaries, or commentaries of commentaries. So much for those who have perceived this present text as too long, heavy or obscure. It is, then, very relevant to consider that Francisco Varela who, phenomenologically and problematically in turn influenced computer scientists, based his position on Eastern thought, especially Buddhism, as revealed in his book Ethical Know-how: Action, Wisdom, and Cognition (Stanford Univ. Press, 1999.)


A book on Philosophical Issues of Information Systems (1997) which, by the way, does not cover the issue of this essay, describes (p. 76) Varela as having developed together with Humberto Maturana "radical theories from a biological perspective, concerning the nature of living organisms, their nervous systems, cognitive capacities [whatever is their definition] and language (my brackets). This is not to mention "neurophilosophy" (that could be accounted for, as mentioned earlier, as "brain mythology"), listed in Wikipedia's article on Maturana. And their findings have been explored in the field of computer science by researcher such as (mainly) Terry Winograd.


To get a taste of in what cognitive world does this "phenomenological Buddhism" operates I will shortly present a presentation of the Chilean engineer, entrepreneur and politician Fernando Flores who worked in close contact and dependence upon the computer world, and appears to be consistent if not harmonious with Varela. In a presentation of Flores in the issue 57 (2009) of Strategy + Business the journalist and consultant Lawrence M. Fisher tries to summarize a work that dazzles academia because its eclecticism and rhetoric does not allow intellectual analysis, and writes (excerpts):

"More controversially, Flores argues that there is no objective reality: that the human nervous system cannot distinguish between reality and perceptions. In practical terms, to Flores, this means that individuals and organizations are never fully trapped in any situation, even one as drastic as imprisonment — if they remain willing to change the way they think and talk about it. 'We human beings are linguistic, social, emotional animals that co-invent a world through language," says Flores. That means that reality is not formed by objects. [...] One uncontested fact is that the years of imprisonment turned Flores toward philosophy. [...] With endless time, he read and reread, devouring the works of the German philosophers Martin Heidegger and Jürgen Habermas; of the pioneering Chilean neurobiologists Humberto Maturana and Francisco Varela; and, perhaps most significantly, of John Searle, a Berkeley professor and former student of J.L. Austin. Searle had refined Austin's concepts into a practical set of phrases, coining the term speech acts to describe them."

Let's observe the words, that there is no objective reality, the appeal to neuropsychology that is aloof from most readers as much as metaphysics and ontology are, that reality is invented through language [which, remember, was downgraded by Brouwer, and that is in turn mediated by computer communication such as in computerized social media], Heidegger and Habermas, Varela, etc. Putting all this against what else was considered in this text of mine up to now it is easy to see that it is a patchwork that evidences how a barely understood "Buddhist" infrastructure does not work, at least in the sense that it is not understood or understandable, and therefore confusingly mind-blowing in our Western Judeo-Christian world.

In view of the above difficulties and as a preparation for further analysis of our main problem I would like to mention also an earlier related paper of Mark van Atten and Robert Tragesser on "Mysticism and mathematics: Brouwer, Gödel, and the common core thesis", in W. Deppert and M. Rahnfeld (Eds.) Klarheit in Religionsdingen [Clarity in religious questions]. Leipzig: Leipziger Universitätsverlag (2003). See also on the net some unidentified author's (alias David Koning <>) commentaries (2006) of the text. Also published (with and interesting list of similar books and articles) in In Robert Tragesser, Mark van Atten & Mark Atten (eds.), Essays on Gšdel's Reception of Leibniz, Husserl, and Brouwer. Springer Verlag (2015).


Atten-Tragesser initially recall that the famous mathematician David Hilbert starts his book Axiomatic Thought (English version included in William Ewald, From Kant to Hilbert: Readings in the foundations of mathematics, Vol. 2, 1996) stating that "the most important bearers of mathematical thought have always [...] cultivated the relations to the domains of physics and the [philosophical] theory of knowledge." And that both went beyond philosophy, cultivating relations to mysticism. Further they add that the distinction between Philosophy and Mysticism is a matter of degree. To me that means also a critical instance to the meaning of logic since for most people the difference between philosophy and mysticism consists in their content of logic (and empiricism).


The authors continue describing how Brouwer and Gödel each relate mysticism and mathematics, and make a comparison. They recall what is for me the ethical dimension in that they see mysticism as a sudden illumination that makes people to see reality in a different light, given that reality is good and mystical practice aims to perceive this Reality as Good. In the details of the article that I will not survey in detail, they claim that this Good, however, is not objective or the same for all varieties of mysticism. This is so despite "the somewhat analogous case in the philosophy of science where scientific realists hold to a common core thesis with respect to scientific theories through the ages" that show massive disagreements and still they all try to express the same objective reality.


We will not follow the article's argument except for some remarks. It reminds Brouwer's observation that the greatest merit of mysticism is its use of language independent of human collusion and of animal emotions of fear and desire. Contemplative thoughts may come through without obscurity since there is no mathematical system that distorts them.

"The mystical writer will even be careful to avoid anything that smacks of mathematic or logic: weak minds might otherwise be easily made to believe and act mathematically outside the domain where this is required either by the community or their own struggles for life and end up in all kinds of follies." [My italics.]


Gödel, on the other hand, is referred (on the basis of a book by computer scientist Rudy Rucker's Infinity and the mind, recalling Zellini's interest for infinity) as perceiving abstracts objects or pure abstract possibilities by having "to close off the other senses" and "to seek actively". "Doing mathematics is one way to get into contact with that Absolute [...] There is, then, no break between mathematics and mystical practice." And referring to Husserl Gödel would have said that "At some point [...] everything suddenly became clear to Husserl, and he did arrive at some absolute knowledge. But one cannot transfer absolute knowledge to somebody else; therefore one cannot publish it. [...] One fundamental idea is this: true philosophy is [arrived at by] something like a religious conversion." Atten-Tragesser, then, observe that it is likely that Gödel tried to experience such an illumination or conversion and mention that Gödel's personal library contained among others books on Christianity, Islam, Buddhism and mystical-spiritual movements, concluding that in contrast to Brouwer he believed (as Thomas Aquinas did?) that the "intellect" (and psyche, which we shall survey below) has a positive role to play in spiritual life.


In comparing Brouwer and Gödel the authors conclude that both were looking for mystical experiences in which "openness of the mind to the Absolute is operative", and that something is disclosed and imparted to the person. They could have mentioned religion and God or Truth instead of the Good and the Absolute" but obviously they find it inconvenient. They state, however, that the imparting is preceded by a preparation or transformation of the person. The SELF must be brought into a condition to receive, support, and appreciate what is to be disclosed. "This preparation we see mentioned by both Brouwer (the abandonment of mathematics) and Gödel (closing off the senses, etc.). Atten-Tragesser conclude the whole article with the sentence: "Both were disgruntled with the materialistic and formalistic philosophies prevalent at their time; both thought that these philosophies could not do justice to the Good."


In the perspective of this present paper of mine, against the background of such conclusions, the computer being an embodiment of mathematics and logic in a materialistic and formalistic frame of mind cannot "do justice to the Good". Its formalism programmatically excludes teleology of "the good", and its mysticism expresses itself in the infatuation that lies at the basis of computer addiction and feeling of awe for its sort of spiritual power, recalling the Biblical saying that not all spirits are the Holy Spirit. Awe is indeed the feeling of reverential respect mixed with fear or wonder that we all have felt in the interaction with a computer until we eventually become insensibilized, albeit often addicted.


Finally, let's return to the above mentioned statement that imparting of higher knowledge is preceded by a preparation or transformation of the person and that the SELF must be brought into a condition to receive, support, and appreciate what is to be disclosed. This vocabulary with emphasis on the Self can be seen as aptly representing a call to prayer and religion. At the scientific level where the concept of science itself in is question, it can be seen as closely related to analytical psychology of the perception of reality, which has been already mentioned on several relevant occasions in this essay.







What is left is a better understanding of some key concepts that have been used throughout our text up to now. As in logic, the conclusions depend upon both the definitions and consequently the stability - as we have seen - of the truth of the basic assumptions. These have been mentioned above in terms of such words, among others, as intellect, (exodus of) consciousness (from its) deepest home, and (or) ultimately self contrasted to the ego. To illustrate and motivate the recourse to analytical psychology let's consider some dictionary definitions of these terms and show their complexity. In what follows I will only lightly edit and select the alternatives that are most relevant for our problem, when several definitions are given. My own comments are enclosed in square brackets, and those words that are in turn most relevant within the relevant selections are underlined; readers who already understand and are in agreement with the spirit of this essay of mine up to now may be able to draw right away some correct conclusions of it all without waiting for my further explanations below in the rest of my text concerning the relevance of analytical psychology as alternative to the related psychoanalysis.


One may start the inquiry trying to be simple starting with "intellect", one of words mentioned most often in our discussion up to now, and that today is often bypassed by the similarly used and misused "cognition", Wikipedia claiming dimly to encompass a a hodgepodge of "knowledge, attention, memory and working memory, judgment and evaluation, reasoning and "computation", problem solving and decision making, comprehension and production of language". A smart engineer colleague of mine who appreciates simplicity as much as logic challenged me to be logical in discussions of ethics while he downplayed a supposedly pedantic need of definitions claiming that one must be able to rely on the intuitive common sense of daily popular discourse. The next step in complicating the issue would be consulting the so called Google dictionary, or rather the definition appearing at the top of the Google search. The source is reported as per June 2017 to be the Oxford American College Dictionary. I start using this Google only for the keyword intellect and the search gives as the most complex result

INTELLECT (Google - Oxford American College Dictionary)

"the faculty of reasoning and understanding objectively, especially with regard to abstract matters" with closest synonyms being "mind, brain, brains, head, intelligence, reason, understanding, comprehension, thought, brainpower, sense, judgement, wisdom, wits." And, going over to the etymological question:


INTELLECT (The Online Etymology Dictionary)

The online etymology dictionary states about intellect that it is a latin-based translation of the Greek nous or "mind, thought, intellect, intellect" in Aristotle. A key word that also appears in the etymology dictionary is "discernment" (along with "a perception" and "understanding"). Discernment appears more clearly in the etymology of intelligence and intelligere i.e. from assimilated form of inter "between" (see inter-) + legere "choose, pick out, read," from the Proto-Indo-European root *leg- (1) "to collect, gather", all this suggesting that it is a question of an "art" of choosing or gathering [i.e. for a purpose in a systemic context.] [My square brackets, as they will stand for in what follows. Let us note that there is no mention of logic or mathematics, and that perception stands together with understanding, as well as mind together with head, brain, judgement and wisdom.]


If we choose a more ambitious source such as Webster's Third New International Dictionary (1971) we find, starting with intellect (from intelligere to perceive, understand):


INTELLECT (Webster's Third New International Dictionary)

1a: the power or faculty of knowing as distinguished from the power to feel and to will [this is the typical main source for separation between supposed cognitive knowing functions from feelings and willing.]

1b Aristotelianism (1): passive reason (2): active reason

1d Thomism (1): the receptive faculty of cognition that makes apprehensible the phantasms of intelligible forms -- called also passive intellect, possible intellect, potential intellect (2): the aspect of the soul that is immortal and constitutes the active power of thought operating upon the phantasms of intelligible forms -- called also active intellect, agent intellect.

2a: a person given to reflective thought or reasoning: a person of notable intellect.


Since there is above a reference to the less common word phantasm, let's consider its definition:


PHANTASM (Webster's Third New International Dictionary)

2: a sensuous idea or impression -- compare SPECIES

2b <all of the sensible qualities are but phantasms of the observer, not properties of the object -- Douglas Bush>. [Can relate to archetype in analytical psychology. So over to the mentioned SPECIES:]


SPECIES (Webster's Third New International Dictionary)

2b (1): a mental image, phantasm, or sensuous presentation (2): an idea or object of thought that is the similitude of an object in nature whether in the guise of a modification of sense or of a purely intellectual correlative of the natural object; broadly: FORM, ASPECT, APPEARANCE


EGO (Webster's Third New International Dictionary)

1: the self especially as inside one as contrasted with something outside (as another self of the world): as a metaphysical philosophy (1) in Descartes: the soul or an underlying mental or spiritual substance (2) in Kant: a transcendentally postulated unity either of apperception or of the morally free person -- also called pure ego (3) in Fichte: pure self-determining activity positing itself -- called also pure ego b empirical philosophy (1) in Hume: a complex of ideas or a system of successive mental states (2) in Kant: the conscious subject of experience (3): the consciousness of an individual's being in distinction from other selves c: SELF

2b: WILL

3 [translation of German ich] psychoanalysis: the largely conscious part of the personality that is derived from the id through contacts with reality and that mediates the demands of the id, of the superego, and of external everyday reality in the interest of preserving the organism.


I (Webster's Third New International Dictionary)

1: someone possessing and aware of possessing a distinct and personal individuality : SELF, EGO. [And since awareness has been mentioned several times, and its closest meaningful synonym in our context is consciousness:]


CONSCIOUSNESS (Webster's Third New International Dictionary)

1a awareness or perception of an inward psychological or spiritual fact : intuitively perceived knowledge of something in one's inner self b: inward awareness of an external object, state, or fact c: concerned awareness : INTEREST, CONCERN

2: the state of activity that is characterized by sensation, emotion, volition, or thought : mind in the broadest possible sense : something in nature that is distinguished from the physical

3: the totality in psychology of sensations, perceptions, ideas, attitudes, and feelings of which an individual or a group is aware at any given time or within a particular time span -- compare with STREAM OF CONSCIOUSNESS [...]

5: the part of mental life or psychic content is psychoanalysis that is immediately available to the ego -- compare PRECONSCIOUS, UNCONSCIOUS 2

It is should be easy to notice the degree of complication of the issue when one compares the common sense "engineering" approach to intellect etc., which roughly equates it to logic-mathematics and empiricism (cf. logical empiricism, also called logical positivism, and hidden in Rudy Rucker's computer-oriented mathematics), with what is implied by the above definitions. In my experience, however, I have noticed that typical engineering minds of people who are gifted engineers or logicians-mathematicians have difficulty to notice this degree of complication, a phenomenon that I will regard as mainly ethical-psychological beyond the materialism of psychoanalysis. In fruitless discussions they persist in reducing the issue to its simplest form, having induced me to write a whole essay on debate and fruitlessness of debates and discussions on complex matters, already denounced by Brouwer, when there is no bond of genuine unconditional friendship.


In order to unravel psychologically the relation between mathematics-logic, empiricism, and the psyche in its totality I will apply the structuration offered by analytical psychology in general, partly summarized in Jung's Collected Works, volume 7 - CW7, and of psychological types in particular, in CW6. A quite correct overview is offered in Wikipedia's article on analytical psychology as per end of August 2017. In doing so I will not enter into a debate about analytical psychology itself, a field that could expand an almost limitless analysis in secondary and tertiary literature that I also have consulted in the latest thirty years.


A preliminary taste of what is all about can be obtained in secondary literature or original quotations on the net, such as in Wikiquotes on Psychological Type. On the basis of the references given here above I will use analytical psychology's concepts in order to draft a hypothesis on the problems considered in this essay, including the shortcomings of Brouwer's approach and the criticisms for and against it.


We can start, however, with a curious and mind-blowing philosophically logical analysis of the first definition of intellect above where both reason and understanding side by side, often recurrent in the whole of our text together with misunderstanding, are given as synonyms. Reason and understanding, however are terribly complex and to be differentiated in the most advanced and "canonized" texts of our modern Western culture. The Blackwell Dictionary of Western Philosophy (Blackwell Reference Online) start summarizing the matter "Reason and Understanding" as follows:

"Epistemology, metaphysics [German vernunft , reason; verstand , understanding, associated with to stand]. The distinction between reason and understanding was first discussed in Kant 's philosophy. Kant claimed that understanding applies its own categories to experience and generates scientific knowledge, while reason moves from judgment to judgment and seeks to go beyond the limits of experience. Reason tries to apprehend the unconditional, but ends with antinomies, in which reason falls into conflict with itself. Hegel offered a different account of the distinction. He considered understanding to be a fixed or mechanical way of thinking, which produces clear analysis and is in general the first stage of logic and science. But understanding isolates things from one another and is partial, finite, and without fluidity. Reason stands in contrast to the absolute fixation of the understanding. It is associated with inference and argument and tries to discover connections among truths. Reason has two forms. Negative reason uncovers and collapses the contradictions implicit in the abstractions of understanding. Positive reason draws positive conclusions from the work of negative reason. The final purpose of reason is to resolve all conflicts and to grasp totality. For Hegel, reason and understanding are immanent in the absolute idea..."

Having tried to understand (with my reason?!) Kant's all three Critiques, I am sure that the matter is complex indeed and it convinced me that if one succeeds in understanding what is claimed then one is stuck in Kant's own framework. The way out for me has been to try to read some of Kant's historical critics such as the psychologist Carl Jung that appears to be conscious and respectful of his Kant-heritage. My attitude (cf. positive reason?) has also been formed by Kant's criticism inbuilt in my doctoral counselor, often mentioned West Churchman, as represented by his two main works The Design of Inquiring Systems and The Systems Approach and its Enemies. They also aim at incorporating Hegel's criticism of Kant as implied in his own view of the very same concepts, as illustrated by Nectarios G. Limnatis' book Reason and Understanding in Hegelian Philosophy. But it does not prevent "analyzers" to apply understanding to everything: to the free will, logic, mysticism, Nietzschean philosophy, (mis)understanding of semantics, etc. So much, then, for intellect seen as reason and understanding, not to mention seen as only logic and empiricism.


And so on, if we were to return to my text in the present essay, from the beginning. For example in the discussion about the view of formalism vs. intuitionism on whether mathematical exactness is to be found in the intellect or on paper we will find that elsewhere the question is expressed as whether it is to be found in the abstract or in the world of experience. And what about the role of understanding in understanding: the intuitional attitude, the abstracting of emotional content, unity and stability of the observer, mental image - phantasm and sensuous presentation, the process of objectifying or externalization, turning into oneself, the exodus of consciousness from its deepest home, the essence of the ego as related to given perceptions, the self vs. you and yourself, and the self that must be brought into a condition to receive. (All this extracted from the text up now.)


I wished to go on now "explaining" the framework of analytical psychology but it would take excessive space and effort in this context, while explanations and summaries are easily found in the literature already referenced in links above, for those who have a serious motivation to do so. I prefer to approach the question by positing hypotheses about the psychology or "psychological type" of Brouwer compared to those who use mathematics and logic, including computers, in the alternative "modern" way criticized by him. In doing so I eschew the dangers of being criticized for Ad hominem by adducing without further analysis two reservations accounted for in Wikipedia's summary about this issue: "Non-fallacious reasoning" and "Criticism as fallacy". As also suggested by several items in Wikipedia's article on ad hominem in the section "See also" (e.g. The Art of Being Right) such further analysis leads recursively to the type of problems that I already considered in my earlier essay on Information as debate.


As mentioned earlier in this essay it is suggested that Brouwer was a introvert-thinking type, now to be seen as supplemented with intuition as auxiliary function. Going for once back to original literature, albeit in translation from the German, we find the following in Carl Jung's Collected Works, volume 6 or CW6. Because of my use of my annotated first Princeton/Bollingen paperback printing with corrections of year 1976, and in order to allow consultation of German or other originals, the references are given to paragraph numbers instead of page numbers: § 578 ff. and 628ff.


Jung writes that

"I am fully aware that our age and its most eminent representatives know and acknowledge only the extraverted type of thinking...I will disregard all those sensations and feelings which become noticeable as a more or less disturbing accompaniment to my train of thought, and will merely point out that this very thinking process which starts from the object and returns to the object also stands in a constant relation to the subject...Now, when the main accent lies on the subjective process, that other kind of thinking arises which is opposed to extraverted thinking, namely that purely subjective orientation which I call introverted...The world exists not merely in itself, but also as it appears to me...If we were to ignore the subjective factor, it would be a complete denial of the great doubt as to the possibility of absolute cognition. And this would mean a relapse in the stale and hollow positivism that marred the turn of the century -- an attitude of intellectual arrogance accompanied by crudeness of feeling, a violation of life as stupid as it is presumptuous. But what is the subject?"

The subject is man himself, but later Jung goes into the relation between ego and self:

"The really fundamental subject, the self, is far more comprehensive than the ego since the former includes the unconscious whereas the latter is essentially the focal point of consciousness...But it is a characteristic peculiarity of the introvert, which is as much in keeping with his own inclination as with the general bias, to confuse his ego with the self, and to exalt it as the subject of the psychic process, thus bringing about the aforementioned subjectivation of consciousness which alienates him from the object."

I understand from this that Brouwer through his study of Christian (and Eastern) mystics was well aware, and shows it in his writings, of the difference between his ego and self but most probably had been struggling through most of his life in order to keep them separate. Then appear some thoughts that recall the essence of "pure mathematical" thinking:

"But no more than extraverted thinking can wrest a sound empirical concept from concrete facts or create new ones can introverted thinking translate the initial image into an idea adequately adapted to the facts. For, as in the former case the purely empirical accumulation of facts paralyzes thought and smothers their meaning, so in the latter case introverted thinking shows a dangerous tendency to force facts into the shape of its image, or to ignore them altogether in order to give fantasy free play...


This kind of thinking easily gets lost in the immense truth of the subjective factor. It creates theories for their own sake, apparently with an eye to real or at least possible facts, but always with a distinct tendency to slip over from the world of ideas into mere imagery. Accordingly, visions of numerous possibilities appear on the scene, but none of them ever becomes a reality, until finally images are produced which no longer express anything externally real, being mere symbols of the ineffable and unknowable. It is now merely a mystical thinking and quite unfruitful as thinking that remains bound to objective data..." (§629 f.)

I see this as a possible description of the mathematical mind, indeed as purely mathematical "free play of fantasy" that allows for the "rich" production of mathematical products that later are used and misused by "extraverted thinker" (§ 584-594) in projections upon external objects in a process that was not intended by the introverted thinker, such as Brouwer. And this would refer to the unforeseen application of his algorithmic thinking to automatize numeric computations and software. And the "mysticism" that is mentioned is not religious mysticism as identified by thhe special case of Brouwer, but rather the mysticism of secularized mathematicians who claim to approach divinity, becoming semi-gods or their priests, since their god is supposed to have written cosmos in mathematical language. Alternatively, it is the "mysticism" experienced in the use and interaction with computers in general, and in computer gaming in particular. All this happens while some Christian self-righteous amateur theologians paradoxically criticize and condemn in brainy analyses Jungian thought, often as they find it in critical secondary sources, for being "gnostic", despite Jung's express declaration and explanation of keeping distinct psychology from religion. Paradoxically these brainy critics can ignore if not their own, at least the gnosticism of modern science and technology in general, and of computers in particular, neverthelelss gladly using them daily or, ultimately, becoming addicted to them.


The more detailed description of the Introverted Thinking Type (§ 632-637, see below) contains further considerations of the social behavior of those who are, always roughly, characterized by it. They remind some of Brouwer's traits as rendered and as I have understood them in his biographical accounts. Some hints follow about extreme traits of the type understood as rather an ideal type, that in no way should be seen as personal. An ideal type may happen to be better understood if put into the context of the tradition of the German cultural sphere connected to other concepts such as Verstehen and Nomothetic vs. Idiographic. This is not a wholesale endorsement of the "correctness" of these concepts but mainly to show that their creation points to an underlying necessity and to the existence of a problem. And now over to some of the traits of the introverted thinking type who shares with his counterpart, the extraverted thinking type, the remarkable need or injunction of "abstraction from emotional content":

"If in his eyes his product appears correct and true, then it must be so in practice, and others have got to bow to its truth. Hardly ever will he go out of his way to win anyone's appreciation of it, especially anyone of influence...He usually has had bad experiences with rivals in his own field because he never understands how to curry their favour...In the pursuit of his ideas he is generally stubborn, headstrong, and quite unamenable to influence...[F]or him the relation to people and things is secondary and the objective evaluation of his product is something he remains unconscious of...In his own special field of work he provokes the most violent opposition, which he has no notion how to deal with, unless he happens to be seduced by his primitive affects into acrimonious and fruitless polemics...The various protective devices and psychological minefields which such people surround themselves with...serve as a defence against "magical" influences -- and among them is a vague fear of the feminine sex."

I repeat that this is a question of ideal type, but it has some value for imagining the psychic structure and environment of Brouwer, the motives for his relative isolation and for his attitude to woman as surveyed above in our earlier text. In order to counteract the wrong impression that introverted thinking is basically negative, it should be compared with the description of the extraverted thinking type (§ 584-594) that typically exploited Brouwer's findings ignoring its motivations, and on which I shall not dwell here. And Carl Jung, returning to "the intellect", terminates his work on Psychological Types stating in the Epilogue (§ 856):

"Whatever we strive to fathom with our intellect will end in paradox and relativity, if it be honest work and not a petitio principii in the interest of convenience. That an intellectual understanding of the psychic process must end in paradox and relativity is simply unavoidable, if only for the reason that the intellect is but one of many psychic functions which is intended by nature to serve man in constructing his images of the objective world. We should not pretend to understand the world only by the intellect; we apprehend it just as much by feeling. Therefore the judgment of the intellect is, at best, only a half-truth, and must, if it is honest, also to admit its inadequacy."

Having dedicated this analysis to illustrate how Brouwer's supposed introverted thinking explains his insensitivity to how the outer world would appropriate his results, I wish now to complete with illustrating how there may exist a flaw in his assumptions about the reason for the ""exodus of consciousness from its deepest home", as well has his assumptions about the undefined "intellect" as in seeing the human intellect as the cause of evil. In Jung's conception an undefined intellect is substituted by the whole psyche as constituted by ego-consciousness and different strata of unconscious with its roots in the suggestions of Carl Gustav Carus and systematized by Eduard von Hartmann. Some basic concepts are presented in Carl Jung's CW7 Two Essays on Analytical Psychology, exemplified by the all-important, crucial self so much mentioned by Brouwer (§ 274 f.) In this context I repeat that in this paper, I do not enter into the possible controversies about different psychological theories vs. philosophy of science, the least so into those controversies caused by possible misuses of depth psychology in the New Age movements. Likewise I do not discuss the misuses of (references to) quantum physics in feminist theorizing as mentioned later, here below. For the rest there is a sizeable secondary and tertiary literature on Jung and even critical of Jung that I have consulted besides all his Collected Words, and is available to all interested. Some examples are V.W. Odajnyk Jung and politics: The political and social ideas of C.G.Jung, E.D. Cohen C.G. Jung and the scientific attitude, P. Homans Jung in context: Modernity and the making of psychology, A. Samuels Jung and the post-Jungians, R. Noll The Jung cult, etc., the latter reminding that Jung himself did not want to build a "school" and still less to leave a cult of personality, he himself declaring "Thank God, I'm Jung, and not a Jungian". In this he resembled West Churchman who personally repeated to be happy for not being a "Churchmaniac".


The most ambitious among many overviews of Jungian psychology I know is not easily available since it is contained in a an older book by J.F. Rychlak Introduction to personality and psychotherapy (Atlanta: Houghton Miffin, 1st ed. 1973, pp. 132-199, refs. 529-538). At risk of making some mistakes, missing some terminological and professional nuances, I will use my own words upon what I remember of my study of Jung's original collected works, starting about 40 years ago. The focus is to explain that the real psychic subject is the Self, a psychic component that is supraordinate to and includes the Ego that in turn interacts with the personal and the collective unconscious. The subject's or initially the Self-realizing Ego's interface with the collective unconscious in the external world is represented outwards by the psychic entity of the persona, as the aspect of someone's psyche that is externally presented to, perceived, expected or required by others or society, analog to a role or character adopted and presented behind a mask as facade, by an author or an actor. The interface with the collective unconscious of the personal unconscious or shadow, with the internal world of inborn or inherited common human characteristics is represented by the "contrasexual" entities of the anima-animus in biological men respectively in biological women. They represent respectively the feminine archetypal image in men and masculine in women, that I see as an unexplored basis for explaining homosexuality and the ongoing LGBT phenomena, especially in the Western world. An unconscious identification of the ego with extraverted persona incurs into the danger of an identification of the ego with the anima and ultimately of a projection of unconscious contents upon others and upon nature or work environment. Such a process may also explain some phenomena of computer addiction and gaming where the primal intuition in time sequences typical of mathematical thinking and music is overtaken by unconscious contents and their automatisms, with the paradox that interaction with a mathematical tool representing abstraction from emotional contents is overpowered by addictive passion. Deeper understanding of such interactions between "musical mathematics" and emotions or feelings may be obtained from an analytical psychological study of what has been learned about the role of music in the historical Doctrine of the Affections, pioneered by Johann Mattheson and originally suggested by Plato. Other works seen to concentrate upon only mathematical aspects, such as in Ruth Tatlow's Bach's Numbers: Compositional Proportion and Significance, disregarding the psychological aspects of numbers suggested e.g. in Jung's Psychological Approach to the Dogma of Trinity (CW 11, p.107-200, §169 ff.) and in


A Self-realization incorporating the God-archetype from the collective unconscious, or individuation in a mature psyche is the result of harmonious sort of "negotiations" (cf. prayer and worse synonyms - meditation, mindfulness) of the Ego with the two aspects of the collective unconscious through its interfaces, by means of one of the particular attitudes (introversion or extraversion) and psychic functions (rational thinking or feeling, and irrational sensation or intuition), which are most consciously available to the individual's (whole psyche's) closest ideal type mentioned earlier. Immaturity as absence or failure of individuation, and reliance upon an isolated thinking function corresponds to Brouwer's warning for that "scientific thinking is nothing but a fixation of the direction of will within the confines of the head, and a scientific truth no more than an infatuation of desire living exclusively in the human head", where the will submitted to desire corresponds to an unconscious underdeveloped feeling-function.


At this point it is necessary to note that the aforementioned psychic components or entities are not imagined out or thin air as it may appear to people who have never studied all aspects of psychology, as the discipline has historically emerged out of philosophy. They and their interactions, in analogy with the entities found in, say, particle or quantum physics, are conceptions that had to be postulated in order to explain and make consistent sense of a wealth of empirical observations of actual human behavior or expression of convictions, historically such as alchemical speculations by e.g. Newton, the personal development of Blaise Pascal, and clinical observations such as in neurosis or schizophrenia.







Several partial conclusions have been distributed above in the form of comments along the various chapters. Not all of them will be summarized below where I will rather focus on formulating some completions besides the basic resuming tenet mentioned at the beginning: the main message of this article is that only if we understand what we are doing with the basic formalisms of mathematics and related logic, geometry and such, we will understand what we are doing with computers. And now I can add my conviction that ultimately it all is a theological and religious question - the meaning and survival of our Judeo-Christian culture in face of techno-science.


I wish also to emphasize that the literature surveyed in this text of mine will always be insufficient in view of the complexity of the issues that were covered. I will illustrate this with only one example, an excerpt from the context of "A blog about religion, science, philosophy" that introduces many other interesting ideas not considered here, and is taken from an article I recommend, with the title Zen and the Art of Science: A Tribute to Robert Pirsig. In an argument that leads to reflection about the never resolved crisis in the foundations of mathematics and basic rules of logic, the author tells about Robert Pirsig noting that

[...] the number of hypotheses could easily grow faster than experiments could test them. One could not just come up with hypotheses – one had to make good hypotheses, ones that could eliminate the need for endless and unnecessary observations and testing. Good hypotheses required mental inspiration and intuition, components that were mysterious and unpredictable.  The greatest scientists were precisely like the greatest artists, capable of making immense creative leaps before the process of testing even began.  Without those creative leaps, science would remain on a never-ending treadmill of hypothesis development – this was the “infinity of hypotheses” problem.  And yet, the notion that science depended on intuition and artistic leaps ran counter to the established view that the scientific method required nothing more than reason and the observation and recording of an objective reality.

Consider Einstein. One of history’s greatest scientists, Einstein hardly ever conducted actual experiments. Rather, he frequently engaged in “thought experiments,” imagining what it would be like to chase a beam of light, what it would feel like to be in a falling elevator, and what a clock would look like if the streetcar he was riding raced away from the clock at the speed of light.

[...T]he nature of mathematical discovery is so mysterious that mathematicians themselves have compared their insights to mysticism. The great French mathematician Henri Poincare believed that the human mind worked subliminally on problems, and his work habit was to spend no more than two hours at a time working on mathematics. Poincare believed that his subconscious would continue working on problems while he conducted other activities, and indeed, many of his great discoveries occurred precisely when he was away from his desk. John von Neumann, one of the best mathematicians of the twentieth century, also believed in the subliminal mind. [...]The Indian mathematical genius Srinivasa Ramanujan was a Hindu mystic who believed that solutions were revealed to him in drea ms by the goddess Namagiri.

Intuition and inspiration were human solutions to the infinity-of-hypotheses problem. But [...] there was a related problem that had to be solved — the infinity of facts.  Science depended on observation, but the issue of which facts to observe was neither obvious nor purely objective.  Scientists had to make value judgments as to which facts were worth close observation and which facts could be safely overlooked, at least for the moment.  This process often depended heavily on an imprecise sense or feeling [...]

To return now to Brouwer I mean that his basic error besides his "introverted" neglect of probable misuse by others of his method and results, is the assumption that the damage caused by an undefined "intellect" (paradoxically used by him in his own work) would come exclusively from diverted attention from a likewise undefined self to the external physical world. All this while there can be as many intrusions, including archetypal images besides the basic intuition of time, coming from the inner world of the collective unconscious. He himself seems to have been struggling with his anima as most clearly exemplified in his apparently bizarre attitude to "women considered collectively" in his early book on Life, Art and Mysticism.


On the other hand it is evident from the material overviewed here and biographical details that Brouwer worked very hard to achieve a spiritual readiness akin to an "individuation" by means of a kind of continuous meditation and a familiarization with Christian and Eastern mystical traditions that imply prayer. This may be more than the atomic bomb's "genius" Robert Oppenheimer with his late awakened conscience after the terror crimes in Hiroshima and Nagasaki is said to have done. Despite his rather introverted attitude Brouwer evidently also struggled to practice some extraverted activity such as participation in the scientific community, its conferences, publications and debates. Nevertheless we also saw indications that he ultimately got somewhat socially isolated even if his results came into the domain of use by other mathematicians, and ultimately in extraverted industrial and military activities. His intuitive sensing of the self was probably not sufficient to allow an integration of his ego with his unconscious, let alone his negative shadow and anima.


In any case he succeeded in identifying, especially in the environmental issues of his time, the danger of extraverted thinking combined with the auxiliary function of sensation that leads to projection of fantasies of power over nature and human communications. The success he envisaged, however, would not come from the limitations of his mathematics but rather from his and especially others' psychological maturity and capacity to understand and integrate in their own work Brouwer's insights. This would have implied consequent research by others along his line of thought and their renouncement to improper applications of a tempting mathematics with its supported technology and its supporting formal logic.


Personally I feel intrigued by the necessity of a better understanding of what may be Brouwer's main insight for our purposes. It is the essence of isomorphism in its relation to formalism, since it lies at a ground to the phenomenon of mental projection and fitness of mathematical, and consequently logical forms into nature and the whole external world including humans and their activities that today are the object of intense computerization. Thereby we might understand why humans find consistency in their understanding of selected and causally structured pieces of formalized world whose common characteristic may be to be object of their often unconscious will to power, "infantile" dreams of omnipotence, which consequently are projected into the environment including nature, constituting the psychological essence of scientific technology. Since it is a question of supremacy of "morphism" or form over content, whatever it is or should be, it is also a question of understanding its essence. It is supremacy of fragmented forms over content that leads the trend of computerization, and this may also clarify the title of this article being the computer as "embodiment", the form of form, or meta-form. It leads many computer and information researchers today to abstract from the essence of computation and communication mentioned at the beginning of this essay, luring them into playing down the systems approach and dealing instead with the fashionable trend of "design" with computer graphics, acoustics, haptic technology and talk about numerous neologisms such as digital materiality being a new materiality, or virtual reality being a new reality.


One hypothesis could be that such a supremacy of form over content may be related to aestheticism in its relation to ethics, already implicit in the role of art in Nietzsche's philosophy or, as defined in The American Heritage Dictionary: "An artistic and intellectual movement originating in Britain in the late 19th century and characterized by the doctrine that beauty is the basic principle from which all other principles, especially moral ones, are derived." [My italics]. On an earlier occasion I had already the opportunity to advise the author Gunnela Ivanov of a doctoral dissertation on "design" (in Swedish, pp. 303-305) that Friedrich Schiller had exceedingly interesting thoughts about the matter in his Textes Esthétiques (Paris, 1998): "Sur les limites nécessaires dans l'usage des belles formes" [On the limits necessary in the use of beautiful forms], pp. 82-83 and 86-87, (corresponding to vol. XXI in Schillers Werke - Nationalausgabe, edited by Helmut Koopmann and Benno von Wiese, pp. 17-18 and 23). If I may guess, a superficial and easy but probably defective way to convey the complex idea as I understand it, would be that since Platonic beauty is the form of the good that in Western Christian culture is symbolized by God, people more or less unconsciously tend to feel themselves exhilaratedly good or "godly" when they are in contact with beautiful forms, in our case mathematical ones, the more so if this is combined with an additional feeling of "divine" power thanks to a divinely intelligent mind that approaches god's own designer-mind. For atheists, all the same: "I am best". The same phenomenon may be active in the overpowering feeling experienced by different psychic constitutions in composing, performing or listening to different kinds of music. My criticism of the Wagner-cult, not to mention modern times' satanic metal music fatidically related to phenomena like Charles Manson's cult, exposes paradigmatic examples, whose understanding would be enhanced by an understanding of Plato's references to music (e.g. Republic III 398 ff., IV 423d ff., Timaeus 47c ff., Laws II 653 d).


The "renouncement" or moratorium mentioned above is a cultural, moral and theological social matter that is portrayed in the difficulties experienced in philosophy of technology with its paradoxical appeals for a "moratorium" in the development of technology. There are studies (see here, and here, recalling Joseph Needham's most famous and pioneering Science and Civilization in China) of, for instance, why and how high civilizations such as in China's history did not develop the Western kind of science. Ultimately it would mean a renouncement to search for certain kinds of "knowledge" to the advantage of others, to renounce working for the development of certain kinds of (mathematically) advanced technology, in our case computer technology, and or a renouncement to the extent of its use. At the theological level some of these aspects can be subsumed under the criticism of Brouwer's implicit quietism we already mentioned, as it may be seen permeating even Kempis' Imitation of Christ. The problematic attitudes and life choices involved in these questions, analog to Brouwer's, are illustrated in the modern world literature, and exemplified in Thomas Merton's famous partial auto-biography The Seven Storey Mountain (1948, 1998 ed. especially pp. 304, 310f., 319, 323, 344-347, 361.)


Ultimately it is not an indictment of mathematics or its usefulness, proudly advocated by those few who feel elect to "understand" it. It is, rather, an appeal to keep to an ethical usefulness as implied in genuine pragmatism as I understand it in the tradition of William James and West Churchman where "use" implies an ethically justified goal and justified "tools" or, rather, instruments, far from the supposed neutrality of techno-science. That means also a mathematics, geometry or logic that are not confined to the thinking sphere as defined in analytical psychology but are kept related to the feeling and intuitive functions as also suggested in, albeit criticized, anthroposophy. In fact Rudolf Steiner explains this attitude as I read it in the Italian translation Nascita e Sviluppo Storico della Scienza [Birth and Historical Development of Science] (Milano, 1982) of the original Der Enstehungsmoment der Naturwissenschaft in der Weltgesichte und ihre seitherige Entwickelung, (Opera Omnia n. 326, 1977, referring especially the third and fourth of the nine conferences held in Dornach between 24 December 1922 and 6 January 1923).


Naïve as it may appear or even be in the eyes of present day's established mainstream mathematicians and scientists (just because they are established), particularly mathematical physicists that were already challenged by the previously mentioned Clifford Truesdell, we should be warned by the fact that there are enormous vested interest and a whole military-scientific-academic "industry" that thrives on the paradigmatic status quo. It includes the cheap rejection of complex psychological and social theories on the basis of the requirement that they should allow their "falsification", while espousing defective thinking about the meaning of falsification. The effects of the creation of e.g. metamathematics or metalogic, although unperceived by the general public, will be analogously mind-blowing as those of meta-ethics as seen by those who subscribe to Christian ethics. The creation and application of the tripartite "syntax-semantics-pragmatics" will focus on formalistic syntax, "magically" explain away pragmatics with valued goals of humans, and tie semantics to extravertedly sensed objects in the spirit of logical positivism and analytic philosophy. They also stand at the basis of scientific physical reductionism that spoils conclusions from the work of quantum phisicists such as Carlo Rovelli (see below). The extent of strange mind-blowing (mis)uses of such creations is and will be analog to the ongoing general computerization of life under controlled (cf. police-controlled terror-safe) laboratory-like forms of user interface and human-computer interaction, and to queerly innovative feminist uses of the abstractly mathematized quantum physics, for instance, in Karen Barad's Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning, (Duke University Press, 2007). The contrast between such approaches and studies of relations of quantum physics to psychology can be appreciated in Suzanne Gieser's ambitious The Innermost Kernel. Depth Psychology and Quantum Physics: Wolfgang Pauli's Dialogue with C.G. Jung (Springer-Verlag, 2005, based on her PhD dissertation, in Swedish, at Uppsala University, 1995, ISBN 91-506-1140-2).


In this context of quantum physics and its (mis)undertandings I must mention more recent publications by the theoretical physicist Carlo Rovelli who has become known for his work on and popularization of relational quantum mechanics. In the book Seven Brief Lessons on Physics, and later in The Order of Time (e.g. in chap. 3 and 6) he popularly relativizes the concepts of object, time and space, and thereby those of presence and existence. These complexities undermine in a fruitful way the common lay (and some theological) objections or considerations about the existence of God. At the same time, however, they also tend to undermine all certainties, giving a taste of postmodernity that also encourages the misuses of references to quantum physics mentioned in the previous paragraph. I have concluded that one if not the main shortcoming in Rovelli's approach is that he does not problematize the concept of the observation as being psycho-logical, i.e. necessarily mediated by the psyche (not to mention Spirit). He understandably does not dwell into this, reducing psyche to the "functioning of the brain", decorated by indiscriminate "emotions" that are said to include passions and "love". They are seen as aesthetically giving colour to human life, as he himself does in his text by means of a rich (but inconsequential) display of erudite, moving poetic references to philosophy and the arts. Paradoxically Rovelli even displays, with reference (chap. 12) to Augustine of Hippo, similar insights to Brouwer's about the nature of music without, however, relating them to mathematics. Rovelli treats the observer and psyche of a human being in the light of physical reductionism (disregarding its controversies), paradoxically related to his deconstructed observer-"object" that he sees as constituted by his (its) interactions with the environment. This, without considering that this observer sees from both the "outside" and the "inside", as body and a psyche that consists of more entities that an Ego. Furthermore, the popularizing language he uses in the book hides that the observations (theoretical physics) are mediated by an extremely mathematical reasoning that ignores Brouwer's message. It is a similar shortcoming undermines the otherwise deepgoing work of Arnold Gehlen's Die Seele im technischen Zeitalter (chap. II, 1-2, Italian trans.). Consequently Rovelli does not reflect upon his own account of the misuse of mathematics in commenting the (apparently?) "long series of successful experiments" supporting physics Standard Model (in Seven Brief Lesson on Physics, p. 32):

It's a theory which looks, at least at first sight, piecemeal and patched together. It's made up of various pieces and equations assembled without any clear order. A certain number of fields (but why these, exactly?) interacting between themselves with certain forces (but why these forces?) each determined by certain constants (but why precisely these values?) showing certain symmetries (but again, why these?). We're far from the simplicity of the equations of general relativity, and of quantum mechanics.

As if "aesthetical" simplicity and elegance would have solved the question. Ignoring Brouwer's message and the essential meaning of a mathematics leads Rovelli into a psychological and theological nihilism, in the spirit of ignored sociopsychological interactionism. It is extreme matemathical reasoning that isolates from social cultural criticism and enables mathematical minds to be popularly perceived as polymaths. It enables Rovelli to create a new synthesis in his relational quantum mechanics, creating a logically well structured and therefore treacherously convincing story with a mixture of hypotheses and empiricism, with (atheistic) references to both buddhism and hinduism, and with e.g. the trivial rediscovery (The Order of Time, chap. 12) of the concept of "system" that, undefined, is used indiscriminately as much as "world" or "nature", as much as sytem is used (in Seven Brief Lessons..., pp. 18, 60, 74), or "objects" are are found "only in abstract mathematical space" (idem, p. 16) but they "work well" (pp. 18 ,38) as Brouwer also notes with regard to misuse of causality. All this while the human "observer" no longer observes (p. 33) or no longer speaks but a word's meaning depends on "where it is spoken" (or heard? p. 57), and "the information that one physical system has about another has nothing mental or subjective about it" (p. 68).


In consideration of Rovelli's obvious proficiency it is motivated to adduce Socrates in Plato's observation (Republic VI, 491d-e, here trans. by Paul Shorey) that it is "natural that the best nature should fare worse than the inferior under conditions of nurture unsuited to it". The stronger impact of misled gifted scientists can cause more damage, and in this case I do know of only one other knowledgeable scientist, Kelley L. Ross, who could review and challenge Rovelli's "Seven Lessons" on his own professional terms.


If all this is not enough let's recall the announced advent of additional "mathematical tools" such as quantum computers. Unfortunately it will not require a better understanding of mathematical "tools" inasmuch they will be welcomed, to begin with, for military applications, recalling Brouwer's warning about scientific methods' basic motivations and their consequent essence. The message of this article is nevertheless that it is better to ask ourselves how this squares with the idea of good mathematics without reverting to the old irresponsible tenet that it is neutral and its goodness depends politically upon its or "our" supposedly good applications, guaranteed by our own goodness or explicit commitment to doctrinal righteousness. The problem is analog to the repeated announcement of advantages of artificial intelligence, while neglecting the lessons of what it implicates, as expounded in the already mentioned The Design of Inquiring Systems. I guess the lessons are that if one cannot create computers that "think" as well or better than humans, it is possible to teach or force humans who do not already do so in culturally deprived work environments, to think and act like rudimentary computers.


Thereafter it will be only logical and natural to claim that it is better to replace them by genuine modern computers, which can handle more complex logic and mathematics with faster processing and larger memory. Much debate about AI is idle play around this sort of tautology while ethics and morals are reduced to "moral rules" seen as the "core of our human values", psyche and mind are reduced to "brain", causal chains are reduced to "goals and means", machine learning is reduced to induction-deduction, and computers are reduced to "algorithms". With such a background it is not a question of whether humans will be replaced by "super-intelligent" machines leading to the absurdities of technological singularity, since they already are being and will be progressively replaced. The question is rather which countermeasures can be taken today to offset or meet the cultural and pragmatic consequences it will bring, reviving the core idea of the Frankenstein novel. The whole hype of "superintelligence" and its apparent criticism, hangs on a basic misunderstanding of the roles of mathematics and logic, and its consequences in the abuse of techno-science, in particular of computer-oriented techno-science.


What about the status of these questions in contemporary literature about tecnology, ethics and responsibility? To my knowledge among the most ambitious approaches there there is the already mentioned Carl Mitcham with his late contribution (2015) Rationality in Technology and in Ethics. In: Gonzalez W. (eds) New Perspectives on Technology, Values, and Ethics. (Boston Studies in the Philosophy and History of Science, vol 315. Springer, Cham.) The connection to my present text is, however, better sensed in his comprehensive article integrally available on the net Technology and the Burden of Responsibility, where he refers for once to mathematics and rationality:

"Modern engineering and technology ... introduce into the making activity an engagement with phenomena via mathematically analyzable forces in a sensorium extended through instrumentation into chemical composition at the level of atoms and molecules…In the techno-lifeworld so constructed by the rational taking into account of more than directly experienced phenomena, it is not surprising that moral behavior likewise must move beyond the primacy of anything approaching natural intuitions. Moral conduct too has to become more conscious, more rational, and take more into account."

It is to be noted, then, that what I have been discussing throughout this present text is the ultimate problem of "via mathematically analyzable forces" in a "sensorium extended through instrumentation". All this, as we saw, is related to the meaning of "more conscious, more rational, and take more into account". In an earlier paragraph Mitcham quotes from William Akin's Technocracy and the American Dream (1977):

"One influential effort at formulating engineering responsibility led to the technocracy movement and the failed idea that engineers more than politicians should wield political power. Henry Goslee Prout, a former military engineer who became general manager of the Union Switch and Signal Company, speaking to the Cornell Association of Civil Engineers in 1906, described the profession in just such leadership terms: The engineers more than all other men, will guide humanity forward [...] On the engineers [...] rests a responsibility such as men have never before been called upon to face."

What appears in the discussions of this present text is also that the task reaches beyond engineering and information science that rely on so-called empirical evidence, mathematics and logic. What is never (dares to be) mentioned is that "consciousness and rationality" in engineered computer systems may be framed, as suggested by Brouwer, through psychology and philosophy into basic religious (in our highly technological Western cultural heritage - Christian) and theological tenets. They are supposed to regulate humans' values, the interactions among themselves and, consequently, with nature. Provocatively: ethics and responsibility may be "follow the Ten Commandments" (or most or at least some of them?) and the rest, the correctly understood "conscious rationality", may follow.


Having come this far I must remind the reader of the reasons for my apparently weird sort of academic writing in this article written as a summarizing "swan song" in a relative hurry at an advanced age (80) with a perceived lack of time left for work. If it were not so, I know how it should have written: e.g. as Jean Guitton recommends in his Le travail intellectuel ["A student's guide to intellectual work"]. I write with an excess of details, without sufficient revisions for improved clarity of expression, in a language that is not my mother tongue, with a profusion of references given in the links associated to the used words, links that are intended for only those who need them, and are only available to be read for those who read this text on the computer. I wish to leave a "heritage" to those few readers who are genuinely interested in pursuing this kind of research, to confirm or revise it on the basis of further detail. The general idea of "why I write as I do write" is to be found elsewhere in my homepage, (and at the concluding end of other pages), a version being also found in my blog. I make frequent use of Wikipedia-references because of their often "good enough" comprehensiveness and easy overview in terms of standardized layout. This is done, however, with full knowledge and evaluation of their possible shortcomings that are partly declared in Wikipedia's own auto-criticism. I have also already accounted elsewhere for my critical attitude to Wikipedia. For the rest I am grateful for any comments I may get as I am grateful for all contributions that formed me during during several decades of work. I intend to read and acknowledge all comments, but I apologize if I will not be able to comment the comments, still less to be available for discussions or debates about the text because of reasons that I partly expose in another essay especially devoted to the frequent sterility of debates, including peer review and publication.