UNIVERSITY OF UME
Institute of Information
Processing - ADB
|
Postal
address: S-901 87
UME (Sweden) |
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(Internet): kivanov@cs.umu.se |
Professor
KRISTO IVANOV
Chair,
Administrative Data Processing
DRAFT, 18 February 1991
This
is the first step of one part of a research program about the meaning of
computerization, and it is cast in the form of a "reader". This is to
be understood as a statement of intent and the purpose is to stimulate and capture
suggestions for the improvement of the research idea.
A
computer can be considered to be, among other things, an electromechanical
machine, an economic capital, a communication channel, a logical symbol
manipulator, or a mathematical machine. In this section we shall dwell on the
last mentioned perspective.
At
least from the historical point of view it is obvious that the possibilities
and the problems of application or use of computers are closely related to the
possibilities and problems of applications of mathematics (Zellini, 1988, the
foreword). It is, the least to say, intuitive that mathematics, in its wide
sense, should be relevant to the understanding of the presuppositions and
consequences of the administrative use of computers, considered as mathematical
number machines and symbol manipulators. It is convenient to remember, in the
age of the fashionable artificially intelligent computerized
expert-support-decision tools, that mathematics, "the queen of
science", by itself and through its controversial relation to logic, is
discipline which in the context of a long history and high reputation has been
the field of many important and still relevant speculations and findings about
the nature of supported human thinking. Even today, in the pragmatist
tradition, we are reminded that in Greek mathematics simply meant learning, and
that mathematics can be seen as a way of learning to decide or to prepare for
decisions through thinking: "In many ways it was unfortunate that
philosophers and mathematicians like Russell and Hilbert were able to tell such
a convincing story about the meaning-free formalism of mathematics.... Set and
classes provide one way to subdivide a problem for decision preparation; a set
derives its meaning from decision making, and not vice versa. (Churchman,
Auerbach, & Sadan, 1975, p vii.)
In
which way all this relates, for example, to the present discussions about the
capabilities of artificial intelligence can be seen by means of the historical
and conceptual bridge furnished by pragmatism (Peirce, (Hartshorne, &
Weiss, 1932-1933, pp.27, 36). Problematic as it may be, such bridge is also an
introduction to the rather more sophisticated branch of pragmatism known as
empirical idealism (Singer, 1924, esp. pp.285-293 on "the mathematician
and his luck"; Singer, 1959) and further to the school of social systems
theory (Ackoff, & Emery, 1972; Churchman, 1971).
Nevertheless
it is not necessary to follow the pragmatist tradition in order to appreciate
the merits of our research proposal of relating mathematics to important
currents of continental European culture (Zellini, 1988). Such cultural
currents are not well known in our Anglo-Saxon tradition but, through the
influence of Charles Sanders Peirce and others, they have many points of
contact with pragmatism (Ivanov, 1984). One important hypothesis of our
proposed research program is that we can gain in depth understanding of what is
happening today with the ongoing industrialization and worldwide societal
diffusion of embodied computer computer science, and what should be done about
it, if we get a better understanding of the historical relations mentioned
above. They are, after all, still alive in today's problems.
Searching
carefully it seems possible to find at least thirty years of
"practical" mathematical background and presuppositions for research
on computers and information. Starting from the mathematical basis of computer
programming in the 50's and 60's (Information processing, 1960), during the
80's some arguments were advanced in favor of the strengthening of the formal
and logical basis of research and applications in the field of computer and
information science. This concerned at least the Swedish scene (Bubenko, 1980;
Bubenko, 1982a; Bubenko, 1982b; Bubenko, 1983). Such late emphasis on increased
formalization of information systems analysis, on mathematical-logical models
for formal analysis of correctness and completeness has been advocated as a
programmatic campaign against inefficiency and low quality of scientific work.
Increased mathematization and strict formalization with strict definitions and
"communicable knowledge" are then seen as a tool for avoiding
pseudoscientific speculations, in analogy with the popular positivistic view of
the advancement of physical science. Displaying a curiously defensive attitude
it is also claimed, at the same time, that mathematization does not imply a
natural-science, technical, and in-human bias, since mathematics is neutral,
and is only a common tool to all disciplines, and a precondition for applied
science, for accumulation, integration and communication of usable knowledge.
Increased mathematization would prevent, it is claimed, the diffusion in the
computer market of "miracle tools" that promise design and
implementation of data processing systems in a fraction of the time required by
traditional methods. Therefore, the highest priority for future serious
research work should be given to the establishment of a "standard
notation" for modeling, seen as a prerequisite for cumulative research and
success on other research issues. For less experienced users a friendly but
still stringent high level specification and interaction language should be
devised.
Such
strong claims are, paradoxically, not based on any stringent or mathematical
argumentation, and, as a matter of fact, there are several researchers who
definitely do not agree with the above claims (Ehn, 1988, p. 148; S¿rgaard,
1988, part 6, pp. 44-45). "Mathematicism", then, seems to be a very
weak philosophy with very strong implications. It is, therefore, very natural
in the frame of our proposed research to inquire into the nature of such new
miracle tools in the form of standard notation for modeling which are supposed
to substitute the older miracle tools.
These
new miracle tools have been lately represented by the confidence in two notable
trends in research and development of methods: the increased used of formal
techniques also in the very early system development stages, and the increased
use of deductive and rule-based techniques (Bubenko, 1988). These trends are
supposed to support "languages for capturing and describing knowledge of
the application domain (its structure and behavior), and of the information
requirements in early development stages": deductive and rule-based
approaches work on the basic idea "to capture and to explicitly express
business rules and constraints in a declarative style, rather than to
implicitly embed them in processing procedures or transaction
descriptions". The additional use of a temporal dimension will allow
"to reason about the state of the system at any point of time",
possibly dealing not only with changes of the contents of a database, but also
with changes of the schema describing the contents of the database and the constraints,
as well as the changes reflecting changes in the applications. Quite different
conceptions of formal and informal handling of the problem of change have been
presented in the literature (Forsgren, 1988; Forsgren, Ivanov, & Nordstrm,
1988; Ivanov, 1972; Ivanov, 1987) and, in particular in the software tradition (Parnas,
1972; Parnas, 1976; Parnas, & Clements, 1986; Parnas, Clements, &
Weiss, 1984; S¿rgaard, 1988, part 6).
This
discussion should obviously be related to that important so called functional
feature of computer aided software engineering, CASE, the feature of design
support. It is envisaged as including support for transformation of
specifications from one "level" to the another, "view
integration" which is required "to combine the local specification
efforts of a number of work teams working in parallel", and where
"the restructuring implies the semantics preserving rearrangement of a
specification in order to improve it according to a set of quality (rules),
performance (rules), or other kinds of rules, or according to a designer's
restructuring directives (Bubenko, 1988, p.6). CASE environments are judged to
be advanced if they allow to develop a CASE tool for an arbitrary method which
is the more advanced the more it can handle advanced computer modeling concepts
such as "constraints, derivation rules, operations, and preconditions...,
rules for checking the consistency, completeness, and quality of the designed
objects and their relationships": The CASE tool building requires the
method's constructs to be strictly and formally defined (ibid., pp, 10-11, 17).
Lately this position has been consolidated in an outline of a program for
research on information systems (Berztiss, 1989).
In
a paradoxical contrast to the above claims of what is required in the future by
means of today's research, stands the acknowledgement of the fact that
experience of the use of this type of tools in projects of realistic size is
still quite limited, and that many of the commercial tools still seem to be
"toys" which are not suited for use in projects of realistic size and
complexity (ibid., pp. 10, 12). Obviously this supports the research view which
opposes this vague program of mathematization based on an unstated view of one
kind mathematics in terms of formal systems. Experience and reason (Churchman,
1971, chap.2; Parnas, et al., 1986) indicate why it is legitimate to embark on
alternative research without the expectation that, for instance, we will be
able to achieve a formal software development process in which the programs are
derived from specifications.
From
what has been said above about the formalist-mathematical position, it seems
that it is not so much mathematical as it is a oversimplified version of a
formal-logical position in the spirit of traditional mathematical logic. In
what follows, however, we will anyway pursue the mathematical interpretation of
the arguments, while the logical or mathematical-logical interpretation is left
to another section of the research proposal. By doing so we acknowledge that
the issue of the relation between mathematics and logic is not settled, that
they cannot be vaguely considered to be identical and therefore cannot be
subsumed under the one same label of one discipline, and that mathematical
logic can, at best, be considered as only one particular type of logic (Church,
1962, offers a contribution to this issue).
The
"mathematizing" trend in the development of computer and information
systems which was described above claimed the purpose and capability of
supporting languages for "capturing" and describing knowledge of the
application domain, its structure and behavior, and for capturing information
requirements in early development stages. Deductive and rule based approaches,
or conceptual modeling, were supposed to capture and to express in an explicit
way, in a declarative style, business rules and constraints. The additional use
of a temporal dimension would allow to reason about the states and the changes
of the system.
Dwelling
on the notion of states and changes of a system it has been noted (Rosen,
1985b) that the Newtonian idea of "state determined system" contains
some problematic basic assumptions about causality. Causation cannot be reduced
to simple mathematical relations between propositions which describe events,
with a segregation of different classes of causation into independent
mathematical structures. If the assumption of independence is relaxed, as it
should be e.g. in the case of biological modeling, mathematical images become
like webs of informational interactions which contain the set of
"state-determined" systems as a subset, and where the behavior of one
of these webs can be approximated, albeit locally and temporarily. But this is
a new notion of approximability which is only local and temporary, and this
"explains a great deal about why we have been able to go as far as we have
with the non-generic Newtonian picture, and why we have never been able to go
further with it". What is required, then, is to develop the mathematical
science of simple systems into a science of complex systems: "Namely, by
loosening the Newtonian shackles, we can introduce a category of final
causation" (Ibid, 1985#,p.175). The matter can be consolidated through the
study of other interesting literature. (Ackoff, et al., 1972, pp. 19-31, 248ff;
Churchman, 1971, chap. 3 and 10; Geach, 1981, pp. 128-138 on
"intentionality"; Grenander, 1983; Rosen, 1985a; Rota, 1973).
We
have here also an interesting connection to the history and theory of
statistics in terms of what we write elsewhere about the role of the
unique-single case in psychological research, versus Buckle and Qutelet's
Laplacean conception of universal determinism for cultural phenomena, making
mass-phenomena the sole object of the science of society (Hayek, 1941, pp.
318-319; Lottin, 1912, esp. pp. 313-317, 397ff, 440ff, 501ff, and the reference
to social mathematics on pp. 374ff.)
Our
proposed research will proceed through a detailed understanding of the
assumptions of the Galileian-Newtonian mathematics which is considered as
origin and model for many common formal mathematical systems, including the
characterization of the realm the biological as an indicator of analog problems
in the realm of the social (Dessauer, 1954; Henshaw, 1986, concerning
correspondence about Rosen; Koyr, 1954; Portmann, 1954; Rosen, 1985a; Rosen,
1985b; Rosen, 1986; Strong, 1957; Wedde, 1984). A hypothesis of the research
proposal would be that if any formal models of biological-social phenomena are
to be processed in terms of knowledge bases and rule systems, the specific
nature of mathematical modeling in relation to "states" and
"time" must be interpreted in the light of the above kind of
criticism. The formal models will then probably have to be expanded in unknown
dimensions, e.g. in terms of finalistic or teleological
"interactivity" (Ackoff, et al., 1972, pp.65ff. 160ff) as in some
suggestions which have been proposed in the context of quality-control of
information, man-computer interaction, interactive systems planning, and
computer supported constructive qualitative conversations (Forsgren, 1988;
Ivanov, 1972; Nilsson, 1987; Nilsson, 1988; Whitaker, &
stberg, 1988).
One
main point in the proposed.research, in fact, will be to relate such detailed
understanding of the assumptions at the roots of mathematics to the kinds of
mathematics and mathematical logic which stand closer to computer science.
Unhappily, many works on mathematics and mathematical logic for computer
scientists (Harel, 1987; Levin, 1974) stand really at a summarizing discoursive
textbook-level, and it is difficult to find discussions in depth of the
mathematics of computer and information science. Even the few classics (Turing,
1963, esp. pp. 24-25; von Neumann, 1956; von Neumann, & Goldstine, 1947)
seem to be quite ahistorical and they do not really touch upon the theoretical
issues of mathematics proper. This is by no means uncommon, as it may be noted
in many types of professional contributions in the formal sciences (Church,
1941; Hoare, 1969; Iverson, 1981). It is also interesting to note that attempts
to develop "an outline of a mathematical theory of computation" for a
supposedly mature field of science were made as late as in 1970 (Scott, 1970).
Contributions that cast some light on the basic scientific theoretical
assumptions of pioneers of computer science are rare, and they seem to become
still more rare in relation to the increasing number of writings on
mathematical formalisms in computer science (Minsky, 1967; Winograd, 1979, see
their bibliographies).
This last mentioned tendency
uncovers the superficiality of much thinking and training in the computer field
which lay at the basis of the Christopher Strachey's disarming definition of
computing science so late as in the year 1977 (Fontana dictionary of modern
thought)
Computing
science. The
study of the use and sometimes the construction of digital computers.... It is
a fashionable, interesting, difficult, and perhaps useful activity.
Unfortunately, in spite of appearing to be a mathematical or physical science,
it has so far a pitiably small body of generally accepted fundamental laws or
principles which are likely to remain valid even for the next 20 years, and
consists instead almost entirely of ephemeral "state of the art"
information. A more appropriate title at this stage of its development would
probably be "computer technology".
As
an example, in a Swedish university in the beginnings of the 80's, after the
advent of object oriented simulation languages but before the advent of
functional and logic programming languages, programming was thaught on the
basis of the concept of algorithm which
was introduced as a word originated by the Arab mathematician Abu Jafar
Mohammed ibn Musa al-Khowarizmi (around year 825), and having the following
definition: "a method for performing a task, where the method is expressed
in terms of a finite sequence of rules, operations". The introduction went
on remarking that this obviously is a very general definition including e.g.
cooking recipes, but: "we naturally are interested in algorithms in the
form of computer programs for processing of digital data, with the closer
definition of 'a sequence of operations which in a finite number of steps leads
to the solution of a data processing task'". It was, then, remarked that
the precise formulation of algorithms requires the development of artificial
languages or algorithmic languages which, through compilers, can bridge the gap
between the original language in which the problem had been formulated and the
computer's internal language. Advices about how to construct algorithms or how
to solve problems were a referral to "thumb rules and experience" as
represented by "heuristics".
So
much for the depth of the mathematical understanding of the nature of
programming. With such a kind of approaches, supplemented by more ambitious
references to obscure "abstract machines" and the like (Doyle, 1982),
a whole generation of computer scientists and university professors has been
brought up with a very particular and limited view of the meaning of formalism (Borillo,
1984; Mathiassen, & Munk-Madsen, 1986; Naur, 1982). This generation usually
is very heuristically based on experience but it does not relate the
understanding of the old algorithmics and of the new programming languages to
any deeper understanding of mathematics, formalisms, functions. Attempts to
reach a deeper understanding, however, are exemplified by many outside the area
of computer science (Dessauer, 1954; Rosen, 1985a; Rosen, 1985b; Stenlund,
1987; Stenlund, 1988; Strong, 1957; Verene, 1982). Neither there is a relation
of mathematics to "reality" or operationalization, in spite of the
frequent references to operations (Margenau, 1962, cf. the original Bridgman's
operationalism as described by Stevens in 1935).
More
fundamentally, the thing which is ignored is the historical ongoing debate
about the treatment of the infinite in mathematics (Zellini, 1985a; Zellini, 1985b), an
issue which is deeply related to the possibilities and limitation of discrete
mathematics, and which is often hidden behind eclectic, popular, playful,
pedagogically attractive but superficial allusions to the "mistique"
of the problem (Brown, 1969; Hofstadter, 1979; Mandelbrot, 1982; Maor, 1987;
Pearce, 1978; Pennick, 1980; Purce, 1974; Rucker, 1987; Zichichi, 1988). Such
state of things leads in turn to a very unfortunate lack of respect for the
discipline of mathematics which has sometimes prompted reactions that, however,
keep paradoxically the same superficial level of debate (Hansson, 1983).
The
nature of the mathematical playfulness that contributes to this superficiality
can itself be made an important object of our research. This idea is an
original contribution of our research which was advanced (Ivanov, 1983; Ivanov,
1989) almost simultaneously with the results of certain
"anthropological" research on the interaction of children and adults
with computers (Turkle, 1984). Research about playfulness, in a rather
different key, is proposed in another section of our reseach program as it
regards the use to which both mathematics and computers are put. The quest, in
our case, will be directed towards the "serious" treatment of aspects
of mathematics as related to computers and computer models, aspects which, when
ignored, encourage unmotivated claims about the applicability of mathematics
and computers outside the limited fields of Newtonian natural science (Marchetti,
1983; Peterson, 1975). In the context of social science and social systems
research it seems that such serious treatments of mathematics are, if possible,
even more rare than in the context of natural science (Bosserman, 1981, in
contemporary production; Cobb, & Thrall, 1981; Lottin, 1912, historically).
In fields which stand closer to formal science and computer science modern
problematic developments of mathematics are sometimes observed, but in mild
uncommitted and optimistic terms (Cohen, 1983). It happens more seldom that
fundamental issues are raised about the relation between computer programming
and mathematics (Chaitin, 1974, Chaitin, 1987 #1065, is a most interesting
contribution in this respect) which could be contrasted with early conceptions (Gorn,
1963; Korfhage, 1964). It is still more rare to find mathematicians who set
aside irresponsible playfulness in order to formulate strong explicit criticism
of the misuses and misunderstandings of mathematics in science in general, and
in computer science and scientific computing in particular (Ingelstam, 1970;
Schwartz, 1962; Truesdell, 1984).
There
are today paradoxically serious (postmodernistic?) attempts to defend the
legitimacy and fruitfulness of playfulness (Ehn, 1988; Papert, 1980). In
another section of our research proposal we suggest the conditions for a
genuine research about playfulness by extending certain recent attempts (Carse,
1986) and relating them to
mythical ritual behavior, which would eventually include the character of the
child archetype and "puer aeternus"(Hillman, 1971; Hillman, 1979;
Ivanov, 1986, pp.135-136; Jung, 1953-1979, CW 9:1, €€ 259ff). In any case there are plenty of historically
important research directions which are not easily related to playfulness. One
interesting direction which is today ignored in spite of being implicit in the
discussions on the use of computers is the school of "economy of
thought" (Jourdain, 1914; Rignano, 1913a; Rignano, 1913b; Rignano, 1913c;
Rignano, 1915a; Rignano, 1915b; Rignano, 1915c); Peano, 1915; Kennedy, 1980].
The idea of economy of thought, besides of its possible connections to
philosophical utilitarism, could be an entrance door to rich historical
material for supporting our research about the contact points between
calculation, computers, mathematics and logic, and about the chaotic
development that in the last decades is leading to a "tower of
mathematical Babel (Davis, 1987).
It
is apparent that the quest for the meaning of mathematization or of
mathematical formalism and symbolism in the quest for knowledge raises many
difficult issues. In part they have already been addressed in past times where
today's computers were simply represented by the question of computation or
calculation. A long run research program which must choose among alternative
degrees of mathematization in its work must therefore incorporate an inquiry
into the debates and arguments about the issue. Nevertheless, we will not
exaggerate in being too "philosophical", and we may disregard, for
the time being, some early speculations about mathematics in general, and
Platonism in particular, the concept of "form" which stands at the
root of the possible meanings of formalism, "the Greek mind", etc. (Boodin,
1957; Kitto, 1957, pp.192ff.; Koyr, 1954).
Some
philosophical precedents of the debates may, however, be particularly
interesting as they touch upon the strivings towards greater mathematization of
inquiry. One example will suffice about one of the early philosophers of the
scientific revolution leading to our age (Hobbes, 1962), who criticizes the
tendency to formalize with symbols and, on one occasion, accuses one of his
contemporaries for mistaking the study of symbols for the study of geometry
(ibid., p. 187). He observes that "the symbols serve only to make men go
faster about, as greater wind to a windmill", and that "no logic in
the world is good enough to draw evidence out of false and unactive
principles" (ibid., p. 188). On another occasion, in the essay
"Lessons on the principles of geometry..." addressed to "the
egregious professors of the mathematics, one of geometry, the other of
astronomy", the use of symbols is also opposed by observing that (ibid.,
pp. 247f, 329):
But
are not you very simple men, to say that all mathematicians speak so, when it
not speaking? When did you see any man but yourselves publish his
demonstrations by signs not generally received, except if it were not with the
intention to demonstrate, but to teach the use of signs?.... Symbols are poor
unhandsome, though necessary, scaffolds of demonstration; and ought no more to
appear in public, than the most deformed necessary business you do in your
chambers....
...Symbols,
though they shorten the writing, yet they do not make the reader understand it
sooner than if it were written in words. For the conception of the lines and
figures (without which a man learneth nothing] must proceed from words either
spoken or thought upon. So that there is a double labour of mind, one to reduce
your symbols to words, which are also symbols, another to attend to the ideas
which they signify.
This
would in our times be echoed by others (Keynes, 1952, p. 19n). An evaluation of
such words and disputes certainly requires that they be set into the broader
context of empiricism versus rationalism, and other contexts such as "the
past struggles between symbolists and rhetoricians in elementary geometry",
touching also upon the function of signs, of intuition, psychological studies
of symbolisms, etc. (Cajori, 1929, vol. 1, pp. 426-431 and vol. 2, pp. 284ff,
314, 327), and of "devices that appealed to the eye and thereby
contributed to the economy of mental effort" (ibid., vol.1, p. 265), i.e.
something which in our age of computer graphics could be called the integration
between geometry, aesthetics, and economy.
It
will, then, probably be noticed that the creative work on programming
languages, computer programming, or software engineering, relies heavily on the
creative use of "the history of mathematical notations" (Cajori,
1929) which, in turn, is interleaved with the history and philosophy of
mathematics supplemented with geometry, logic, statistics in the early history
of computers in the form of calculating machines, planimeters, integraphs, and
other mechanical aids to calculation (Cajori, 1980, pp. 485f; Smith, 1951-1953,
vol. 2, pp. 156-207).
It
could be suggested that these historical aspects are something which can and
should be studied apart from the substance of the discipline, in our case
mathematics and computer science. This runs, however, counter to one
influential interpretation of the nature of history, which leads us to be sure
that "no subject loses more than mathematics by any attempt to dissociate
it from its history" (Cajori, 1980, quoting a statement by J.W.L. Glaisher
on the title page). Our hypothesis is that it is treacherous to attempt to
avoid the supposed "genetic phallacy" (Toulmin, 1977, shows the
phallacy of this phallacy) and to strive for a higher degree of mathematization
or formalism in computer science without combining the study of mathematics
with the study of the history of mathematics. It is such a combination that
would allow an adequate understanding of the meaning, possibilities and dangers
of mathematization. It would also contribute to the avoidance of the risks associated with the delivery
of a "powerful tool" in the hands of immature scientists, in analogy
to the delivery of machine guns or bulldozers in the hands of children.
Unfortunately
it is nowadays possible to study the history of anything, and in particular of
mathematics, without coming in contact with the important problematic aspects
of such history when it is conceptualized as "accumulation" of
knowledge, where debates are glossed over. According to such a view of history,
in the spirit of a kind of "social Darwinism", it appears as obvious
that the "right history" in the one which contains only those
developmental steps of the discipline which lead to its present interpretation:
the rest is only a superfluous story of past mistakes which show how smart our
generation is in comparison with prior generations of scientists. In such a
perspective it becomes important to know how to choose of build up an adequate
historical account of mathematics. Certain approaches are mainly summarizing
and, therefore, only superficially informative for our research purposes. At
any rate, they do not enhance, and still less interprete or take position about
the controversial aspects of the historical development, even if they sometimes
offer extensive bibliographies and exciting overviews of curiosities, and are
presented in an attractive elegant style of writing (Bell, 1945; Kneale, &
Kneale, 1965, concerning logic; Struik, 1959). We believe that the study of
administrative, organizational or social use of computers requires a wider and
deeper approach and the inclusion of a more detailed and comprehensive history
of mathematics. It should also include "elementary" aspects in the
sense of including commercial, actuarial mathematics oriented towards
accounting, bookkeeping, auditing, and classical 18th century statistics (Smith,
1951-1953, vol. 2, pp. 552ff). What seems to be elementary may give insights
into the nature of modern problems. It is suggested by the fact that
"relations between algebra and geometry (Smith, 1951-1953, vol.2, pp.
320ff) remind us of the seldom recognized scientific basis of today's graphic
computer processing (Krner, 1960, p. 105, on "diagrammatics").
It
is obvious that the historical quest will unavoidably lead our attention to
certain questions of philosophy of mathematics. In the Anglo-Saxon tradition
such a philosophy has a relatively low historical content in spite of touching.
upon the important relation between mathematics and reality, tools versus
machines, conceptual thinking, etc. (Krner, 1960, pp. 29, 101, 176ff; Krner,
1979, pp. 38-91). In this tradition, history of science gives way to philosophy
of science or theory of science. The loss os historical detail and historical
spirit is perhaps compensated by a clearer focus on certain issues, e.g. the
question of logicism and formalism versus intuitionism (Krner, 1960), which
are critical for our quest about the function of mathematics in computer and
information science. It will then be noticed that the research problem tends to
"inflate", motivating a temporary strategic retreat into simpler
overviewing literature (Mathematics [-as a calculatory science -foundations of -history of], ), 1974
#440]. Such a strategic retreat should then be completed by diving, also at an
"encyclopedic" overview level, into particular issues which have
become relevant because of the late pressures towards mathematization or rather
formalization and mechanization of systems development. Among such issues we
may count the relativity of standards of mathematical rigor (Wilder, 1973) and,
more broadly, mathematics in cultural history (Bochner, 1973) which illuminate
fundamental questions about the meaning of form and symbol processing. In
Sweden this type of studies has had a very weak tradition, but there some
attempts have been made (Olsson, 1988a; Olsson, 1988b; Stenlund, 1987;
Stenlund, 1988; Wallin, 1980).
The
proposed object of research stands already at the frontiers of what
"method" should mean in its relation to mathematical and empirical
reality. The idea itself of having a method constitutes a particular chapter in
the history and study of science (McRae, 1957). It is therefore rather
paradoxical to ask which should the method be for furthering the studies which
have been suggested here. In such a situation it will be legitimate to continue
our proposed research by shifting gradually from the field of so to say
impersonal objective philosophy and history to particular essays and
testimonies of people who have reflected on the nature and development of
mathematics and of the study of form. In its extreme aspects it could be called
a shift from system towards biography. It may concern works which range from
philosophical-historical reflections (Davis, & Hersh, 1981, pp. 34ff on
"the ideal mathematician"; Kline, 1985; Melzi, 1983; von Wright,
1983; Weil, 1970-1974, italian ed. 1978, pp. 76ff, 113ff, 200ff.; Weyl, 1949;
Weyl, 1985; Whitehead, 1911; Wittgenstein, 1978) to more popular and
computer-focused versions of such reflections (Pagels, 1988), and to
biographical notes (Hodges, 1983; Johnson, 1977; Quillet, 1964, pp. 61ff,
207ff; Reid, 1970) including the extremes of noting chronical "don
juanism" and even homosexuality in particular mathematical minds (Hodges, 1983;
Wilson, 1988, pp.215-229). Such extremes may, however, be relevant to the study
of relation of mathematization to the cognitive and emotional functions of the
mind. We have also fiction literature and essays in the original broad sense of
the word leading the thoughts to an European continental tradition of where
science sometimes is still integrated with philosophy, politics, literature,
art and religion (Carse, 1986; Nyman, 1956, pp. 239 on Alfred Korzybski's
"general semantics" movement of the Non Aristotelian Society;
Zellini, 1985a; Zellini, 1985b; Zellini, 1988). To the latter area belong also
such approaches as the "logic of poetry" (Larsson, 1966) with its
probable connections of logic to rhetorics and dialectics (Reichmann, 1968;
Weil, 1970-1974) elaborated in fiction (Pirsig, 1974), and recently revived in
the philosophical literature (Barilli, 1983; Fisher, 1987). To the same last
area belong also those contributions which stand at the frontiers of cultural
criticism proper (Gunon, 1982, chapters 13-14, concerning the postulates of
rationalism, and mechanicism-materialism"), and of critical semi-fiction (Musil,
1952; Rnyi, 1973) including finally some literary works of Poe (Poe,
1969,p.302).
Such
contributions are often the work of people having varied degrees and qualities
of knowledge about present academic formal science but displaying a concern for
the deeper meaning of mathematics in the context of human thought. As such they
are therefore relevant for our own quest about the importance of the mathematical
aspect when studying presuppositions and consequences of the massive use of
computer technology, the dependency of the answers upon the particular field of
application, what should be understood as being a "field", etc.
The
direction of our research, following the above reflections, points out that
mathematics can be seen as a culturally contingent way of thinking having
particular psychic dimensions. In the mainstream of today's research this is
explicitly recognized in the reference to such concepts as "human
information processing" (Attneave, 1959, is an early exponent),
"artificial intelligence" and "cognitive science",
especially when dealing with expert or (decision) support systems. Up to now,
however, there seems to be a tendency to avoid historical fundamental issues (Ivanov,
1988). In the history of the development of mathematics many of the
psychological or psychic dimensions, whenever they were noticed, were
associated to a vaguely understood "creativity" or
"intuition" (Lerda, 1988, is a late expression of such an
understanding). These dimensions have been generally considered to be
intellectually intractable, and therefore they were judged as just interesting
or, at best, potentially useful only from the educational point of view. As
such they could be important as a source of expedients for making the study
more enjoyable or easier for people who lack the motivation or the capability
to follow the details of the analytical arguments. In this sense the intuitive
dimension has been considered in certain works (Hilbert, & Cohn-Vossen,
1952, cf. pp. iii-iv).
It
is natural to see the same attitude and argument today lurking behind the
conceptions of the scientific status of graphic data processing and the
processing of computerized visual images on the high resolution screen of work
stations. An intangible and evanescent so called tacit knowledge or personal
knowledge stands there for the relationship between computer mathematics and
screen geometry. It is seldom recognized that this relationship between
mathematics and geometry stood at the center of the debates around the romantic
"Goethian" conception of science (Bortoft, 1986; Steiner, 1926/1988;
Steiner, 1937/1982) and around Newton's versus Goethe's theories of color (Goethe,
1970).
The
romantic approach, however, or at least the literature about it, does not cover
in sufficient detail one central question for our research, i.e. the reasons and the particular form for the ongoing dissociation
between sensations and emotions in mathematical thinking, whether it is only a
question of these two categories of sensations and emotions or whether there
are other ones, where these categorization comes from, etc. What seems to be
required in the context of computer science is a more rigorous approach which
shows clearly what impact these categorical presuppositions have on the
development of mathematical theories, in a way that is similar to the
difficulties which characterized the birth of the intuitionistic school of
mathematics (van Stigt, 1979). Some help in this respect could be also obtained
from studies of the historical and psychological nature of numbers, formalism
versus axiomatics, presuppositions of the theory of sets, etc., paving the road
from philosophy of science to analytical depth psychology (Bachelard, 1975a,
chap.2, esp. pp. 19ff and 41; Bachelard, 1940, about "O mathmatiques
svres"; Bachelard, 1975b, chap.1, pp. 62ff; Bortoft, 1986, pp. 79f about
multiplicity vs. unity; Meschkowski, 1975; Spengler, 1981-1983/1918, esp.
vol.1, chap.2, pp. 53ff on the meaning of numbers, and pp. 8ff, 426ff on
nature-knowledge; von Franz, 1974). Some approaches lead in particular to
anthropology and religion, treating the role of experience and intuition, the
ritual origins of geometry and counting etc. (Freudenthal, 1962; Pennick, 1980;
Seidenberg, 1962a; Seidenberg, 1962b). They are interesting in the context of
our research since it is obvious that many "irrational" attitudes to,
and uses of, computer support can be interpreted as gambling and ritual
behavior (Ivanov, 1983; Ivanov, 1989; Turkle, 1984).
Among
works which more explicitly refer to psychology and to the problems of
mathematical thinking, intuition, and the "unconscious" there are
some more technically-biologically oriented (Klir, & Lowen, 1989), as well
as well known pioneering classics (Hadamard, 1954). There are also works within
the more general current of the psychology of personality and of learning (Rychlak,
1977) which stand epistemologically very close to the systems theory adopted in
our research program (Churchman, 1971).
It
can be seen that all the above gradually transforms, or rather widens the
original mathematical issue of our research turning it into an issue that today
would be considered as more legitimate under the label of cultural criticism.
In the academic community today such label is often considered as an expression
of contempt. Labeling something as being cultural criticism, journalistic
essaism, or even philosophical, are ways of explaining away problems by
claiming that even if they happen to be important, nevertheless they do not
belong to the realm of science and should not be institutionally supported by
the universities. This appears also to have been the paradoxical fate of all
metaphysics after Kant. It is one aim of our research to contribute to the
understanding why the philosophical implications of problems and crises in the
development of mathematics have, today less than ever, practical and
theoretical impact. So late as in 1972 a well known and respected scientist who
had worked at the frontiers of mathematics claimed the great undecidability
theorem of Gdel had not yet been absorbed by philosophy, though its ultimate
impact would undoubtedly prove to be shattering" (Morgenstern, 1972).
In
contrast, it can be said today that the mentioned crisis in the foundations of
mathematics has not left any trace, for instance in the ongoing research in
computer and information science. Maybe this is an echo of H. Weyl's suggestion
that there is a "Darwinist" line of argument lurking behind the basic
conjectures of D. Hilbert's mathematical program: Hilbert's trust in the human psychological
propensities
which he took to be embodied in the procedures of modern classical mathematics,
and which he directly argued for by appealing to their "practical
success", appears ultimately to be based on faith, in "...the
reasonabless of history, which brought these structures forth in a living
process of intellectual development" (Weyl, 1927/1967, quoted by
Deftelsen, 1986, p.37n) . Perhaps it was this very same requirement of faith
that led Hilbert's contemporary, the mathematician Paul Gordan, to comment on
Hilbert's solution of a problem in the theory of invariants by announcing in a
loud voice "This is not mathematics. This is theology" (even if it
was later tempered by the additional graceful concession that "I have
convinced myself that theology also has its merits" (Reid, 1970, pp. 34,
37).
It
can be the case that the matter has no "practical importance" today
when the computer enhances constructive procedures. Such procedures may,
however, rely implicitly on a trial and error experimentation which has given
up the controllability of the long run consequences of the logical network
build-up (Churchman, 1971, chap.2). It has been noticed that numerical
mathematics has recently undergone an enormous "development" in terms
of study of efficient algorithms in the discrete in order to solve
"approximately" problems defined in the continuum. This is based on
the great foundational premise of the end of the nineteenth century, i.e. the
conviction that analysis could be arithmetized as matter of principle (Zellini, 1985b, p.255). In this
way we come to witness how practical efforts of computer-based mathematics,
that is now mushrooming into efficient algorithms for parallel machines, is
based on foundational-philosophical premises and on a concept of approximation
which is violently challenged in some professional districts of the discipline (Rosen,
1985b; Truesdell, 1984) even if sometimes this is done in a much more limited
"technical" sense (Grenander, 1983). In the context of our research
we might need to explore the hypothesis that the perceived triumphs of
computerized discrete mathematics are essentially late superficial byproducts
or spin-off effects of the real earlier triumphs of Newtonian physics. Such
mathematics and related object-oriented thinking today, however, should raise
many difficult and painful questions in what concerns their applicability to
the biological and social fields of research (Chargaff, 1971; Oppenheimer,
1956) including the related moral aspects in Kass, 1972; Schwartz, 1962].
In
other words, the apparent utilitarian triumphs of recent applied mathematical
science enhanced by computer technology may be mainly a meretricious way for
profiting of past achievements. It would be a using up the
philosophical-scientific heritage after having freed oneself from the peers'
scrutiny and social control thanks to the extreme degree of specialization (Cohen,
1983) and thanks to the independent financing obtained from big industry and
big military state. Consequently, the "improvisational heuristics" of
degenerating research programmes, including even the "ability to suspend
judgement in the face of disconfirming evidence", will allow
"proposals to be made without regret even when they have highly
implausible aspects, or when tests are not likely to be possible in the
foreseeable future" (Holton, 1984).
If
the above turn out to be true, then, we should question the commonplace
statements about the advantages of a mathematization program for information
science, and perhaps of computerization in general. The essential, enduring,
central mathematical issues of the infinite, sets, multiplicity versus unity,
order-complexity-chaos and randomness, are certainly too important to be
gambled away in the recent playful type of mathematics (Hoffman, 1988;
Hofstadter, 1979; Pagels, 1988, all in varying degrees). These issues should be
rescued for the purposes of our research program, in a serious sense,
synthesizing what has been called different categories or types of
mathematics (Browder, 1976), a
seriousness obtained through a further development beyond the attempt of
conceptualization by means of games (Carse, 1986; Eigen, & Winkler, 1985).
In
any case, some important efforts have been made in order to understand instead
of explaining away, e.g. regarding why so called crises, as in the foundations
of mathematics, do not necessarily lead to deeper reflection (Zellini, 1988). A
possibly fruitful analogy, of course, is that crises in the effects of the use
of computers will not necessarily lead to reflection and better understanding
of the underlying problems. It may, therefore, be necessary in the course of
our research to share knowledge of cultural criticism of mathematics and of
"quantitative knowledge" (Bochner, 1973; Gunon, 1982; Ingelstam,
1970; Schwartz, 1962; Spengler, 1981-1983/1918; Tuchel, 1982, less specifically
mathematical). Of particular interest will be that kind of criticism by authors
who are practicing or scholarly mathematicians and attempt to focus their
attention on the details of the discipline itself (Rosen, 1985a; Rosen, 1985b;
Rota, 1973; Zellini, 1985a; Zellini, 1985b; Zellini, 1988).
The
cultural criticism of mathematics may be coming. Two recent, apparently well
diffused and easily available works that have not yet been integrated in our
own work (Barrett, 1987; Davis, & Hersh, 1986) indicate that the interest
for these matters may be on the rise. This obviously does not relieve us from
responsibility, but rather encourages us to contribute to this strife from the
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