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DOGMAS AND THE CHANGING IMAGES OF FOUNDATIONSJosé FERREIRÓS Abstract. We offer a critical review of several different conceptions of the
activity of foundational research, from the time of Gauss to the present.
These are (1) the traditional image, guiding Gauss, Dedekind,
Frege and others, that sees in the search for more adequate basic systems a
logical excavation of a priori structures,
(2) the program to find sound formal systems for so-called classical
mathematics that can be proved consistent, usually associated with the name of
Hilbert, and (3) the historicist
alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to
perfect available conceptual systems with the aim to avoid conceptual
limitations and expand the range of theoretical
options. I shall contend that, at times, assumptions about the foundational
enterprise emerge from certain
dogmas that are frequently inherited from previous, outdated images. To round
the discussion, I mention some traits of an alternative program that
investigates the epistemology of mathematical knowledge. Sumario. Ofrecemos una revisión crítica de
varias concepciones de la investigación sobre los fundamentos de la matemática, desde los tiempos de Gauss hasta el
presente. Se trata de (1) la imagen tradicional, que guió a Gauss,
Dedekind, Frege y otros, y que ve en la búsqueda de sistemas básicos
más adecuados una excavación lógica de estructuras a priori, (2) el
programa de encontrar sistemas formales
correctos para la llamada matemática clásica que puedan demostrarse consistentes,
habitualmente asociado al nombre de Hilbert, y (3) la alternativa
historicista, que guió a Riemann, Poincaré,
Weyl y otros, la cual busca perfeccionar los sistemas conceptuales disponibles
a fin de evitar limitaciones conceptuales y ampliar el abanico de
opciones teóricas. Defenderé que, en ocasiones, se encuentran supuestos
acerca del trabajo sobre fundamentos que emergen de ciertos dogmas,
frecuentemente heredados de imágenes previas ya superadas. Para completar la
discusión, menciono algunos rasgos de un programa alternativo, que investiga la
epistemología del conocimiento matemático. __________________________________________________________________ Confronted with my title, the reader may have thought that it is an
unhappy idea to put together the idea of
foundations and the word "dogma." After all, foundational research consists
of logical and/or mathematical results formulated and proved in the most
rigorous possible way. Thus we are talking
of a domain of objective results, unaffected by beliefs or vogues.
To put it simply: we are talking about logic, not about any aspect of culture
that may be affected by dogmas or by historical shifts. Only the fact that we
have been living in this intellectual atmosphere of postmodernism - you
may have reflected - can explain why such
titles are taken even a bit seriously. What can I reply? The difficult thing for me would be to convince you
that those points include some subtleties
that are far from being "evident," and, at the same time, that granting
this does not throw us in the arms of postmodern thinking. That, as I say,
would be a difficult argument, and I will
not try to make it here. So let me begin in a different way. Let me
underscore that the word "images" is in my title for some reason. My aim is to reflect on the activity of
foundations research. Once we conceive of it as an activity, a practice, it is
automatic that foundations research does not come down to a bunch of theorems - not even a multilayered set of problems, methods and
theories. Following Leo Corry [1997],[1]
let us call those theories, methods and results the body of
foundational knowledge. As all
practitioners know, the practice of foundations research is also guided by
certain images of this enterprise, images which may vary from
researcher to researcher, and certainly from time to time. These images
tell us about the goals one pursues when doing foundational
research, about important and irrelevant problems, acceptable and unacceptable,
promising and unpromising ways of approaching these problems, and so on.
What my title suggests is not
that the body of foundational results is affected by dogmas, just
that some images of foundations are. Put this way, I am sure more than
one reader will be relieved. In fact, examples of dogmatic
attitudes abound. A noteworthy one can be found in Quine's
writings on set theory. I do not mean his distrust of the Zermelo-Fraenkel
system for axiomatic set theory (ZFC) and his desire to consider alternative
systems. What seems dogmatic to me is his philosophical evaluation of the
paradoxes, the so-called "bankruptcy theory."
Here, Quine was fully in agreement with Russell: the paradoxes had shown commonsense
logic to be contradictory. This view is put forward, not only in his Mathematical
Logic [Quine 1940, § 29], but
also in the much later Set Theory and its Logic [Quine
1963, Introduction]. Here he writes that the "only natural attitude"
towards the notion of class, which
is so fundamental to thought, is the Comprehension Principle: that every
open sentence in one variable determines a class. The effect of the paradoxes
was to discredit this natural attitude, to
show that "commonsense is bankrupt," and "intuition" is not
to be trusted. Quine was dismayed to find that
most logicians were "retraining their intuition" by immersing
themselves in the system of ZFC set theory. In his view, one ought to consider
the
whole variety of possible systems (type theory, ZFC, von Neumann-Bernays-Gödel
set theory, Quine's own systems known as
NF and ML,[2]
and so on), treating none of them as standard.
For it would be imprudent to consider one single system as the standard,
natural one [Quine 1963, pp. viii, 1,5]. In my opinion, Quine's views on what he calls "commonsense" and
“bankrupcy" are worthless as a
philosophy. There is no rationale for thinking that the Comprehension Principle is more "natural" as a hypothesis than any other, say
Zermelo's Axiom of Separation. As a piece
of history, however, Quine's views are very interesting and revealing - they display before us the picture of a generation of
logicians whose "intuitions," to use his word, had been trained in
the logic of classes based on Comprehension.[3]
One is tempted to apply here Kuhn's terminology, speaking of a paradigm shift (from the Comprehension logic of classes to axiomatic systems of
set theory), of Quine as a member of the old
generation, unable to absorb the impact of the revolution, and so on. But this
is not our topic today. 1.
Generalities about dogma. In what follows I intend to discuss several different images of
foundations, and a few dogmas that are in
fact interrelated. Let me open my mind and mention at least one of these right
away. We all have some image of what the relations are between
foundations, logic, and human reason (a
noteworthy triangle). This frequently belongs to our cherised intimate convictions,
and it may well happen that we do not want to discuss them with strangers. I beg
your pardon, but as a philosopher it is my business to thematize such intimate
convictions and reflect about them, eventually trying to see if they
depend too much on traditional beliefs. Just in case somebody doubts that
the connections between logic, reason and foundations
can really be found in noteworthy contributions to this field, let me offer an
example. Recall the well-known Hilbert
program. It is interesting to reflect on some of the more philosophical opinions that Hilbert expressed (and actually he was
generous in offering such opinions).
For instance, in his address 'Axiomatisches Denken' of 1918 we can
read: the most important mathematical thinkers … have
always cultivated the relations to the neighboring sciences, especially the great empires of physics and
epistemology, above all for the benefit of
mathematics itself. The essence of these relations and the reasons for their fruitfulness will be most clear [...by] describing […] the axiomatic
method. [Hilbert 1935, 146] In recent years, historians such as Volker Peckhaus [1991] and Leo Corry
[1997] have made clear the extent to which
this was not propaganda, but the expression of a serious concern
of Hilbert's with physics (especially its axiomatization) and with
epistemology. In the celebrated 1900 lecture on 'Mathematische Probleme', Hilbert
already expresses his concern with deductive finitism, and says as
follows: This requirement of logical deduction by means
of a finite number of processes is simply the requirement of rigour in reasoning. Indeed the requirement of rigour,
which has become proverbial in mathematics,
corresponds to a universal philosophical necessity of our understanding {Verstand). [Ewald 1996, vol. 2, 1099] As Michael Hallett [1994] has made clear in a paper devoted to
connections between Hilbert's axiomatic method and the
"laws of thought," his finitism was thus meant to express a key (though weak) principle about the workings of the mind. In
1928, Hilbert made the point by saying nothing
less than the following: The fundamental idea of my proof theory is none
other than to describe the activity of our understanding, to make a protocol of the rules according to which our
thinking actually proceeds. [van Heijenoort 1967,
475] In
these sentences, the word Verstand seems to have been chosen to comply
with Kant's terminology,[4]
but, being a bit less literal, we could perfectly well translate it by
"reason." In
sum, according to Hilbert there is a very close link between foundational
research and the study of thinking and reason, i.e., epistemology. To be sure,
Hilbert did not pretend that his Beweisstheorie would exhaust that
study and supplant epistemology. Many important epistemological matters would
remain open even after the satisfactory completion of his program
(see Peckhaus [1991]). But the foundational results of Beweisstheorie certainly
had, in his view, quite direct epistemological implications. Is this a conception that one must share? Or, does it
depend on certain assumptions that are
open to questioning? A short general reflection may suggest the latter. In
both scientific and mathematical research, ten years are a very long time,
but, as you know, in the realm of human beliefs and intellectual traditions a
century may not be a very long time. It is only in the last century and
a half that we have started to think about human beings and minds in terms
of evolution.[5]
We belong to a culture that has almost always regarded men and women
as having an eternal essence, and most of us have been raised in this belief. Different
names for that essence, obviously with different connotations, have been
"the soul" and "reason." Kant's philosophical ideas, to be
sure, were formulated long before any need could be felt
to even faint that such a belief should be questioned. So one might well
employ the following rule of thumb: whenever
you see a foundational researcher that is deeply influenced
by Kant, beware and consider the possibility that he may be equating
epistemology with foundations in a way that is unwarranted, i.e., that is
based on questionable assumptions. Intimately entangled with the idea
of a special link between logical foundations and human reason, one can find other noteworthy beliefs. When we do research
on the foundations of mathematics, we look for systematic
foundations. Of course it is very healthy and
interesting to search for unification and system, and I could have
nothing against it. But, when coupled with the
above-mentioned belief in a transcendent essence, one can easily be led into
thinking that the systematic foundations one obtains tell us something about human reason. The very idea of systematicity may thus,
coupled with other beliefs, give rise to a
second dogma. We might go on trying to locate and reconsider
beliefs that could be dogmatic. Some come to my mind rather quickly. The
idea that mathematics and logic are marked out by enjoying absolute
certainty, by consisting of results built on the most solid rock (not
"on sand," as Weyl once wrote).
Or perhaps our habit of taking logic to be a (even the) kernel of
human knowledge, and our associated tendency to assume that any act of
thinking ought to correspond to some kind
of logic (say, e.g., inductive logic). Also the high expectations we
place on the capacity of formal systems to capture all kinds of subtle
concepts and relations. But enough. At this point we shall stop talking generalities and begin to
examine some particular, historically
given images of foundations. This will allow me to review some key stages
in the evolution of foundational ideas, and also to analyze the emergence and
persistence of some dogmatic beliefs. We shall concentrate on the first two
mentioned above: the idea of a special link between foundations and epistemology,
and the epistemological
interpretation of systematicity. 2. Two
traditional images: Gauss to Hilbert. Let me begin with Gauss, just 200 years ago. As a young man, Gauss was
interested in philosophy and read Kant and other philosophers with some care.
We are fortunate to have
a few statements of his, expressing thoughts about the philosophical
foundations of mathematics - what he called,
using now obsolete terminology, the "metaphysics of mathematics."[6]
Occasionally, Gauss felt the need to enter into a discussion of such philosophical topics in order to clear the way to novel
mathematical developments. Thus he ended
up touching on the "metaphysics" of number, of magnitudes, and of
space. Particularly important were his parallel reflections on the
epistemological foundations of geometry and
of arithmetic. These reflections were rich in mathematical content, since they
related to Gauss's evolving views on non-Euclidean geometry and differential
geometry, on number theory and
function theory. Their final outcome was synthesized by the learned man
in the form of a Greek motto: 9O qeoj a0riqmhti&zei "God does arithmetic,"
his thoughts consist in numbers and number-relations, even when we cannot follow them. The significance of this statement can only
be understood if we take into account that Plato is
reported to have said, "God does geometry." The Gaussian motto documents the end of the millenary domination of geometry
in Western images of mathematical knowledge,
and starts a historical period in which pure mathematics
would be "under the sign of number" [Hilbert 1897]. The idea was
spelled out in letters of 1817 and 1830; I
quote the 1830 letter to Bessel: According to my most intimate conviction, the
theory of space has a completely different position with regards to our knowledge a priori, than the pure
theory of magnitudes. Our knowledge of the
former lacks completely that absolute conviction of its necessity (and therefore of its absolute truth) which is characteristic of the latter.
We must humbly acknowledge that, if number is just
a product of our minds, space also has a reality outside our minds, and that we cannot prescribe its laws a priori. [Gauss
1900, 201] The language is strikingly Kantian, certainly more so than the language
we find in Poincaré's allusions to Kantian epistemology. Gauss was not doing "metaphysics" in
our current sense of the word. What he did was to search for new systematic accounts of several mathematical topics, to
elaborate new mathematical theories, hoping to find thereby the philosophical
and epistemological basis of mathematical
knowledge. But his results and theories did not possess philosophical meaning
in and off themselves. They acquired epistemological significance because Gauss viewed
them against the background of Kantian (and Leibnizian) epistemology. He was
guided by the old image of human knowledge as a combination of elements derived
from two sources: Reason and the
senses, the rational and the empirical. In Gauss's view, agreeing
with Leibniz and Kant, mathematical knowledge has a strictly rational core,
which is an a priori product of pure Reason. Geometry did not
belong in that core, which he now identified
with "arithmetic" in a broad sense - the theory of the complex number
system in all its aspects. Though some of these reflections became
available to the public as of 1831, for the most part
they remained in the hands of a few friends until they finally were published in
the 1860s. From this time on, it became customary
among relevant German authors to conceive of
"arithmetic" as another name for pure mathematics, and to exclude
geometry from this domain.[7]
To give just an example, when Dedekind gave expression to the view that pure mathematics
is logic, 4 years after Frege but independently of him, he wrote:
"arithmetic (algebra, analysis) is just
a part of logic." Interestingly, Dedekind chose to synthesize his new
view with a motto that prolonged the Plato-Gauss tradition: Aei _ o( a!nqrwpoj a0riqmeti&zei i.e.,
"man ever arithmetizes." With this new move, the pure theory of
numbers and their relations ceases to be a godly
matter, to become human, very human. In the preface to his epoch-making Was
sind und was sollen die Zahlen?, Dedekind made
clear his opinion that arithmetic and pure mathematics are a matter of logic, an
immediate product of the reine Denkgesetze, the
pure laws of thought. They rest solely on the
notions of set and mapping, and therefore on primitive abilities of the mind
without which no thinking at all is possible. Thus they are available to anyone
in possession of a gesunder Menschenverstand, a sane common
understanding. Dedekind went as far as to propose
the view that the chain of proofs that he presented in detail takes place
actually in the mind of his readers as soon as they employ numbers; only
it takes place unconsciously, and so our consciousness extends merely to some
byproducts, complex arithmetical truths that we usually mistake for simple,
intuitive evidences. To me, the situation with Dedekind
is reminiscent of what we have seen apropos of Quine. I admire very much
Dedekind's work, but his belief that the notions of set and mapping are primitive ones, that they are engraved in our minds from the
time of conception - this I can only regard as an
unwarranted belief, and in all likelihood a false one. It is surprising to find that so many authors, including good
philosophers of mathematics in recent years, still
consider the notion of set as an intuitive one, as epistemologically primitive.[8]
Of course this point is easier for us to grasp with hindsight, thanks
to our knowledge of issues like the polemics surrounding the axiom of choice,
the possibility of predicative and other deviant conceptions of sets, and so on. The work of Weierstrass, Dedekind
and others on the "arithmetization" of pure mathematics led to modern
systems of logic and set theory, which started to become explicit in work of Dedekind, Peano, and Frege during the 1880s. But
these men were still immerse in the old conception of human
knowledge, and through them it influenced other authors like Hilbert himself. As
we have seen with Dedekind [1888, p. iv-v], they believed that
actual human knowledge, as historically given, is (partly at least) the product
of unconscious rational, logical thinking activities.[9] Their search for deeper
systems of mathematics was for them a logical excavation in the hidden
structure of Reason. This constellation of
ideas we shall call Image 1. Image 1 was also in Frege's mind
when he compared arithmetic "with a tree that unfolds upwards in a multitude of techniques whilst the root drives into the
depths" [Frege 1893, xiii]. Frege's main goal was
epistemological: he wanted to prove most strictly that the laws of
arithmetic are a priori, indeed that they are purely logical laws. It was
only as a means to obtain full
control of his assumptions and developments, to check systematically that his goal had been attained, that he became interested in formal systems of
logic. As I have said before, under the assumptions of Image 1, it was natural
to expect that the systematic search for
sounder and broader logical bases would amount to a search for epistemological foundations.
But images of foundations and foundational research have changed greatly since the nineteenth century. * * * The search for logical foundations
culminated in the new axiom systems proposed during
the decade 1899-1908 for geometry, for the arithmetic of both the natural and
the real numbers, for set theory. The gain in terms of unification,
systematization, and freedom to work in
modern mathematics, was undeniable and certainly wonderful. These new systems were perfected and made fully precise some twenty years
later with heir strict
formalization (due in good measure to Weyl, Skolem, Hilbert and Bernays)[10].
Around 1920, Hilbert conceived of a change in perspective that would give
rise to a new foundational program and a new image. The new Image 2 inherited
many of the traits of 1, unreflectedly to a
good extent. Hilbert was still attracted by Image 1, but he was aware of the
difficulties involved in trying to show the a priori (indeed, the
logical) nature of any of the above-mentioned
systems, and particularly aware of the need to refine and develop logic
to suit the needs of modern mathematics. Hilbert and Bernays started the new conception when it
finally became clear that the former's hopes
for a revival of logicism, in the wake of Principia Mathematical,[11]
had foundered.
This had a neat effect on the goals of the whole foundational program, for now it
was no longer a matter of establishing the truth of the propositions
belonging to pure mathematics, or the sources
for such truths, but merely a question of establishing the acceptability of
classical mathematical systems by a strict proof of consistency. The shift was
not voiced very much aloud, but it is very noteworthy - it involved acceptance
of the hypothetical character of
pure mathematics (what some call, ambiguously, its 'quasi-empirical'
status), and thus a deep reform of received images of mathematical knowledge.[12] Within Image 2, the main goal was to find
sound formal systems, sufficiently powerful to
derive all of classical mathematics within them (a requirement that was simply
equated with completeness), but such that
they could befinitarily proven to be consistent. Recourse to formal languages, which in Frege had been merely a means to check the
sufficiency of the proposed
axioms/principles,[13]
now became an essential trait of the foundational program. But many authors continued to believe that the
formal systems would somehow uncover the hidden logical structure of
mathematical Reason, which supposedly had always acted behind the course of
historical events. Some even hoped that the chosen system would be
all-embracing, in such a way that new mathematical developments would remain
within its bounds. As if our historical experience did not show mathematics to
be a creative human activity, and mathematical theories the temporary outcomes
of an open-ended process of development. When Gödel's incompleteness results forced
foundational research to shift its main target, from consistency to goals such as relative consistency (and
others), a very interesting branch of mathematics
came to maturity - indeed, a handful of branches. We might go on here and attempt a finer analysis of several new and different Images
2.1, ... 2.n of foundational research, among which programs like those
of proof theory, model theory, or reverse
mathematics, are particularly noteworthy. These programs have made available very
interesting results, such as those of Gentzen and his followers on the
consistency of arithmetic and other systems, the results of Cohen on the
independence of the Continuum Hypothesis,
etcetera, and more recently the results obtained in predicative mathematics and
reverse mathematics. From the standpoint of my present review of the
connections between foundations and epistemology,
however, we must emphasize that one can no longer see in any of these projects
and results a fully-fledged program to establish the foundations of mathematical
knowledge. Their epistemological relevance is
not obvious, which contrasts strongly with Hilbert's
desire to "eliminate from the world once and for all the question of the foundations
of mathematics" by establishing the absolute "freedom from
contradiction" of the classical theories with his Beweisstheorie.
(To be sure, the idea I am now presenting is not new, it has been emphasized, e.g., by Feferman and Sieg; but too many
others are not yet aware of it.) I would not wish to go on without one further
comment. To say that foundational studies, in
their present shape, are not immediately relevant to the epistemology of mathematics, is by no means the same as saying that
they are (or even worse: that they must be) irrelevant. Many of the classical results in foundational studies are
highly illuminating as to the nature and characteristic traits
of classical mathematics. A paradigmatic example was the increasingly clearer realization of the contrast between
(process-oriented) constructive
mathematics, and classical mathematics (object-oriented,
"platonistic"). Another
was, of course, the discoveries about possibilities and limits of the
formalization of mathematical theories. While many present-day results on
foundational matters will probably be of no consequence
to future epistemological debates, some are of a different kind. From this standpoint,
I would like to warn against a wrong interpretation of my words, and I would like to make a call for renewed interactions between foundational studies
and the philosophy of mathematics (to
which I must add the history of mathematics). 3. A historical
alternative: Riemann to Kitcher. Let us come back to the presumed convergence of systematic foundations
with epistemological roots. I have suggested (very
sketchily) some ideas about its origins and
development, and how it became increasingly dissolved
within the transformations suffered by foundational research in the
twentieth century. Today, the old static image of knowledge is deeply outdated,
judging it against the framework of present-day philosophy or, for that matter,
of contemporary scientific knowledge. Once we discard the a priori belief
in the existence of a transcendent human
faculty that goes under the name of Reason; once we take
into account scientific discoveries about the biological and cultural evolution
of humanity; once we consider
historical studies of past mathematical theories and practices; it becomes
quite doubtful that a hidden structure may have been present throughout. For reasons like these, in recent
decades new kinds of historicistic and naturalistic conceptions of mathematical knowledge have emerged. It is interesting to
realize that these recent trends are akin to a very different approach to
the foundations that existed since the mid-nineteenth
century, coexisting with Images 1 and 2. We may speak of tradition starting
already with Riemann, and continuing through a good number of twentieth-century
authors. Interestingly, Poincaré counts among them. Due to his philosophical beliefs, Riemann
consciously avoided the image of Reason as the
a priori source of knowledge. In his view, all knowledge arises from the
interplay of "experience" broadly conceived {Erfahrung) and
"reflection" (Nachdenken) in the sense of reconceiving
and rethinking. Human knowledge begins in everyday experiences and proceeds
to propose conceptual systems which aim to clarify experience by going beyond the
surface of appearances. Reason in the old sense is found nowhere, there is no
hidden a priori structure
- those elements in our theories which do not simply arise from sense-data are
just of a conjectural nature, hypotheses like the axioms of geometry [Riemann
1854].[14]
To give you at least a superficial
impression of Riemann's turn of mind, let me quote a fragment
from the last page of his famous lecture On the hypotheses upon which
geometry is founded: A decision regarding these questions [about the
validity of geometrical assumptions at different levels of physical reality] can only be taken by starting from
the previous conception of phenomena, whose foundations were laid by Newton and
which has been confirmed by experience, and by
reforming it gradually, considering facts that cannot be explained from it.
Investigations which start from general concepts, like the one developed here,
can only serve to avoid that such work may be
hindered by conceptual limitations, and that the progress in our knowledge of
the connections among things may be limited by prejudices handed down by tradition. [Riemann 1854, 286] For Image 3, the point of foundational studies is to perfect
available conceptual systems by spotting
conceptual or theoretical inadequacies, to expand the range of available concepts
and avoid conceptual limitations, to strive for greater generality, and to
eliminate traditional prejudices. Thus, Image 3 is strongly diachronistic. It is
no longer a matter of excavating hidden
structures, but of going beyond traditional ideas in order to gain a deeper
grasp of reality (though perhaps not a perfectly realistic one). New forms of developmental
understanding of mathematical knowledge would later be found in Weyl, in Piaget, in French authors like Cavaillès, even in one
of Hilbert's closer collaborators, Paul Bernays [1976].
More recently, philosophers such as Lakatos and Kitcher
joined this group and tried to analyze the fine structure of historical
processes of mathematical development. But, should we come to the conclusion that mathematical
knowledge is absolutely undetermined, except
for the constraints imposed by tradition and history? In my opinion, not
at all. 4. Towards a new
image of the roots of mathematics. Human history, including mathematical history, is an expression of human
activities. Its diversity and degrees of freedom
will therefore be limited by any strong constraints acting upon human activity. Here, I must limit myself to a
rough sketch of what seem to be key constraints (though I am obviously aware of
the controversial nature of these issues, and the resulting
need to discuss carefully and substantiate each of the following points). Human beings are members of the human species,
"linguistic animals" as Aristotle said, at once biological and social.
A world of physical objects, biological abilities and needs, sense-perception
and motor action, the use of language within a web of social life and common activities - these are some of the constraints bounding human
activity and therefore history. That constitutes an invitation to explore and formulate
varieties of so-called naturalism that may
fit with biology and history at a time. To avoid confusions generated by the
trendy and equivocal term "naturalism," it might be preferable to
describe the kind of viewpoints I am thinking about through some other
label. Perhaps one might use the phrase genetic epistemology, but in a
sense divergent from orthodox Piagetianism. Indeed the adjective "genetic"
suggests the biological genesis of human knowledge, its emergence from our natural
abilities, but also the historical genesis of human knowledge. At this point, we are not talking about an "image 4" of
foundations, because this kind of viewpoint incorporates a deep shift, which we
could try to make explicit by distinguishing roots
from foundations. Now we
are not analyzing the foundations of mathematics, looking for a purely
rational or (at least) a perfectly systematic framework within which to develop
current mathematics. Instead, we aim to explore the epistemological roots of
mathematical knowledge. This is what I mean to
connote by changing the biological metaphor of roots for
the architectural metaphor of foundations. But, again, this is not the place
to attempt an original theoretical development. So let me close with some remarks on a most noteworthy proponent of naturalism,
none other than Quine [1969]. The writings of this great philosopher and
logician radiate with the joy of engaging in
the search for systems. Quine was a great system builder both as a logician, as
a philosopher of language, and as a naturalist. But when he transferred this
trend of mind from logic to epistemology, I believe he was making a
characteristic mistake. In line with the old epistemological tradition,
Quine believed that the search for systematic foundations converges
with the quest for epistemological roots. In this latter context, however, systematicity
may well be a trap rather than a virtue. Let me give a telegraphic example. There is not
the least reason to believe that arithmetic as practiced by a 10-year-old
child "must converge" with the foundations of our set
of natural numbers, or that fractions as used by different cultures must be
systematically explained together with the rational number system [Benoit et ai, 1992].
Elementary, commonsense arithmetic can
be understood from a purely constructive standpoint, while the step to N
as a set, harmless as it may seem to present-day mathematicians, involves of course
the introduction of actual infinity. Peano arithmetic involves quantification
over infinite domains, and thus what Hilbert and his followers called a
"transfinite axiom" [Hilbert
1925, 382]. What a genetic epistemology (in the above sense)
should do, is to analyze the epistemological roots
of these different practices and theories, to understand the links between them (be they cognitive, historical, social, or what not). It
should also emphasize the shifts and
displacements which distance them, and search for the factors that help explain
those shifts. In doing so, one must carefully avoid the temptation of being
over-systematic. So let me take exception to
Quine's dogma, and invite you again, in the spirit of Riemann, to
reconsider received ideas on all of these issues. 5.
References. Benoit, P., Chemla, K. & Ritter, J. (coords.) 1992. Histoire de fractions,
fractions d'histoire, Basel, Birkhauser. Bernays, Paul 1976. Abhandlungen
zur Philosophie der
Mathematik, Darmstadt,
Wissenschaftliche Buchgesellschaft. Corry, Leo 1997. Modern algebra and the
rise of mathematical structures, Basel/Boston, Birkhauser. Dedekind,
Richard 1888. Was sind und was sollen
die Zahlen?, Braunschweig, Vieweg. Reprinted in Gesammelte mathematische Werke, New York, Chelsea, 1969. English
translation in [Ewald 1996, vol. 2]. Ewald,
William, ed. 1996. From Kant to Hilbert, Oxford
University Press, 2 vols. Ferreirós, José forthcoming. 9O qeoj a0riqmhti&zei The rise of pure mathematics as
arithmetic with Gauss. In C. Goldstein, N.
Schappacher, J. Schwermer, eds., The Shaping of Arithmetic: Number theory
after Carl Friedrich Gauss's Disquisitiones Arithmeticae, Springer, Berlin. Frege,
Gottlob 1893. Grundgesetze der
Arithmetik, vol. 1, Jena, Pohle; reprint: Hildesheim, Olms, 1969. Gauss, Carl F. 1900. Werke, vol. 8, Göttingen,
Dieterich; reprint Hildesheim, Olms, 1973. Hallett, Michael 1994. Hilbert's Axiomatic Method
and the Laws of Thought, in A. George, Mathematics and Mind, Oxford
University Press, pp. 158-200. Van Heijenoort, Jean.
1967. From Frege to Gödel, Harvard Univ. Press, reprinted in
2002. Hilbert, David 1897. Bericht iiber die Theorie
der algebraischen Zahlen, Jahresbericht der DMV4. Reprint in Gesammelte Abhandlungen, Berlin, Springer, vol. 1, 1932. 1935. Gesammelte Abhandlungen,
vol. 3, Berlin, Springer; reprinted in 1970. Maddy, Penelope 1992. Realism in mathematics, Oxford,
Clarendon Press. Peckhaus, Volker
1991. Hilbertprogramm und
kritische Philosophie, Gottingen, Vandenhoeck & Ruprecht. Quine, Willard Van O.
1940. Mathematical Logic, New York, Norton.
1963. Set theory and its logic, Harvard Univ. Press. 1969. Epistemology naturalized, in
Ontological Relativity and other essays, Columbia Univ. Press. Riemann, Bernhard 1854. Über die Hypothesen, welche
der Geometrie zu Grunde liegen, in Gesammelte Werke, Berlin, Springer, 1991. English
trans. in Ewald, op. cit., vol. 2. (See also Riemann's Fragmente philosophischen Inhalts, in Gesammelte
Werke.) Sieg, Wilfried 1999. Hilbert's
programs: 1917-1922, The Bulletin of Symbolic Logic 5. [1]-
Corry talks about the modern structural approach in the images of
mathematics and in the body of mathematics (e.g., with Bourbaki, or with
category theory). [2]-
The abbreviations come from the titles of the works in which he proposed
those systems: 'New Foundations' (1937) and Mathematical Logic [Quine
1940]. [3]-
The
notion of intuition that emerges here is very far from the Kantian one, but
perhaps close to the teachings of Felix Klein. [4]-
His distinction of the more dry faculty of the Verstand (understanding)
from the idealizing faculty of the Vernunft (reason). [5]
- I mean here the fact of biological evolution on earth, and leave
open the thorny question of the right theory of evolution. [6]-
Fragments from 1800, 1816-17, 1825, 1831. See [Femims, forthcoming]. [7]-
There is an early exception, Martin Ohm, who identified Zahlenlehre with
pure mathematics from as early as 1819. He
is certainly relevant because his textbooks and his views enjoyed wide
diffusion in the Gymnasien. [8]-
Choosing only among first-rate authors, an example can be found in Maddy
[1992]. [9]-
Frege is a different matter, for his fight against psychologistic logic
distanced him from this standpoint, but he had little success until well
into the 20th century.
[10]-
Of course, here one must also remember Frege, Peano, Russell and
Whitehead, but when I say "strict" formalization I must
refrain from citing them (especially the last three). We are talking about
first-order formal axiomatizations. [11]-
I am referring to the 1917 Zürich address Axiomatisches Denken (in
[Hilbert 1935]) and to some of Hilbert's courses at Gpttingen. On this
topic, see Sieg [1999]. [12]-
From Plato to Kant, from Descartes (perhaps even Euclid?) to Frege, the
propositions of mathematics were taken to be truths simpliciter. [13]-
“...thus we obtain a basis for judging the epistemological nature of the law
we have proven" (preface to Frege [1893]). [14]-
It may be convenient to remind the reader that Riemann's
differential-geometric axioms or "hypotheses" are quite different
from, and deeper than, Euclid's.
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