FOM: Quasi-empiricism and anti-foundationalism
Patrick Peccatte
peccatte at club-internet.fr
Fri Sep 11 06:50:39 EDT 1998
This is an attempt to understand the commonly accepted correlation
between quasi-empiricism and anti-foundationalism which is one of the
topic of the thread 'foundationalism and anti-foundationalism' (see for
example: Stephen Simpson 1 Feb 1998 22:17:36 and 21 Jul 1998 18:16:13,
Martin Davis 10 Mar 1998 17:18:42 and 12 Mar 1998 12:35:50, Reuben Hersh
11 Mar 1998 15:43:40, Jeremy Avigad 15 Jul 1998).
[I apologize for my poor English, but fortunately, this posting is
filled with lots of quotations].
As far as I know, the term 'quasi-empiricism' has been coined by Imre
Lakatos in the sixties. It actually designates a very heterogeneous
stream which contains at least two distinct perspectives. To explain
this distinction, let me try to draw a sketch in 7 steps of this/these
viewpoint(s) in which I emphasize on the anti-foundationalist opinions:
===> 1. Prehistory: mathematics and empiricism
Many mathematicians and philosophers have compared mathematics and
empirical sciences in order to understand the mathematical
characteristics using empirical concepts (fact, experience, induction,
challenging the _a priori_ feature of the mathematical truth, etc.). In
the seminal article mentioned below, Lakatos quotes mathematicians as
Russell, Weyl, Curry, etc. This kind of comparison between mathematics
and other sciences is an old inspiration which is present in John Stuart
Mill's _System of Logic_ for example. But the modern idea seems largely
independent and could have been 'rediscovered' by many authors; in
particular, Emile Borel and Jacques Hadamard have implicitly referred to
an essay on the foundations of mathematics published circa 1880 by Paul
Du Bois-Raymond (_Metaphysik und Theorie der mathematischen
Grundbegriffe: Größe, Grenze, Argument und Funktion_) in which two
fictitious persons are dialoguing: the Idealist and the Empiricist. And
of course the views of the 'so-called French empiricists i.e. Borel,
Baire, Lebesgue, Lusin, et al.' as they are qualified by Stephen Simpson
(15 Dec 1997 18:46:59) show that it is possible to conjugate
foundational or foundational-like researches with an empiricist feeling.
===> 2. Forerunners
- Stephan Körner, 1965, _An Empiricist Justification of Mathematics, in
Y. Bar-Hillel (ed.), Logic, Methodology and Philosophy of Science_,
Amsterdam: North Holland, 1965, pp. 222-227. _Proceedings of the 1964
International Congress of Logic, Methodology and Philosophy of Science_.
For Körner, "the scientific theories embedded in mathematics function,
and are justified, together with their mathematical framework as
syncategorematic constituents of empirical propositions" (p. 222).
Körner had further given a communication to the 1965 London Congress
(_On the Relevance of Post-Gödelian Mathematics to Philosophy_, see ref.
below, pp. 118-132) where Kalmár and Lakatos were presents, and again he
talked about the empiricist justification of mathematics (p. 130).
- László Kalmár, 1967, _Foundations of mathematics - Whither now ?_ in
I. Lakatos (ed.). _Problems in the Philosophy of Mathematics_,
Amsterdam: North-Holland, 1967, pp. 192-193. _Proceedings of the
Colloquium in the Philosophy of Science_, London, 1965.
Kalmár has argued i) "that the research in the Foundations of
Mathematics has hitherto been pursued with the supposition that we shall
show it to be a _firmly founded_ pure deductive science"; ii) "this
hope has never, in fact, been realised, and [
] the axioms of any
interesting branch of mathematics were originally abstracted more or
less directly from empirical facts, and the rules of inference used in
it have originally manifested their universal validity in our actual
thinking practice"; iii) "the consistency of most of our formal systems
is an empirical fact; even where it has been proved, the acceptability
of the metamathematical methods used in the proof (e.g. transfinite
induction up to some constructive ordinal) is again an empirical fact."
(p. 192).
So, in the confused trend called 'quasi-empiricism', Kalmár seems to be
one of the most critical thinker towards the foundations of mathematics.
===> 3. The main philosophical quasi-empiricists
The following two articles are the classic foundations of the genuine
philosophical 'quasi-empiricism', each using this neologism which can
also be found in the well-known book _Proofs and refutations_ by Lakatos
(about this book, cf. fom-list: Jeff Zucker 13 Dec 1997 22:00:05, Moshe'
Machover 14 Dec 1997 18:26:56, and a bibliography on Lakatos in Julio
Gonzalez Cabillon 15 Dec 1997 02:55:03):
- Imre Lakatos, 1967, _A Renaissance of Empiricism in the Recent
Philosophy of Mathematics ?_ same reference as Kalmár above, pp.
199-202.
This article is the longest response to the Kalmár paper (other
responses are signed S.C. Kleene, A. Heyting, Paul Bernays, Y.
Bar-Hillel). It has been completed and improved in: _British Journal for
the Philosophy of Science_, 27, 1976, pp. 201-223, and this version is
the chapter 2 of : _Mathematics, Science and Epistemology :
Philosophical Papers, vol. 2_, J. Worrall and G. P. Currie (eds),
Cambridge: Cambridge University Press, 1978.
The paper has been published after the Lakatos' death (1974) and Lakatos
had probably the project to write a book on the philosophy of
mathematics starting from this paper.
I don't summarize here this well-known article but only review few
sentences in relation to anti-foundationalism. So, Lakatos quotes Gödel
and von Neumann:
Gödel is reported to have said:
" the role of the alleged 'foundations' is rather comparable to the
function discharged, in physical theory, by explanatory hypotheses
The
so-called logical or set-theoretical 'foundations' for number-theory or
of any other well established mathematical theory, is explanatory,
rather than really foundational, exactly as in physics where the actual
function of axioms is to _explain_ the phenomena described by the
theorems of this system rather than to provide a genuine 'foundation'
for such theorems." (Mehlberg M. _The Present Situation in the
Philosophy of Mathematics_. in. B.M. Kazemier and D. Vuysje (eds.),
_Logic and Language_, Dordrecht: Reidel, 1962, 69-103. p. 86).
And von Neumann, in 1947, concluded that :
"After all, classical mathematics, even though one could never again be
absolutely certain of its reliability
stood on at least as sound a
foundation as, for example, the existence of the electron. Hence, if one
was willing to accept the sciences, one might as well accept the
classical system of mathematics." (Neumann J. von. _The Mathematician_ ,
in R. B. Heywood (ed.), _The Works of the Mind_, Chicago: Chicago
University Press, 1947, 180-196. pp. 189-190).
Recall that these two quotations are in Lakatos' paper.
And in part 3 of his paper, Lakatos begins to explain why 'Mathematics
is quasi-empirical' in these terms:
"By the turn of this century mathematics, 'the paradigm of certainty and
truth', seemed to be the last real stronghold of orthodox Euclideans.
But there were certainly some flaws in the Euclidean organization even
of mathematics, and these flaws caused considerable unrest. Thus the
central problem of all foundational school was: 'to establish once and
for all the certitude of mathematical methods' (Hilbert, 1925, p.35).
However, foundational studies unexpectedly led to the conclusion that a
Euclidean reorganization of mathematics as a whole may be impossible;
that at least the richest mathematical theories were, like scientific
theories, quasi-empirical. Euclideanism suffered in its very
stronghold."
- Hilary Putnam, _What Is Mathematical Truth ?_ in _Philosophical
Papers: Mathematics, Matter and Method, vol. 1_ , Cambridge: Cambridge
University Press, 1975
Putnam supports in this article a realistic conception of mathematics
and he challenges the current opinion that mathematical truth is _a
priori_. The paper does not contain any very explicit
anti-foundationalist views but it ends with a suggestion about a modal
conception of mathematics which was developed in a famous previous
article:
- Hilary Putnam, _Mathematics without foundations_ (_The Journal of
Philosophy_, LXIV, I 19 January 1967). Reprinted in: _Philosophical
Papers: Mathematics, Matter and Method_, see above.
This article is often considered as a source of anti-foundationalism
(see for example Martin Davis in the fom-list, 10 Mar 1998 17:18:42: "
In a paper that has become something of a
classic, Hilary Putnam has informed us that mathematics doesn't "need"
foundations."). But challenging foundations is not the main aim of this
article despite its title and the fact that Putnam wrote: "My purpose is
not to start a _new_ school in the foundations of mathematics (say,
'modalism'). Even if in some contexts the modal-logic picture is more
helpful than the mathematical-objects picture, in other contexts the
reverse is the case. Sometimes we have a clearer notion of what
'possible' means than of what 'set' means; in other cases the reverse is
true; and in many, many cases both notions seem as clear as notions ever
get in science. Looking at things of many different 'equivalent
descriptions', considering what is suggested by _all_ the pictures, is
both a healthy antidote to foundationalism and of real heuristic value
in the study of scientific questions." (p. 57).
Charles Parsons has correctly noticed the authentic aim of this article
in the fom-list, 20 Jan 1998 08:40:53:
" There Putnam sketched a translation of the language of set theory into
a modal language, where the idea was to replace the existence of
mathematical objects with, roughly, the
possibility of structures. But Putnam made clear that the possibility
involved was not physical but distinctively mathematical. I have never
seen how to work out Putnam's sketch, but the idea has been pursued by
others, most fully by Geoffrey Hellman in _Mathematics without
Numbers_ (Oxford 1989)".
It is interesting to note that the modal structuralist and nominalist
system proposed by Hellman cannot be qualified as quasi-empiricist nor
anti-foundationalist.
===> 4. Lakatosian studies
Many studies on Lakatos' philosophy suggest corrections and improvements
to the quasi-empiricist conception of mathematics; for example Gianluigi
Oliveri proposes the concept of _sophisticated falsifiability_ and Teun
Koetsier develops the notion of _weak falsifiability_. It is not our
question to discuss here the pertinence of these innovations, but we
remark that these studies are not really concerned with the
anti-foundationalist features we can find in Lakatos' thought. Several
of these studies are not strictly lakatosian and may challenge certain
lakatosian ideas, in particular the assimilation of the foundational
investigations in mathematics to a 'repulsive' Euclidean methodology.
For example, Brendan P. Larvor writes:
"Sometimes mathematicians may turn such a picture into a foundational
research programme. Mathematics seems to differ from contemporary
physical science in this respect. Scientists may have views about what
it means to be scientific which may inform their research, and the
fortunes of that research may count for or against the philosophical
assumptions. However, physical science cannot usually study itself
directly. In short, mathematics differs from physical science in that
its self-image impinges on its development by supplying explicit
standards of rigour or by inspiring foundational research programmes.
The _general_ self-image of science, by contrast, has comparatively
little effect on its progress nowadays (things may have been different
at the dawn of the scientific revolution). The challenge for anyone
hoping to develop a methodology of mathematical research programmes is
to capture the reflexivity of modern mathematics without losing the
'Hegelian' quality of Lakatos's thought. An authentically Lakatosian
'methodology' for mathematics needs the idea that explanatory
historiographic units are internally complex dynamic wholes." (Brendan
P. Larvor, _Lakatos as Historian of Mathematics_, _Philosophia
Mathematica_, vol. 5, 1997, pp. 42-64.).
===> 5. Extensive quasi-empiricism: Tymoczko (I)
First part of Thomas Tymoczko's anthology _New directions in the
philosophy of mathematics_ (Birkhäuser, 1986; revised and expanded
edition: Princeton University Press, 1998) is entitled _Challenging
Foundations_. It contains the previously mentioned articles Lakatos' _A
Renaissance_ and Putnam's _What Is Mathematical Truth ?_ but not
Putnam's _Mathematics without foundations_.
This introduction is a clear anti-foundationalist manifesto despite the
fact that Tymoczko "attacks "foundationalism", but he never defines that
concept" as Simpson said (1 Feb 1998 22:17:36). Reuben Hersh has
proposed to define the word in the fom-list 15 Jan 1998 16:07:20: " I
use Lakatos term "foundationnism" for Frege, Hilbert and Brouwer -- all
people who sought to repair or rebuild the foundations in a
philosophical sense, in order to recover the lost certainty of
mathematical knowledge". This is the most commonly accepted sense of the
term in the elastic version of quasi-empiricism starting after Lakatos
and Putnam and defended by Tymoczko, Hersh and others. But in fact, for
Tymoczko, the word 'foundationalism' seems to have a more conceptual
sense, briefly exposed at the very beginning of his anthology; it is the
reduction and assimilation of the philosophy of mathematics to the
foundational studies: "Many [
] readers, I suspect, will acknowledge
dissatisfaction with the foundational approach to the philosophy of
mathematics. More would do so if they felt they had a choice, but many
people assume that 'the philosophy of mathematics' simply means
'foundational studies'. 'Foundational studies', in turn, is practically
equivalent to 'mathematical logic'. We have to work to disentangle the
major schools of foundationalism - platonism, logicism, formalism and
intuitionism - from the major branches of mathematical logic - set
theory, proof theory, model theory, and recursion theory. Such
identifications are worth fighting against, for they consign the
philosophy of mathematics to an extremely small group of experts."
(p.1).
However, the main peculiarity of the Tymoczko's version of
quasi-empiricism is its proclivity to annex many thinkers who are not
really engaged in the same ways than Lakatos and Putnam. These
mathematicians, philosophers, sociologists, computers scientists never
used the expression 'quasi-empiricism' to qualify their ideas, and even,
they are using moderately some terms derived from 'empiricism'; most of
them would probably be better qualified using other -isms as
'structuralism' (Resnik), 'naturalism' (Nicholas D. Goodman),
'sociologism' or 'social-constructivism' (Hersh), 'ethno-relativism'
(Wilder), 'computationalism' (Chaitin), or even 'neo-aristotelism'
(Thom), etc.
===> 6. 'Humanistic mathematics': Tymoczko (II), Hersh, Ernest
Recently, specific admirers of Lakatos have introduced the expression
'humanistic mathematics' to call a "growing movement [
] that seeks to
make mathematics intelligible to students as on ongoing humanistic
discipline" (Tymoczko, posthumous Afterword of the second edition of his
anthology, p. 396).
See also many different mails from Reuben Hersh in the fom-list, for
example 22 Dec 1997 14:58:15.
Paul Ernest uses a similar terminology, speaking of 'The Human Face of
Mathematics' (in _Mathematics, Education and Philosophy: An
International approach_, ed. by P. Ernest, London: The Falmer Press,
1994). For him, "a revolutionary new tradition in the philosophy of
mathematics has been emerging which has been termed quasi-empiricist
(Lakatos, Kitcher, Tymoczko), maverick (Kitcher and Aspray) and
post-modernist (Tiles)" (p. 1). In another place, he says that the
'human face of mathematics' consists of "new faillibilist,
quasi-empirical or social-constructivist views of mathematics" (P.
Ernest, _The Legacy of Lakatos_, in _Philosophia Mathematica_ vol.5
(1997), p. 125). Following Ernest, we must admit to this 'human face of
mathematics', in addition to Lakatos, Putnam and Tymoczko, authors like
Wittgenstein, Bloor, Harding, Rotman and even Latour, Derrida, Serres.
Quasi-empiricism in this acceptation becomes a subsidiary to relativism
and is annexed to a great social-constructivist or post-modernist
melting-pot.
===> 7. 'Computational quasi-empiricism'
It is remarkable than the most known mathematicians engaged with
extensive use of computers are considered by themselves in the spirit of
Lakatos. See for example:
G. Chaitin, _The limits of mathematics_, Springer, 1998, p. 24.
J. Borwein, P. Borwein, R. Girgensohn and S. Parnes: _Experimental
mathematics: A discussion_, October 16, 1995, _CECM_.
See also some provocative views by Zeilberger (all these articles are
available on the Web).
However, these mathematicians do not really attack or contest the
foundational research in mathematics.
Comments:
-------------
It could be possible to refine this sketch, but it is sufficient for
trying to ask the questions raised by Stephen Simpson in fom-list 21 Jul
1998 18:16:13, in response to Jeremy Avigad's posting:
"What I find truly bizarre about the self-styled "quasi-empiricists" is
that they refuse to acknowledge the current standards of rigor in
mathematics. Jeremy, do you have any idea what is eating these people?
Why are they so bitterly and irrationally opposed to f.o.m.? Are they
angry at f.o.m. for disparate reasons, or is there a common
denominator?"
- First of all, it is almost impossible to consider 'quasi-empiricism'
as a well-defined and coherent direction in the philosophy of
mathematics. I propose to distinguish at least two sense of the term:
'core quasi-empiricism' proposed by Lakatos and Putnam in few writings,
and 'elastic quasi-empiricism' initiated by Tymoczko and adulterated by
elements from relativism, social-constructivism, post-modernism, etc.
- Opposition to fom is not a common denominator of an
empiricism-inspired philosophy of mathematics, and, again, is not a
coherent and well-defined feature. It is an obvious part in Kalmar
article, for example, but not essential in Lakatos writings and in
lakatosian studies; it seems like an ideological position in part of the
'elastic quasi-empiricism' but the famous Putnam's _Mathematics without
foundations_ has been largely misinterpreted in a sense of hostility to
fom; finally, the philosophical background to a computer-oriented
philosophy of mathematics or to experimental mathematics do not deal
with anti-foundationalism.
- A plausible interpretation of the origin of anti-foundationalism is a
kind of 'revolt' against the reduction of the philosophy of mathematics
to foundational researches. As Smorynski says: "People seem finally to
have grown tired of the 'foundational crisis of the turn of the
century'. Witness, for example, the occasional recent attempts to
'revive' the philosophy of mathematics. My explanation of this
revivalist fervour is that the difficulties in set theory were so
interesting and important that they (and the ensuing battle between
Brouwer and Hilbert) monopolisted the Philosophy of Mathematics for
decades and it is only recently that everyone has bored of that crisis
and, looking around, has discovered this monopoly to have left the
Philosophy of Mathematics without any other leadership or direction."
(_The Varieties of Arboreal Experience_, p. 186).
- In addition, I argue it is possible to philosophically refresh the
concepts which would be the very substance of quasi-empiricism (i.e.
'mathematical experience', 'mathematical fact', and maybe above all
'mathematical tool') without to accept quasi-empiricism as a whole (but,
what is it really). Shifting the analogon from empirical sciences to
computer theory *and* to computer practice could be useful for this
expecting 'revival' of quasi-empiricism disengaged from the ideological
form of anti-foundationalism.
Regards
---
fom member digest:
Name: Patrick Peccatte
Position: Software engineer and philosopher
Institution: Independent private company and Paris 7 University
Research interest: History and philosophy of sciences and mathematics
(specially quasi-empiricism, experimental mathematics, ontology)
More information: http://peccatte.rever.fr (in French)
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