FOM: defining ``mathematics''

Matt Insall montez at
Sun Jan 9 04:21:05 EST 2000

This is part three of my reply to professor Sazonov.

Matt Insall

> Jan Mycielski <jmyciel at euclid.Colorado.EDU>
>     Date: Wed, 29 Dec 1999 13:00:08 -0700 (MST)
> wrote:
>         Indeed if we regard the problem of defining mathematics as a
> problem of natural science (that is mathematics is viewed as a physical
> process just like other physical processes), then the answer is:
> mathematics is the process of developing ZFC, i.e., the process of
> introducing definitions and proving theorems in ZFC.
>         [As every theory of a real physical phenomenon this definition
> is
> not complete. Indeed we ignore here the rare phenomenon of addition and
> uses of new fundamental axioms beyond ZFC (e.g. large cardinal
> axioms).
> <snip>
> ]
>         My prefered formalism (for ZFC) is not first-order logic, but
> logic without quantifiers but with Hilbert's epsilon symbols. In this
> formal language quantifiers can be defined as abbreviations. This has
> the
> advantage that the statements in such a language do not refer to any
> universes. So this does not suggest any existence of any Platonic (not
> individually imagined) objects.
> Thus, at least some extensions of ZFC should be also considered
> as describing what is (contemporary?) mathematics. Why to consider
> only extensions of ZFC?

I guess I agree with you here.  Some fragments of ZFC are just as valid for
describing most mathematics as is ZFC itself, or any proper extension
thereof.  You should be careful, though, to include enough formal systems in
your ``definition of mathematics'' so that work on large cardinal axioms,
CH, etc, fall within the realm of your definition.  For instance, a
``definition of mathemics'' as the theory of finite graphs is terribly
restrictive, even though just about everything we do can be encoded in
finite graphs somehow.

>Why not arbitrary formal systems? Also
> the last paragraph shows that in principle these formalisms may
> be based not on only the ordinary first order logic (FOL).
> In principle we could consider formalisms even not reducible to
> FOL. Why should we restrict (also future) mathematics to some
> special formalisms such as ZFC (or its extensions) or to
> those based on FOL?

Well, I can think of a few reasons to restrict mathematics to systems
formalizable in FOL:
1.  FOL is a very natural model of the practice of mathematics.
2.  FOL is (demonstrably) sound.
3.  FOL is (demonstrably) complete.
4.  I feel I understand FOL, at least reasonably so.  :-)
5.  Every argument I have seen presented in a meeting or a research article
appears to be formalizable in FOL.  (This is true even of all the papers on
``fuzzy reasoning'', in which claims are made that the fuzziness of the
reasoning system somehow improves the results obtained because physical
reality is actually ``fuzzy''.)  That is, behind every argument, there seems
to be an FOL lurking about.

>         Also this point of view shows that there is no qualitative
> (ontological) difference between ZFC and PRA. (Integers such as 10^10^10
> and sets such as a well-ordering of the continuum seem equally
> imaginary,
> i.e., without any intended outer physical interpretation.)
> Completely agree! This is confirmed also by a very nice results of
> Professor Mycielski from 80th on "isomorphism" of any formal theory
> (say, ZFC) with some its locally consistent version (s.t. each
> finite set of axioms has a finite model, probably unrealistically
> large one). Thus, say, infinite cardinals may be treated as large
> finite objects! I consider this result as having a great philosophical
> value for f.o.m.

This appears to me to be merely a problem of language.  Again, if one theory
interprets the word ``infinite'' as a blue unicorn, but turns out to be
consistent, that does not mean that what I mean by the word ``infinite'' is
the same thing you mean by ``blue unicorn''.  It also does not imply that
either one is nonsense.  (Blue unicorns may exist. I know, though, that I
have never seen one.  Yet, I believe I have come to understand \aleph_0 well
enough to distinguish it from 10^10^10, if I must do so.  I also believe
that both of these numbers exist.)

>         I have not seen in the literature any clear exposition of the
> philosophy stated above. All Platonists reject it. Their definition of
> mathematics (a description of a Platonic universe independent from
> humanity) assumes more but it does not seem to explain more. Hence it is
> inferior. [Of course the Platonic definition puts mathematics in the
> realm
> of science, while the ZFC definition puts it in the realm of art in as
> much as it is independent of any intended physical meaning. This may
> have
> some negative political implications for mathematicians, but it seems to
> me that truth is more important.]
> With my additions presented above this kind of philosophy could be
> called *formalism* (however, Professor Mycielski wrote that he do
> not like this term; this seems only terminological disagreement) or
> *formal engineering* (cf. also my previous posting to FOM).
> Really, was formalism described by anybody as a consistent point
> of view on mathematics, not as an shameful brand or a kind of
> soul-less bureaucracy? (By the way, bureaucracy organizes the life
> of a society. This seems not so bad, and, moreover, inevitable.)
> I would say that Platonists pretend to "put mathematics in the
> realm of science" without any real grounds for this. Science
> about our vague illusions?

If you must say it so, yes.  Part of the science of mathematics is related
to a part of psychology, which studies the reasonings, musings and
rationalizations of human beings, ``vague illusions'', as it were.  In fact
the part of mathematics which you seem to think comprises all of mathematics
(the study of formal systems) is modeled mainly upon ideal reasoning systems
(the devices you mention), and was developed long before there were physical
realities (computing devices, etc.) associated with it.  Thus the only
``physical'' model for the development of formal systems when they were
originally designed was an ideal version of the rational human mind.  (To
wit Boolean logic, etc.)

Matt InsallMatt Insall
montez at

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