FOM: Definition of mathematics
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Wed Dec 29 15:00:08 EST 1999
DEFINITION OF MATHEMATICS
Recently a number of authors on f.o.m. have discussed the problem
of defining mathematics. I do not understand why this problem is viewed as
one which deserves some discussion, and not one which has been
definitively solved over 70 years ago. (Although my "definitevely" seems
to cover all the issues appearing in the correspondence on f.o.m. to which
I am referring, still it should be taken with a grain of salt. See below,
where the remaining problems are mentioned.)
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). No
doubt those additional events are caused by our brains and experiences,
and presumably they are not preditable, i.e., this process is not RE.
Hence, the phenomenon of addidion of new fundamental axioms must be left
undefined. Even the process of addition of new definitions does not seem
to be RE, since it is often stimulated by outer or inner physical
experience (by inner physical experience I mean thought-experience). But
in this case we have the theoretical abstraction: we may consider the
definitionally closed extension of ZFC.]
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.
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.) Likewise all
the literature based on the distinction between concrete and abstract
objects (going back to Hilbert and then carried on by the constructivists
and the Platonists) makes no philosophical sense to me. It appears to be
an analysis of words and ideas without any ontological or scientific
significance.
[Category theorists tried to achieve a better definition of
mathematics (other than ZFC). But their definition seems to be more
complicated and hence inferior. Notice that the only primitive concept of
ZFC is the membership relation, hence it is difficult to imagine a simpler
theory in which mathematics can be formalised.]
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.]
Some philosophers seem to attach a special significance to PRA. It
seems to me that the only distinguishing quality of PRA is that PRA is a
natural level in the classification of mathematics (in Reverse
Mathematics). Of course PRA talks about imagined integers (or about
hereditarily finite sets) while ZFC talks about imagined sets. But what
are imagined sets? My answer is:
They are imaginary containers intented to contain other
imaginary containers (one of them, called the empty set, is to remains
always empty).
[This view of sets probably goes back to Cantor. His definite
(or "consistent") sets could have been called containers (so that it does
not make sense for a container to contain itself), it also seems to be
implicit in Poincare, and it is well expressed a paper of Hilbert of 1904
("On the foundations of logic and arithmetic" (the assertions I, II and
III), see the collection of J. van Heijenoort "From Frege to Godel", pp.
135 - 136). Hilbert writes there that "general objects" can explain
quantifiers (although he introduced his epsilon-symbols much later) and
he writes that sets are mental objects which can be created prior to their
elements.]
In conclusion let me mention the following unsolved problem.
Although we know what is the stucture of mathematics, we do not know how
we construct it. More concretely, we do not understand the mechanism by
means of which mathematicians invent proofs of fully stated conjectures
within well defined axiomatic theories. I believe that in the present
state of knowledge the main challenge of Mathematical Logic is to explain
this mechanism. A solution of this problem will give us a deeper
definition of mathematics.
[The literature which I am criticising in this letter is well
represented e.g. in the collection "The philosophy of mathematics", Editor
W. D. Hart, Oxford, 1996, and it appears in many other books and papers.
If the reader wanders about the rationale of criticising such a large body
of literature, let me add this remark. Philosophers are the people who
are the most responsible for the intellectual catastrophy described by A.
Sokal and J. Bricmont "Fashionable nonsense". This catastrophy and waste
of human energy, time and money (especially in the academia) would have
been avoided if the critics did their job. It seems to me that a similar
phenomenon is happening in the philosophy of mathematics. Of course
mathematicians (like Godel) and philosophers (like Russell or
Wittgenstein) have caused this lack of critical thinking (by ignoring
published and readily available knowledge). The latter seem to have
overlooked the philosophical significance of the ideas of Skolem and
Turing (it is known that Turing attended some of Wittgenstein's closed
seminars).]
Jan Mycielski
More information about the FOM
mailing list