FOM: What is mathematics?

[Harvey Friedman]

Reply to Mycielski, Wed, 29 Dec 1999 13:00.

>	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.)

My view is that a truly satisfactory definition of mathematics would
include some illuminating criteria as to what "good" or "important"
mathematics is; at least some criteria as to how to evaluate it.

An indication of just how wide open this question "what is mathematics is"
is that there is virtually nothing of substance written about criteria for
the evaluation of the quality or importance of a piece of mathematics. In
fact, there is extreme disagreement and controversy over the evaluation of
even specific well known mathematical developments. E.g., even Godel's
incompleteness theorems.

>	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.

This cannot be a significant step in the definition of mathematics for
several reasons. One is that the vast preponderance of makers of
mathematics do not even know what ZFC is. Another is that no published
mathematics by ordinary mathematicians is done within ZFC in any explicit
way. Still another is that no serious mathematics can even be done in any
practical sense in ZFC - for example, it does not support abbreviations.
And of course, this definition also does not even remotely indicate how to
evaluate mathematics as to quality or importance. In fact, almost
everything that can be done in ZFC is completely worthless - completely
unpublishable.

>	[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.

Since our brains and experiences are finite, I am not convinced that
notions like RE (recursively enumerable) make any sense here.

>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.]

Same remark.

>	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.

This makes ZFC even more impossible to do mathematics in in any practical
way, or in any way that mathematicians generally would find convenient or
reasonable. I do not think that anything is gained by introduction of
Hilbert's epsilon symbols for this purpose. Mathematicians do not introduce
Hilbert epsilon symbols, except in the sense of introducing abbreviations -
only when convenient.

>	Also this point of view shows that there is no qualitative
>(ontological) difference between ZFC and PRA.

This are enormous qualitative and ontological differences between even
arbitrary infinite sets of integers and integers themselves. This is
reflected in what mathematicians are interested in, feel comfortable with,
and know about, as well as obvious distinctions related to informal
categoricity, clarity of mental images, etcetera.

>(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.)

They are extremely different from each other. One can be defined, uniquely
generated, and the like. The other cannot even be defined, certainly not
uniquely generated, etcetera. All mathematicians are completely familiar
with the difference, and it is reflected in what they work on, are
comfortable with, etcetera.

>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.

It has clear ontological and scientific significance, and is closely
related to actual mathematical practice, as well as natural philosophical
ideas.

>	[Category theorists tried to achieve a better definition of
>mathematics (other than ZFC). But their definition seems to be more
>complicated and hence inferior.

It is not inferior just because it is more complicated. My view is that it
is not sufficiently philosophically coherent to form an autonomous
foundation for mathematics. Almost all mathematicians are familiar with
basic category theory, a great many use it to good effect, and almost all
view it as ultimately being defined in terms of set theory.

>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.]

But it is the fact that the axioms come out to be so natural and simple
that makes it work, not just that the relational type is so simple.

>	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.

The Platonist view or realist view does explain more - or at least attempts
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.]

For instance, it attempts to explain why ZFC works so much better than
alternative formal schemes.

>	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).

The endless series of natural numbers from left to right is incomparably
clearer and closer to physical experience than any of this talk about
imagined arbitrary infinite sets of arbitrary rank, etcetera.

Of course, I do adhere to the view that there is a natural compelling
nature to all of this imagined sets stuff that ZFC codifies. It's just that
I make a big distinction between it - however naturally compelling - and
the far more naturally compelling imagined endless series of natural
numbers. So would practically anyone else.

>	[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.]

But I don't think that there is any published analysis of this concept that
really generates ZFC in a completely clear way, or generates small large
cardinals in a completely clear way. More analysis and more distinctions
are needed for this.

>	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.

The Platonists and Realists would say that this is done by a combination of
direct perception, experimentation, and deduction, just as one does this in
physical science.

>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.

That is one thing that the Platonists and Realists like to think that they do.

>...Philosophers are the people who
>are the most responsible for the intellectual catastrophy described by A.
>Sokal and J. Bricmont "Fashionable nonsense".

Why philosophers? Do you mean professional philosophers, analytic
philosophers, continental philosophers, who?

>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).

Are you saing that Godel and Russell have caused this lack of critical
thinking by ignoring published and readily available knowledge? Give us
some examples.

>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).]

Please explain more about the "latter".