FOM: Ontology of Mathematics

[Harvey Friedman]

Reply to Mycielski 6/24/00  3:09PM:

I find this posting extremely misleading. In fact, In fact, its effect
would be to discount much of the most interesting and important work done
in f.o.m.

>  	I believe that the solution of this problem is:

This is already misleading. Age old philosophical problems are just not
rationally subject to statements like this.

>	A rational ontology of pure mathematics tells us that the finite
>structure of mathematical ojects (say sets as containers designed to
>contain other containers) which are actually imagined by mathematicians.

This sentence does not seem to me to be grammatical.

>A mathematical counterpart of this structure is a finite (growing) segment
>of the algebra of epsilon terms (of the Hilbert epsilon extension of the
>first-order language of set theory) modulo the equations which have been
>proved (that is those about which we know).

This is a technical construction that simply serves to complicate the usual
model of mathematical reasoning through axiomatic set theory. Nothing is to
be gained, foundationally, by creating these epsilon terms, as they would
very rapidly become unreadable and unusable already in extremely elementary
mathematics.

>  	I believe that this is so simple and so compelling, that I do not
>see any significance of the intuitionistic critique of classical
>mathematics.

It is complicated and uncompelling. In no way does it bear on the
significance of intuitionism.

>Indeed the above ontology is ferfectly finitistic and fully
>constructive.

It is infinistic and fully nonconstructive.

>It explains also our feeling of concreteness of mathematical
>objects, since they are things which we have imagined or at least named.

Giving a name to an object does not make it more concrete. E.g., you can
give a name to a well ordering of the reals, but that doesn't make it
concrete.

>And it explains our feeling that, say, ZFC is consistent, since those
>finite segments grow in such a regular way that we cannot imagine that the
>process of their construction (which is described by the axioms of ZFC)
>could lead to 0 = 1.

It does not add any confidence that we might have that ZFC is consistent.
The epilson terms are a very general process that applies equally well to
any first order theory, and therefore has nothing whatsoever to do with set
theory in particular. What happens when you apply it to an inconsistent set
theory like ZFC + j:V into V?

>  	The above ontology is briefly mentioned by Hibert in 1904, however
>at that time he does not have yet his epsilon symbols. He introduced the
>latter no later than in 1924, but in a paper of 1925 he does not mention
>his ontology of 1904. (My readings are from van Heijenoort and a book
>(thesis) by Leisenring.)

It is a nice proof theoretic idea that has yielded results in proof theory
through the so called epsilon substitution method. And it can be made
closely related to cut elimination. But you are trying to use it for
something that it is not suited for.

>I have not seen anywhere a clear (as above)
>definition of this ontology, but I have read a lot of confusing and
>confused (I believe) papers and books. It looks to me as if everybody
>forgot about Hilbert's 1904 paper, and although his epsilons are
>remembered, their fundamental significance (as tools for describing the
>structures which are really present in human imaginatioins) is never
>recalled. Am I right about this silence?

I think so, and perhaps you can see some good reasons for this silence.

>	Now it is also obvious to me (long ago in a conversation with R.
>M. Solovay we were in agreement about this) that there is no known
>distinction between the degree of abstractness of any imagined objects.

This is ignoring most of what we have learned from f.o.m. That there is
indeed a robust hierarchy of levels of abstraction - proof theoretic,
definable theoretic, interpretability theoretic, etcetera. And these are
remarkably related and robust. In dismissing the fundamental significance
of this hierarchy, you are dismissing much of f.o.m.

And where do you get the idea that this is "obvious"? By the way, I doubt
very much if Solovay would defend such a statement on the FOM.

>All are equally imaginary untill they are applied in descriptions
>of physical reality.

Why "equally"? Again, you are dismissing vitally important robust
hierarchies. And why do you take descriptions of physical reality as having
special significance? Such descriptions may be- and frequently are - even
more murky than abstract set theory.

>Thus, contrary to Brouwer or H. Weil or E. Bishop or some statements of
>Hilbert (where wanted to distinguish in mathematics the concrete from the
>abstract), we think that in pure mathematics nothing distinguishes the
>integers and their algebra from other mathematical objects and their
>"algebras".

E.g., the integers are normally distinguished from the space of continuous
functions from the reals to the reals.

>We can only say that not all of the latter are used in natural
>science and some (like a well ordering of the real line) are unlikely to
>get such uses.

Why does natural science have such special status for you? If you are going
to make special distinctions, why not adopt the usual ones, say, between
integers and continuous functions on the reals?