FOM: Ontology of Mathematics
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Sat Jun 24 17:09:36 EDT 2000
ONTOLOGY OF MATHEMATICS (J. SHIPMAN'S INQUIRY)
Dear Joe,
Your inquiry about the ontology of mathematics (fixing its
concept) seems very natural to me since I thought also about this problem.
In fact I wrote already on FOM (long ago) about this question. So let me
repeat those thoughts.
I believe that the solution of this problem is:
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.
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).
I believe that this is so simple and so compelling, that I do not
see any significance of the intuitionistic critique of classical
mathematics. Indeed the above ontology is ferfectly finitistic and fully
constructive. It explains also our feeling of concreteness of mathematical
objects, since they are things which we have imagined or at least named.
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.
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.) 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?
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.
All are equally imaginary untill they are applied in descriptions
of physical reality. Of course the above ontology agrees with this.
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". 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.
Regards
Jan Mycielski
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