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FOM: response to Mycielski's definition of mathematics



In a message dated 12/30/99 5:32:40 PM Eastern Standard Time, 
jmyciel@euclid.Colorado.EDU writes:

<<           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....
    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.
>>

There is much to be said for this definition, if your orientation is to look 
at mathematics formally rather than as what mathematicians actually do; but 
does this definition also apply to the mathematics done by Euclid, 
Archimedes, Newton, Euler, Gauss, and Riemann?


<<  [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). 
>>

You don't need to ignore it; even the results involving large cardinal axioms 
can be stated as ZFC-theorems.

<<  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. 
>>

Can you elaborate on this, please?

<< 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.>>

Surely there is significance related to the issue of algorithmic definiteness.

<<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.>>

On the contrary, it explains the unity (mutual consistency and 
interpretability) of almost all the mathematics developed by thousands of 
mathematicians over the centuries.

<<  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>>

This seems right to me (although the "container" model is not intuitively a 
perfect fit for ZFC).

<<  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.>>

I agree with this part of the paragraph....

<< 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).] >>

But I disagree strongly here re Godel and Russell.  Russell's work was well 
before Skolem and Turing, and Godel's work was philosophically very precise 
and unambiguous, though some later interpreters certainly muddled it.  Can 
you please state precisely what mistakes you think Godel made, with citations 
from his work?

-- Joe Shipman




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