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*To*: jmyciel@euclid.colorado.edu*Subject*: FOM: response to Mycielski's definition of mathematics*From*: JoeShipman@aol.com*Date*: Fri, 31 Dec 1999 10:23:56 EST*CC*: fom@math.psu.edu*Sender*: owner-fom@math.psu.edu

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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