FOM: response to Mycielski's definition of mathematics
JoeShipman at aol.com
Fri Dec 31 10:23:56 EST 1999
In a message dated 12/30/99 5:32:40 PM Eastern Standard Time,
jmyciel at 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
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
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
[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
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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