FOM: Tennant's re-invention of the numbers

Solomon Feferman sf at Csli.Stanford.EDU
Mon Feb 2 02:09:40 EST 1998

In his posting of 27 Jan 12:15, Neil Tennant took issue with my comment
the day before (7:59) on the quotation from Wigner's "The unreasonable 
effectiveness of mathematics".  Recall:

>I would say that mathematics is the science of skillful operations 
>with concepts and rules invented just for this purpose.  The principal
>emphasis is on the invention of concepts.

>This seems to point to the subjective origin of mathematical concepts
>and rules.

>Fundamental question:
>Why should the invention of concepts be a *subjective* matter?

Answer (American Heritage Dictionary, 3d edn.):
"Invent, tr.v. 1. To produce or contrive (something previously unknown) by
the use of ingenuity or imagination. 2. To make up, fabricate."
Neil, what definition do you use? Perhaps Wigner didn't mean what he said,
but taken at face value it's hard to read it other than the way that I
did.  Note that I spoke about the subjective *origin* of mathematical
concepts.  This does not imply that once invented, mathematical concepts
have no objective status.  On the contrary, they gain that status through
their intersubjective communication.  Inventions have a subjective origin
and objective products: witness e-mail and post-its.  (I have no doubt as
to the objectivity of the internet--but what exactly is it?)

The main part of Tennant's riposte is devoted to a neo-Fregean
re-invention of the number concept.  Basically this comes to the
(contextual) introduction of such numbers as 2 in a conservative extension
of logic, with #xFx = 2 equivalent to "there are exactly two F's".  Is
this really a re-invention?  More importantly, is this an explanation of
the number concept?  NO.  It is only a partial explanation for each
natural number n, as to how the equation #xFx = n may be reasoned with.
It does not explain the general notion of natural number.  If that is to
have the properties of Peano's axioms for 0 and successor, with induction
expressed for arbitrary properties, much more has to be done.  Doing so
may indeed "command rational assent", but first one has to have the idea
how that is to be done (Dedekind, Frege being two approaches), then it has to
be communicated.  

My own view is that our *usual* conception of the structure of natural
numbers is the ordinal one descended from Dedekind via Peano.  Frege
instead tried to extract the notion from the more general notion of cardinal
number.  We should not conflate the two.  However explained, once you and
I have a meeting of minds what we are talking about, it's in an objective
arena.  What this involves is no more mysterious than the objective
status of chess or go, at least not as far as the fundamental conception
is concerned.

Sol Feferman  

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