FOM: social construction?

[Martin Davis]

At 02:47 PM 3/19/98 -0500, Charles Silver wrote:
>
>To Martin Davis:
>
>	I find the following remarks interesting, but I don't fully
>comprehend them.  Could you please say more about what you mean? 
>
>Charlie Silver
>
>*****************************************************************
>
>Martin Davis said:
>
>> For me the crucial
>> aspect of "refering" or "existence" of the objects of mathematical discourse
>> is the OBJECTIVITY of their properties. 
>> 
>> It is a FACT of mathematical
>> experience that we come to have knowledge of mathematical entities and that
>> we have no ability to alter what we discover about them. Why and how this is
>> so is the main business of f.o.m. But what we mean by existence is no more
>> than that FACT.
>
>
>
I don't know how to put it much better than I have. I'm suggesting that when
we say that 7 or sqrt(2) or sqrt(-1) or the real number system "exist" what
we (as mathematicians) mean (or at least ought to mean) is that we know how
to determine some of their properties as definite and have good reason to
think of those we can not decide as definite problems to work on. So:

7 is a prime.
sqrt(2) is irrational
the real numbers form a complete ordered field

and of the second kind

all non-trivial zeros of the zeta function are of the form 1/2 + h sqrt(-1),
h real?

When we ask whether measurable cardinals exist what we need to know is not
whether they are to be found in Plato's never-never land, or whether they
inhabit the mind of God, or certainly not how mathematicians at any given
time would vote. What we need to know is whether they have objective
ascertainable properties. It may well be that in cases like these we will
have to more and more be satisfied by empirical evidence in the form of the
consquences we have been able to draw, fitting together in an illuminating
picture; we certainly know that traditional methods (in particular ZFC)
won't suffice.

Of course earlier in this century, perhaps many would have said the same
about the continuum. This is a fun time!

Martin