FOM: Reply to Shipman on conclusiveness

[Moshe' Machover]

J Shipman asks:

> Are you really sure with a subjective probability greater than
> 0.999999999999999999999999999999999999999999999999999999999999
> 999999999999999999999999999999999999999999999999999999999999999
> 999999999999999999999999999999999999999999999999999999999999999
> 9999999999999999999999999999999999999999999999999999999999999999
> 99999999999999999999999999999999999999999999999999 that you are
> not actually a crank who thinks he has a proof but is really
> just invincibly ignorant?

Actually, in my incorrigibly blinkered dogmatism I maintain that--although I
*may* well be a crank, and although I *know* I am invincibly ignorant--the
proof of the proposition that for any natural n there is a prime p > n
leaves absolutely no room for doubt.  In the appropriate (subjective?)
probability space, the event that the old proof is correct not only has
probability 1, but has an empty complement.

But beyond this half-tongue-in-cheek exchange there is a very serious
question: is there no such thing as objective logical/mathematical validity?

The social constructivists are in my view right to point out that, as
social animals, our ideas--including those about mathematics--are socially
conditioned, or even socially constructed.  But is the relativistic
conclusion that they draw from this really warranted?  If it is not, then
there must be some mechanism, some process, or whatever, that causes the
socially constructed notion of logical/mathematical truth to converge to
some objective truth.

Analogy: although--as the empiricists and Kantians claim--our notions of
material-physical reality are constructed in our own heads, mediated by our
subjective perceptual and conceptual apparatus, realists nevertheless
maintain that these notions can still converge to an objective truth.  A
possible mechanism, in this case, is our interaction with the external
world, plus evolutionary selection.


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