FOM: Mathematical certitude; response to Shipman

[Moshe' Machover]

Shipman says:

> What I have been trying to elicit here is an "internal" characterization
> of mathematics (which I might hope to recognize as a definition) to answer
> Hersh's social constructivist claims with.

This is more or less the point I was trying to make, perhaps less eloquently.

He also says:

> Machover's insistence on the unique certitude of properly
> derived mathematical results is not enough, because you can't distinguish
>the
> mathematicians from the chessplayers by the degree of certitude

Of course it is not enough, but not for the the reason given by Shipman,
which has to do with *subjective* feeling of certitude rather than
objective grounds.

Allow me to have another go at explaining the point I was trying to make.

Let us take it for granted that *theorems* of mathematics possess a special
quality of certitude, manifested in a high degree of consensus. But it
cannot be this feeling of certainty or the high degree of consensus which
enable us to distinguish mathematical statements from non-mathematical
ones. In order to recognize that a given statement is mathematical, we do
not have to wait until it is proved (or disproved) and observe whether the
proof is utterly convincing and elicits general consensus among
mathematicians. We knew that FLT was a mathematical statement even before
Wiles proved it. And even if Wiles' proof turns out to be faulty after all,
FLT will still be a mathematical *statement*; only its status as a
*theorem* would be affected.

Exactly  the same applies also to Shipman's proposition about chess:

> White wins in chess if you remove Black's Queen from the initial position.

If by `wins' we mean *has a winning strategy*, then this is a mathematical
proposition par axcellence, irrespective of how certain chess players feel
about it. For all I know it may even be a *false* proposition; but it is a
mathematical one all the same. (If by `wins' we mean something empirical,
then the proposition is definitely not mathematical.)

This is why Hersh's characterization of mathematics is useless. It provides
no criterion for distiguishing a mathematical statement from statements in
other socially constructed mental domains (law, religion, literature--you
name it). Such a criterion must apply equally to proven and unproven
propositions, so it cannnot depend on the degree of consensus about their
truth value.

In so far as [proven] mathematical theorems elicit a high degree of
consensus, this must be a *consequence* of what, using Shipman's
terminology, I would call the *internal* character of mathematics. It is
something that a useful characterization of mathematics ought to explain.
Hersh's characterization simply assumes what ought to be explained.

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