FOM: Hersh's dubious doubts/Davis' examples

Robert S Tragesser RTragesser at
Mon Jan 5 14:17:39 EST 1998

        Likely Reuben Hersh must fail to appreciate 
Davis' examples exactly because he (Hersh) 
proceeds by the method of (dubious) doubt rather 
than by aiming at positive characterization 
(through thought experiments with living 
mathematical thinking).
        I've been trying to say that Reuben Hersh is 
building his humanistic/consensual philosophy of 
mathematical knowledge on dubious (= 
unreasonable = irresponsible doubts -- 
"irresponsible" because unresponsive to the 
problem of capturing postiviely what is so 
distinctive and peculiarly cogent about 
mathematical knowledge).
        [It was pointed out to me that most people 
on FOM would not have picked up on the 
reference of 'skeptical terror' to Cavell,  so that 
my remarks seem vapid and vague.--Sorry,  if 
        Hersh has built his humanistic philosophy 
of mathematics and his view of mathematics as 
sustained by consensus on blanket (and dubious) 
doubts -- that all mathematical theorems are 
dubious,  even the most elementary.
        When Hobbes (quoted by Lakatos) and 
Locke call mathematical knowledge "certain", 
they are using that word to indicate that there is 
something distinctive about mathematics (in 
contrast to empirical science,  "natural 
philosophy"),  NOT that mathematical 
knowledge is absolutely infallible.   There 
remains the task of characterizing what is so 
distinctive about mathematics.
        Feferman had objected to Lakatos' excesses 
of falliblism -- that,  against Lakatos,  there is an 
end to guesswork,  that there are "successful 
struggle[s] to solve a problem or complete a 
proof".   Rota insists that there are abiding 
mathematical facts.  (Both Feferman and Rota do 
point out that -- as Feferman puts it --
"results are viewed in changing perspective 
over historical periods.  Their significance is 
reassessed,  they are generalized and 
understood in wider settings. . . .But this is 
quite a different picture from that given by 
Lakatos of endless guesswork.")
What is there about mathematical that it supports
definitive solutions?

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