FOM: Foundationalism

[John Mayberry]

I want to make a couple of brief points in reply to recent postings by 
Colin McLarty and Reuben Hersh

To Colin McLarty's posting: It would be silly to say that Descartes, 
Newton, and Riemann were not doing mathematics. But it is nevertheless 
true that standards of rigour in proof and in definition have been 
enormously sharpened since they did their work. Clearly simply going 
back to their standards is not a serious option. But we don't have to 
deny Newton (along with Leibniz) credit for the proof of the 
Fundamental Theorem of Calculus just because the proofs of that result 
contained in our Analysis textbooks go far beyond theirs in point of 
rigour.

To Reuben Hersh's posting: Hersh claims that "foundationalism" is "the 
demand for a secure foundation to make mathematics indubitable". I 
should have thought that the demand for a "secure foundation" for a 
subject that purports to deal in rigorous proof is not an unreasonable 
one. But "indubitable"? Surely any scientist worth his salt should be 
willing to submit even his most cherished convictions to scrutiny, and 
that involves, "doubting" them, at least in some sense. But what is 
required here is serious, particular doubt, not just general, nominal 
doubt. When we encounter a proposition that, as a matter of 
(sociological) fact, is taken by mathematicians as a first principle, 
we should ask ourselves whether we can raise a *mathematically* 
intelligible and usable objection to its validity, or to its status as 
a *first* principle (e.g. by deriving it from something more 
fundamental). If we cannot raise such an objection, then we have no 
*serious* doubt, though we may express *nominal* doubt, perhaps on the 
grounds that absolute certainty is an unattainable ideal, or perhaps by 
way of reserving the right to call the principle in question should it 
subsequently occur to us how this might be done in a mathematically 
significant way. However, I fear that views of the foundations of 
mathematics such as Hersh's "Humanism", precisely by calling 
*everything* indiscriminately into doubt, discourage us from taking ANY 
foundational principle sufficiently seriously to subject it to the kind 
of doubt that can be mathematically fruitful. To pursue "indubitable" 
foundations is, no doubt, to pursue a will-o-the-wisp; to pursue 
"secure foundations" is an essential part of mathematics.

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John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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