FOM: Well-known historical remarks

Moshe' Machover moshe.machover at
Mon Nov 3 19:04:54 EST 1997

Two historical (and probably well-known) remarks.
J Shipman writes:

> The old philosophers felt Euclid's parallel postulate was "necessary" but 
> would have admitted they were wrong after reading Lobachevsky -- but 
> could have also been corrected by physics though historically they 
> weren't.

I think this confuses two senses of `necessary', and is historically 
inaccurate. Before Riemann's epoch making 1854 paper `On the hypotheses 
that lie at the foundation of geometry' everyone--not only philosophers, 
but also mathematicians--thought that geometry was the study of real 
physical space, albeit in an idealized form. Riemann made the revolutionary 
point that there are really two `geometries'. One is the study of 
manifolds, which is pure mathematics; the other is the study of real 
physical space, which is physics. 

Lobachevsky and Bolyai had shown that Euclid's 5th postulate was not 
`necessary' to `pure' geometry. This is a logical fact. (They did not 
actually *prove* it but simply assumed it. The relative consistency of 
their geometry was proved much later.)

This did not cut much ice with a philosopher such as Frege, who did not 
take Riemann's distinction on board, and continued *despite Lobachevsky* to 
maintain, with Kant, that geometry is synthetic a priori, (and the 5th 
postulate presumably necessarily true).

What `physics' showed was that the 5th postulate is not `necessary' or even 
true in *physical* geometry (in which straight lines are eg idealized light 

2. A propos of the Neil Tennant/Vaughan Pratt exchange:

     `So set theory appears to us today, in logical respects, as the proper 
     foundation of mathematical science, and we will have to make a halt 
     with [or: to keep with] set theory if we wish to formulate principles 
     of definition which are not only sufficient for elementary geometry, 
     but also for the whole of mathematics.'
               Hermann Weyl(!) 1910

     `I believed that it was so clear that axiomatization in terms of sets 
     was not a satisfactory ultimate foundation of mathematics that 
     mathematicians would, for the most part, not be very much concerned 
     with it. But in recent times I have seen to my surprise that so many 
     mathematicians think that these axioms of set theory provide the ideal 
     foundation for mathematics; therefore it seemed to me that the time 
     had come to publish a critique.' 
               Skolem, 1922

Note: Both of the papers quoted clarified Zermelo's notion of *definite 
property*. They both gave essentially the same answer (= first-order 
definability in terms of membership). Skolem's paper is mainly 
concerned with his eponymous paradox; but also proposes the axiom of 
replacement, and conjectures that it  would `no doubt be very difficult' to 
prove the consistency of Zermelo's axioms, and that the CH would be 
independent of them.

Weyl was to change his mind about the status of set theory, and became much 
more of a constructivist.

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