FOM: Well-known historical remarks
Moshe' Machover
moshe.machover at kcl.ac.uk
Mon Nov 3 19:04:54 EST 1997
971103mm.txt
Two historical (and probably well-known) remarks.
------------------------------------------------
1.
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
rays).
.......................
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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