Subject: Re: [HM] Kant and non-Euclidean geometry
From: Jeremy Gray (j.j.gray@open.ac.uk)
Date: Sun Mar 12 2000 - 14:09:11 EST
Dear List members,
The evidence that Kant knew of Non-Euclidean geometry through his
correspondence with Kant is elusive. They corresponded, indeed Kant had a
great respect for Lambert, ever since their independent realisation of the
nature of the Milky Way in the late 1750s, and would have dedicated his
Critique of Pure Reason to him had not Lambert died unexpectedly of
pneumonia in 1777. But Lambert’s work on Non-Euclidean geometry was
inconclusive (which is probably why is was only published posthumously, in
1786 when Johann III Bernoulli edited Lambert's papers) and ends with
suggestions about how Non-Euclidean geometry might be shown to be
self-contradictory. I am not aware of any explicit reference to
Non-Euclidean geometry in letters from Lambert to Kant.
Kant may have used Lambert as a guide to the subtleties of geometry, and it
could have been that he learned from Lambert not to regard as logically
impossible a geometry based on the denial of the parallel postulate (and
leads to Non-Euclidean geometry). However, in his Critique, Kant argued that
it is an inbuilt characteristic of our minds that as they struggle to make
sense of physical space they perceive it as Euclidean. He did not say that
space itself must be Euclidean, but only that we intuit it to be so. Thus,
he claimed, a geometer can prove that the angle sum of a triangle is 180°.
Obviously, Lambert would not have agreed! More precisely, a philosopher
cannot prove this, but a geometer can, because a geometer can make the
appropriate construction.
Kant’s and Lambert’s views about geometry (for an English translation see A.
Zweig (ed.) Kant’s Philosophical Correspondence 1759-1799, Chicago U.P.
1967) make a fascinating pair of readings. The sophisticated ideas of Kant
about how we can know about things contrasts with Lambert's more naive
realism (he observed that he had not been able to deny all reality to time
and space); yet Kant's view of what we might know is much more restricted
than Lambert's. This was not to be the last time philosophical subtlety was
countered by mathematical bluntness.
Best wishes
Jeremy Gray
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