Subject: Re: [HM] Kant and non-Euclidean geometry
From: John Conway (conway@math.Princeton.edu)
Date: Sat Mar 11 2000 - 12:43:02 EST
On Sat, 11 Mar 2000, Bill Oates wrote:
> I tried to locate the information concerning the relationship between
> Lambert and Kant, but could find no relevant details quickly. It is
> clear that Lambert was a friend of Kant's. Kant actually taught
> mathematics from time to time and Lambert was aware of non-Euclidean
> geometry. The conclusion seems inevitable but I can not demonstrate
> from Kant's own writings.
I agree that it's quite likely from this that Kant learned something
of non-Euclidean geometry from Lambert, but to say that "the conclusion
is inevitable" is too strong.
It hardly matters, because although Lambert did indeed develop a
fair amount of non-Euclidean geometry, he can't be said to have a
clear idea of why it was consistent, let alone compatible with our
intuition...
> Kant does indicate that non-Euclidean
> geometries are wrong, not because of logic, but because they do not
> fit into our intuition of space.
... and so quite probably had exactly the same view that you here
ascribe to Kant. The convincing proofs of consistency only came in
the nineteenth century, and I think it's fair to say that so did the
realization that this hyperbolic geometry was in fact quite compatible
with our intuition.
> This is part of the argument that
> geometry is synthetic not analytic. So for us to agree that
> non-Euclidean geometry is possible and in fact may represent
> reality (whatever that means), I would argue is not a refutation of
> the synthetic a priori.
Indeed not. Did anyone say it was?
> I believe that Kant was not per se interested in geometry but in
> showing that synthetic a priori are possible.
I cannot understand how you can read Kant's words and think he
was uninterested in geometry. He clearly was.
> Euclidean geometry is chosen as an example.
This suggests that you think that's the only reason Kant discusses
geometry. It's possible to maintain that, but only, I think, by reading
his words in a decidedly unlikely way. Kant certainly did regard
Euclidean geometry as his prime example of the synthetic a priori...
> If the example fails it does not mean that synthetic a priori are impossible.
... and in this he was wrong, because it indeed does fail. Of course
you are right that this doesn't imply the impossibility of the s.a.p. -
that's another matter.
On that matter, let me say that 20th-century developments in logic
have still further reduced the number of things that one used to think
"must" be considered as s.a.p. I'm not myself wholly convinced, for
instance, that even the integers are such; and to go out on a limb,
I suppose that means I'm not wholly convinced that anything is. But
I wouldn't go so far as to say that "the s.a.p. is impossible" -
especially since I don't have a clear idea what that means.
John Conway
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