Subject: Re: [HM] Kant and non-Euclidean geometry
From: Gordon Fisher (gfisher@shentel.net)
Date: Sun Mar 12 2000 - 17:51:46 EST
Jeremy Gray wrote:
[deletion]
> 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.
In my view, what Kant was considering was this -- at least this is what I
concluded from my rather intensive study of Kant some years ago:
Kant's Proposal 1: there is one and only one way that humans can *perceive*
space and time. This will be an interesting and significant matter to
investigation, or assumption to proceed on.
Kant's Proposal 2: the one and only one way that humans can *perceive* space
was discovered by Euclid about 300 BC, and written up by him in a work called
(in Greek) "Elements*.
Fisher's Proposal 1: Kant may have been right or wrong as each of these
proposals.
Case 1. Suppose Kant is wrong about his Proposal 2. This doesn't imply
that his Proposal 1 is wrong, since we might be able, for example, to substitute
an elliptic or hyperbolic non-euclidean geometry, or some other formal geometry,
instead. The way we perceive spatial objects may not have been formulated by a
gentleman named Euclid about 300 BC (basing his work on a number of his
predecessors), nor later by Lobachevsky or Bolyai or Riemann or Hilbert or Pasch
or Veronese or other 19th & 20th century geometers.
It may be that *none* of our formal or formulated geometries -- euclidean,
non-euclidean or some other kind -- successfully describe the way we *perceive*
spatial objects, or objects in "space". It may even be that no such consistent
formal geometry has yet been found, or even -- perish the thought -- can be
found. Does any of this destroy Kant's idea that there is one and only one way,
or better, one conceivable distribution of ways, that humans can perceive
objects in space? ("Conceivable distribution" is meant to allow for distortions
due to disease or drugs, and also for a range of differences among "normal"
individuals.)
I don't see how.
Question: Did Kant ever base anything to such an extent on Euclid's parallel
postulate, or an equivalent, that his entire thought, or even just his Proposal
1, is destroyed because he thought through a given "point" not on a given "line"
one and only one line parallel to the given one can be *visualized* lying on the
given point? Does Kant discuss the necessity of making this assumption about
parallels for grounding his analyses? Wouldn't he have been able, or aren't we
able, to modify his
thought rather easily to accommodate to the fact that we mathematicians now
speak of two basic classes of non-euclidean geometries together with euclidean
geometry, or of a continuum of geometries characterized by curvature (with
euclidean geometry having curvature zero), or whatever?
Does anyone want to claim that we humans can *perceive* spatial objects in
whichever of these geometries we choose to? As we know, "models" for
non-euclidean geometries suffer from certain difficulties when one tries to
"embed" them in our perceptual space (which I take to be a different thing from
3-dimensional spaces of the sort considered by geometers and physicists).
Case 2. Suppose Kant is right about his Proposal 2. How could this come
about? Suggested answer: It may be that as far as *perception* is concerned,
"normal" humans do indeed *perceive* in Euclidean geometric terms, although they
have, by exercise of their reason and imagination, made and learned other
geometries which differ from Euclidean geometry as to the parallel postulate.
This seems odd to me, since to me, from some standpoints, Euclid's parallel
postulate seems counterintuitive when it comes to human perception. Ditto for
his notion of "line" (maybe "point", too, but that's a separate matter). A
"straight" line going "to infinity"? What's this "straight" business, at least
not without some theory of curvature which itself builds on the notion of a
Euclidean line (e.g., when you interpret differentiation as producing "tangent
lines"? And how about a line which "goes to infinity"? Lots of people have
thought in these terms, with gret success in many respects, but who ever
perceived such a thing?
So I conclude that Kant was wrong about his Proposal 2. However, I conclude
this not on the grounds that people found some other systems like Euclid's
except for the parallel postulate, nor on the grounds that it is possible to
think about our universe as a 3-sphere, spatially speaking.
How about Kant's Proposal 1 above? It seems to me that if one wants to show
that Kant was wrong here, it's (1) not sufficient to point to non-euclidean
geometries as mathematicians and physicists and maybe some others consider them,
and (2) the existence of non-euclidean geometries is not very central with
regard to issues of how our perceptions affect our ability to find out how
things are, independent of our minds, and how our reason and imagination and
maybe emotion, too, can or cannot be used to deduce or imagine or feel how
things "really" are.
Gordon Fisher gfisher@shentel.net
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