Subject: Re: [HM] Kant and non-Euclidean geometry
From: Gordon Fisher (gfisher@shentel.net)
Date: Fri Mar 10 2000 - 19:44:36 EST
I hadn't hear about Kant knowing something about non-euclidean geometry, but I
can well believe it. The idea that Kant's views on euclidean geometry somehow
undermined his whole system strikes me as a kind of academic "urban myth". If
one reads his works, I think you will see that while he may have made an error
in this regard, it is a relatively minor one, and that one can adjust it
without doing any real damage to his thought. I read a whole book many years
ago which was devoted to just this project, I thought quite successfully, so
this is not a new idea with me, although I came to my own way of thinking
about in when I read through Kant's major works, some parts more than once.
For one thing, to state it roughly, one can argue that we humans are confined,
when we "image" internally, to a particular kind of geometry, be in euclidean
or not, whereas other geometries may be arrived at by some kind of reasoning
and analogical thought.
This isn't to say that Kant's system can't be and hasn't been severely
criticized, and depending on your views, even undermined in basic ways. This
has been done or attempted many times, but not merely, in my view, on the
basis of his views about the role of a unique kind of geometry being employed
in intuitions (Anschauungen, cf. his "transcendental aesthetic") of space, or
about his attitude toward human minds being able to cope with "completed"
infinities without involving themselves in paradox.
Speaking of which, is everyone here completed satisfied about the role of
Russell's paradox and the like in set theories? How many of us, as practicing
mathematicians, just largely ignore this paradox nowadays? When I was a
graduate student, I was taught to make some kind of maneuver, such as
distinguishing between classes and sets in an explicit way, or using theory of
types in a rigorous way, when doing topology. Hasn't this practice rather
died out, beyond, perhaps, distinguishing between a member of a set and a
subset of a set? Does that mean that we should conclude the whole of Russell
and Whitehead's *Principia* should be rejected on the grounds that the theory
of types is a pain in the neck? Shouldn't we, as practicing mathematicians
and maybe practicing logicians stay away from it, and works of the same genre,
on some more substantial grounds (as numerous people have, not counting those
who have never thought much about this work at all). Does this mean
Russell's paradox is no longer a paradox, or only that most of us have got
used to going ahead without worrying about it, or, on the other hand, worry
about it, but employ some devices to avoid it or render it harmless?
Gordon Fisher
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