Re: Choice sequences, intuition, etc
- From: Bill Taylor <w.taylor@xxxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 13 Oct 2008 21:05:59 -0700 (PDT)
Some more detailed comments on Neil Rickert's post,
which is, BTW, a definite keeper.
Certainly, the *development* of mathematical intuition is a topic
that far too little time has been spent on. OC it would need
expertise in both math and cognitive psych,
which is probably rather rare!
I'm inclined to doubt that cognitive psychology can tell you much.
On second reading, I suspect maybe we could be at crossed
purposes here. It seems that Neil was speaking more of
the development of cultural intuitions. I had been speaking
more of individual, personal intuition(s). We teachers must
always be on the alert not to crush students' intuitions or
overbully them too definitely along one track. Even when dealing
with absolutely routine basics of analysis etc, one must tread
carefully, wording things to allow for a constructivist or other
interpretation. That's my feeling, anyway.
I believe there are two sources of intuition involved here, and
...I shall say that the first of these is synthetic. It is our
intuition about counting.
...
I shall refer to the second source of intuition as analytic.
It derives from geometry.And, roughly speaking, Euclidean geometry
is based on an idealization of the process of measuring with
the use of a movable measuring rod. I call it "analytic",
I agree virtually 100% with all this, as regards our cumulative
cultural intitions. And very well expressed indeed.
Using the synthetic approach (i.e. counting),
we never get beyond finite sets of numbers.
When we idealize, we never get beyond countable sets.
...analytic approach we come up with something like Dedekind cuts.
But these never get anything [larger] than sets of cardinality c.
An intriguing view of the "naturality" of set sizes up to c,
and thus the "supernaturalness/surrealness" of higher set theory.
I'm suggesting that much of our mathematical intuition comes
from uniting these two approaches, forcing what we construct
synthetically to be identical to what we construct analytically.
Certainly, for basics, this has been a common view ever since
Descartes showed the world the amazing efficacity of combining
algebra and geometry.
And I am inclined to think that mathematics is useful in physics
precisely because have united the two approaches.
Hmmm. Maybe that's a bit over-speculative? But maybe you're right.
[AISI], constructivism wants to stick to the synthetic approach,
yet still pretend that what is done is relevant to physics.
I think that may be now definitely OTT. I can't think what it means.
Perhaps you might elaborate, Neil?
-- Badgering Bill
** "The math is done right, but is the right math done?
.
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