Re: Choice sequences, intuition, etc

Bill Taylor <w.taylor@xxxxxxxxxxxxxxxxxxxxx> writes:

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).

I'm not sure that's a clear distinction. We derive our personal
intuition within a culture. And what could "cultural intuitions"
even mean, other than typical personal intuitions of members of
the culture?

We teachers must
always be on the alert not to crush students' intuitions or
overbully them too definitely along one track.

We university teachers mostly see students after their mathematical
intuitions are already well formed. And if they are not well formed
by the time the student enters the university, the chances are that
the particular student won't go very far in mathematics.

Of course, intuitions about axiom of choice won't be formed in high
school. But more basic intuitions about numbers, about "exist" in
the mathematical sense, about the infinite (at least the infinitude
of integers) will already have been formed. These days, they don't
study Euclidean geometry in high school as much as used to be the
case, so I do wonder how much that change affects intuitions.

...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.

Sure, I agree that sets of size larger than c are, in some sense
surreal. So is the set of ordinals less than the first uncountable
ordinal. And the Stone Cech compactification of the reals always
seemed highly surreal (this, of course is larger than c).

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?

A physicist wants to consider the point of intersection of two
idealized light rays. I cannot think of any reason to consider that
point constructible. Well, okay, you could say it is physically
constructible. But why should it be mathematically constructible?
Why should the path of a light ray be mathematically constructible?

It seems to me that physics applies mathematics in ways that there
is no reason to assume possible under constructivism.

Now maybe we could redesign physics to avoid this problem. But to
do that would challenge the way physicists think, and maybe detract
from their use of physical intuition.

.

Relevant Pages

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  • Re: Why real numbers and points cant model continuum
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  • Re: infinity and intuition
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