Re: Choice sequences, intuition, etc

I have held off posting in this thread. But maybe it's time. So
I'll comment on a little bit of Bill's original post.

Bill Taylor <w.taylor@xxxxxxxxxxxxxxxxxxxxx> writes:

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.
However, as a mathematician, and an heretical dabbler in cognitive
science, I'll add my comments.

I believe there are two sources of intuition involved here, and
they are really both intuition about mechanism and its consequences.

I shall say that the first of these is synthetic. It is our
intuition about counting. We think of a counter, and idealize that.
Much of our intuition about the natural numbers comes from this.
The numbers themselves originate as labels for the counts, but we
reify those to treat numbers as entities in their own rights. I use
the term "synthetic" because we construct (synthesize) the number
system based on the process of counting and its natural extensions.
However, it is the process of counting, rather than the reified
entities, that mostly concern us.

Remembering back to graduate school, one of our professors
(S. Kakutani, if I recall) would say: "Think of a large number.
Square it. There's a theorem that has never been published."
And his point was that mathematical facts ("x is a square") are
not in themselves inherently interesting. What is valuable is
the process we follow to get those facts.

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", because in
some sense we start with the world as a whole, and we divide it into
the two sides of a line that we draw. By dividing, we add features
(the parts into which we divided it), and the process is one of
analysis into smaller and smaller features.

When we draw two lines in the physical world, these are thick lines,
and they intersect (unless parallel). But once we idealize to
infinitely thin lines, there is no obvious reason that two lines
should intersect. In terms of a common world, there is no reason
that there should be a common particle that is on (or touched by)
the two non-parallel thin lines. So, in a sense, we have forced the
idea of intersection. This forcing makes sense, because the aim
is to idealize our experience with measurement and our experience
with thick lines, where there is an intersection. Moreover, that
intersection is important to us in our physical use of measurement
and of thick lines. So we needed to force the idea of intersection
for it to idealize what we wanted to study.

Using the synthetic approach (i.e. counting), we never get beyond
finite sets of numbers. When we idealize, we never get beyond
countable sets.

Using the analytic approach we come up with something like Dedekind
cuts. But these never get anything smaller than sets of cardinality c.

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.
And I am inclined to think that mathematics is useful in physics
precisely because have united the two approaches.

As I see it, constructivism wants to stick to the synthetic approach,
yet still pretend that what is done is relevant to physics. And I
question whether that can actually work.

.

Relevant Pages

  • Re: Lesson learned
    ... intersection and for some reason I slow down... ... Woman's intuition? ... Glad to hear about the non-accident that almost was that didn't happen. ...
    (rec.motorcycles.harley)
  • Re: Lesson learned
    ... intersection and for some reason I slow down... ... Woman's intuition? ... Glad to hear about the non-accident that almost was that didn't happen. ...
    (rec.motorcycles.harley)