[FOM] Convincing math-blind people that math is different
joeshipman at aol.com
joeshipman at aol.com
Tue Dec 23 13:13:24 EST 2014
The right way to do this is to actually teach them some elementary mathematics in the way mathematicians experience it rather than the way it is taught in school. Even if they are "math-blind", I will presume that they have at least average intelligence. Here are some examples:
Proposition: any shape with straight edges is cut-and-paste equivalent to a square. Constructive proof: draw diagonals to get a bunch of triangles, convert each triangle to a rectangle with an easy 3-piece dissection, show lemma that a rectangle can be converted to a shorter rectangle up to twice as wide by an easy 3-piece dissection, successively widen all the rectangles until they are as wide as the widest one, stack them on top of each other so they make a single rectangle, then widen that one to make a square (if it is already too wide turn it on its side first).
Any child who has played with scissors and tape or glue can get the ideas here. The "math-blind" person who can follow this construction and see that "IT WORKS" now has a good example of mathematical knowledge -- a very general and non-obvious proposition made certain by construction.
Proposition: no square number is twice another square number. There are many proofs of this, and an intelligent math-blind person is likely to be able to follow at least one of them (though different math-blind people might understand different ones depending on their modes of thought).
Even if they can't follow any of the proofs, they can get the general idea is that there is a PARAMETRIZED fact which is really an infinite number of individual facts (25 is not twice 9, 100 is not twice 49, 64 is not twice 36, etc). If they can grasp the concept that it is possible to know a TEMPLATE statement such that any INSTANCE is true, they will have a valid notion of a specifically mathematical mode of knowledge.
-- JS
-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
To: fom <fom at cs.nyu.edu>
Sent: Tue, Dec 23, 2014 12:18 pm
Subject: [FOM] Convincing math-blind people that math is different
I have recently been mulling over a certain unusual angle on the
philosophy of mathematics that leads to what I think is an interesting
question. Roughly speaking, the question is whether it is possible to
convince a math-blind person that mathematical knowledge is qualitatively
different from other kinds of knowledge.
By a math-blind person, I mean someone who lacks the internal experience
of mathematical proof and the feeling of certainty and objectivity that
mathematical proof (usually) confers. If we want, we could even take the
extreme Kripkensteinian route: the math-blind person can't comprehend what
a syntactic rule is, thinks "grue" is a natural concept and is baffled by
"green," etc. But we'll assume that the math-blind person is a normal
person in non-mathematical areas, and in particular has a deep
understanding of human social interactions and activities.
Those of us who are not math-blind are (mostly) convinced that there is
something qualitatively different about mathematical knowledge compared to
other kinds of knowledge. My question is, is there a way to convince a
math-blind person that this qualitative difference exists?
Math-blindness blocks perhaps the most natural route. Instinctively, most
mathematicians want to talk about the *certainty* of mathematical
knowledge. Now, there are familiar difficulties with the concept of
mathematical certainty, having to do with the difficulty of being certain
that there is no error in a proof, or the question of which axioms, if
any, are certain. But even leaving those difficulties aside,
math-blindness presents a much simpler obstacle. The math-blind person
has no idea what we're talking about when we talk about the subjective
experience of certainty. All the math-blind person can see is that
mathematicians *claim* to be certain. But religious fanatics also claim
to be certain, and most mathematicians want to distance mathematical
knowledge from religious fanaticism. The math-blind person can, of
course, test this alleged certainty by seeing whether the community ever
changes its mind about the "certain" facts. But on this score, the
religious fanatics will probably outdo the mathematicians, who must
publish errata on an embarrassingly regular basis.
Similarly, *unanimity* doesn't seem to be a good litmus test either. Is
there unanimity about whether an arbitrary angle can be trisected using
compass and straightedge alone? Sure, say the mathematicians. But wait,
asks the math-blind person. What about those angle-trisectors over there?
Oh, they don't count. Well, if it is permitted to exclude some groups of
"wackos" in order to achieve "unanimity," then it seems that there are
plenty of other areas of knowledge where there is "unanimity" among the
"in group."
The best candidate I have been able to come up with so far is that in
mathematics, it is possible on rare occasions for an "outsider" with no
credentials among the "in group" to be able to make a very brief statement
about an alleged error in the accepted body of knowledge, and have it be
immediately acknowledged to be correct, even when the error has
significant consequences. I feel that I'm on the right track here, but
I'm not sure that I've really "nailed it." For example, an error in, say,
Google Maps can survive for a rather long time, and anybody can point out
the error and have it be universally acknowledged after a rather brief
verification process. One response might be, "Whether something is a
one-way street *is* a mathematical fact." This point of view seems to put
us squarely in the fictionalist camp, where mathematical facts are
agreed-upon conventions. But even if you are a fictionalist, there seems
to be a distinction between definitional conventions and chains of logical
inference. Google Maps is perhaps mathematical insofar as one regards all
definitional conventions as part of mathematics, but intuitively there is
still something different about mathematical knowledge and traffic
conventions that is not captured by the outsider-can-find-an-error
criterion.
I am wondering if someone can come up with a better criterion, or
alternatively, argue convincingly that there is no way to convince the
math-blind person that mathematical knowledge is qualitatively different
from other kinds of knowledge.
Tim
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