[FOM] Convincing math-blind people that math is different
JoeShipman at aol.com
Thu Dec 25 11:24:47 EST 2014
I think I reject the premise of your argument that a person can be both intelligent and "math-blind".
In my experience, every adult who is intelligent enough to speak English clearly and make *any* kind of mental effort is capable of understanding certain simple ideas which can serve as a basis for genuinely mathematical insight. For example:
1) ask them to start writing the positive whole numbers in an increasing sequence: 1,2,3,... and ask them if there is a largest number or if they could continue writing the numbers forever. If they cannot grasp that these are different and incompatible propositions, then I claim that they are not only math-blind, but that they are unintelligent.
2) show them the definition of a prime number and get them to the point where they admit that, for any given number, it is possible to tell if it is prime or not by trying divisors successively starting with 2.
3) start writing out the prime numbers 2,3,5,7,11,...and ask them to verify that the first part of your list is correct, then ask them whether they think it is a meaningful question whether there is a largest prime number or whether the prime numbers go on forever like the positive whole numbers do.
Note that at this point you have not asked them to understand a proof that the prime numbers go on forever, you're just getting them to affirm that that is a meaningful statement. If they do not know a proof of it, you can them get them to affirm that they "know" that there is no largest whole number in a different way than how they "know" there is no largest prime number.
4) This is already enough to establish what is special about mathematical knowledge, but you can go further and assert that you "know" that the primes go on forever in the same way that they "know" that the whole numbers go on forever, not by authority but by understanding. They may or may not be able to follow the proof, but they should be able to grasp the distinction about types of knowledge.
5) if they get this far, you can also point out that you "know" the twin prime conjecture in a lesser sense--you believe the twin primes go on forever from numerical evidence but you don't have an understanding of why they must do so, so that belief is not the same kind of knowledge as your knowledge of the primes going on forever.
Sent from my iPhone
> On Dec 24, 2014, at 11:59 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
>> On Wed, 24 Dec 2014, Auke Booij wrote:
>> So here's a radical idea: mathematical knowledge is not qualitatively
>> different from most other kinds of (academic) knowledge.
> I am glad that you have taken the time to articulate this point of view. I think you have pinpointed the crux of the matter, that anyone who wishes to argue for the distinctiveness of mathematical knowledge must address.
> However, first I would like to back up one step. It seems to me that you might agree that there is at least a tangible distinction between what I'll call "mathematical/scientific knowledge" and other kinds of knowledge, even if you deny any kind of sharp boundary within that category (namely, between mathematical knowledge and scientific knowledge). In any case, whether or not you believe this, I think I can sketch a way that such a distinction could be demonstrated to a math-blind person pretty convincingly. I will do this now. The description may also help to clarify, by way of example, what kinds of capabilities I envision the idealized math-blind person to have.
> The demonstration is simple to describe. I build a computer and implement an algorithm that prints out, on paper, a million digits of some constant that hasn't been explicitly computed before---say, sqrt(12523599347). Then I build a completely different kind of computer and implement a completely different algorithm. I announce that my new system will print out exactly the same million digits. Then I hit the "go" button and the machine duly churns out the predicted million digits. The math-blind person can verify that the million digits are indeed the same.
> This sort of demonstration would seem to have no analogue in other fields of knowledge. For example, we might be able to find two people who are able to recite the entire Koran word-for-word, but this is because the Koran has already been written out explicitly for all to examine. The first million digits of sqrt(12523599347) have not been written out before, as far as the math-blind person can see. All non-scientific examples I can think of that involve agreed-upon conventions (e.g., laws, works of art) require that a community spend considerable time drawing up the conventions explicitly, and explicitly disseminating that knowledge. The way in which an algorithm encodes an enormous number of digits seems to be a uniquely mathematical/scientific phenomenon.
> Although I find this to be a convincing demonstration of the qualitative difference between mathematical/scientific knowledge and other kinds of knowledge, I am not sure that it serves as a demonstration of the distinction between mathematical knowledge and scientific knowledge. We are, after all, still in the realm of finite predictions of finite experiments with finite results. If, as I would like to propose, a math-blind person lacks the ability to *extrapolate* or *abstract*, then it seems to me that we are stymied at this point.
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