I am reading Eli Maor's

*Trigonometric Delights*, a book supposedly aimed at the "intelligent layperson." In it, I find this step in a demonstration to the solution of a problem proposed by Regiomontanus:

((x / a) * (x / b) +1) / (x / b - x / a) = (x / a - b) + (ab / ((a - b) * x))

Now, I have little doubt that this equality is true. ("Little" because mathematicians

*do* make mistakes.) And what maneuvers led from the first expression to the second one may be very obvious to some. But after playing around with each side of the equation for a while and not managing to turn one side into the other, I brought the book to a friend who is just finishing his Master's in mathematical education. He spent about 15 minutes toying with it, and also could not see how to move from one expression to the other.

So, while the equality of these two expressions, again, may be completely obvious to some of my readers, the point I am hoping to make is that it is curious to find the "obviousness" of the equality simply assumed in a book specifically aimed at non-mathematicians.

Now, one explanation of my perplexity here could just be, "Well, Gene, you have no aptitude for mathematics." But I don't think that will wash: I scored 800 on my mathematics GRE, and an 'A' in every college mathematics course I have ever taken.

No, I believe the real issue is that a particular mode of mathematical understanding is focused upon in most mathematical texts, while another is neglected. An example: I was recently watching some lectures on number theory. The lecturer set out to prove that there are infinitely many

Pythagorean triples. He offered an algebraic proof -- this one I followed with no problem -- but as I watched, I wondered why he was going about this in such a complicated way: if we simply spin a hypotenuse of length one around a unit circle, it was obvious to me that every time the length of the other two legs of the right triangle it spawned were rational numbers, we would have a Pythagorean triple, and clearly there must be infinitely many such cases. And even that previous sentence does not really capture what I thought, since it is a verbal formulation of what was a purely visual experience: I could "see" the hypotenuse spinning around the circle, and could "see" that again and again, the other two legs of the right triangle it formed could be measured by rational numbers.

In the very next lecture, it was explained how Pythagorean triples map to "rational points" along the unit circle. I had never even heard of a "rational point" before then, but it is the technical term for one part of a relationship that was obvious to me as soon as I considered the question at hand. "Symbolic" mathematicians go through a whole mess of symbolic manipulation to prove this point, but I could visualize the conclusion they reached as soon as I saw that hypotenuse spinning around.

So, my contention is this: standard mathematical education is (unduly? -- yes, I think so) oriented around symbolic manipulation -- no doubt an important skill! -- and thus poses an unfortunate hurdle to those who "get" mathematics primarily by visualizing what is going on. And this point, of course, ties into earlier posts about "

thinking in pictures": many symbolically oriented thinkers seem to assume that

all thought is symbolic thought, e.g., "no creature which lacks language (in the relevant sense of 'language') can be said to think or reason in the strict sense." Well, sorry,

Temple Grandin,

Albert Einstein*, and I all beg to differ.

The advocates of "symbolic mathematics" might very well respond, "OK, your visual and physical intuitions are all well and good, but our symbolic mathematics can

*prove* theorems." However, as Lewis Carroll

demonstrated long ago, every supposedly "formal" proof must ultimately rest on the intuitive understanding of the person expected to accept the proof. There are, after all, infinitely many possible systems of symbol-manipulation rules, and it is only human judgment that can discriminate between the multitudinous hoard of them that produce nonsense, and the much more limited number which produce truth.

UPDATE: Shonk corrected my parenthesizing of the equation above.

* "For example, to begin with, words and language do not play any role in his thoughts. They must be sought for laboriously later." and "Virtually all of these achievements depended upon a very astute form of physical thinking. Einstein then dressed it in mathematical clothing, seeking where ever possible to keep the mathematics as simple as he could."