Explaining Mathematics for the Layperson
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."
((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."
Yes, I 100% agree with this.
ReplyDeleteA great deal of my educational time (especially in math) I simply came to accept that class time would be wasted because I would not understand anything & would just have to make stuff up for myself. The few professors I really learned from were almost always unpopular and poorly rated. The one time I attempted an engineering math course (calc 3) I got creamed & had to drop. Took it in the math department, easy A. Exact same content, different presentation.
To this day, I still cannot do chemical stoichiometric calculations the way it is taught (which was really awkward when I was teaching and tutoring...) I have always just muddled along with basic interconversions.
Yes, it's false (what you wrote), in general. In Mathematica:
ReplyDeleteIn[7]:= Simplify[1/(x / b) - 1/(x / a)== -1/(x/(a - b))]
Out[7]= True
In[8]:= Simplify[((x / a) * (x / b) +1) / (x / b - x / a) == (x / a - b) + (ab /
(a - b) * x)]
2
a b + x 1 ab
Out[8]= b + --------- == (- + -----) x
a x - b x a a - b
"Yes, it's false (what you wrote), in general."
DeleteHuh?
In fact, not only don't I understand your first sentence, I also have no idea what the Mathematica you're showing me is supposed to mean.
DeleteYou're missing some parentheses in the equation. It looks to me like it should be this:
Delete((x/a)*(x/b) + 1)/(x/b - x/a) = (x/(a - b)) + (ab/((a - b)*x))
But your larger point is absolutely correct: mathematics communication is far too dependent on symbol manipulation. Partially I think this is the result of the philosophical stance of Hilbert and other influential mathematicians in the early 20th century, but much of it can probably be explained simply by the incredible difficulty and expense of communicating visual arguments, especially in the pre-computer era. My guess is that symbolic arguments were considered more trustworthy because they were much easier to communicate unambiguously and to verify for correctness.
Your proof of the infinitude of Pythagorean triples is surely the better proof, not least because anyone who really understands it could surely convert it into symbols if necessary, but if the only goal is to allow the reader to mechanically verify its "correctness", the algebraic proof is definitely easier.
Yes, two pair of parentheses are missing. I think the reason algebra displaced geometry was it was easier to extend to higher dimensions where our intuition fails us and doing it enough leads to its displacement.
DeleteI meant your equation is false for arbitrary values becuase Simplify resolved it to another expression, instead of True. As opposed to the always true first equation.
DeleteYes, I missed some parens.
DeleteAnd we have to know that a is not equal to b.
DeleteFor what it's worth, I attach a link showing the intermediate step. Please excuse the image quality; I simply wrote it out and took a photo of the paper.
ReplyDeleteYour broader point, on mathematical writing, is an interesting one. A large part of the problem, I think, is that when writing for publication mathematicians tend to prize elegance above all else, even in textbooks. There's almost always a large amount of intuition (including visualisation), guessing and scratch-work underlying any piece of mathematics, but that's typically all tossed aside when it comes time to commit to paper, so that the exposition can be rendered as succinct and elegant as possible. The expectation being that the reader will go through the same process themselves - and if they're not willing or able to do so, they're probably not suited to reading it anyway!
Gauss, I think, summed this attitude up very well with his famous statement that "no self-respecting architect leaves the scaffolding in place after completing the building."
"The expectation being that the reader will go through the same process themselves - and if they're not willing or able to do so, they're probably not suited to reading it anyway!"
DeleteThat may make sense in a text for PhD students. But it certainly doesn't in one for the intelligent layperson!
Thanks for the image.
I can appreciate this, but my beef with the situation is that when you leave out the intuition, visualization, etc, you are leaving out large parts of the core of the idea. How is it that a relation between a bunch of abstracted symbols is considered 'complete'? In the field of, say, biochemistry, you are expected to demonstrate the veracity of what you are saying by showing that it holds under many different test conditions. (I used to call it the "six different ways from Sunday" structure when I saw it in a paper. It took me forever to figure out this was what the authors were doing, and not six or seven independent experiments.)
DeleteI understand that math is more about logic and proofs (so if you've proven it just once, it's true, not as with science), but it still seems incomplete without the intuitive part of things. I am tempted to say it has to do with defending what you say -- the less you say the easier to defend. Which is understandable.
But whatever the case, it seems the accounting is being put down incomplete, knowledge is being lost and having to be rediscovered/made-up new over and over, and it is systematically knowledge of a certain disposition (which is even worse).
I've heard that one of the big mysteries of ancient architecture is that nobody knows how many structures were built anymore, in large part because nobody bothered to document the scaffolding. It was deemed unimportant, and now we have no idea how these things were built. Very smart people are spending their lives trying to figure this stuff out (which is probably kind of fun, but still...)
No-one looks at this and says, yes true. They manipulate and apply rules.
ReplyDelete