This post has nothing to do with Michael Artin Algebra! Artin of this post is my son, and this is the story of him learning algebra. He is 11 years old now, and I guess one of few students on the planet that still has a positive mentality towards mathematics!
We are reading together Gelfand’s’ Algebra, page by page, solving its problems one by one. The section “Letters in algebra” includes the following “Magic trick”.
Choose the number you wish. Add 3 to it. Multiply the result by 2. Subtract the chosen number. Subtract 4. Subtract the chosen number once more.
I asked Artin to choose a number without revealing his number to me. It was his first encounter with this kind of algebraic trick. I read the rule, and at the end, I knew the result is 2. Interestingly, when I asked what was his number, I realized that even right from start he had tried to “beat me” by choosing a big number, one million. Of course, It didn’t work.
Artin: Does it work for every number?
Then he said, “okay, I beat you” and chose another number, that was 1.5, and again the result was 2.
Artin: It also works with these. God damn it!
I knew that it takes time until his every number embraces different categories of numbers that he knows. Fortunately, he automatically was choosing different kind of numbers. Then, he chose a fraction, and this time, the result wasn’t 2. As I mentioned in the paper Specularity in Algebra, in this case, getting the result wrong is a better algebraic opportunity than getting it right. First of all, I said that I am sure he had some miscalculations, and that is why he hasn’t got 2 as the final result. To prove my point, I used the idea of specularity. I asked him to write his number on a piece of paper. I folded the paper to avoid seeing the number and touching it (using it in calculations). Then, we followed the rule, keeping the number intact. Here something amazing happened that is the reason that I am writing this post. You know, it is well-known that one of the difficulties of moving from arithmetic thinking to algebraic thinking is accept something like \( x+3 \) as it is. To see the difficulty, try to think of say \( 4+3 \) without thinking of 7. In arithmetic we are pushed into the result, but in algebra, the operation is the result. Simply speaking, you cannot do anything with\( x+3 \).
Artin’s number was written on a folded piece of paper (let us suppose that \( [1/4] \) denotes his number in the folded paper) and he could not do anything with it. The first rule change \( [1/4] \) to \( [1/4]+3 \) . Suddenly, Artin got to the heart of algebra, but at the same time, found it strange and arithmetically useless.
Artin: You have the answer right here (pointing to \( [1/4]+3 \) ). What is the point of this if this \( [1/4]+3 \) is the answer?
Look at this equality:
\( (a + b) + c = a + (b + c) \) , or this one:
\( a . (b + c) = a.b + a.c \) .
They are true structurally. In principle, You can just replace one side of the equality with the other side without any extra comment. However, we usually like to describe these in terms of processes. For the first equality, we might say that adding the first two and then the sum to the third is the same as adding the first to the sum of the last two. For the second equality, we might say that from left to right it is multiplying through a set of parentheses, and from right to left, it is factoring something out. The italics are in the language of processes. Thus, when does the structure matter? Here is a strange example that I observed when teaching differential equations.
There is a technique in which you multiply both sides of a first order equation by something called “integrating factor”. The point is to write one side of the equation as the derivative of a function. For that, you need the product rule for derivatives: The derivative of a product of two functions is the derivative of the first times the second, plus the derivative of the second times the first. In symbols, we can write the product rule as \( (fg)’=f’g+g’f \) or many other ways. When moving from left to right we are doing something, we find the derivative, we multiply, we add. However, it seems that when moving from right to left, we are not doing anything. We just replace \( f’g+g’f \) with \( (fg)’ \) . Indeed, the mere fact that we can just replace the right side of the product rule with the left side was quite unacceptable for one of the best students of my class (hence he couldn’t see the logic behind the integrating factor). I would like to see more examples in which understanding structure really matter.