Mathematics is largely a process of symbolic abstraction. That is, we create symbols that abstract a physical thing or process into a character whose properties we can work with on the page. The mental thing we do with the symbol is to remember on the surface of the brain somehow what the symbol means physically. That is the abstraction from physical thing to symbol. At the simplest level, a mathematical symbol represents a single scalar value. It is either known or unknown. A physical thing is governed by laws that relate this quantity to other quantities. Abstraction implies that we know not only what the quantity represents but also all the physical laws that can be applied to it Abstraction is not fundamentally a difficult concept to grasp but the process of abstraction does scale up to some of the most profound and difficult concepts man has ever recognized. It was a Newton's profound realization that force and acceleration were proportional to mass and that given a known force over time, a straightforward, purely mathematical operation could predict velocity and position for the same time period.
So what does this have to do with aging? As I have gotten older, I find that the number of things available for ready recall on the very surface of my brain has gone down a bit. It is hard to say how much but definitely down and not up. Solving a mathematical problem, I have to maintain a large number of abstractions and equations to see how they might be manipulated algebraically to effect a solution. Of course, it is not useful to write down an equation that is more than a few lines long so it is always necessary as problems become more involved to abstract the symbols even further to compress the representation into a manageable size equation. The more complex the problem, the more complex the symbols become, and the more of the problem has to be carried in the head rather than written out on the page. Herein lies the problem. There is less space up there than there used to be. Another problem is that one has to pick a direction to try from the available directions to move the current problem closer to a useful form. It is like being in a maze and having many directions to choose, I have to make a good guess since exhaustively trying every possible thing would take far too long. The instinct for doing this is born of experience which I have but also the ability to play it out a few steps ahead like a chess master does when considering a range of moves. This takes some serious memory skills to visualize the steps very rapidly without writing anything down. I am pretty sure I would be a poorer chess player now than when I was younger just as a I am poorer at mathematical problem solving
I used to say my IQ or problem solving ability was limited by the size of the piece of paper I had to work things out on and in fact some of my most useful derivations were written out on blotter size quadrille ruled sheets. I also said it was related to the size of the desk to open up books I needed for reference. Now the limitation is similar but is internal.
The same difficulty is not present if I simply read someone else's work. The author has reduced the solution to a linear, step-by-step process. New abstractions are generally introduced at the point that are needed with appropriate references and explanations. One can focus just on a single equation at a time to follow along. One does not need to have the encyclopedia of mathematics immediately available from within memory to see how a problem has been worked out. Taking things one item at a time requires less mental surface area.
The wheel problem is getting more complicated with each refinement of the derivation with additional terms in the equations and additional coupling between equations. I am still able to solve the equations but it goes slowly and I have gone back to discover where I missed something. I am copying equations over on a new piece of paper to renew the terms in the mental image of the equation.
As some one observed, "Getting old is not for sissies."
No comments:
Post a Comment