Thursday, July 30, 2009

Confessions of a Math Idiot

I wasn't always a math idiot. I took the ‘advanced’ math classes in grade school. While I was in high school I took Calculus II over at the community college (the professor was a Harvard grad with a Ph.D. in theoretical physics from Cornell). I took the AP Calc test and my math SAT score was nothing to sneeze at. I went as far as differential equations and linear algebra in college.

But then I found out about computers. On the first day of 6.001 the professor wrote this on the board:

                f(x + dx) - f(x)
f'(x) =  lim  ------------------
        dx->0        dx

(define dx 0.0001)

(define (deriv f)
  (define (f-prime x)
    (/ (- (f (+ x dx)) (f x))
       dx))
  f-prime)

(define (cube x) (* x x x))

(define g (deriv cube))

(g 4)
;Value: 48.00120000993502
It was the beginning of the end.

I started to unlearn math. I began seeing math through the eyes of a computer scientist. Things that I believed in stopped making sense. I used to believe that integration and differentiation were opposites: that one ‘undid’ what the other ‘did’. But look:

(define (deriv f) 
   ...)

(define (integrate f low hi)
   ...)
The arity of deriv is 1 and the arity of integrate is 3. They can't be opposites!

I used to think you couldn't divide by zero. But look:
(/ 2.3 0.0)
;Value: #[+inf]


I started seeing ambiguities in even the simplest math. Is ‘x + 3’ an expression, a function, or a number? Is ‘f(x)’ a function or the value of a function at x? When I see log (f(x)) = dH(u)/du + y, are they stating an identity, do they want a solution for y, or are they defining H?

I started reading weird things like Gosper's Acceleration of Series and Denotational Semantics by Joseph Stoy and Henry Baker's Archive. These look like math, but they seemed to make more sense to me.

But I drifted further away from mathematics. A few years ago a fellow computer scientist explained to me the wedge product. It is a straightforward abstraction of multiplication with the only exception being that you are not permitted to ‘combine like terms’ after you multiply (so you get a lot of irreducable terms). Then he explained the Hodge-* operator as the thing that permits you to combine the like terms and get answers. It's really not very complicated.

But then I looked it up online.
The exterior algebra ∧(V) over a vector space V is defined as the quotient algebra of the tensor algebra by the two-sided ideal I generated by all elements of the form x⊗x such that x ∈ V.
And I discovered that I'm now a math idiot.

I'm probably going to have to live like this the rest of my life. I'm trying to cope by relearning what I used to know but translating it to a style I'm more comfortable with. It will be a long process, but I'm optimistic.
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