Why Don't They? Math Vision No. 1

I envision a complete [anyway extensive] non-redundant mathematical encyclopedia. Could search engines and AI help? What does Wolfram already have? Could the math part of Wikipedia provide a base? It should be extensively linked.

How much can you teach a computer?

How much math can you teach a computer? Consider finding an expression for the sum of the numbers from 1 to n.

I. Proof by induction: The computer should be able to do what students are asked to do, but this is not finding an expression, only verifying it.

2. Assume a quadratic form f(n) = an^2 + bn +c
A "little algebra" gives

f(n+1) - f(n) = 2an + (a+b)

the left-hand side of this equation, being the sum of the numbers from 1 to n+1 less the sum of the numbers from 1 to n is n+1. This gives us 2a =1 and a+b = 1 so a = 1/2 and b = 1/2. Our equation is now

f(n) = (n^2)/2 + n/2 + c

Setting n to 1 (or any number including 0) sets c to 0 giving our final equation

f(n) = n(n+1)/2

3. Manipulate the numbers using the distributive law. There are two cases depending on whether n is even (the simpler case) or odd. For the even case:

The sum of the numbers from 1 to n is equal to the sum of the numbers from 1 to n/2 plus the sum of the numbers from (n/2 plus 1) to n. The sum of the numbers from (n/2 plus 1) to n is the sum of the numbers from n to (n/2 plus 1). i.e. reverse order. Using summation pseudo-notation:
Sum(i=n/2 + 1 to n)(i) = Sum(i = 1 to i)(n+1-i)
Then
Sum(i=1 to n)(i) = Sum(i=1 to n/2)(i) + Sum(i=1 to n/2)(n+1-i)
=Sum(i=1 to n/2)(i+n+1-i) = Sum(i=1 to n/2)(n+1) = n/2 times n+1 . . . QED
The odd case is slightly more complicated or one could apply induction to the even case.

4. Graphical solution:

Take n vertical bars numbered 1 to n and set them side by side in ascending order. Each bar should be one unit wide and as many units high as its number. Draw a diagonal from the lower left corner to the upper right corner. The area under the diagonal is n squared over 2. Each of the n bars has half of a square unit extending above the diagonal making an area of n over 2. So n squared over 2 plus n over 2 is n(n+1)/2.
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How much of this could the computer come up with. I'd like to see an AI expert teach it to learn math.
　

Math & AI - Part I

Saturday, June 13, 2009

Math and AI

First, a little background;

I am 82 years old. I got my masters in math in 1950 with courses in analysis and probability but no group theory. Working in insurance, I passed actuarial exams with the self-study methods common for actuaries at the time. I learned to program from IBM seminars (a whole week) and the manuals.

My reading hobby covered such things as Simon and Newell's work on computer problem solving and later the four-color problem, an interest I return to periodically to this day.
My interest in advanced math was recently reignited by The Problem That Couldn't Be Solved which led me to other books including Abel's Proof and Galois Group Theory and books on symmetry (The Monster) and Lie Groups.

I am curious about how much math can be taught to a computer and whether it could follow or even rediscover the proofs in these books. After all, they have been taught high-school geometry and symbolic logic.