### Friday, January 13, 2006

## Accounting for mathematicians

For the “things I wish someone showed me in high school” department:

I just found out how to do long math problems without getting lost! I’m elated. This discovery hasn’t just made things easier; it has made possible things that I was previously incapable of.

It’s just an accounting system. First, label every initial proposition with a number, in order:

X > 3 __________________ (1)

Y > 3 X ________________ (2)

Z > 3 Y ________________ (3)

etc.

Then start deducing stuff, labeling each new deduction and indicating how you got there:

Y > 9 by (1) and (2) ___________ (4)

Z > 27 by (3) and (4) __________ (5)

etc.

That’s right, my discovery is: write stuff out just like it would be in a math book. I knew that such a format was a good way to present an answer, but I didn’t realize it's useful before the answer's in hand.

To make it obvious what I’m talking about, I’ve had to choose an example that’s easier than the sort of problem for which you would actually need to do any accounting. But for complicated problems, the difference that a labeling system makes is just ungodly huge. For one thing, when something goes wrong, it’s much easier to find out where the error lies. For another thing, you can see what you’ve already tried, without which benefit I and billions (just a rough guess) of others can be prone to going in circles.

I feel like I’ve thought I was blind for years, and I just discovered that I’d been trying to see with the lights off. My inability to do long math problems has shaped to a large extent the areas of math I’ve explored – I took on topology, abstract algebra, combinatorics, etc., and stayed away from differential geometry, optimization and whatnot, because the proofs in the first set of things I listed were short, and the proofs in the second set were long. Whole new career options may have opened up for me today.

I just found out how to do long math problems without getting lost! I’m elated. This discovery hasn’t just made things easier; it has made possible things that I was previously incapable of.

It’s just an accounting system. First, label every initial proposition with a number, in order:

X > 3 __________________ (1)

Y > 3 X ________________ (2)

Z > 3 Y ________________ (3)

etc.

Then start deducing stuff, labeling each new deduction and indicating how you got there:

Y > 9 by (1) and (2) ___________ (4)

Z > 27 by (3) and (4) __________ (5)

etc.

That’s right, my discovery is: write stuff out just like it would be in a math book. I knew that such a format was a good way to present an answer, but I didn’t realize it's useful before the answer's in hand.

To make it obvious what I’m talking about, I’ve had to choose an example that’s easier than the sort of problem for which you would actually need to do any accounting. But for complicated problems, the difference that a labeling system makes is just ungodly huge. For one thing, when something goes wrong, it’s much easier to find out where the error lies. For another thing, you can see what you’ve already tried, without which benefit I and billions (just a rough guess) of others can be prone to going in circles.

I feel like I’ve thought I was blind for years, and I just discovered that I’d been trying to see with the lights off. My inability to do long math problems has shaped to a large extent the areas of math I’ve explored – I took on topology, abstract algebra, combinatorics, etc., and stayed away from differential geometry, optimization and whatnot, because the proofs in the first set of things I listed were short, and the proofs in the second set were long. Whole new career options may have opened up for me today.