October 25th, 2006

Invented or Discovered?

5 minutes is an awkward time to kill- not enough time to do anything, too much time to just go to wherever you have to go to next.

One thing I've been doing with those blocks of time is perusing Yahoo Answers .  People ask a bunch of questions, and people answer them. 

I usually stick to science and math, and occasionally gaming.  I know I'm just helping people with their homework, but maybe if I explain things well, someone will actually learn something.  It's the teacher in me.

Anyway, today they had a pretty cool question about whether things in math are "invented" (in the sense that you create a proof that wasn't there before) or "discovered" (in the sense that you notice something that has always been a universal truth).

I'm kind of proud of my answer:

 I think that's like saying that Edison "discovered" the lightbulb by recognizing that if you hook all of the wires and stuff in the right way, you'll get something that can light a room.

I think mathematics is a language, and things like proofs or calculus are created using statements in that language. The whole language is based on a few elementary concepts (axioms), and it's possible that we can change the axioms we get another, equivalent way that can also be used to describe things (see, for example, non-Euclidean geometry).

Calculus, specifically, creates new definitions (for limits, derivatives, and so on), and then uses those definitions to prove other concepts that are consistent with how the world works. But that doesn't mean that we couldn't "invent" some other language that does the same thing calculus doesmbut has totally different ways of doing things..

Of course, we can extend the argument as well- there obviously are facts that are objectively true or not by the rules of math.  Goedel tells us that there are also facts that can never be proven correct.  Doesn't that mean that a proof is just a discovery of a fact?

Perhaps, but I think you still need to invent a method to show that you're correct.  You don't discover a persuasive argument to convince someone that you're right- you create one.  Similarly, you need to create your proofs.  And, in the case of calculus, invent a whole new language to frame what you're talking about.

Someone else said "I think he discovered certain phenomena and then invented a technique (calculus) to describe them."