It's said that you can be certain of only two things in this world - death and taxes. Well, you can add all of mathematics to that list. That's because when you've proven something in mathematics, you can be absolutely, totally, one-hundred percent sure it's true. Proof - after this on Earth and Sky.
JB: This is Earth and Sky, on the subject of proof. Generally, something is true if you can prove it. But proof - even in much of the scientific world - is a somewhat shadowy concept, and just what constitutes "proof" can be hotly debated. Not so in mathematics. We spoke with Dr. David Harbater, a mathematician at the University of Pennsylvania, about mathematical proof:
(Tape 0:21:07-0:22:07) Well, mathematical proof is definitely different from what one means in other things. I mean, often, when people talk generally about proof, you just mean something that's convincing - that the average person would tend to find it plausible. There could be proof in a legal case, there could be proof by the preponderance of evidence, proof beyond a reasonable doubt, and so forth. There's scientific proof, which means you've run an experiment lots of times and it keeps coming out a certain way and so then you're convinced that it's true. But mathematical proof, mathematical proof is different from proof in everyday life or proof even in the experimental sciences. It involves proving something conclusively by means of deduction in which everything you say follows from everything you've said before. So you start with things that are known, and then you build up from one step to the next. And in the end you have a demonstration of an assertion which you can be absolutely sure of.
JB: Thanks to Dr. David Harbater for speaking with us. And with thanks to the National Science Foundation, I'm Joel Block, for Deborah Byrd, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual for aiding in the preparation of this script:
Dr. David Harbater
Department of Mathematics
University of Pennsylvania