I like to draw inspiration from the Google search terms that bring people to this blog. And someone recently arrived by way of the following philosophical doozy, a question close to the heart of every frustrated 9th-grader and every soul-searching tenure-track mathematician:
It’s helpful here to contrast math with science. That’s because they’re completely unalike. Sure, they share a symbiosis, and fans of one are often fans of the other, but their definitions of “truth” couldn’t be further apart.
Science’s measure of truth is the experiment. A theory may be perfectly consistent, wonderfully elegant, and intellectually satisfying—but if it fails to predict what happens in the world, then we toss it on the compost heap and move on. To a scientist, “truth” is whatever we see happening in reality, and all our explanatory frameworks must ultimately answer to the cold, hard facts of nature.
Mathematics’ measure of truth is the proof. Frankly, mathematicians don’t care about physical reality. They ask only one type of question: “Assuming that A is true, can we conclude that B is also true?” A proof is an airtight argument, deploying logic to show exactly why B must follow from A. It shows that the fact could be no other way—logic simply would not permit for alternatives.
Thus, proof doesn’t merely support a fact. Proof immortalizes a fact. Proofs, like diamonds, are forever.
Look back 200 years. Much of the science from 1814 has fallen out of favor, supplanted by new and superior ideas—the periodic table, the theory of evolution, general relativity, the modern atom. Even some cutting-edge science from as little as 20 years ago has been discarded as new evidence came to light. (Remember “junk DNA”?) [Edit: Apparently “junk DNA” is still a thing. See the helpful comments section below. My bad!]
By contrast, math never goes stale—not in 20 years, not in 200, not even in 2000. We’re still teaching kids about Euclid’s geometry, a two-millennia-old system that’s as fresh and true as the day Euclid sighed and said, “All done.”
So why do we prove things in math class? For the same reason that we run experiments in science class: because that’s how we pin down the truth.