Topics Mathematics

Question How do we know modern day math is correct? An example would be one is equal to zero point nine repeating. You can divide them both by three, and get point three repeating, but if you times point three repeating by three you can only get point nine repeating... another question could be, where does the rest of it go?

Accepted:
December 17, 2005

Comments

For the answer to the question about 0.999..., see Question 181.

Mathematicians try to ensure the correctness of math by never accepting a mathematical statement as true without a proof. Of course, it's always possible that a mathematician will make a mistake when writing or checking a proof, so even if a mathematician has proven a statement and the proof has been checked by other mathematicians, there is still a small chance that there is a subtle mistake somewhere in the proof. (It has occasionally happened that flawed mathematical proofs have been accepted for years before someone finally spotted the flaw.) So if you're looking for an absolute guarantee of correctness, I don't think you're going to find one.

But even if we ignore the problem of careless errors, there are other questions one could raise about whether or not a proof of a mathematical statement guarantees the correctness of the statement. Usually a proof of one mathematical statement makes use of other mathematical statement, so one could ask how we know that those statements are correct. Of course, they have proofs too, but those proofs depend on other statements, and so on. How does the whole process get started? Most mathematicians today consider the starting point to be the Zermelo-Frankel (ZF) axioms of set theory.

Can we be sure of the correctness of the axioms? Some people regard the axioms as so intuitively clear that it is reasonable to accept them as true without proof. Another approach would be to simply define "correctness" in mathematics to mean "provability from the ZF axioms". But almost everyone would agree that if the axioms are found to be inconsistent, then they can't be correct, so one step toward ensuring the correctness of mathematics might be proving the consistency of the axioms. Early in the 20th century David Hilbert tried to find such a proof in an attempt to establish the correctness of mathematics. Unfortunately, Godel showed that it couldn't be done.

Despite all these worries about the certainty of mathematics, I think it should be said that math does seem to have a higher degree of certainty than any other field of study. Newton's laws of motion are no longer considered to be correct, but Euclid's theorems are still considered correct after more than 2000 years.