What is the difference between a formula and a proof?

As I understand it, a formula is a method for solving a mathematical problem expressed using alpha numeric characters like the quadratic formula is a method for solving quadratic equations when factoring will not work. I understand a proof to be a logical argument that may or may not produce a formula, but will produce a statement that something is true or false mathematically.

Take for example: $$x^2+13x+22=0$$ This equation will not factor, so we would use: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ to solve for x.

However, does the fact that this formula always allows you to solve for x constitute a proof or "scientific proof" that this formula speaks to mathematical and thus scientific truth? Or, must a formula have a proof in order to really be considered valid scientifically?

• Formulas are typically proven. One proves the general case, e.g. the quadratic formula produces solutions to $ax^2 + bx + c = 0$ regardless of choice of coefficients. – Kaj Hansen Oct 26 '16 at 2:04
• But are formulas and proofs essentially the same thing? – Mr. Concolato Oct 26 '16 at 2:05
• the proof proves that the formula works – janmarqz Oct 26 '16 at 2:07
• Incidentally, it looks like you're conflating scientific and mathematical reasoning in a context where they really shouldn't be. Theories in science are built from limited and imperfect evidence - while that's true of mathematics as well, to some degree, in math we tend to have the goal of replacing that evidence with solid proof, while in science that's really not the case. (And at this point I should stop philosophizing about science, since I'm not a practitioner - but there are plenty of texts on the philosophical differences between science and mathematics!) – Noah Schweber Oct 26 '16 at 2:31
• @NoahSchweber I know. That was deliberate so that those that think that mathematical proof and scientific proof are the same can read your comment above. Thank you for that. Have you completed your PhD? – Mr. Concolato Oct 26 '16 at 2:35

The only way that I know a formula works is via a proof. I might be able to verify by hand that it works for a number of cases, but that doesn't mean it always works; see here and here for some examples of this. This is not to say that experimental evidence is worthless - quite the contrary. But the special role of a proof is something which cannot be ignored.

Now, there are subtleties here. In order to prove something, I need to begin with axioms. What axioms are "acceptable?" The standard axiomatic foundation of mathematics is ZFC (but see here), but there are some "concrete" problems which can't be proved using these axioms alone (see e.g. here). The existence of such problems - and Goedel's theorem more generally - shows that ultimately, the notion of "proof" is more nuanced than we might think at first. For example, there could be a formula that always "works" for a given concrete problem, yet can't be proved to always work inside ZFC.

However, this situation tends to be the exception rather than the norm. And the answer to your question is no - formulas and proofs are quite different!

• This is helpful, thanks. – Mr. Concolato Oct 26 '16 at 2:20

In general, a proof is what leads to the formula. While on occasion, people are able to come up with an equation out of thin air, most of the time formulas are carefully derived through a proof.

Specific to your questions: Just because it works does not make it a proof, but lack of a proof does not make it more or less true. However, it is only accepted within the math community as truth with a proof.

• I appreciate the answer, but I can't help but wonder if a formula comes out of a series of experiments in which a consistent method appears to produce the correct solution to a problem. One then tries to see if they can prove that the observed method will always produce the correct answer. Am I wrong on this? – Mr. Concolato Oct 26 '16 at 2:16
• @Mr.Concolato You're right in general, but I don't see how that's relevant to your claim that "formulas and proofs [are] essentially the same thing". (And in fact I don't believe you're always right: many formulas are found via their proofs, that is, via derivations - the quadratic, cubic, and quartic formulas, I believe, fall into this category. – Noah Schweber Oct 26 '16 at 2:19
• @NoahSchweber I did not make that claim. I asked that question above in a comment. I am sensing that one can confidently say that proofs and formulas are not the same thing. – Mr. Concolato Oct 26 '16 at 2:23
• @Mr.Concolato Ah, I misunderstood. In that case: yes, one can. They are definitely not the same thing. – Noah Schweber Oct 26 '16 at 2:27