3

I know, proof is the most crucial part of mathematics, it makes all the things be rigorous and keeps mathematics from contradiction.

In real life, there's things that we know to be true, for sure. Such as in physics, objects with opposite charges attract each other, or in biology, human can't live without oxygen or water. Of course, these are based on experimental result.

What I'm thinking is that, is there possibly something outside (or above) mathematics, that can tell us that "..." is true, in a way very different from normal mathematical proof, and we can totally believe this. I know this sounds unscientific, like religion, but seems that science haven't prohibited this to happen. Can there actually be another definition of "True" in mathematics, that we can know something to be true without really proving it?

Git Gud
  • 31,356
JSCB
  • 13,456
  • 15
  • 59
  • 123
  • This raises the epistemological question: what does it mean to know something? – Git Gud Jan 05 '14 at 12:30
  • Maybe something like "I know I exist" is the one thing I know for sure, but I'm not sure anything qualify as a proof for this certainty. – Denis Jan 05 '14 at 12:48
  • In mathematics, the only things whose truthfulness is assumed without any proof are the axioms. For instance, we might know that the Euler-Mascheroni constant $\gamma$ is irrational and transcendental without actually being able to prove it, but knowledge and proof are two different things. Obviously, proofs are a very small and restricted subset of explanations, and it is well-known that not everything we know can even be explained in the first place - let alone proven! – Lucian Jan 05 '14 at 12:51
  • @Lucian That's not exactly true. A realist thinks every statement is either true or false. For instance the $\sf CH$ is either true or false for a realist, even though it's independent from $\sf ZFC$. The realist argues that $\sf ZFC$ doesn't have enough axioms. Gödel defended this position in What is Cantor's Continuum Problem. – Git Gud Jan 05 '14 at 12:56
  • @GitGud: I don't recall saying that the number of axioms is finite, or that it has to be finite, or that it can even be so in the first place; nor did I say that all things must be either true or false (Russell's paradox, anyone ?). – Lucian Jan 05 '14 at 13:01
  • @Lucian I don't know why you're mentioning finitess of axioms, I didn't bring that up. You're stating the only thing mathematicians assume to be true is the axioms and I'm saying that's not correct, a realist sees it differently. – Git Gud Jan 05 '14 at 13:05
  • @GitGud: I'm afraid I have no idea what you're talking about. The reason I mentioned the finiteness of axioms is because you said that The realist argues that ZFC doesn't have enough axioms. – Lucian Jan 05 '14 at 13:20
  • @Lucian ZFC has an infinite number of axioms. When someone says it doesn't have enough axioms, he doesn't mean that the number of axioms is too small, he means that some essential axioms are missing. Like, before zero and negative numbers were invented, there were not enough integers, although they had $\aleph_0$ integers then just as we do today. – bof Jan 05 '14 at 13:31
  • @bof: Can this infinite number of axioms be grouped into a finite number of classes, families, or categories ? – Lucian Jan 05 '14 at 13:38
  • @Lucian Yep. Using things called axiom schemes (or schemas, or schemata, I'm not sure) which contain an infinite number of similar axioms. – bof Jan 05 '14 at 13:45
  • @bof: That's what I meant by infinite. That there's the possibility that we might never be able to restrict them in any way; e.g., by a generating function of all necessary and sufficient (groups and sets of [other generating functions of]) axioms. – Lucian Jan 05 '14 at 13:53

1 Answers1

4

I am personally of the opinion that the claim $2 + 2 = 4$ is a scientific conclusion of repeated experiment, to do with combining collections of size 2 and observing the size of the result. In particular, if I'm building a foundations for mathematics and after much effort I devise an axiomatic system that proves $2 + 2 = 4$, it is completely absurd to say “oh, thank goodness, it is true after all”.

Such a proof, I claim, is purely and only a proof of the adequacy of your logical system for proving true facts, which after all you'd very much hope a logical system could do. Imagine what would happen if you devised a system with axioms that you believed were true, and after all it happened that in this system $2 + 2 = 5$. You would certainly not revise your opinion of arithmetic! You'd re-inspect your axioms, or your rules of inference, or the whole idea of logical foundations sooner than you'd decide $2 + 2$ was anything other than $4$.

So, yes, I claim that there are certainly truths that precede any proof. They are perhaps not systematic or easily-identifiable, but that is after all the whole purpose of formal logic and proof, to introduce some order to the mess.

Ben Millwood
  • 14,211