how do we know for sure that our mathematics is right?

Question

do we know with certainty that it is right? does our mathematics fail in some instances?

Answer 1

How do you know that you are really alive, and not just a thought in someone else's dream?

Answer 2

Hehehehehehe.

We do not, actually. David Hilbert had delivered his Decidability lectures when a puny little Austrian, probably the greatest logician who ever lived, published his Theorems of Incompleteness. It was Kurt Godel.

All of mathematics works on a few assumptions that are taken without question to be right. They're axioms; they can neither be proved nor disproved. For eg. Parallel lines meet at Infinity. What the hell is infinity? Will they really meet? How do we know for sure? How can we confirm? Can't, right? And therein lies the vulnerability of mathematics and thus its incompleteness.

First theorem - For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Second Theorem - For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Simply put, the first theorem says, that we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods. But the thing that jolted mathematicians out of their coats, is the rephrasing of the second theorem which has deep consequences when you define the limits of human knowledge - If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

It made John von Neumann, probably the greatest mathematician ever, to remark that his career in mathematical logic had come to a precipitous but merciful end.

Answer 3

First, I'd like to point out that I believe people tend to misuse Godel's Incompleteness Theorems. The way I understand it, all it says is that no logical system can "prove itself." But we wouldn't want it to, otherwise it would involve some circular arguments, which we always want to avoid.

Here's an example that I remember thinking about as a child:

Someone once told me that anything is possible. I then reasoned that if ANYTHING were possible, it would be possible for something to be impossible. This is clearly a contradiction.

All logical systems (mathematics, as an example) are based on a set of assumptions, or axioms. All proofs that follow proper logical arguments starting from those axioms are correct.

So we don't know "for sure" that mathematics is "right;" but we do know that it logically follows from a set of basic axioms, so it is correct with respect to those axioms.

Also, I think you'll notice 2 basic types of answers here. Some talk about mathematics in its true form: in terms of proofs and logic. Others are referring to the game of arithmetic, application to observable situations, and approximation. These are 2 very different things.

Answer 4

Mathematics are tools to solve certain problems.

Is a hammer right? If you want to bang nails, sure. if you need to remove a screw, not the best. If you need to apply paint on a wall, the hammer may fail.

Mathematics will fail if you are trying to make them do things they can't do.

However, in most instances, the tool should not be blamed. It's the user.

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If you are using real numbers and you need to find the square root of a negative number, mathematics will fail.

However, if you use a different tool (complex numbers), then they might work.

Answer 5

This is a great question.
Math is based on assumptions (called axioms)
and logical reasoning. So, the answer is that
a mathematical statement is true only if the
assumptions are true and we have no errors
in our reasoning. There are no absolutes here,
it's all based on assumptions and logical reasoning.

I have a ba in math and taught HS math.
But, do NOT believe what I said unless
it makes sense to you.

Answer 6

Who decides right or wrong when it comes to math? Your teacher? Engineers? Einstein? Maybe... but not really. Over thousands and thousands of years, humans have found a way to do what we call "math". This method works for what we need it to work for. That's all that matters. Let's say that some Alien species uses letters instead of numbers, and divides the way we add. Does that make them "wrong"? No. They'd think that we'd be "wrong" also. The letters work for them, so that's all that matters. As long as there is some way to compute things that need to be computed -- it's "right".

Am I "right"... or am I "wrong"? ;)

By the way, I made up the thing about the aliens (didn't I?).

Answer 7

We know because we write proofs for our theorems, and publish them in the literature.
Even on the undergraduate level, if the instructor in a Calculus Class writes some balony on the blackboard, some of his students will point out his error.

Answer 8

beign a mathematician and a teacher of it- everythign in math is logical and therefore requires a proof.
there are cases when we know things tend to something wth a 99% certainty. and other things that are completely full proof. without a proof you cant say anything- i wont believe it.

Answer 9

Use math to predict physical properties or what will happen. Then verify the prediction. This has shown our math and predictions to be good.

Example: How long will it take a rock to fall from 30 feet before it hits the ground? Calculate the time and then measure it. It works out!

Answer 10

do v have ne option other than believing wat v have been taught?