Definition of the mathematical proof

Question

How do we define a mathematical proof?

Is it a series of arguments?

Is it a series of conclusions?

Is it manipulation of formulas?

Is it a mixture of laws of logic and axioms,theorems or definitions?

Answer 1

I'm going to try to pitch my answer toward the way (it seems to me, at this point in my development as a mathematician) proof fits into the process of learning and communicating mathematics.

You play around with numbers, shapes, quantities, and other mathematical objects for a while, and you'll start to see patterns, things that look true based on what you've seen.

But how do you know that these patterns aren't just figments of your imagination, or the result of unconscious bias or misinterpretation in your investigations?

To be sure that you're not just misleading yourself, to convince yourself and other mathematicians, you take what you know about the things you're playing with and try to use them to verify that what you think is true, is actually true. So you take the properties of those objects and try to show that they must behave the way you think they behave.

I recommend Bill Thurston's excellent essay "On Proof and Progress in Mathematics." It has had quite an impact on my perception of proof as a tool of the mathematical community.

Answer 2

A proof is simply a process wherein you start with axiomatic or already proven statements that are either known to be true, or taken to be true, and, by means of logical arguments, produce a result.

There are many different methods of proof, such as direct proofs, proof by contradiction, proof by induction etc, but in general, all they consist of is, initially, a set of statements known to be true, a bunch of logical arguments based on these truths, and then some logical conclusion that follows from the argument.

Edit (Example proof). Claim: If $x$ is an even integer, then $x^2$ is an even integer.

Proof. Let $x$ be an even integer, hence $\exists k\in\mathbb{Z} : x=2k$. Now,

\begin{align} x^2&=(2k)^2\\ &=4k^2\\ &=2(2s^2)\\ \Longrightarrow \exists l\in\mathbb{Z} : x^2&=2l. \end{align} Hence $x^2$ is an even integer. $\square$

What we did here was begin with a known rule, namely that an integer is any number that can be expressed in the form $x=2k$ for some integer $k$. Then, after some logical reasoning, we showed that $x^2$ is also an even integer.

Note that when first being taught proofs you would have to justify each step. Most of it follows from basic algebra, but you would need to mention things such as the closure of the integers under multiplication.

Answer 3

I think the best way to define a proof is as follows. This proof is taken from the 1996 documentary on Fermat's last theorem by Simon Singh:

"In mathematics, there is the concept of proving something; of knowing it with absolute certainty, which is called rigorous proof. Rigorous proof is a series of arguments based on logical deductions which build one upon the other, step-by-step until you get to a complete proof. That's what mathematics is about."

For example, Take Euclid's theorem: There are infinitely many prime numbers.

Proof: 1) Suppose, by contradiction that there are FINITELY many primes $p_1,...,p_n.$

2) Multiply all the primes together and add or subtract one, ie, $P=p_1...p_n\pm 1.$

3) Then $P$ is natural number with $P>1,$ but $P$ is not divisible by any of the prime numbers.

4) This is a contradiction since it violates the fundamental theorem of arithmetic. Hence we have infinitely many primes.

What we have done is provide a series of logical steps that build one upon the other until we get a conclusion. In this case, we were looking to violate a well-known results to conclude that our question must be true if we assumed it did not hold. Often, mathematicians may not be sure where there work will lead them. There is a joke that says the difference between mathematicians and philosophers is that philosophy has a goal with no rule, whereas mathematics has rules with no goal.