What are the main tactics in proofs of theorems?

Question

Can You please help me to answer the following questions. I never was thinking about such questions since they are not in the core of mathematics. However, from the educational point of view they are very important. I have my own answers, but they don't look exhaustive for me.

1) Why in the proof of a theorem we must use only the facts which are given in the conditions of this theorem?

2) Why in the proof of the theorem we should always understand what do you want to have in the end and sometimes make the proof going from this end?

This questions look to obvious to answer seriously. However, for students they are not really obvious, what make all possible answers interesting.

Answer 1

The basic idea behind 1) is indeed pretty obvious: don't assume that something is true if you don't know it is true!

A good example are logic puzzles. So here you are given pieces of information like "Bob's shirt is red" or "Sara's shoes are white". Clearly, in a context like this we cannot assume that "John's socks are blue" if that's not given to us. Only work with what's given to you.

Another clear example is formal proofs: here, you are very explicitly given certain premises, and clearly you cannot ad any further premises to that.

In practice, though, it is not always clear what we can assume or not. In fact, look at most of the questions posted here on Math.SE. Often they are something like "Prove that for any natural number $n$ it is true that $n^2+n+3$ is odd". Well, exactly what are we allowed to use as givens here? Can we assume that every natural number is even or odd? Or do we have to go all the way back some Peano axioms? Or ...? Still, even in these cases you common sense and experience will give you a pretty good idea what you can or cannot assume. That every number is even either or odd seems pretty safe. But that $n^2+n+1$ is odd probably not. And you certainly can;t start with: "Let $n$ be some natural number. $n$ is odd. ..."

Regarding 2): I would put this idea even stronger. It is not just that sometimes we 'make the proof go' from the conclusion backwards, but that we do this pretty much every time! That is, the nature of the conclusion will tell us whether to set up the proof as a universal proof, a conditional proof, a proof by contradiction, a proof by cases, a proof by induction, etc. And those proof techniques are what is going to provide the all important 'proof plan' or 'proof skeleton': it provides the structure of the proof to which you then add the details.

Answer 2

Proofs have different purposes in different places in mathematics. When you're publishing in a research journal you must convince your readers that your claims are true, using the hypotheses in your theorem and any other tools it's reasonable to expect professional mathematicians in your area to know and accept.

When I teach students about writing proofs I know the theorems are true, so they are not trying to convince me. What I want from them is an argument that shows me that they have convinced themselves of the truth for good reasons. Those reasons should include the hypotheses, of course, and any other tools taught in class, or accepted at the level of the course.

Often rather than asking students to prove something, I ask them either to prove it or provide a counterexample, in hopes that will persuade them to think about the mathematical truth rather than the particular shape of the argument.

Answer 3

Regarding your first question $(1)$, sometimes that is not the case. For instance, Euclid proved that there are infinitely many values $a, b, c$ such that $a^2 + b^2 = c^2$. This has been a major theorem, and we can prove it a different way from an approach that is not mentioned in Euclid's proof or in the theorem itself.

We know that for a value $x$, $$(x+1)^2 = x^2 + 1 + 2x;$$ $$(x-1)^2 = x^2 + 1 - 2x.$$ Therefore, $$\begin{align} (x+1)^2(x-1)^2 &= (x^2 + 1 + 2x)(x^2 + 1 - 2x) \\ \Leftrightarrow \big((x+1)(x-1)\big)^2 &= (x^2 + 1)^2 - (2x)^2 \\ \Leftrightarrow (x^2 - 1)^2 + (2x)^2 &= (x^2 + 1)^2.\end{align}$$ Substitute $a = x^2 - 1$, $b = 2x$, $c = x^2 + 1$ and we have proven the theorem. $\qquad\qquad\qquad\quad\,\,\,\Box$

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