# Proving vs. explaining in mathematics

Imagine a high school level maths exam in which students are
requested to solve the following simple equation: *x*^{2} -
3*x* + 2 = 0. How would you grade the two following
answers:

- We proved in class that such an equation has at most two solutions.
[A few lines of maths in which the student computed them.] Therefore,
the solutions to this equation are
*x*= 1 and*x*= 2. - We proved in class that such an equation has at most two solutions.
We check that 1
^{2}- 3×1 + 2 = 0 and 2^{2}- 3×2 + 2 = 0. Therefore, the solutions to this equation are*x*= 1 and*x*= 2.

Many teachers dislike the second solution, because they think that the student must have cheated. Indeed, the solution seems highly suspicious, because we cannot guess where the solutions come from. (Alright, those were easy to find, but you get the idea.) Notice however that, from a logical standpoint, the second solution is perfectly valid: it justifies the fact that they exists at most two solutions, and explicits them in a perfectly rigorous fashion.

[Interestingly, I had at least one maths teacher who preferred solutions of the second type, because he only wanted us to prove that our answers were correct, and believed that the process leading to the solution was none of his business.]

The point I would like to draw attention to is the following: whereas
both solutions *prove* what they state, only the first one
*explains* where it comes from. Of course, there are more complex
examples of this distinction: while some math papers, books and teachers
make a great job of explaining things, others just prove theorems
mechanically without any effort to show what's going on behind the
scenes.

This distinction isn't limited to numerical solutions to equations either. Even with no equations involved, there are proofs in which you can see that things work out as planned but can't understand why they do; proofs in which you build ridiculously complicated concepts with which you unexpectedly manage to prove simpler claims; proofs in which, to put it simply, the author writes things but does not explain what gave him the idea to do things this way. Proofs which, in a way, look like a program with no comments.

The important fundamental difference between explaining and proving
is, in my opinion, the following. As far as proving is concerned, you
can, theoretically, write proofs which are undoubtedly correct, using
only the axioms and core deduction rules. Of course, you never do that
in practice, but it means that you *could* reach perfection if
you wanted to. However, when you explain things, you have no reason to
believe that it is possible to write something which is undoubtedly
understandable. To do this, you would need to make each step seem
natural, that is, not only do things, but also explain, at each step,
what gave you the idea to do so, and why it gave you such an idea, and
so on, and so forth. [Philosophically, the idea is that you cannot
understand well enough what's going on in your head when you think to
explain *why* you are thinking in this way...]

Keep in mind that this issue of writing proofs which explain things
in addition to proving them should not be confused with the well-known
problem of writing proofs with the right level of detail, ie. the
tantalizing fact that when you prove something, justifying every step is
infeasible in practice but just writing "Trivial from the axioms." isn't
acceptable either so you have to find a compromise between these two
extremes. The issue I am dealing with has nothing to do with this: the
second answer in the example above could formally justify every logical
deduction (even going back to the axioms) and *still* manage to
cook up arbitrary values for *x* and *y* without
explaining how they were found.

It is quite sad that many math books and courses seem to be written for machines rather than humans in that they prove things but don't really try to explain what they do. In my opinion, things would be better if the two were present and clearly distinguished, with a machine-readable formal proof, and an informal discussion explaining how the proof is built and why it is built that way.