Hello,

I just registered to this forum to ask a question I couldn't get answered elsewhere.

Let me sketch the situation: I'm a dad, I'm an experienced physicist with quite some affinity for mathematics, but I was baffled recently by something that happened to my son, and I wanted to find out whether there was a fundamental piece of elementary knowledge about maths that I succeeded in missing for several decades, or whether it is something cultural, or whether the situation is simply weird.

I went to a teacher's forum in the country, but essentially I got insulted away from it, because I dared questioning a practice, and I don't accept an argument of authority as an explanation - but I'm totally open to any argument that holds water.

The question is this: "is there a good reason to sanction the writing down symbolically, of expressions, that turn out not to correspond to mathematically existing objects, during the phase of exploration of, exactly, their existence ? Or is this a generalized cultural thing in mathematics ? Or is this just a local quirk ?"

Level: towards end of high school.

Let me explain. My son had to find out whether a given function had a derivative in a given point a.

He started out writing: f'(a) = lim_{h->0} (f(a+h) - f(a))/h = ..... to find the value of the limit, concluding that given that the limit exists and was equal to something, f was derivable in a, and its derivative was equal to the number he found.

This was barred. He had to first work out (f(a+h) - f(a))/h, show that this function existed, only then be allowed to write lim before it, and in the end, conclude that given that the limit existed, that f was derivable, and write down f'(a).

Now, I have no souvenirs of ever having done things "backwards" that way. It seems to be forbidden, to write f'(a) if one isn't yet sure that it exists. But that is, to me, a strange thing to do, if the question to be answered is, exactly, to find out whether it exists.

I don't see the problem in writing f'(a) = lim .... ; to eventually conclude that, if ever the limit doesn't exist, f'(a) doesn't exist, or, as was the case, if the limit exists, that this is the f'(a) looked for. I always did so.

Of course, as long as an object isn't proved to be existing, one cannot USE it, but I didn't know it was FORBIDDEN to write it down. Simply, because if it is forbidden to even be written down, how do you even write down the question symbolically ?

So, is this generalized practice in maths to forbid writing down the symbolic expression of objects that might not exist (NOT calculate with them !) in the phase of exploration of their existence ?