I have a question concerning a proof that two negatives makes a positive :
Here is the outline of this proof: Let us prove first that 3 . (-5)= -15. What is -15? It is a number opposite to 15, that is, a number that produces zero when added to 15. So we must prove that 3 . (-5) + 15=0

Indeed, 3 . (-5) + 15= 3 . (-5) +3 .5 = 3. (-5+5)=3.0=0

(When taking 3 out of the parentheses we use the law ab+ac=a(b+c) for a=3, b=-5, c=5;we assume that it is true for all numbers, including negative ones.) So 3.(-5)=-15

....
One thing which I do not follow is, why the need to prove that -15 is really -15 ? We know that, (-5)+(-5)+(-5)=-15

Logically, I don't see what else it could be But in the above text, are we really trying to show that the number -15 is really -15 by adding it to positive 15 ?So if it really gives 0, we are sure to be in presence of negative fifteen(3.-5=-15) ? Or is there anything that I've missed about it ? ANy help would appreciated ! Thank you.

This all depends on which axioms and number system you work with.

You seem to be under the impression that ##3\cdot (-5)## is defined as adding ##-5## three times. This is fine when working with natural numbers, but this definition breaks down pretty fast. For example, how would you interpret

$$\pi\times \sqrt{2}$$

There is no natural interpretation here as addition.

Usually, in the real numbers, addition and multiplication are introduced separately. Multiplication is not defined as addition.

It would help us if you would mention what axioms you're working with or how addition and multiplication are defined.
Furthermore, given a number ##a##, how did you define ##-a##?

Well, there is no real definition given, it's just an explanation given out by IM Gelfand in his Algebra. Here's the rest :
(The careful reader will ask why 3.0=0. To tell you the truth, this step of the proof is omitted- as well as the whole discussion of what zero is.)

(-3)+3=0

and multiply both sides of this equality by -5 :

((-3)+3).(-5)=0.(-5)=0
Now removing the parentheses in the left-hand side we get

(-3).(-5)+3.(-5)=0,
that is, (-3).(-5)+(-15)=0. Therefore, the number (-3).(-5) is opposite to -15, that is, is equal to 15. (This argument also has gaps. We should prove first that 0.(-5)=0 and that there is only one number opposite tp -15.)
But about what I was saying that 3.-5=-15
He wanted to prove that -15 was really the answer ? Didn't he ? The reason why I ask this is because it's the first time that I see someone trying to prove -15 as the answer.

Yes, he wanted to prove -15 is the answer. But it's a bit a failed attempt since he doesn't really state what he's starting from. I know what he means, but I do realize it looks strange and unmotivated for you.

His point is that there is no a priori reason why a negative times a negative should produce a positive. Teenagers memorize this as a rule, but it really is quite arbitrary. If we wanted to make a mathematics where negative times negative is negative, then we could. Gelfand tries to explain why we don't take this convention.

Yes, he wanted to prove -15 is the answer. But it's a bit a failed attempt since he doesn't really state what he's starting from. I know what he means, but I do realize it looks strange and unmotivated for you.

His point is that there is no a priori reason why a negative times a negative should produce a positive. Teenagers memorize this as a rule, but it really is quite arbitrary. If we wanted to make a mathematics where negative times negative is negative, then we could. Gelfand tries to explain why we don't take this convention.

Ah ok, thank you for your answer. I just found it weird, because like you said(I only have high school education), I've never questionned myself for trying to prove that 3.-5=-15. It just distabilized me a bit xD
Thank you again.