In what sense are proofs that rely on the Law of the Excluded Middle really proofs?

Question

As a non-mathematician, it seems to me that the Law of Excluded Middle is merely an axiom when it comes to proofs that employ it.

Can we really rely on such proofs? In what sense are they valid?

EDIT

In the light of comments, my point is that yes, all proofs depend on axioms (I believe - I have never proved anything myself!) but, if we want to prove something in a specific field, how can we be certain that a proof is valid?

For example, I understand that there were mistaken proofs concerning factorisation because it was not realised that it was necessary to take into account complex numbers.

Therefore the main part of my question (i.e. In what sense are they valid?) really means, How does one decide whether or not this axiom is relevant in a particular field.

Answer 1

That depends on what you define as "valid".

Most mathematicians work with classical logic, i.e. logic in which the law of excluded middle is an accepted axiom. As David pointed out in the comment, there is also intuitionistic logic, in which the law of excluded middle is not accepted as axiom.

Since most mathematicians work with classical logic, most modern mathematical results are proved with the law of excluded middle assumed as an axiom. Thus, these proofs are "valid" in classical logic. On the other hand, if a proof uses the law of excluded middle somewhere (e.g. proof by contradiction), then it is not a "valid" proof in intuitionistic logic.

In summary, there is no single property determining whether a proof is "valid". Instead, we can only ask if a proof is valid in a certain class of logic.

Answer 2

Answering,

" How does one decide if the law of excluded middle, is relevent to one's field"

There is quite a bit of work, often done by logistician-mathematicians that attempts to show what is provable with the least assumptions possible.

finding the weakest set of axioms needed to prove x.

I would say there isn't a general rule, for knowing if one needs a certain axiom or not. It's case by case.

Law of Excluded middle lets you employ proof by contradiction arguments, which work even when you can't build an example of what you are proving.

So, if there is some theorem saying you can't ever construct x, but you want to prove something about it.... there's a decent chance you will need the Law of Excluded Middle.