Since natural languages (e.g. English) are prone to ambiguities and misunderstandings due to their constant evolving nature and lack of rigorous formalization, and given an arbitrary philosopher X who wants to show the validity of his argument, would it be possible for that philosopher to translate his argument into a formal, non-ambiguous mathematical proof written in some kind of formal mathematical language in order to convincingly and indisputably show everyone else his argument's correctness?
What you describe is the general structure of a mathematical paper. Words regarding the mathematical discovery are paired with precise mathematical notations to demonstrate correctness. However, a general version of this is tricky. If a philosopher says "Here's natural argument A and here's formal argument B. As the author of both, I say they are intended to be the same" that's one thing. However, if we try to say "here's a process that lets you turn the crank on A to produce B," we have to question whether the process works right.
Indeed, proofs by folk like Tarski showed very particular limits on processes like you describe. It's very hard for a language to prove it's own semantics, and most of the languages we are interested in when talking about proofs are simply not capable of it.
Of course, there is a way to do this. What defines a "correct logical reasoning in natural language?" If the definition is "there exists a corresponding formal logical proof," then the answer to your question becomes yes!
You might be interested in looking at Attempo Controlled English (ACE). ACE does the dual of what you seek. It is a way to write precise formal logic in a format which native speakers of English can read as though it were natural English. If you read it naturally, you get the right intuition. If you read if formally, you get the right formal meaning.
Hypothetically, yes. For any non-trivial argument though, the logic you’d employ would be much more complicated than the textbook logic you learn in a first course in formal logic.
Most important philosophical arguments will employ concepts like conceivability, parthood, necessity, causation, and so forth. To formalize your argument, you have to give a precise logical definition of how these concepts are to be used.
But that is only the first step. What one has “proven” by formalizing such an argument is syntactic, this argument is well formed, the conclusion follows from the premises. Whether the concepts one has formalized accurately capture reality is a semantic question—and that’s where the real philosophy happens. Put a different way: it is easy to invent concepts and show how they can be combined and what follows from those combinations. The secret sauce is figuring out whether those concepts describe reality.
In my opinion, formalizing an the argument can be done — and it can even be done straightforwardly.
Recall, however, that an argument consists of
- A list of hypotheses
- A conclusion
- A proof leading from the hypotheses to the conclusion
In the (claimed) straightforward process of formalizing an argument to make it completely rigorous, what is going to happen is that all of the disputable parts are going to appear in the "list of hypotheses" section, rather than the "rigorous proof" section.
So, what's going to happen is that you will have an argument that is convincingly and indisputably valid, but you haven't really gained much — you now have the problem of convincing people that the list of hypotheses are satisfied by something someone cares about.
Mathematics is somewhat unusual in that there is a wealth of notions for which you can write down a list of hypotheses that will generally not be disputed, but yet are complete and precise enough to enable formal proof of interesting conclusions.
See Stephen E. Toulmin’s The Uses of Argument. He claims that the standards for valid arguments are field specific and arguments in some fields need not be reduced to analytic arguments to have validity.
In particular he writes (page 235):
What has to be recognised first is that validity is an intra-field, not an inter-field notion. Arguments within any field can be judged by standards appropriate within that field, and some will fall short; but it must be expected that the standards will be field-dependent, and that the merits to be demanded of an argument in one field will be found to be absent (in the nature of things) from entirely meritorious arguments in another.
Should one get a position that validity of substantive arguments in all fields can “be translated into a formal mathematical proof”, that is, that all substantive arguments can be written as analytic arguments with entailment justifying validity then one should test that position against what Toulimin offers to counter it.
if it is possible to translate an ordinary language statement into mathematics then it must have been precise in the first place. This was the flaw in the idea of translating philosophical statements into symbolic logic and the reason Russell's attempt didn't prove useful. Where it's possible it isn't necessary. One cannot make a statement in English more clear by translating it into French.