Hmm. Great.
You have not studied philosophy Poly. One doesn't have to study everything, but one has to have humility to know that everyone is not lying and making things up.
Actually, I *have* studied philosophy. Don't make assumptions.
If you think people are making up things like analytical truths and mathematical truths in philosophy right now in this thread, it could be that you just don't know much about it.
No, I have studied this and I know that there is a wide range of definitions for these concepts. And different definitions lead to different conclusions. That is why I ask for the specific definitions you are using.
Saying these are new terms in philosophy you have never heard of is unbelievable. Unless you have not studied anything in philosophy.
When I have I said these are new terms in philosophy or that I have not heard of them? What I said is that modern discoveries show that the old definitions are no longer tenable. I also know that most of these concepts have more than one definition in the literature. Often different definitions lead to very different statements. For example, Plato's understanding of mathematical truth is quite different than Benacerraf's, which is different than Decartes', etc.
Are you of the opinion that all of philosophy has agreed upon the definition of 'mathematical truth'?
In mathematics one could make a proposition that could be argued for, you know this, since you have a Phd in maths. Someone arguing for a parabolic with a compact set or something in order to come with a formula, that's not a mathematical truth in philosophy. A mathematical truth is a logical deduction that has to be and will be true in all possible worlds. I think I gave you an example and you made a bizarre argument just for the sake of making an argument about it being reliant upon an axiom. It's absurd. It's not reliant upon an axiom which is just a cooked up nonsense, it's just simple logic that anyone can relate to.
And this is simply false. Yes, it depends on the specific axioms of math (and of logic) you choose to use. Logic alone does not and cannot reproduce mathematics, in spite of Russell and Whitehead's attempts.
For example, even to define the number 2 requires going beyond mere logic. Addition is a bit beyond that. So even to write the claim that 2+2=4 is going farther than logic alone. To say that x+y=y+x for all natural numbers x and y is way beyond simple logic.
In particular, to even define the concept of 2, you need *at least* to have some initial number (usually 0 or 1) and the concept of successor. The latter involves functions, which usually means some sort of set theory. To define and use addition requires some sort of ability to define operations recursively. None of these are trivial assumptions.
But you will argue against it. You will argue against anything. If someone says I am alive will just argue about it for whatever reason.
I argue against it because I know it to be false.
If you are a formalist, you should be able to understand at least the basics of formal logic. It's very basic, not like solving some major mathematical problem. You can't even understand the most basic logic. Either that or you just wish to argue about it.
I assure you that I understand formal logic, almost certainly in more depth than you do.
But, for example, do you understand that there are alternative versions of formal logic in which the law of non-contradiction is false? or where the law of excluded middle is false?
Tell me. How could you breathing because you are human and all humans breathe be wrong because an axiom could structurally change? It's nonsensical. Why not actually read up on these terms rather than making such arguments?
It is an observation that I am a human and a very complicated set of observations along with induction to say that all humans breathe.
But yes, I usually adopt the *axiom* that
For all x, P(x) implies that P(a) where a is any term.
You do understand that this is either an axiom or a result proved from some axioms, right?
Maybe *you* are the one that should be reading up on these things. Might I suggest Kunen's book, The Foundations of Mathematics? you might just learn a few things if you read this *basic* book.
You did not present a single philosophical argument though asked several times while claiming you had studied it all. All you have done is like a cut and paste say the same thing from a mathematical point of view. Is that even related to Descartes thoughts on infinity? It's a completely different category. You are presenting a strawman. 