I also have a philosophical objection to this, which is that it is completely false. Mathematics has a very strict axiomatic foundation, but has no logical foundation. There is no ontology there to support or refute (even mathematically) the proposition that one equals zero.
I suspect that this confusion arises from this fact: the word “prove” has two distinct meanings. In mathematics, a mathematical proposition, or axiom is (1) true, (2) not a logical tautology, and (3) can be proven. A mathematical proof, on the other hand, is a proof of a mathematical theorem. For example: “1+1=2” is a mathematical theorem, but cannot be proven in terms of other mathematical propositions.
In the philosophical literature, it is common to use the word “prove” in the sense of “establish” or “demonstrate”. It’s not too hard to see how, if one accepts this definition, one might get confused.
This would imply that if we prove a theorem, then it is true, and that a proof is a way of getting at the truth.
I’m not so sure that’s what mathematical logic suggests. For example, the logic that underlies arithmetic can’t tell you that the real numbers are uncountably infinite. Similarly, you can’t talk about the truth value of a logical truth in its “lone existentiality” or “truth” – as you know, a logical truth is something that can be “logically deduced”. (I guess the usual way of talking about it is that a “tautology” is a logical truth which cannot be “deduced”.)
You’re quite right to point out that there is a sense in which we “know” mathematical facts – for example, we “know” that there are infinitely many primes, and that every arithmetic truth can be proven. But these truths don’t correspond to any truths about the physical world. This point is quite important; we shouldn’t think that mathematical truth is just analogous to physical truth.
As for why these mathematical truths are independent of physical evidence, that’s quite simple. The reason we can use mathematics to model the natural world is because we can find mathematical patterns in it. Thus, a mathematical theory that doesn’t fit the data will tend to be modified or thrown out. The theory of General Relativity, for example, was put forward on a much more primitive basis than we now have, because we know how to apply it to the actual evidence (of course, at the same time, we also know how to apply it to the evidence, but we didn’t yet know that). Thus, our mathematical theories of the universe aren’t, in the first instance, founded on empirical evidence.
The most interesting philosophical question about the existence of mathematical objects is the question of how they relate to physical objects. One idea, called “naïve realism” or “physicalism”, is that there are non-mathematical, non-abstract physical objects that are identical with the mathematical ones. Another, called “conceptualism”, holds that the concepts themselves are physical and non-mathematical, and the mathematical objects correspond to the concepts. Most people think that naïve realism is false, but there are arguments for it, based on analogy.