logic, negation, set theory, algebra I spoke to a physicist/mathematician, about the sign '~'. He said that it is a sign for equivalence in mathematics, and I mentioned that it is often used to express negation in logic. He replied "Mathematics does not negate. Only metamaths." Not being a mathematican I did not know how to respond, but it seems to me obvious that negation is used by mathematicians, and outside metamathematics (existence denials of the form "there are no numbers such that..."). I said this to him, and he replied "that's not operational negation. They use the E cross (?). That's not an operator though, as negation in boolean algebra. But if you meant that, ok, fair enough." He went on to say "There's a clear difference between what mathematics studies and the study of mathematics from the point of view of Logic. I agree with you that maths can be legitimately studied by using logical methods. I disagree about Maths having the attributes of Logic. If that were so, they'd be the same science." My question is, is it true that as he said, "mathematics does not negate" and that apparent uses of negation in mathematics do not express negation as it is understood in logic? Does this conclusion change if we suppose a sharp distinction between logic and maths?
 A: First of all, logic and set theory are part of mathematics. In fact a large part of "meta mathematics" is actually just mathematics. It's just that instead of talking about integrals and linear operators, we talk about formulas and theories and proofs. But we do it just as mathematically as those people who solve partial differential equations do.
Secondly, $\sim$ is a symbol. That's it. It can be used for anything you'd like. It is, however, common to denote with $\sim$ an equivalence relation. But as you said, we sometimes use it to denote negation in propositional logic, Boolean algebras and so on. However we also use $\lnot$ for negation in logic, and $-$ for the negation in Boolean algebra.
Finally, we negate in mathematics. Often. Just writing $-1$ is negating. This is not subtracting because it's not an equation, there's no quantity from which we subtract. We negate the value of $1$. And as I said, set theory and logic are an integral part of mathematics (and yes, logic has other parts which are... less mathematical). The difference between set theory and analysis is just about the same as the difference between category theory and algebraic geometry and partial differential equations. Those are different fields of mathematics, and they look different. But the underlying reasoning, the requirement for rigor, and writing proofs... those are all pretty much the same all across the board.
A: Perhaps by "E cross" he meant "$\not\exists$" as in the sentence "$\not\exists x\,P(x)$" meaning that there is no object $x$ with the property $P$.
As a logician I would think of $\not\exists x \, P(x)$ as being shorthand for $\neg \exists x\, P(x)$ where "$\neg$" denotes negation.  I would argue that the use of either notation, or even the use of natural language statements such as "it is not the case that there is an $x$ with property $P$" or simply "no $x$ has property $P$" all use the concept of negation.  Because these statements appear in mathematics all the time, I don't think it's correct to say that there is no negation in mathematics (even if we don't count mathematical logic as part of mathematics.)
It seems to me that the use of $\neg$ or $\sim$ or $-$ as an operation (by which he probably meant an operation on Boolean algebras) is a separate issue.  Certainly the study of Boolean algebras belongs to mathematical logic, which it sounds like he does not consider to be part of mathematics.  But the negation operation (more usually called complementation) on Boolean algebras is not the main meaning of negation, nor even the most fundamental meaning of negation, so it seems like a red herring as far as the question of "is there negation in mathematics" is concerned.
