And the Dirac delta is the limit of a function with integral 1, not a function.
Maybe you need to hang out with a better class of physicists? There are certainly plenty of mathematicians who believe odds things, but I'd be leery of tarring the whole lot with the same brush.
On the other hand, it really is a useful result. The negative part is, for example, why the Casimir force is attractive.
Here's an article about the whole thing, with links to the various Numberphile videos and Phil Plait articles that have been done about it: http://physicsbuzz.physicscentral.com/2014/01/does-1234-112....
> In the Wikipedia article, they have an equation that looks like this \zeta(-3)=1/120, but the stuff on the left hand side is just another way of writing 1^3+2^3+3^3 +4^3+...
If you can't explain why the above claim is screwy, then every time you claim, "the sum of all positive integers is -1/12," you need to qualify it with "I have no idea why, but I know there's some complicated mathematics that removes the word 'sum' and 'positive integers' and replaces them with more complicated concepts I'm don't have the time to learn about, and we physicists abuse it to say what I just said."
> If QED is correct ... then I would argue that the things that go into QED calculations must be just as correct.
Also not how logic works.
Physics today does have some remarkably talented mathematical thinkers, like Edward Witten, but most of them are involved in very fanciful theories like string theory.
Sure sure, but do know that a lot of all that hacky regularization stuff can be made mathematically sound after all --see for instance [1,2].
[1] http://www.maths.qmul.ac.uk/~tp/talks/casimir.pdf#page55
[2] http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin...