Math thread.

Ask about:

analysis

calculus

topology

graph theory

game theory

abstract algebra (math over fields of functions or other weird objects that aren't normally thought of as numbers.)

vector spaces

predicate logic

lambda calculus

regular expressions

computability

set theory <-- actually probably the best starting point for learning math, even easier than algebra

number theory

I know a little about fractals and chaos theory, and next to nothing about knot theory, but I might be able to help with articles intended for a lay audience about them.

Stats are useful and therefore not real math, and outside the scope of this thread.

I swear to god do not even talk about quantum physics. Quantum physics math is bullshit. I don't mean the goofy physical interpretations that imply magic not-particles or whatever. I mean physicists make up axioms as they go along. "Okay, we have a raising and lowering operator. But we must have a bottom rung of our energy ladder or else we'd get particles with negative energy, and that would be *silly.* So a step down from the bottom step must have a non-normalizeable square integral, which means it [lower of]a_{0} is the constant zero function. Working backwards we can derive all the permissible energy states... yes GA? Well, *mathematically* yes, there are a lot of other functions that aren't normalizably square integrable but those don't happen in physics. Now, we return to our square well, where we have PE(x) = 0 for x in [0, 1] and PE(x) = Infinity for x not in [0, 1]...."

And then the very next chapter they are suddenly completely okay with a particle having negative spin, but negative energy is so preposterous that they can't even think about it. But they have no problem with "infinite" energy, and will happily tell you that the integral of f(x) = Infinity if x = 0, 0 otherwise, is 1. Like, the number one. They just integrated a point discontinuity at infinity and got 1. That's not even on the real number line anymore. If your target spaces is "The reals + a number larger than all of the reals" then every nice algebraic property you're used to explodes. a + b = a + c doesn't imply that b = c any more, for starters. Infinitely wide sin waves are a well-defined square integrable function that doesn't break math at all, and are sometimes allowed in quantum (e.g., as part of an orthogonal base) and sometimes not okay. ( sin(x) is a perfectly good replacement for constant 0 in the bottom rung argument, except that then you don't derive the right things so unnnnnnnnnnngh we declare that the world doesn't work that way.)

Two thirds of the way through the semester, I finally figured out that all of these "wavefunctions" in "Hilbert space" (the thing physicists call "Hilbert space" is one particular Hilbert space with a whole bunch of goofy extra rules, but they don't care that they're forking nomenclature) are not functions at all, but Cauchy series of equivalence classes of functions under some kind of strange distance metric that I think was degenerate for most pairs of equivalence classes. Which is a fine, if unusual, space for a mathematician to work in -- you don't have functions that map from from the reals to the reals anymore so it's a bit Twilight Zone-ish. But that's apparently "too abstract" for physicists, so they turn around and *pretend* that the limit of a Cauchy series of equivalence classes of functions is itself a function. Maddening and in defiance of all sense, I tell you.