Are hyperreal numbers actually useful? Other than making calculus a lot more intuitive by eliminating the need for limits and other shit, do they actually help mathematicians solve other problems? I'm not a mathematician, so I have no idea what kind of problem or proof would require hyperreals.
>tl;dr: uses of hyperreal numbers?
You can use them in physics to perform calculations involving infinite sums where it's not obvious that the sums are going to cancel out to produce something finite. The classic example is in QED, where you need to perform summations over all possible paths that a particle can take between two points. Most of those paths cancel out, but without hyperreals, formal proof that they do is rather difficult. For systems with many particles, you're essentially forced to use hyperreals or some equivalent.
Yes. Nonstandard techniques can go way beyond simple calculus stuff, some results have been proved for the first time using this framework in functional analysis, probability, even in algebraic topology.
But you got it wrong, it doesn't eliminate the need for limits, it's just another way of looking at them.
Cool shit you can do with it:
nice characterization of compact sets - a set K is compact if and only if all x in K is infinitely close to a point in K (when I saw this it seemed quite natural to me that balls in infinite dimensional Banach spaces aren't compact)
the Dirac delta function is actually a function and not a distribution - if w is infinitesimal, define f(x) = 0, if |x| > or = w and f(x) = 1/w if |x| < w
defining the real numbers: *Q is the set of hyper-rational numbers, K is the set of limited hyper-rationals and N the set of infinitesimals, then R = K/N
But all this is just applied model theory, and who the fuck doesn't want to learn model theory?
Looked at the source for OP's image. Maybe it's a bad source but hyperreals just seem to be a way to sidestep the use of formally using limits. Kind of disappointing because, on the surface, they seemed similar to the concept of complex numbers in the sense that they extend the real number line.
That's like saying you don't like irrationals because they're just a way to get around using geometric constructions. The hyperreals are a legitimate number system derived from a simple axiom schema, in much the same way as the reals or rationals. There are some theorems that connect them to differential calculus and make some uses of them akin to limits in some way, but they're far more general than that.
Conway's construction of the surreals, which are an even bigger ordered class, is one of the most beautiful things in math.
There are applications to physics and finance. Nonstandard Analysis: Theory and Applications (ed. Henson et al) is a nice survey of its applications and connections with other mathematical fields.
Keisler's elementary calc book will present nothing sophisticated or interesting and probably convince you it's useless, but he did contribute a nice section on stochastic DEs to the book i mentioned above
Hochschild cohomology can be used in mirror symmetry to see that different Ainfinity categories are equivalent. And thus prove that string theorists are not just bullshitting their computations.
>hyperreals just seem to be a way to sidestep the use of formally using limits
Except it isn't so much sidestepping as it is taking a detour to climb Mount Everest when travelling from California to Arizona.