Almost every area of mathematics that’s been around for any amount of time has real world applications. The notion of Turing degrees that I posted earlier is one of the very few exceptions.
One exception? There’s a few more than one! Various type theories, for instance, were invented for purely mathematical reasons. System T was invented by Goedel to prove the consistency of Peano arithmetic. System F was invented by Girard (and reinvented by Reynolds) to prove the consistency of second order arithmetic. The lambda calculus and simple type theory was originally part of a much larger system (which turned out to be inconsistent) intended to form a functional foundation of mathematics in opposition to Cantor’s sets.
Today similar type theories are used in proof assistant software to verify hardware and software designs. Both AMD and Intel, as well as smaller manufacturers like Rockwell Collins, employ whole departments using this software to verify the floating point units of their processor designs before they are manufactured.
gee whiz, I didn’t know that…
My only grasp of what mathemeticians do comes from reading “Fermat’s Last Theorem” by Singh. Spending your entire career solving that theorem seems to me to be pretty much the definition of useless, pure research.
There are certainly areas of chemistry where the “look what I can make” quotient is a very high. Ultimately, even those have the potential to develop techniques for making something useful, but most of them don’t. They do however, serve as a vehicle to train scientists that often go on to work in commercial industry.
Natural product synthesis is almost entirely this. No company is going to use a 30 step non-convergent synthesis with a yield of 0.00003%, but commercially employed chemists absolutely must develop their technique in grad school. Textbooks can’t teach that.
Wiles proof of FLT entailed proving the Taniyama-Shimura conjecture, which is very closely tied to some of the math underlying cryptography.
It’s hard to say any path of basic research can’t lead to commercial applications. The whole point of research is discovering things we didn’t know and nobody can predict what those discoveries will be.
A hundred and fifty years ago, nobody knew what an electron was. If you had explained the concept to somebody, it probably would have seemed about as useless a concept as could be imagined. But now ninety percent of our technology is based on it.
I haven’t reviewed many big $10million level grant proposals, but I have reviewed a bunch of smaller ones, and a proposal stating potential commercial impact is a lot different from a proposal with actual potential commercial impact. And I do engineering things. Grant writers are very creative. There is a requirement that the grant describe how it will forward social equity, and it is pretty amusing to see how people asking for small business money to help paint yachts better answer this one.
I’ve spent a lot of effort trying to enable university researchers to make their research be more relevant to the real world by providing real test cases, and for the most part they would rather go with their 25 year old trivial examples. There are some exceptions.
I’d say that saying cosmology research may have commercial applications is really pushing it - with the exception of Penzias and Wilson.
Knowing that the Universe has a background thermal radiation of microwaves at 2.7 K is practical and usable information, since (as Penzias and Wilson found) it can be one source of noise in communications. Knowing the detailed structure of the microkelvin anisotropies in that radiation, though, is (so far, and in the foreseeable future) useful only for the sake of knowledge.
Back in the 1920s or so, G. H. Hardy, one of the giants of number theory stated that number theory, at least, could never have any practical application. Shows what he knew. It is the basis of most modern cryptography. If basic research knew where it was going, it would already be there.
Another example. In the mid 19th century, matrix theory was also considered the purest of pure math that could never have any practical application. Could Riemann have ever imagined that his differential geometry would provide the basis for the physical theory (general relatively) that is the basis for navigational systems?
You’re missing the forest for a tree, there. Matrix algebra is used in everything in physics. I’m skeptical, though, that it was ever seen as useless-- I would have expected that it would have been developed specifically for its many uses.