My point is that when the consumer application became apparent we already had the required concepts to build the technology on top of. In mathematics it still hasn’t happened that an LLM system has invented a conceptual framework. In most if not of the major AI announcements they’ve worked within known frameworks and assembled ideas across frameworks.

Moreover, it’s not clear that if (and when as I believe) they do, creating technologies with no human understanding of the framework is possible or desirable.

> Did the theoretical math lead to the invention of GPS, cryptography, lasers, etc?

For cryptography, perhaps you would enjoy reading the paper of Diffie and Hellman that proposed public-key crypto: https://ee.stanford.edu/~hellman/publications/24.pdf

You will find they were inspired by the NP-hard knapsack problem, and inspired a bunch of later research that led to RSA.

I think the tapestry of history would suggest the answer to the question "is math responsible for this invention" a lot more complicated than it appears. For lasers, Einstein proposed the idea based on purely theoretical physics, and it was made possible in 1960. Is that "theoretical math leading to the invention of lasers"? Surely he was at least relying on a lot of additional theoretical work for that. On the other hand, much theoretical that came out of Bell Labs were responses to needs for better vacuum tube technology, better amplifiers, etc., which were a deep collaboration between theory, practice, and tradesman with a strong intuition for how to build with various materials and at varying scales.

A lot indicates the answer to your latter question might just be 'no'

>If we run into a real physical problem today, can we just have AI invent the math on the spot to solve the problem without a human having done the theoretical math in the past?

This depends on unanswered questions on what math actually is and it's causal connectivity.

Imagine we have problem A that needs to connect to math solution Z.

The problem is the A -> Z route can only occur in polynomial time in which you need to burn the visible universe to solve. So, that itself is not workable.

As you look at the problem space of A there are a potentially infinite number of paths you could take in the problem topology so again you'd have to brute force the path... mostly unworkable on a lot of problems.

The breakthroughs tend to occur when somewhere in between A and Z there is another mathematical construct M that can link them together. M was very likely discovered something so completely and wildly different you would never link them by brute force. By M existing you narrow the problem space to NP time. M might have sat in the toolbox 100 years unused before that point.

You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture.

The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect.

I may have misinterpreted either or both of them though!

So my question is this:

Would this slow technological progress? Or make it go faster?

That math problems are found and solved as we run into real physical problems.

Oh some few. No right triangle with rational sides has area equal to a perfect square depends on N=4 for instance. And it can be used to form other theorems that are terribly actionable. Every elliptical curve over Q is modular, which has consequences throughout number theory.

But yes, none of those are very tangible, until applied to problem solutions that are tangible.

That depends on the solution, right?