Suppose I have a practical problem in my business. How do I discover the right mathematical branches/theorems/concepts/models relevant to my practical problem?

There is no natural mapping among the vocabulary of my hypothetical practical problem and the vocabulary used to name mathematical concepts (usually combinations of names of prominent mathematicians).

A lot depends on what you want to measure. If you want to measure the pervasiveness of mathematics in the economy, you'll notice that it has been soaring. Someone has to develop the algorithms that control automobile engines, delivery scheduling, industrial processes, farm machinery and so on.

Something like a thermostat was high tech in its day, but optimally using a thermostat to control a coking oven was high tech in its day. It ended the expensive batch processing of coke and allowed continuous production. Back in the days of the USSR, mathematicians like Kolmogorov did theoretical work, but he also worked on problems in lubrication. It was considered remarkable at the time, but no one remarks nowadays when disposable diaper companies hire mathematicians.

You can try measuring mathematical progress in some abstract sense of advancing, but one problem with mathematics is that one often doesn't know whether one is advancing or not. Mathematicians do all sorts of mathematics for the beauty of it, and only centuries later does their work turn out to be useful in proving some intractable theorem or solving an applied problem.

Excellent subject. From a non-mathematician perspective, I put together this series of four essays on the importance of math to civilizational progress throughout history, if anyone’s interested:

Goldman Sachs will pay in the high six figures for a couple of years of grad school, escalating quickly into seven figures if you're good. Hanging around for a PhD is deadweight loss opportunity cost. Maybe have a look at the number of grad students or MAs in Math?

Also, if you're just looking at Pure Math, you're missing much of the picture. When I was in grad school in the '70s, Applied Math and Stats and Comp Sci were for your idiot cousin who couldn't cut it in Real Math. Data Analysis, Financial Math, Computational/Applied Economics, Computational Physics, didn't exist in the sense they do now.

Of course, those things reinforce the idea that Pure Math may be in decline: People with strong quantitative abilities have a lot more career paths than proving ever-more-esoteric theorems.

tell people that when I started grad school, there were three things you could do with a Math degree: actuary, NSA, academic. None of those appealed to me, so I ditched Math for Business School. What they did with Math was appalling, but it's given me a comfortable retirement.

On the other hand, if I'd hung on in Math for another couple of years, I could have been one of the first quants and retired with a fortune at age 30.

I am very interested in the following problem.

Suppose I have a practical problem in my business. How do I discover the right mathematical branches/theorems/concepts/models relevant to my practical problem?

There is no natural mapping among the vocabulary of my hypothetical practical problem and the vocabulary used to name mathematical concepts (usually combinations of names of prominent mathematicians).

A lot depends on what you want to measure. If you want to measure the pervasiveness of mathematics in the economy, you'll notice that it has been soaring. Someone has to develop the algorithms that control automobile engines, delivery scheduling, industrial processes, farm machinery and so on.

Something like a thermostat was high tech in its day, but optimally using a thermostat to control a coking oven was high tech in its day. It ended the expensive batch processing of coke and allowed continuous production. Back in the days of the USSR, mathematicians like Kolmogorov did theoretical work, but he also worked on problems in lubrication. It was considered remarkable at the time, but no one remarks nowadays when disposable diaper companies hire mathematicians.

You can try measuring mathematical progress in some abstract sense of advancing, but one problem with mathematics is that one often doesn't know whether one is advancing or not. Mathematicians do all sorts of mathematics for the beauty of it, and only centuries later does their work turn out to be useful in proving some intractable theorem or solving an applied problem.

Excellent subject. From a non-mathematician perspective, I put together this series of four essays on the importance of math to civilizational progress throughout history, if anyone’s interested:

https://paultaylor.substack.com/p/maths-path-part-1?s=w

https://paultaylor.substack.com/p/maths-path-part-2?s=w

https://paultaylor.substack.com/p/maths-path-part-3?s=w

https://paultaylor.substack.com/p/maths-path-part-4?s=w

I can hypothesize the decline in Math PhDs:

Goldman Sachs will pay in the high six figures for a couple of years of grad school, escalating quickly into seven figures if you're good. Hanging around for a PhD is deadweight loss opportunity cost. Maybe have a look at the number of grad students or MAs in Math?

Also, if you're just looking at Pure Math, you're missing much of the picture. When I was in grad school in the '70s, Applied Math and Stats and Comp Sci were for your idiot cousin who couldn't cut it in Real Math. Data Analysis, Financial Math, Computational/Applied Economics, Computational Physics, didn't exist in the sense they do now.

Of course, those things reinforce the idea that Pure Math may be in decline: People with strong quantitative abilities have a lot more career paths than proving ever-more-esoteric theorems.

tell people that when I started grad school, there were three things you could do with a Math degree: actuary, NSA, academic. None of those appealed to me, so I ditched Math for Business School. What they did with Math was appalling, but it's given me a comfortable retirement.

On the other hand, if I'd hung on in Math for another couple of years, I could have been one of the first quants and retired with a fortune at age 30.

Do you really consider Space Travel a mathematical problem on par with Cryptography and machine learning?