There are two different things people mean when they say "mathematics". (Probably lots more, but I'll focus on two.) There's mathematics that is invented, and then there's mathematics that is discovered. The "math that is discovered" was already there. It must be true. It was true before our universe existed, and would be true in every other conceivable universe. It is very difficult to believe that any of this math that we discover could possibly not be useful. It's what reality is made out of. How could a deeper understanding of reality not be useful? And I assert that it is simply not possible to know whether a given discovery would be useful before you discover it; how could you possibly evaluate a discovery's usefulness before you even know what it is?
The "math that is invented" is attempting to describe this more fundamental math. These are the terms we come up with and write down in papers, the symbols we use, etc. Sometimes we discover that The X Theorem and The Y Theorem are actually describing the same fundamental math in different ways. Clearly this descriptive math can vary in usefulness. How simply and clearly does it describe the underlying "ideal" math? Does thinking of it this way lend itself to immediate application to make human lives better? Does it help us discover even more math?
Almost everyone who criticizes math is missing this distinction: they're criticizing the writing, the models, even the institution of math. Some of those criticisms have merit; most don't. Either way, they tend to overlook the fundamental math that was already true, already there long before our universe. You simply cannot identify whether or not a given mathematician is "wasting their time", because you cannot know what underlying fundamental math they might be about to discover, and what its uses might be.
If I were in charge of giving out grants for general-purpose math research, I would not attempt to quantify "usefulness" at all. The main metric I would try to optimize for would be breadth. I'd prioritize the math that seems to be exploring areas that haven't already been thoroughly explored, or making connections between areas of "described math" that currently seem only distantly related. In our current institutions of math, there are forces that push toward increased breadth and forces that push toward conformity (decreased breadth). It might be enough to simply work against those conformity forces -- demanding "usefulness" is a conformity force.
The "math that is invented" is attempting to describe this more fundamental math. These are the terms we come up with and write down in papers, the symbols we use, etc. Sometimes we discover that The X Theorem and The Y Theorem are actually describing the same fundamental math in different ways. Clearly this descriptive math can vary in usefulness. How simply and clearly does it describe the underlying "ideal" math? Does thinking of it this way lend itself to immediate application to make human lives better? Does it help us discover even more math?
Almost everyone who criticizes math is missing this distinction: they're criticizing the writing, the models, even the institution of math. Some of those criticisms have merit; most don't. Either way, they tend to overlook the fundamental math that was already true, already there long before our universe. You simply cannot identify whether or not a given mathematician is "wasting their time", because you cannot know what underlying fundamental math they might be about to discover, and what its uses might be.
If I were in charge of giving out grants for general-purpose math research, I would not attempt to quantify "usefulness" at all. The main metric I would try to optimize for would be breadth. I'd prioritize the math that seems to be exploring areas that haven't already been thoroughly explored, or making connections between areas of "described math" that currently seem only distantly related. In our current institutions of math, there are forces that push toward increased breadth and forces that push toward conformity (decreased breadth). It might be enough to simply work against those conformity forces -- demanding "usefulness" is a conformity force.