I have sort of a contrarian take on this, at least by traditional academic standards. I think applicability is extremely undervalued in modern mathematics. Tao references algebraic geometry in the interview, but the people who do algebraic geometry in applied settings ("applied" is also being quite generous here, in my opinion) are doing math that is basically unrelated to algebraic geometry as practiced by theoretical mathematicians. Even in connections to computer science, the people studying algebraic circuit complexity-which a priori should benefit a lot from algebraic geometry-basically can't use any modern algebraic geometry theory because (to my understanding) the theoreiticians choose to work in a setting that is so elegant it rules out interesting algebraic circuits. And circuit complexity is still quite far removed from applications the way I think of it.<p>I think the best and most beautiful mathematics is driven directly by applications, and made better for the pressure to apply to a real problem (not a problem that just mathematicians care about).<p>A quote from Milind Tambe's 2018 AAAI keynote talk sums up why:<p>> Deploying the algorithms in the field often gives us new insights into what might be wrong with our models and provides us new research directions.<p>If you've never heard of Tambe's work, you should really check it out. He literally sends his grad students to do field work to study how their algorithms are performing.<p>Too many mathematicians consider "applications" as "can be used in other, purely theoretical academic papers," and based on my research for my next book (pmfpbook.org) I find that large swaths of what is claimed to be applied math is actually not useful for solving the real world problems they claim to be useful for. A math thing will be tried in production, to much acclaim and fanfare, and then it will be quietly discarded in favor of simpler, more holistic, or less complicated solutions that end up performing better. But the academic publications praising it persist for decades, and nobody seems to get the message that it's not used.<p>One thing Tao gets right is that the best math is interdisciplinary. Connecting ideas across different theories is one great way to demonstrate the power of a theory. I would add that, to be great a theory needs to _grounded_ in a practical setting, where it's not just mathematicians judging the math for its beauty but an external reality pushing back. For my aesthetic, a pretty theory is just not enough.
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 <i>invented</i>, and then there's mathematics that is <i>discovered</i>. 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 <i>any</i> of this math that we discover could possibly <i>not</i> 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?<p>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?<p>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.<p>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 <i>breadth</i>. 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.
Good or interesting mathematics imho is about finding the exceptions. Like, why something works for a particular case when otherwise it does not, or the converse. An example is why a general quintic cannot be solved with radicals, or finding non-trivial quintics that can be solved with radicals.