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Well, it depends on what you think a "public good" is, and your own personal tolerance for ambiguity, I suppose.
Do you think a paper on Frobenius exact symmetric tensor categories is an unambiguous public good? Hell, do you think Frobenius exact symmetric tensor categories are even real? Because this is a representative exemplar of the sort of thing that the university does. And this is from a STEM department, the "useful" half of the university.
I do think it's a public good - it's just a public good of a non-utilitarian kind. This kind of research isn't going to help you cure cancer or feed more people or build the next killer app; but I view it as a public good nonetheless, because it's an integral part of what makes life worth living in the first place.
Perhaps diverting your tax dollars to causes you find nonsensical is itself a public good, if it serves as an impetus for you to reflect on your own preconceived notions of value.
Oh yes, my preconceptions of value and what is worth dispensing violence over are regularly challenged by bluehair tier academia.
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I have good news! I am willing to perform a good deed for you - in exchange for a large sum of money, I'll provide you with an impetus for you to reflect on your own preconceived notions of value. Hell, I'll even be generous and throw in one or two bonus impetuses at no extra cost. If you think this is a good worth spending other people's tax dollars on, why not spend some of your own money on it?
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...I do not think this is going to be a persuasive argument for most people, but I must admit that it certainly seems likely to be a public benefit, at least from my perspective, for you to continue to make it.
[EDIT] - ...More charitably, I think it behooves supporters of the Universities to recognize the degree to which the vast public costs of these institutions have been justified by long-standing claims of public benefit far more concrete, immediate and utilitarian than "an impetus for you to reflect on one's own preconceived notions of value". The level of appalling smuggery in that last phrase, in particular, is... quite something.
I do not recognize why "Frobenius exact symmetric tensor categories" are an unambiguous public good, and I emphatically reject the argument that they are any part, integral or otherwise, of what makes life worth living in the first place. Maybe this is because I'm too stupid to understand the vital role such categories play. I'd certainly be interested in seeing an actual argument as to why their pursuit is worth trading away sizable and utterly irreplaceable chunks of my life, though, should you deign to provide one.
I can elaborate on my views a bit, at any rate.
It will be helpful if we continue to confine ourselves to mathematics, rather than university studies as a whole, since mathematics is less politically fraught than the humanities, and certain peculiar properties of mathematics help bring questions of usefulness vs uselessness into sharp relief.
I don't know how much you know about research-level mathematics, so I'll give a brief summary. Mathematicians primarily concern themselves with writing proofs of theorems - deductive arguments in support of statements about mathematical objects. These mathematical objects can be ordinary and familiar, like numbers, or they can be extremely exotic, like hypervector spaces over fields of characteristic p. It is an open philosophical question as to the precise relationship between mathematical objects and what one might call "reality". Mathematicians simply speak them into existence with flat declarations like "Define a hypervector space as any set of points that meets such and such requirements...", the way one speaks the rules of Chess into existence with statements like "Define the Queen to be a piece that moves in such and such a way...", which naturally raises questions about the ontological status of these objects. After a proof has been written and certified by peer review, it will be published in a journal where it will be largely unread, largely incomprehensible to all but ~15 mathematicians who specialize in the author's sub-sub-field, and it will pass largely unnoticed into the dustbin of history. Such is the peculiar enterprise we concern ourselves with.
Fields medalist Timothy Gowers gave a nice lecture on the value of mathematics, and why it's deserving of public funding. The most substantive part of the argument is when he points out the obvious fact that mathematics does indeed have practical real-world applications that everyone can agree are important; otherwise we couldn't have science and engineering and computers and all the rest of it. The twist though is that you can't know a priori which field of mathematics, let alone which individual results, will turn out to be "useful" in a utilitarian sense. The classic example is how number theory turned out to have applications in cryptography, but this application of number theory only became apparent over a century after Gauss and Galois inaugurated the modern study of the subject. Unlike fields of mathematics that developed in close relationship to concrete problems in physics, like differential equations, no one could have guessed that trivia about things like quadratic reciprocity would ever be useful to anyone - and yet it did turn out to be useful, in the context of the technology of the distant future.
This argument should be enough to convince even the most hard-nosed utilitarian that funding some level of speculative mathematics research is worthwhile. In the short term it may look like a lot of waste and false starts, but the potential payoff is so great, and the proven track record of past results is so strong, that I think the expected ROI is positive.
My own views go beyond this though. I think even if we had God's guarantee that some branch of mathematics would never have any practical use whatsoever, it would still be deserving of public funding, as long as it met the specifically mathematical criteria for mathematical relevance, of course (desirable traits that mathematicians will point to in a theorem like aesthetic beauty, "naturalness", impact on other results, etc). This I believe simply because the purpose of civilization itself is to enable higher types of minds to do higher types of things. Society should not be a playground arranged so that atomized individuals have maximum freedom to "do as they please" - rather it should be arranged with no other goal in mind than to support the flourishing of humanity's most meaningful achievements. Lower types of activities should subsidize higher types of activities. A paltry <1% of the federal coffers is a small price to pay in pursuit of this goal.
I once witnessed a conversation that went as follows:
I think B was getting at something quite important. The standard criteria that people use to determine whether something is "useful" is precisely backwards. Infrastructure, commerce, defense, all types of "efficiency"... these things have no value in and of themselves. Their only justification is to serve as the foundational support structure for authentically human activities, the things that separate man as such from the mere animals.
As for why I give mathematics specifically such an exalted place, I could elaborate on my reasons further, but they would quickly become poetic and unpersuasive to you. I will simply say that direct contact with the subject is the best argument. I don't see how a genuinely sensitive and inquisitive mind could understand the statement and history of problems like the search for an exact quintic equation, or Fermat's Last Theorem, or the Riemann Hypothesis and remain unconvinced that there is something of immense value there. Such apparently simple problems, such deceptively childish questions, and yet they resisted the continuous assault of humanity's greatest minds for centuries, their solutions had such surprising and far-reaching implications, or they remain unsolved even today despite us bringing the full weight of our resources to bear on them, such as the RH and an infinite sea of even further and loftier problems...
Not all mathematical problems are interesting of course. Many are boring and trite and have no real relevance to anything or anyone. But similar to the argument given above about practical utility, you can't know a priori where great ideas will come from. A random paper about combinatorics published by a professor at a third-rate state school probably won't lead to a proof of the Riemann Hypothesis, but at the same time, it might. The history of mathematics is full of such surprising connections. Which is why it's better to simply let mathematicians do what they do and not try to prune the search space too early.
To be clear, nothing in this argument is self-serving. I am not employed by or affiliated with a university, and never will be.
Similar considerations will apply to most university STEM fields. They may change somewhat if we were to consider the humanities, but the basic thrust is the same.
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