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G. H. Hardy's A Mathematician's Apology provides an answer as to why one would do mathematics, but I'm unable to find an answer as to why mathematics deserves public funding. Mathematics can be beautiful, but unlike music, visual art or literature, much of the beauty, particularly at a higher level, is only available to the initiated. And while many great scientific advances have been built on mathematical discoveries, many mathematicians make no pretense of caring about any practical use their work may have. So when the taxpayer or a private organization provides grants for mathematical research, what are they expecting to get in return?

I'm asking not just so I can write more honest research proposals, or in case it comes up in an argument, but so I have an answer for myself.

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@Asaf: Those are quite big jumps you're making there. It's not clear to me that saving people's lives could rely on measure theory, topology or large cardinals –  godelian Jul 11 at 14:27
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Some mathematicians fall into the trap of justifying the spending to themselves. Witticisms and weak connections might seem ok at reassuring us that mathematics has positive value. However, these are not convincing to someone weighing some proposed abstract research against someone proposing to study how to deactivate a gene involved in a disease killing hundreds of people each year. It's not enough to have some positive value. –  Douglas Zare Jul 12 at 4:54
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@bubba: You know, it's easy to say that. But you don't take into account the parts that the 90% remainder of each field help a lot to create an environment and influence that progress that 10%. And that it has a big factor on the fact that these 10% even there. –  Asaf Karagila Jul 12 at 9:52
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Take a look to the answers of the following post: Why fund the research in pure mathematics? –  Sébastien Palcoux Jul 15 at 11:11

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Benson Farb answered this beautifully in his University of Chicago commencement address. Here is an excerpt:

Since I am a pure mathematician, Dean Hefley suggested as a possible topic for this talk: “Why the square root of negative 1 is necessary”. I could take up this challenge of justifying pure science on its vast applicability; indeed the square root of negative 1, the basic “imaginary number”, underlies a huge swath of modern technology, from the design of circuits, airplanes and skyscrapers, to the construction of economic and financial models, to robotics. I have decided, however, to take the opposite point of view. I want to defend the value of basic science for its own sake...

...the purpose of pure mathematics, of basic science, is not the quick harvest. It is nothing less than an attempt to bring human thought and understanding to a higher level. It is an attempt to change not just what we think about the world, but how we think about it. The importance of this for human evolution is incalculable. As British physicist JJ Thomson said: “Research in applied science leads to reforms, research in pure science leads to revolutions.”

Benson Farb, 2012

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Wow, that's a reminder that it's not the individual who is important, but the species. Interesting. –  godelian Jul 11 at 14:48
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This is the same sort of answer that I get when I ask about the value of literary criticism. "It is nothing less than an attempt to bring human thought and understanding to a higher level." This is vague and I don't see how this justifies math research. "The importance of this for human evolution is incalculable." What exactly is this incalculable importance? Through what mechanism does it give us this value? This sounds more like cheerleading than an argument to convince a skeptic. –  fhyve Jul 12 at 18:19
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To me, this answer is a compelling argument for funding curiosity-based and not just applied-driven research. I don't know much about literary criticism, so I can't comment on it. But I can say that, in mathematics, finding worth-while direction is a difficult and complicated pursuit that is best done as a collective effort (see Thurston's link above). You might also enjoy reading Benson's entire speech: tinyurl.com/bfgs2012 –  Khalid Bou-Rabee Jul 12 at 18:50
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This is essentially what I reply to people asking why I do research. I do biological research, so I could easily say, as many do: "my study will help cure <insert your favourite disease here>" (please note the use of will instead of could). However that is not why I do research. As a basic researcher my goal is to understand how things work. If someone finds a practical application for my work, all the better, it just is not what I do, and I am not ashamed of saying it. I think Feynman explained this really well: youtube.com/watch?v=lmTmGLzPVyM –  nico Jul 13 at 10:32
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My reaction (not quite an answer) to this question is related to this answer: let's ask a better (more honest) question, namely why should the "mathematical community" get funding/ resources. If one thinks about all of the value added by our community, to everyone we directly impact (other scientists, graduate and undergraduate and (to a lesser extent) K-12 students) as well as to everyone we indirectly impact, one comes up with better answers, most of which revolve around "brining human thought to a higher [I would say deeper/ more insightful] level." –  Dev Sinha Jul 15 at 23:51

I will suggest a "proof by contradiction" here.

First, let us make a very quick (and poor) statistical analysis on the amount spent on Mathematics research. Let us say that the research agencies spend 2% of money on mathematics*. Excluding the other basic sciences, this means that more than 90% of research is being spent on "helping deactivating a gene that kills hundreds of people" (to quote some comment above). Of these 2%, a small portion produce more applied results, that maybe will be used in high-standard technology in a soon future. The other portion provides comprehension of a very specific field, helping bulding an environment for eventual disruptive breakthroughs ("on the shoulder of giants", right?).

Hence, as an investor, I would have no doubt in putting 2% of my money in mathematics. In return, I get some quick and applicable results that may lead to patents or products - (e.g. statistical models, industrial operations research models, etc). Eventually I get big breakthroughs with various applications, which come with the bonus of helping improving human comprehension of the world's very basic and profound questions (e.g. Complexity Theory, Population Dynamics, Information Theory).

Of course I could take the 2% of pure research and redirect it to more applied research. But, is it worth it? Is it worth NOT funding mathematics and loosing all the potential advances of science? And here is, as promised, the "proof by contradiction". NOT funding mathematics is dangerous, and may block very profound advances. Ergo, agencies should give money to research in mathematics.

Furthermore, although I am an applied mathematician, I completely disagree that you would get more "useful" results in redirecting all the efforts to the problems that come from real world applications (medicine, engineer, etc..). This can give you quicker results. This probably can help you downloading a video 1.3x faster. This probably can help a doctor with a better MRI equipment. Nevertheless, the real breakthroughs occur when one investigates more profound questions.

*Here in Brazil 2% is a very good approximation for the percentage of math scholarships, in comparison to scholarships granted to all areas. If anyone has more accurate information on the US agencies (for example, this NSF link is useful), I would be happy to know.

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+1 I think this is the most reasonable and pragmatic answer (which seems to be what the question was asking for). Two percent of the research budget is indeed not that high of a price for the chance to make a breakthrough. –  Paul Manta Jul 12 at 20:35

A mordant response from Bertrand Russell, Human Society in Ethics and Politics (Simon and Schuster, New York, 1955), 43:

In universities, mathematics is taught mainly to men who are going to teach mathematics to men who are going to teach mathematics to ...  Sometimes, it is true, there is an escape from this treadmill.  Archimedes used mathematics to kill Romans, Galileo to improve the Grand Duke of Tuscany’s artillery, modern physicists (grown more ambitious) to exterminate the human race.  It is usually on this account that the study of mathematics is commended to the general public as worthy of State support.

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I came, I read, I was answered. –  user89 Jul 16 at 20:24

As a mathematician I go along with the classical Jacobi-Poincaré line exemplified by Farb's

...the purpose of pure mathematics, of basic science, is not the quick harvest. It is nothing less than an attempt to bring human thought and understanding to a higher level.

However, I think that this is a dead end in justifying the public support of mathematics. In a world where people still starve and lack basic needs (and this is true of some sections of the population of "rich" countries as well), I don't think that exaltation of the human spirit will get us very far. A better strategy is perhaps to compile a very long list of all the scientific and technical progress that mathematics has stimulated. Tomography and Radon transforms, Fourier analysis and circuit theory, etc. Coarsely speaking, if people become convinced that without mathematics they would not have an iPad, then we do not need to fear for our funding.

Another good question is how much funding does mathematics REALLY need and how to use it most effectively. The current trend of mega-grants to a selected few (who are then burdened by their management) to the detriment of institutional funding does not seem to me to be the best way to go about funding mathematics.

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A century old opinion: http://www.jstor.org/stable/2972761 (The Significance of Mathematics, by E. R. Hedrick)

Shall we not search our own house? Shall we not ask if our own collegiate and graduate courses in mathematics demonstrate to students the real significance of the theory they cover? Have we denatured each subject until insight is eliminated and only formalism and logical tricks remain? So long as this blight remains, we must expect and we shall deserve public disdain and sincere doubt of our value to humanity

and the modern one: https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf (The Importance of Mathematics, by W. T. Gowers)

I don’t think this philosophical question has been satisfactorily answered, but we can be grateful that in our world it is possible to use simple mathematical models. These can describe, or even explain, the great complexities of physics and to a lesser extent the other sciences. Once again, complexity arises from simplicity, and, once again, beauty reveals itself to be important. Thanks to this piece of good fortune, we can be confident that mathematicians, if they are given the freedom to pursue the subject that gives them so much pleasure, will continue to produce a body of work that is important in every sense of the word.

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Gowers's lecture was filmed and it is available on YouTube here: youtube.com/watch?v=BsIJN4YMZZo --- well worth watching. –  Nik Weaver Jul 11 at 17:58

Mathematicians are constantly in the habit not just of determining mathematical truths, but continually rewriting their results, getting at the right level of generality, developing useful notations, absorbing theory into well-chosen definitions, etc. Over time this continuing process makes mathematics evolve into particularly usable and general forms for future generations of mathematicians and other scientists to pick up, understand, and readily use (cf. "unreasoable effectiveness"). Examples might include the language of differential forms and tensor calculus, the theory of connections and fiber bundles, and the use of string diagrams and circuit diagrams à la Penrose, Joyal-Street, and being pushed into new directions and applications by John Baez and his coworkers (control theory, electrical circuits, chemical reaction networks, etc.).

Jean-Yves Girard said somewhere (perhaps in Proofs and Types) that computer science is the great consumer of mathematical logic; flexible and general forms of logical languages have huge potential pay-offs in the design of flexible and efficient programs and programming languages. In short, mathematicians are in the business of getting the language right, and the resultant economy of thought gradually but inevitably works its way into the language of other sciences.

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In my personal experience as a programmer, programming is exceedingly complicated; anything to make it easier would be welcome; but I suspect, as a technology deployed in the real-world it will remain so; any gain in efficiency is going to be offset by having to interface with a wide & disparate range of already existing software technologies. –  Mozibur Ullah Jul 15 at 6:10
    
@ToddTrimble: I didn't find it in Proofs and Types -- perhaps it's in The Blind Spot? –  Neel Krishnaswami Jul 16 at 10:39
    
I find the motif "unreasonable effectiveness" a sleight of hand. What's so unexpected about the products of an intelliigence embedded in and evolving with the world being useful in dealing with that world? Cf. Philosophy in the Flesh. –  Tom Copeland Jul 19 at 1:24
    
@TomCopeland Okay. It was intended more as a literary reference, not as agreement with the choice of word 'unreasonable'. –  Todd Trimble Jul 19 at 1:30
    
@NeelKrishnaswami I somehow missed your comment before. It could easily be my misremembering, coupled with a projection. The closest I found in Proofs and Types is, "The disaster was averted because of computer science -- that great manipulator of syntax -- which posed it some very important theoretical problems" (page 4 from Paul Taylor's pdf: paultaylor.eu/stable/prot.pdf). –  Todd Trimble Jul 19 at 1:44

What a good question! My attempt at an answer in one line is that I think that a lot of fields that we consider very important were off shoots of mathematics at some point in the past.

If one goes far enough back I don't think there were many "mathematicians" as we label them today. I think people were "natural scientists" and mathematics was a powerful tool that they developed for either its own sake or for use in their other studies.

As we gained more knowledge and things became more specialized then things started to branch off. For example, I think you would be hard-pressed to find a pure "physicist" or a pure "mathematician" pre 1940s. They did exist, but the two fields of study were much more entwined. With the rise of industrial physics due to the war we commercialized physics and created physics departments (again, they may have existed in the past but not as they do today).

Another more recent example is computer science. A lot of computer science theory was (and still is) developed in mathematics departments. The creation of new computer science programs and departments is a relatively new thing and is again the commercialization of a mathematics off shoot.

I think it would have been near impossible to have predicted the commercialization of physics or computer science in the 1800s, just as it is near impossible for us to predict what will come of it now. If the past is any predictor of the future, though, there are going to be fields of study that don't necessarily exist today that may be born out of mathematical research.

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+1. As a software developer I certainly am grateful for the efforts of mathematicians like Turing and von Neumann whose ideas foresaw and helped create my own industry. –  Jirka Hanika Jul 22 at 13:37

This is not meant to be a fully convincing answer, but to argue for one possible way that pure mathematics research is important. I believe that mathematics has some special features that make it good for promoting international cooperation and friendship, building bonds between diverse peoples.

First, as is clear to us who do it, pure mathematics is interesting all on its own. One manifestation of this is that it doesn't seem to depend on cultural background or political ideology whether the ideas in mathematics are found to be interesting by a sufficiently curious mind. The same ideas can be appreciated regardless of one's overall perspective.

Second, mathematics is (pretty much) objective. One can recognize the value of a mathematician's contribution, and if one has the bare minimum of honesty, it will trump all other prejudgements about the other party. In this way, it has the ability to break through prejudice or adversarial political commitments, without first requiring subjective emotional change to get started.

Third, mathematics only requires time and thought. In modern times, once mathematicians with close interests are introduced, collaboration can begin easily and will not necessarily require costly investments by governments or others to get going, in contrast to experimental science.

Of course, all the pure intellectual pursuits of mankind are valuable in themselves. But mathematics, though practiced by a small number of us, seems to have a special ability to transcend cultural barriers and perhaps thus contribute in some small way to peace.

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The Mathematicians Apology's author is a great argument for why pure mathematics is "useful".

Hardy argued that pure mathematics was worth doing for its own sake. Hardy's particular category of pure mathematics was number theory, a kind of mathematics so removed from the practical world that it is now the foundation for the world's electronic commerce.

Billions of dollars spent every year, protected by our knowledge of Number Theory.

Category Theory, a system of abstracting mathematical structures so we do not have to talk about "concrete" things like fields and other (abstract to anyone else) mathematical structures when doing proofs has now inspired entire new programming techniques in Haskell, and from that it is diffusing into other corners of the information revolution.

Group theory, the abstract study of the "simplest" algebraic structure, is heavily mined by theoretical physics. The last great revolution in theoretical physics resulted in nuclear power and bombs. There may be surprising yields in fundamental physics yet unseen, and study of E8 and similar groups was long a pure mathematical endeavour.

Mathematics is unreasonably effective at describing the world. Even the purest corners of mathematics end up running into applications. Yes, the payoff may not be immediate, but that does not mean the payoff won't be large.

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Have a look at our Knot Exhibition, which aims to explain how mathematics gets into knots. It explains some of the methods used: representation, classification, invariants, analogies, laws and applications. The applications come after one has developed the necessary concepts and methods, and may also be motivated by such potential applications.

Mathematics develops rigorous language for expression, proof, analogy, verification, falsification, calculation. As an example, the language of abstract group theory was developed by many pure mathematicians, and was found necessary to determine all the 230 crystallographic groups, and also to develop the theory of quarks.

The abstract language of mathematics is really about analogy. Much modern pure mathematics is about describing abstract structures, relating them, often via category theory, and the difficult task of describing their interaction. For this, numbers are not enough.

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It's a very good question, and it takes a brave and honest man to ask it.

Personally, I think that public funding of pure mathematics research is pretty hard to justify.

I think all the arguments about possible future applications are weak, at best. Certainly you'd be more likely to get results that are useful in engineering or medicine if you worked directly in these fields themselves, rather than in pure mathematics.

You can point to pure mathematics developed in past centuries that found applications in modern times, but this is the exception, rather than the rule, I think. And my impression is that mathematics is much more abstract today than it was in the past, so applications are even less likely.

It seems to me that many people choose to study mathematics because it's beautiful and enjoyable. So, in a sense, they treat it as an art form. It's an art form that can be appreciated only by a tiny fraction of the population, but the same is true of avant-garde jazz and some other art forms. And, lack of appreciation by the masses (arguably) does not diminish the art. But these are reasons for doing mathematics, not reasons for funding it. Public money ought to be spent on things that somehow improve and enrich our lives. The arts are funded for this reason, presumably, and one could perhaps use the same reasoning to justify funding for pure mathematics. But the amount would be small, I suspect.

In my day job, I do mathematics that solves problems in engineering and manufacturing. I get paid a lot of money for this. In my play time, I do mathematics because it's fun and I find beauty in it. But I don't expect to get paid for this playing. And I don't want my tax dollars to pay for other people's mathematical play, either.

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So you are saying that all great mathematicians in very pure areas do "mathematical play"? Would you say that Gauss, Euler and big names in the past that influenced many applications were doing "mathematical play"? –  Campello Jul 12 at 16:29
    
@bubba: so the simple idea of binary arithmetic, which is mathematicians playing around with varying how we express numbers in different bases and an essential basis of software wasn't, according to you worth pursuing/playing with? Its a necessary but not sole ingrediant as to why I can read what you've written a continent away and comment on it. –  Mozibur Ullah Jul 15 at 6:14
    
@Campello. I don't know which parts of Gauss' and Euler's work were motivated by fun or pursuit of beauty. Those parts that were (if any) qualify as "play" in my book. As I said, I would support some small amount of funding for "play" mathematics, anyway, and this would presumably be enough to cover Euler and Gauss. But the vast majority of mathematicians working today are of a far lower calibre, obviously. And if these folks want to do mathematics "for fun", then I'm personally not willing to fund this. –  bubba Jul 19 at 23:39
    
@MoziburUllah. Binary arithmetic is a miniscule part of software and computer technology. And it would have been invented by the first computer engineers, anyway. You misread what I wrote. I didn't say that all pure mathematics is useless,I just said that it's less likely to be useful than mathematics and other science that's developed with applications in mind. –  bubba Jul 19 at 23:45

One has to note that in many countries there is a ministry that combines art and science. Clearly, society has decided that pure science research should be funded using taxpayer's money for similar reasons why art is funded. But funding research is only one way of promoting it. Another task of the government is to make sure that the public is educated well enough to be able to enjoy and benefit from the results of the research. Here most countries treat science and art differently. The educational program is quite minimalistic as far as science and math is concerned, you only learn what you need to know to get a job. If you want to learn more you need to study at university. We don't take this attitude when it comes to art or literature.

This difference in attitude toward art and science explains why the beauty of math isn't as easily accessible to the wider public compared to the beauty of literature. But beyond not being able to appreciate the beauty of the subject, this has negative consequences for society. How can a democratic society choose what it should do to curb climate change if most people don't have enough scientific skills to separate expert opinion from nonsense? So, perhaps we are now paying the price of not having taken math as serious as other subjects.

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Classic question on the lines of "someone should do something", and easily incorrectly answered by an argument of why the respondent would do something.

The question is about a "taxpayer or a private organization ". The latter is vague; it could be just because the private organisation is a foundation chartered to fund mathematics.

But, if by "taxpayer", we really mean a government, I will assume to start with that we mean government of a rich country.

Let's look at a few different departments:

  • Science, let's ask Roger Bacon:

"“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.”

This has been true throughout the history of science and remains true today; genetics is pushed forward by mathematicians who may not have originally commenced their career in that area at all.

  • Industry

The modern information economy is based on technologies whose key founders such as Turing and von Neumann were mathematicians. In the first instance their research (prior to the war, see below) would surely be classed as pure rather than applied. In addition of course, countless areas have made use of applied maths.

  • Arts

As covered elsewhere, mathematical truth is beauty.

  • Defence/Defense/War or however referred to in the said country.

Maths had a major role in WW2 in both code breaking and in the modelling supporting the Manhattan project, as well as many other areas. In the future, code breaking is even more important as it underpins more and more of our infrastructure. In the US, how does the NSA's budget compare to federal funding of mathematics? To misquote Dr Strangelove, can any country really afford a "maths gap"?

  • Finance/ Treasury/ Economics

Well maybe most Western governments could use a few more mathematicians there?

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Like any other area of basic (i.e., non-applied) science, mathematics may not appear to have any use. However, there's no telling where some bit of basic "pure" research will lead to. What if research on Quantum Mechanics (a useless and esoteric playground for underutilized physicists if I ever heard of one) had not been funded in any way in the 1920s - 1940s? We likely would not have the transistor and all the improvements (such as affordable computing power) that flow from that. Likewise, some areas of pure mathematics that may seem like a waste of funding (to the William Proxmires among us) may turn out later to have great value, such as in the applied fields of communications theory and cryptography. In short, there's no telling when basic research could lead to applied benefits that no one could have dreamed of, very often returning the investment manyfold.

Applied research usually only leads to near-term improvements in products and methods (think: improving photolithography by moving from visible light to ultraviolet). It takes basic (pure) research to open up new fields (think: QM leading to transistors, leading to computer chips). Mathematics is no different. Who knows what obscure corner of math will yield something of great applied importance?

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As a respite from boredom.

I'm half serious. Certainly many people do math and other activities for recreation, but more importantly boredom as an evolutionary strategy stops us from spending too much of our resources on one activity and motivates us to pursue other activities that are or may become important for our present or future well-being and even survival whether we are aware of that or not. Some speculate that it was homo sapiens sapiens (twice wise man) superior innate curiosity and imagination that allowed us to survive while Neanderthals became extinct (except in the NFL). It is the ensuing eclecticism that allows us to adapt and survive in an ever changing somewhat unpredictable world. Of course, one is still left with the dilemma of how to rationally allocate available resources, but it is clear from the progress emerging from the synergism between the sciences and technology and mathematics that exploratory math plays an important role in expanding these resources. Drives (productive or not as the circumstances dictate) generally trump principles, and we should foster the productive ones in the appropriate situations just as we foster the play of our children not just for our own and their pleasure but also for developing skill sets, knowledge, and other attributes important for their future prosperity and happiness. Thanks to evolution the pragmatic and pleasurable are often entwined, frequently with unforeseen, wonderful consequences.

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Neil Armstrong: ... there's a fundamental truth to our nature, Man must explore ... –  Tom Copeland Jul 17 at 2:23
    
There's a visceral aspect to doing mathematics for many if not most mathematicians (Thurston comes to mind) that can imbue their lives wtth meaning, a sense of purpose, a calling, a passion, a depth. When we talk about saving lives, we should not forget to consider for what kind of life. We underwrite religion, sports, the arts, and other endeavors that can give meaning, direction, a centering in the chaos or inane that might otherwise engulf us, so given the resources why not support exploratory math? –  Tom Copeland Jul 18 at 16:48

One should ask "How does one justify funding?". If the money is spent or donated voluntarily for research or anything else for that matter, there is nothing to justify.

If the money is acquired from people involuntarily, then you must make the presumption that what is being funded is a better way to spend the money than the people the money was acquired from would have spent it. However, if you are satisfied with such a justification, then you don't have much room to complain if someone thinks they know how to spend your money better than you do and spend it on something that you don't agree with.

Overall, it would be far better to promote the opportunity and desire for private funding.

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This answer has nothing particularly to do with mathematics, does it? –  Todd Trimble Jul 15 at 19:32
    
The question is overspecialized. It would be like asking "How does one justify funding research on Homological Algebra?". One might be tempted to discuss the merits of Homological Algebra. However, that doesn't change the fact that my answer and presumably any correct answer to the posted question would answer this specialized question. –  Alex Williams Jul 15 at 20:38
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(A correct answer?) The question was about justifying funding for mathematics, which properly would take a different form to how one would justify funding for cancer research or for the fine arts, say. As I read it, your answer is an umbrella response on the merits or demerits of public funding in general (and insofar as it is not really about mathematics any more, I think it is arguably off-topic). Meanwhile, your assertion that the question is overspecialized is, of course, an opinion and one which is I think more in the purview of MO meta, if you really want to take it up. –  Todd Trimble Jul 15 at 21:00
    
I understand your point of view. The last line of this posting's description "not just so I can write more honest research proposals, or in case it comes up in an argument, but so I have an answer for myself" lead me to believe that my post was relevant. Sometimes things become clear when you attempt to answer them from a more general point of view. –  Alex Williams Jul 15 at 23:56
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I see you are relatively new to MO (and so welcome!). But maybe I should say at this point (wearing now my moderator's hat) that socio-political opinions -- and your post could be read this way -- are considered strictly off-topic. Quite typically they are deleted. I'll let it go (and will comment no further on this), but let me refer you to mathoverflow.net/help which you can read to get a sense of the norms and guidelines here. –  Todd Trimble Jul 16 at 2:54

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