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I post this question on politics.se, as suggested by this answer of Bjørn Kjos-Hanssen.

The research in pure mathematics has (generally) no other choice that being funded by public organizations and universities, so that the money allocated to it (for new positions, new institutes, conferences...) depends on decisions of people with high responsibilities (generically politicians), who are (generally) unfamiliar with this research area.

Question: Why fund the research in pure mathematics?

Through this question, I would like to know and understand, on one hand, the main arguments of the politicians (or eq.) for deciding to allocate a particular proportion of their budget to the research in pure mathematics, and on the other hand, the main arguments of the mathematicians themselves.
I'm also interested in new (or not sufficiently developed publicly) arguments.

Remark: In order to be fair, the arguments in support or against (either some or more) fund the research in pure mathematics, are likewise requested (see this comment of Steven Landsburg).

Edit : See also the new mathoverflow post How does one justify funding for mathematics research?

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  • "and also the arguments of the mathematicians for convincing to have enough funding" is awkwardly phrased. Do you mean to refer to the arguments that mathematicians use? – Sam I am says Reinstate Monica Apr 8 '14 at 22:19
  • @SamIam: Yes. The arguments of the politicians on one hand, and of the mathematicians on the other hand, are not strictly the same. – Sebastien Palcoux Apr 8 '14 at 22:35
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    I would ask "why do we fund politicians?" instead. They seem useless compared to Mathematicians. – CsBalazsHungary Apr 9 '14 at 9:30
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    @CsBalazsHungary - because 53% of population in USA is too dumb to NOT believe politicians when they say that they are more useful than anyone/thing else. – user4012 Apr 12 '14 at 12:10
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    probably a deeper question is "why should govt fund science & research in general" where math is just another basic/ fundamental science. the answer must be highly related to the economic system. ie capitalism vs socialism vs communism all have different answers in theory & practice. capitalism in particular increasingly seems to have little investment to offer scientific advances that are not immediately monetizable. could record levels of wealth inequality exacerbate this issue? this is a complex topic also tied with R&D investment by govt/ industry. – vzn Jul 31 '14 at 5:41
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The most obvious politician's argument against funding research in pure mathematics is that it costs money and it isn't useful. I'm aware that there is a point of view that holds that the state should not fund research of any kind, but it is very alien to my milieu, so I won't presume to explain it. As I'm closer to mathematicians than to politicians, the bulk of my answer will take their point of view.

The very definition of research is that you don't know in advance what you'll find. If you take researchers and tell them “invent something useful”, what they'll only find very small increments to what already exists. The only way to make significant advances to science is to have enough researchers and tell them to search in whatever direction they feel their efforts will lead to something. Research is like art production in that way: you can't have all books be bestsellers.

Even mathematicians themselves cannot reliably know how useful their work will turn out to be. This is true not only while they're searching, but even after the fact. G. H. Hardy was famously proud that his work in number theory was particularly pure mathematics, where pure is to be understood as both noble and useless. Yet some years later number theory turned out to have important practical applications to cryptography, with considerable military and economic importance.

Additionally, science proceeds faster when scientists share their ideas. (I don't have a citation for that, but it is a commonly shared sentiment among researchers.) In order to maximize the productivity of scientists, there has to be some who do work in more theoretical fields with no obvious applications, and who are there to discuss with more applied scientists, teach them, validate or refute their ideas, etc. Diversity pays.

This drives the conclusion that in order to be efficient, research requires a large enough number of cooperating researchers, at least a fraction of which have no imposed goal. This requires a sufficiently large organization that applies policies that encourage sharing rather than competition and do not cull the least productive units. This is better suited to a state than to a private enterprise (but it can also work in a patronage system such as practiced by private universities).

On a different track (but related to the remark above about cooperation between ivory-tower scientists and economic-value scientists), having a reputable pure mathematics department attracts students and fellow researchers. The best students tend to be attracted to the best professors. Some of these students will end up being pure mathematicians themselves, but others will produce economic value in a more direct way (applied researchers, engineers, …). In terms of economic competition between countries or other environments, there is an advantage to being the place with the best mathematicians.

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    +1. I would also add about national prestige. Science is like sports: if a scientinst from a country earns a Nobel prize, takes Fields medal or makes another brand achievement, this is like an athlete taking the gold on the olympics. – Anixx Apr 9 '14 at 9:21
  • You can try to ask on CogPsy or Skeptics SE about citation for "exchange of ideas" benefit. I remember several research papers on the topic I saw before. – user4012 Apr 10 '14 at 20:34
  • Also, the second part of your paragraph #5 is directly contradicted by paragraph #6. Colleges can easily be private entrerprises for the purposes of attracting faculty that will attract good studends AND grantrs; and most of the best ones were always private. – user4012 Apr 10 '14 at 20:36
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    @DVK “It costs money and it isn't useful” is a fairly common remark. A specific politician doesn't come to mind with respect to mathematics, but France's former president (in)famously said something similar about literature. – Gilles 'SO- stop being evil' Apr 11 '14 at 9:15
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    I would stress the correlation between high-level teaching and research opportunities. For a university, as a public teaching institution, it is essential to offer its faculty good research opportunities, if it wants to attrack the best teachers. The students who trained by such top-level mathematicians will then be able to to learn and apply new mathematical results in other areas, such as applied mathematics, computer science, physics, computational biology, economics, etc., and thus stimulate scientific progess also in "applied" sciences. – UwF Apr 12 '14 at 10:51
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I wish to provide an answer with a concrete example of why funding pure mathematics is a fruitful endeavor.

I use an application on my Android phone called "Google Maps." It allows me to obtain directions to any place in the world. This application exploits what is called a Global Positioning System (GPS) in which we have satellites that orbit earth while broadcasting their positioning and the time they broadcasted. There is one problem though: Einstein demonstrated that time is a relative phenomenon. Time can be affected by changes in gravity and by increased velocity. If we don't correct for this issue then the satellites give faulty readings and eventually Google Maps stops working all together.

The concept of general relativity was necessary in order to correct the time that the GPS satellites were reading. Albert Einstein's work in Theoretical Physics turned out to have a very real application and the precise mathematical equations he worked out did have a very useful application.

I wasn't asked to defend Theoretical physics. Why bring up Einstein? Because Einstein didn't work in a vacuum. He took heavy influence from a (Pure?) Mathematician by the name of Bernard Riemann whose construction of Riemannian Geometry laid the groundwork for general relativity. I am pretty sure Riemann's motivations for studying these questions were academic. In other words a project that was once pure mathematics became an application which helped society.

This is my favorite story but by far not the only example of how pure mathematics impacts the research infrastructure. I remember the poster bringing up Hardy's quote about number theory. Well sorry to say but number theory is now applied in encryption software to help us fight against the data breaches. Pure mathematics lays the groundwork for future projects to succeed.

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    Another invention is the one I am typing on right now. About 100 years ago a Logician by the name of Alan Turning was looking at questions related to paradoxes in mathematics. He stumbled on a concept known as a "Turing Machine" which we now call a computer. – Daniel Parry Apr 16 '14 at 5:50
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    100 years ago, Alan Turing was starting to talk... He was born in 1912... – Bernard Massé Mar 15 '15 at 0:03
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First of all, you're under the misconception that math requires tons of funding. It may be true for modern physics; but most branches of pure math require a trivially insignificant amount of money (basically, a living stipend for a couple hundred mathematicians who are great enough that you don't want to waste their time on teaching-for-a-living; some office supplies; and if you're really forward thinking, a larger one-time outlay for a fund whose interest pays for prestige prizes ala Fields Prize).

In practical terms:

  • NSF's FY 2014 budget request is $7.626 billion (src)

  • Division of Mathematical Sciences (DMS) funding in 2013: $237.77 million (src) - that's 1/20 th (5%) of total NSF budget; basically a statistical error in the grand scheme of US budget.

    • Even that's a gross overestimation, since DMS covers applied mathematics and computational mathematics which naturally have much higher outlays due to computer hardware needs. I wouldn't be surprised if real pure math outlays are under $50M.

    • Then, you subtract money wasted on non-math related BS that politicians hid in there ($5 Mil for "clean energy" - what's THAT have to do with math?; $3.5M in sustainability).

      • Plus $17Mil for "diversity initiatives" which have NOTHING to do with "pure math", and another $5-6M for other educational initiatives.
  • All in all, you'd be left with pure math spending being $20Mil... that's 1/200th or 0.5% of entire NSF budget... and a rounding error of a rounding error of a typo for US budget.

Based on that, and given the practical need for pure math research as being the driver for much of technology innovation (all of our computer revolution is based on pure math work of preceding couple of centuries; all of modern finance and economics is based on previous "pure" math; I won't even go into cryptography and number theory dependency since Gilles covered that. So it's a pretty easy sell to spend $20Mil with not much political gnashing of teeth by pretty much anyone. The only reason that there's not more being spent is probably because there's no need for more. There aren't all THAT many mathematicians on a level that warrants supporting their lively-hoods so they can do pure math research.

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  • Thank you for all these details. If the recruitment in the USA is similar to that in France, then grossly 1/10 of the PhD in pure mathematics (who want to continue the research) find a permanent post, so if the budget in pure mathematics is so little, why not just multiply it by 10 for engaging everybody (because the cost in pure mathematics is meanly human resources)? I ask you this question. An answer could be that this is feasible, but useless because there are enough mathematicians for the reasons (you cite) why we fund pure mathematics research: I question these reasons and this argument. – Sebastien Palcoux Apr 10 '14 at 22:31
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    @SébastienPalcoux - I seriously doubt that there are so many Math PhDs who are SO talented that it'd be a waste of their talent to teach or hold some applied job vs doing pure research full time (having almost been one of those "not all that great" ones, I feel entitled to such an attitude). – user4012 Apr 11 '14 at 4:11
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    I agree with most of what was said there, but feel obliged to point out that 237M is about 3% of 7626M. Still just a blip on the radar, but not quite as insignificant... – Martin Hairer Apr 12 '14 at 0:37
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    @SébastienPalcoux - you'll have to present evidence that 90% of Math PhDs (1) Want to do research [I personally dropped out of Math PhD program specifically because I didn't want to, so it's not a facetious point] ; and (2) That they are good enough math researchers that they need to be funded as such. Ability to write a PhD thesis doesn't mean that they will produce over their life the research that makes them worth supporting as math researchers. – user4012 Apr 12 '14 at 12:35
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    @DVK: I have been involved in several hiring committees at the associate professor level (in France). The typical number of applicants for a position in pure mathematics without specific profile is 170 to 200. This year, we have over all France about 20-30 such positions (CNRS+universities). From my experience, about half the applicants have a solid record and make me feel reasonably confident they could stay good mathematicians for a career long. So, the ratio is probably not 1/10, but it is certainly more than 2 or 3. – Benoît Kloeckner Apr 12 '14 at 19:14
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To make it obvious, take an example: if somebody finds a very efficient prime test then all the computer safety (banking, military data, corporation safety) is dead. If somebody publicly finds the mathematical method, it will be a well known fact, if not, that person will have the power (let's be optimistic, that person wouldn't use it) to crack the best known cryptographies. Theoretical mathematics is one of the most important sciences, since it gives new methodologies and points of views for all other sciences. It rarely gives new breakthroughs, but whatever we call applied science such chemistry, biology, etc... are basicly build on pure mathematics.

The conflict between politics and pure mathematics might come from emotions. From rational perspective, it is obvious, mathematics need funding - is it private or state, might be the subject of debate - but present day mathematics need good support of calculation capacity sometimes to verify their theories, and to be fair: Mathematicians are also human, they need to eat, live somewhere, and their work is usually not something which can be immediately sold into industry use.

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    We have efficient prime tests and they are widely used to find big primes (e.g. to be then used as private keys in cryptographic protocols). I suppose you are thinking about integer factorisation which if we could solve efficiently then e.g. the RSA would be breakable. – David Herskovics Aug 11 '15 at 7:52
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From the politicians point of view, there could have the following aspect (I'm not too much happy with):

It could be important for the politicians to maintain a corps of people trained to intensive thinking (by solving any new mathematic problems), as an insurance in the case of a new major conflict (or any unforeseen problem). In fact, such people can be useful during a war, see for example the recent biopic of Alan Turing during WW2.

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I disagree that there's "no other choice" than to be funded by public organizations. I think you meant to say that it's difficult to make it pay a profit in a reasonable timescale.

But plenty of private organizations do mathematics research, including MIT, CalTech, Harvard, etc. Private universities can easily do pure research on whatever topic they want, tempered by the demands of academics, students, and donors.

Moreover, there's no reason that other organizations could not also donate to pure research in mathematics. This could be anything from a kickstarter campaign to a corporation (maybe a tech company?) seeking to find creative ways to market itself. It could also be a few billionaires who wanted to invest in something new and innovative and nerdy - just as so many of them have poured money into space-related and energy-related causes. Charities could also be formed to pursue grant money to give to mathematicians (like a much bigger Fields Medal award or a MacArthur grant solely for mathematicians).

If anything, the government is least suited to funding mathematics and least interested in doing so. The only difference is that a government has a ready funding source. But the politicians who dole out money aren't particularly interested in funding mathematicians because that isn't going to win any appreciable number of votes - but wasteful spending will lose votes, as will cuts to welfare, nutrition programs, education, etc. Math takes a back seat in government spending because it's just not a very popular priority among voters.

I think the problem is not that it must be done only by the public. The problem is that most people think there are much more pressing priorities for money. Millionaire nerds and geeks are focused on things like space travel. Charities focus on diseases and nutrition. Tons of activists are tied up working on the environment, energy, microfinance, and other charitable fields.

The low level of support for pure mathematical research is not a failure of institutions to plan ahead. It's the success of institutions to obey the aggregated choices of the public. If only a small number of people care about a particular social goal, then it makes sense for that social goal to only receive a small amount of funding. It's incumbent on mathematicians and other groups in a similar situation to start convincing people that their social goal (i.e. pure math) is meaningful enough to be funded. If it were me, I'd focus on landing a big endowment from somebody like Zuckerberg or Musk, rather than the Congress.

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  • Thank you for your answer. About public/private I see the confusion: I'm French and in France all the universities are public (except very few religious universities, but I don't think there are pure math. institutes there). When I wrote "public organization" I had mainly in mind "universities", and you're right, there are private universities (out of France) with high level pure math. institutes. I will improve the post about that. Anyway, my question is not "who can fund?" but "why fund", and I'm looking for detailed arguments. – Sebastien Palcoux Apr 8 '14 at 23:33
  • Note that in France there exists a public organization of research separated from universities, called CNRS. – Sebastien Palcoux Apr 8 '14 at 23:51
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    The research at all of the institutions that you mention is still largely funded through government research grants (NSF, NIH, DOE). Private money at those institutions is typically used to pay for facilities and scholarships. – Chris Mueller Apr 9 '14 at 15:40
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    "If only a small number of people care about a particular social goal, then it makes sense for that social goal to only receive a small amount of funding." = From a 'common sense' POV, yes, that's true. Though it probably should be noted that from a true benefit to society POV, that's not true by any means. It's often imperative that minority causes become elevated above and beyond the percentage that are advocating for it. It's, of course, EASIER to fund causes when there's a large amount of backers. – user1530 Apr 9 '14 at 17:45
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    @DVK it's all relative of course, but in the context of the question, investing in research typically means investing in human resources, which costs money. Yes, investing in Math should be cheaper than investing in astrophysics. – user1530 Apr 10 '14 at 20:45
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The other answers have all made good points. A few other additional points:

  • Funding mathematics employs mathematicians and the grant money that they spend has further economic impact in the community as a result of the "multiplier effect." Creating jobs and boosting the economy with spending is something that typically makes politicians happy and helps win the support of the people who benefit.

  • One answer notes that G. H. Hardy's work in pure mathematics ended up having national security benefits. Another notes that having people like Alan Turing in your society can provide a resource when you need it as his skills were used in WWII. There is another related argument too: The skill set needed to do pure mathematics is a dual use technology. Someone who does pure mathematics can also do a lot of nuclear physics and other national security relevant cryptography with their skills if they aren't doing pure math. And, if you don't fund them, someone else might. By funding pure mathematics you are preventing hostile competitor nations from securing the services of this talented mathematician who might otherwise be doing applied math for your enemies, driven there by an inability to find employment in his or her true calling at home.

  • Pure mathematics research is overwhelmingly conducted by university professors and is historically a field pursued by the "best and the brightest" in the field. By supporting pure mathematics, you are increasingly the ranks of people who are, in general, top quality professors in advanced math topics, which furthers math education, which creates human capital that increases the productivity of the country.

  • As noted in another answer, pure mathematics funding is one of the least expensive forms of research to fund that there is. Basically, those dollars are going towards hiring graduate research assistants for professors, funding trips to conferences to discuss math with other mathematicians, and paying for mathematics journal subscriptions. A $50,000 grant would be a big one in pure mathematics. So, for $1,000,000 you can support 20 pure mathematicians in their research. For the cost of a single jet fighter (about $100 million), you can support 1000 pure mathematicians for twenty years. The odds that 20,000 lead investigator years (perhaps 1,000,000 mathematician-hours) of research will produce something, sometime that has value comparable to a single jet fighter out of many hundreds (despite the fact that there have been only a handful of dogfights in the last twenty years and there is a good likelihood that this will also be true in the next twenty years), is decent. As the example of the pure mathematics that made general relativity which made GPS and satellite phones possible illustrates, you never know when an obscure idea can produce immense, incalculable benefits.

  • A corollary to the fact that pure mathematics sometimes has tangible economic benefits, even though this is very rare, is that it doesn't make sense for anyone else to fund pure mathematics as a business proposition. So, if you don't fund it, it won't happen to nearly the same degree. So, funding pure mathematics makes possible black swan type outlier discoveries that otherwise simply wouldn't happen at all. The logic is somewhat akin to that of a lottery player saying, "If you don't play, you can't win.", only no one knows for sure what the odds are of a desirable result from your investment in pure mathematics research.

  • Pure mathematics can provide national or regional prestige even when it has not direct economic value, in much the same way that commissioning a work of art, having an Olympic athlete win a gold medal, or producing a grandmaster chess champion does. When your country solves Fermat's Theorem, or some new major unsolved problem in pure mathematics, your country looks good. In a way, it is a form of conspicuous consumption in the same vein as historic investments in opera houses, stadiums, art museums, historic landmarks, cathedrals, clock towers, or Mars exploration missions. The fact that your country has money to burn on a profound human achievement with no intrinsic applied value in the short term is a way of signaling that you are a prosperous country and that makes voters feel better about their country and you as one of its leaders. Also, while many symbolic achievements of this kind in cultural spheres are often politically controversial, nobody is going to be offended if you sponsor a mathematician who proves the Riemann Hypothesis or finds a simpler proof to Fermat's Last Theorem, so there will be no backlash when you get symbolically notable, but economically useless results.

  • Further, in a national security sense, if you demonstrate that your mathematicians can do very hard things that no one else in the world can in public in benign pure mathematics research, you are sending a message that you may be doing something equally sophisticated in secret, that has military applications that your adversaries don't know about. This could add an element of uncertainty to the calculations of anyone plotting against you that might discourage them from taking harmful actions against you because you have provided credible evidence that you might have a "secret weapon" up your sleeve.

  • Maybe you are a corrupt politician, and your son or nephew or sister is a pure mathematician and you want to divert economic benefit to them. Or slightly more strategically, maybe someone you need the support of, like a key swing vote in the U.S. Senate, has a relative who is a pure mathematician or has a pure mathematics hobby, and this will help you curry the favor of that person for sometime that you need a favor from them in the future.

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I think it's important to imagine a world without funding for pure mathematics (the counter factual) when looking at whether or not to fund pure mathematics.

Without math funding, the general public would have minimally more tax dollars, but more importantly they would have many more brilliant minds in the workforce. Genius mathematicians probably also make fairly amazing pharmaceutical researchers, entrepreneurs, or engineers, so conceivably there could be more advances in the applied sciences and wealth creation in a world without as much math research. Maybe some people who would accomplish wonders in the math world would fall apart in a more practical setting, but some might also create much more wealth and happiness in a field funded by investors trying to maximize creating things consumers want.

Unfortunately, current statistical understanding seems quite inadequate to satisfyingly answer the question of what the impact of math research is relative to this counter-factual world. Even if we could adequately explain past math research's impact relative to this counter-factual, who's to say we haven't already passed the point of diminishing returns to math research?

I don't think we can say the answer to this question is obvious to one side or the other. Perhaps we will be able to someday if there is more research into pure mathematics.

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Mathematics research is compared to particle physics research is very cheap. They don't have to build large accelerators encircling small cities.

Pure mathematics research is helpful to the larger scientific community given how often mathematics is implicated in scientific research. Here's a simple example:

Category theory arose from questions of covariance in algebraic topology. Both subjects seem far removed from the typical mathematics that are used, say in engineering. However, category theory is playing an increasingly large part in helping to organise the vast amounts of mathematical research already done. This is important given how much young mathematicians are expected to know to engage in research.

Moreover, research in subjects that aren't directly utilitarian is part of aimimg towards social goods and ideals. For example, there is also research in anthropology or art history.

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  • Being cheap isn't a reason to fund it. – Joe W Apr 21 at 23:35
  • @Joe: Why not? Anyway, I point out also that helps underpin research in other disciplines even in such basic subjects as calculus used in engineering texts. – Mozibur Ullah Apr 21 at 23:42
  • Just because something is cheap doesn't mean you should do it. There are many other reasons to fund it but the price tag isn't one. – Joe W Apr 22 at 2:44
  • @Joe W: Did you notice the two other paragraphs that I wrote? (Three now, since I made an edit since) where I describe how pure mathematics is of value to the wider scientific community? This is a reason, then to attach costs to it is to conduct a cost-benefit analysis. Hence I'm saying that pure mathematics is cheap considering the benefits it brings to the wider scientific community and surely that the latter requires supporting goes without saying giving the technological sophistication of our civilisation. – Mozibur Ullah Apr 22 at 10:12
  • Your answer seems to center around it being cheap as that is how it starts. Being cheaper shouldn't really be part of the answer at all. – Joe W Apr 22 at 12:29

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