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On January 19 2012 17:32 Muirhead wrote:Show nested quote +On January 19 2012 17:22 munchmunch wrote:
Don't forget about Riemann. We would never have had relativity if Einstein hadn't learned some Riemannian geometry. But could Einstein have expressed enough of his ideas that physicists could have developed a language like Riemannian geometry themselves? Certainly it would have been a lot slower than having a ready-made language in place! But does the existence of the occasional ready-made language really justify the work on mathematics that could be spent working on other things?
We don't spend that much on high end pure maths as it is, and every time it speeds up a new super important advancement by 10-20 years it is worth more than all the money ever spent on maths together. Almost all of the GDP of the world today has its foundations in quantum physics, imagine if that development took 10 years longer then we would basically be 10 years behind in technological advancements overall.
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Studying math is more like studying art, and gets funded more like art than science. Some of the post-classical mathematicians were funded by benefactors, and things like NSF grants work like that today.
It just so happens it can be useful, and interesting questions are posed by actual problems, but it can exist perfectly well on its own.
Is it a valuable use of time? There are much more "valuable" uses of ones time than most areas of interest in science. String theory, searching for dark matter, general relativity, even a lot of the more practical applications in space exploration is largely useless to the great majority of people. A lot of new medicines being developed are for highly specialized circumstances and are so prohibitively expensive that most of the world will never be able to use it.
Overall, engineering is much more useful. And social sciences may be even more useful to more people, and politics more so (potentially).
We fund math because we are curious about the world (or in the case of math just curious), which has been part of a reason why science has been funded. Science being useful is a more important reason, which is why its research value is higher. If there is ever a very important application in "theoretical" math, an influx of money will be there.
Most mathematicians are paid much less than CEOs, lawyers, other businessmen, and professional athletes (which competitively might be comparable), so the field is getting about what it's valued.
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On January 19 2012 17:26 PolskaGora wrote:Show nested quote +On January 19 2012 17:10 Muirhead wrote:On January 19 2012 16:53 PolskaGora wrote:
Well, my opinion kind of agrees with that of Argument 1. Consider the fact that Isaac Newton invented differential and integral calculus way ahead of his time. Even though it wasn't really all that essential back in his time, his development of this math paved the way for HUGE engineering development, and practically invented methods for other, new (I'm speaking from an aerospace engineer's point of view) engineering fields to be possible that were essentially impossible to practice prior to calculus. I'm sure the same holds true for mechanical engineering.
So basically, to sum it up, without Newton we wouldn't have any efficient methods of transportation developed yet (or at least it would be significantly delayed), and as we all know transportation is required for trade, and trade drives innovations, which drives human development. As for your friend's rebuttal, keep in mind what I said about transportation being significantly delayed without the early development of calculus. If we waited around until we needed calculus to develop it, human development would have been significantly delayed. We don't know what new innovations will be fueled with a new system of doing math. Perhaps it will help Physicists make sense of how to travel near the speed of light? Their research could then help us aerospace engineers develop new rocket engines that have high thrust but maintain a high specific impulse. Why not invent a new method of doing math early on that could make engineering even more efficient? There's no point in waiting around. I agree with you to a large extent. I know how my friend would react though. He would say that randomly developing mathematical theories would indeed occasionally yield an application to engineering etc. However, he would propose that: (a) The engineers would invent the math themselves when they needed it, perhaps with a lot of work. (b) The utilitarian hit to society coming from engineers' taking a while to invent this math is exceeded by the waste of talented thinkers we have working on parts of math that will (for whatever random reason) never be applied. a) There wouldn't be enough time for this in the line of work. Engineers have very strict deadlines, and most of the time they are working their asses off to meet them. They don't have enough time to be bothered with inventing new math. That's the mathematician's job. The only way that could be realistically possible is if we were to gamble our time by choosing to use it on developing new math instead of doing our job in the hope that in the small chance that if we are successful it will help out in the long run, which is unrealistic and could have pretty bad consequences for our careers if you know what I mean. b) Hmm, I'm not really sure how to respond to this, it makes sense logically but I don't think it would practically. But as I mentioned above, engineers usually only create tangible things, not new methods of doing math, so if we want to keep inventing new math efficiently it would be best if we had minds working on that separately.
So we need mathematicians because the engineers don't live in a society that lets them sit back and develop math when they need it! In this view, the random nature of mathematical progress is an effect of the people who are in a position to develop the math not being in the position to understand the applications. I agree... perhaps the institution of "mathematician" is not really efficient unless we realize that society works through people, who only have finite amounts of expertise. It takes constant, slow, two-way communication across cultures to get things done, and that is inherently inefficient.
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On January 19 2012 17:10 Muirhead wrote:Show nested quote +On January 19 2012 16:53 PolskaGora wrote:
Well, my opinion kind of agrees with that of Argument 1. Consider the fact that Isaac Newton invented differential and integral calculus way ahead of his time. Even though it wasn't really all that essential back in his time, his development of this math paved the way for HUGE engineering development, and practically invented methods for other, new (I'm speaking from an aerospace engineer's point of view) engineering fields to be possible that were essentially impossible to practice prior to calculus. I'm sure the same holds true for mechanical engineering.
So basically, to sum it up, without Newton we wouldn't have any efficient methods of transportation developed yet (or at least it would be significantly delayed), and as we all know transportation is required for trade, and trade drives innovations, which drives human development. As for your friend's rebuttal, keep in mind what I said about transportation being significantly delayed without the early development of calculus. If we waited around until we needed calculus to develop it, human development would have been significantly delayed. We don't know what new innovations will be fueled with a new system of doing math. Perhaps it will help Physicists make sense of how to travel near the speed of light? Their research could then help us aerospace engineers develop new rocket engines that have high thrust but maintain a high specific impulse. Why not invent a new method of doing math early on that could make engineering even more efficient? There's no point in waiting around. I agree with you to a large extent. I know how my friend would react though. He would say that randomly developing mathematical theories would indeed occasionally yield an application to engineering etc. However, he would propose that: (a) The engineers would invent the math themselves when they needed it, perhaps with a lot of work. (b) The utilitarian hit to society coming from engineers' taking a while to invent this math is exceeded by the waste of talented thinkers we have working on parts of math that will (for whatever random reason) never be applied.
But to develop new maths you have to already understand actual maths. Engineers are already really busy applying the maths we constructed for them, if they had to develop maths themselves, they'd have to learn so much things they would turn into mathematicians... As for (b) I don't think your friends realizes the difficulty of what he proposes.
Some poster made the point that we can't really predict which part of maths will be "applied" (in a sense of being useful enough to be applied). Also what do you mean by "applied maths" ? Because there are practical applications (like statistics, or effective computation, or cryptography) and theoretical applications (physics, chemistry, computer science...). The example of physics is maybe the most interesting : theoretical quantum physic is nowadays 90% highly non-trivial and delicate mathematics, but physicians know how to interpret what they are writing, whereas mathematicians know how they can write more thing.
To me, math is a language. As a physicist I need math to "speak"/prove how the world works. Theoretical mathematicians come up with new ways of speaking that can be applied to understand the world better.
I kind of agree with this. Every time I describe maths to other people I say this "we build the tools for other to use them. Without us, you could not develop technology ; without you, we would still live in caverns." By tools, I mean cerebral tools and language is imo exactly this.
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I hate math. It makes me rage. I sure am glad that I took calculus since I use it so often in my daily life.
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On January 19 2012 17:28 Muirhead wrote:Show nested quote +On January 19 2012 17:20 HaNdFisH wrote:
I think one of the big purposes of theoretical mathematics is to provide rigour for other areas of study. As a physics PhD student, I don't have to fully understand the work theoretical mathematicians have put into differential geometry, but can use applications of their work to do general relativity, knowing that the results and concepts I am using have been rigorously proven. Statistics is another example of this, people doing applied work generally want to just plug their numbers in and get a result. It is the job of the mathematician to provide the method they are using and to make sure it is valid.
In terms of pursuing the development of maths for it it's own sake, the Binary numeral system and Boolean logic were for instance studied before the development of computers and this work helped significantly with the development of the first digital circuits. Group theory was being studied for its own sake at the same time physicists were struggling with trying to reconcile all the new elementary particles that had been discovered, collaboration here ended up producing quantum field theory and the standard model as we know it.
It is difficult to say what use a mathematical idea will have in the future, most of the work done may never be used practically. This along with slow forced incremental advances, where people have to meet quotas of papers rather than publishing only when they have a brilliant idea does result in inefficiencies. That really is the nature of research though, small incremental progress until a breakthrough or sudden change is found which results in a flurry of new work in that area.
The funding provided to theoretical mathematics is not an excessive amount, being a research mathematician is generally not a high paying or sought after position like a lawyer or doctor. There is provision for those few people who really want to study mathematics and have an aptitude for it to do so for a living. Well the last paragraph isn't very inspiring  . Any given person trying to maximize his use to society should not go into a field just because "there is provision for it," but because he believes it is more valuable than other things he can do. Now whether value should be derived from utilitarian or personal concerns is another matter... We all agree that smart people working hard developing theories will eventually derive useful things, but much of their stuff won't be useful. My friend would say that means they should be doing something else which is more immediately useful. The physicists could have developed group theory on their own, much more slowly if it didn't already exist in math, but perhaps that societal hit is not as bad as a bunch of really smart people spending all their time on more (unpredictably) useless branches of math. I think the value of math is not to provide a ready made language, like group theory, to physicists (though that is a useful side-effect of doing math). The main point is to study, rework, connect, digest (and part of that is your "rigorizing") the languages that are already there.
My point with the last paragraph was that the people who really want to, and are suitable for it can pursue it. The fact that it isn't a glamorous high paying job means that if someone thought their time was better spent elsewhere they would leave and earn more money in industry or similar. If you stick with the field it means you really want to be there, and it shouldn't matter whether this is because you believe you are most useful to society in that position or because you simply love doing mathematics.
The line between mathematicians and theoretical physicists is often a non-existent one, I imagine if there were no mathematicians doing group theory then some of the physicists would have essentially become mathematicians and worked on group theory as a subject in its own right. I guess a better point here would be if some areas were felt to be useless, the funding bodies that give grants to mathematicians can change their policies to force research away from "pointless" areas. How you decide what areas are useless though is a difficult question, if you ask any random person on the street they will probably say that any maths beyond addition/subtraction and multiplication is pretty useless. If you ask a practicing mathematician I doubt they will say their area is useless, otherwise why would they be studying it. The current system seems to me to be the best way of dealing with this.
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If the only measure of the importance of a field is practicality, there would be no University -- only trade schools. All the stuff we have now would work great, but there would be no progress.
Also, all the theoretical math people do it because they are good at it and they love it. Theoretical math is not a field to enter looking for high pay and fame. You put those eggheads to work fon anything but what they love and they will wilt like flowers. They will be taking the jobs of the people who actually want to do the practical work.
People doing what interests them, challenges them and makes use of their talents results in greater total happiness -- isn't that a value worth preserving?
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...Are you kidding me? Math is the only pursuit that yields truths. I don't want to derail with a philosophical side-battle, but I don't know what you want out of the universe if not knowledge.
Aside: Now, granted, we discover theorems "at random", and machines can enumerate all the theorems anyway, albeit in a different order than we would come by them. However, the purpose of mathematical research is to assemble theorems in a way that provides meaning, whereas machine assembled theorems inherently lack any meaning, and might not find the ones we want any time soon. End aside.
Anything science can muster is fundamentally bereft of the same epistemological security math has. I don't believe the universe will ever be at a point where intelligent agents would do better to hold off on pure math, because that group will always be far outnumbered by those who simply live by their tricks and happenstance knowhow in a happenstance environment.
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Aotearoa39261 Posts
As a masters student in pure math, we study math in the hopes that one day someone discovers a use for it. At the end of the day, computer science is just math, economics is math, engineering is math, physics is math, there's math involved with some areas of genetics and protein folding, there's some math involved in chemistry etc. Math provides the framework for all of these ares to use and so their own progress isn't hindered by a lack of understanding.
Without the rigor of maths then the method of 'developing things as we go' wouldn't yield the same results. Some of the results in math are very technical and rely on the rigorous framework that math provides. Sure, physicists might have 'guessed' that the theorem was true based on experiments/trial and error but they probably would have guessed some things which weren't true as well.
On January 19 2012 18:00 Mr. Black wrote:
You put those eggheads to work fon anything but what they love and they will wilt like flowers. They will be taking the jobs of the people who actually want to do the practical work.
Yeah, no. The people studying math generally aren't people who are 'egg heads' and we certainly won't 'wilt like flowers'. Most of us are multi-talented and could easily transfer to any math related field. I've also seen some very talented mathematicians choose not to study math and instead pursue another field with success.
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On January 19 2012 17:32 Muirhead wrote:Show nested quote +On January 19 2012 17:22 munchmunch wrote:
Don't forget about Riemann. We would never have had relativity if Einstein hadn't learned some Riemannian geometry. But could Einstein have expressed enough of his ideas that physicists could have developed a language like Riemannian geometry themselves? Certainly it would have been a lot slower than having a ready-made language in place! But does the existence of the occasional ready-made language really justify the work on mathematics that could be spent working on other things? Actually, the real question is, would Einstein have had his ideas (about relativity, I'm sure he would have done great with Brownian motion) if he hadn't known about Riemannian geometry?
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By the same token, why do people allow their fellow humans to exist on arts, sports, or video games? The answer is why not.
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Why do you work out? Will you ever find a situation where you are forced to lift weights, otherwise you will die? No. But we work out to make our bodies stronger and fitter in our everyday lives.
Same thing with advanced math. Maybe it doesn't have any real uses (yet), but by studying it you are effectively exercising your brain. 
Also, if we only studied things that were practically important we wouldn't have any art.
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The sheer beauty and elegance of mathematics is reason enough to hold it in high esteem. It is pure logic, a celebration of some of our best and most rigorous thinking. It is remarkable that reality can be modelled using maths, but even if it somehow had no "practical" application ever, we should still support it just for its capacity to let us glimpse clear, brilliant, pure and eternal truth, even for a moment. A noble discipline if there ever was one and with immeasurable esthetic and cultural value.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty —a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. - Bertrand Russell
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I actually opened this thread thinking this was bringing to Team Liquid some form of debate about why Americans call Mathematics 'Math' while we in the UK call it 'Maths'. I was willing to throw something into that debate, but I'm far too tired to make a proper comment about this now lol :D
I will say to answer:
On January 19 2012 16:29 Muirhead wrote:
So TL why, honestly, do we fund theoretical math?
I'd go with:
On January 19 2012 20:00 writer22816 wrote:
if we only studied things that were practically important we wouldn't have any art.
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The first argument doesn't seem to actually understand what pure math is. There are plenty of pure math problems with immediate applications to real world questions, like computing the cover or blanket time of a graph or analyzing certain PDE. 'Pure math' is not synonymous with 'math which is not currently useful'.
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Yes.
For one key reason.
Algorithms and Computer Science.
Computers are the way of the future. More powerful algorithms need to be developed, and mathematics is key to this.
The universe needs further exploration, and mathematics is key to this.
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I completely disagree with his answer to argument 1. A lot of the stuff he lists was actually developped from a pure maths perspective, or with very little care of their application.
Plus plenty of mathematician are working on stuff which have known application.
Finally, we do fund research in much less useful fields, so why not for pure maths.
Anyway reminds me of this article : http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
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Seriously, such a question is unheard of, almost disrespectful.
"Should we even bother with theoretical mathematics?"
My goodness, if such a thing were to ever happen I'll lose faith in all Humanity. Which is pretty low as it is if such a question is being pondered.
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I don't even understand the argument. Why do we have to justify ourselves as mathematicians? The true value of math doesn't lie in its applications, that's only the economic value. Other fields don't have an inherent economic value but they don't have to explain why it is okay for them to do research.
The true value of math is the acqusition of knowledge. We do math because we are curious about ourselves and our surroundings. Every little bit of knowledge, regardless of subject, adds to the pile. If you question why we do math you should question everything you learn. 90% of all knowledge you gain in school is useless in your day to day life. I know people who get by perfectly fine without any education at all. And I'm fairly certain that's how it's been for a long, long time. Being curious is an essential part of human nature. Satisfying this craving for knowledge is a worthy pursuit no matter who you spin it.
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I can't understand mathematicians as I'm pretty bad at math and I don't have any interest in it. But I do understand that studying what you love is generally the right thing to study. May it be a language, may it be math, may it be biology or management. The question if studying math is useful is asked, you have to ask if studying any science is useful. And the reason is simple. Science can only go as far as mathematics are already. One cannot use the m-theory if the underlying math is not developed enough. Think of science as a tree; Math is the clade and physics, chemistry, biology and even computerscience are branches. No branch can grow longer than the clade has grown. Without progress in math, there is no progress in any science.
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EPT
