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A very kind person, who I respect very deeply, posed a question: Is it a valuable use of time to study theoretical math today? Does it help humanity? Are there better ways to use one's time?
There are many types of math, and I can't understand most well enough to even approach the question. But I can try a little, at least with the sort of math I'm interested in...
Argument 1: Much math is practically important today. The math I am working on is not practically important today, but maybe it will be the math that is practically important tomorrow. How can we predict what will be useful? It seems like pushing math generally forward is the best response to this uncertainty.
My friend's rebuttal: + Show Spoiler +If we really want to evaluate this argument, it is important to understand the conditions under which the important math of today was done. In the case of calculus, differential equations, statistics, functional analysis, linear algebra, group theory, and numerical methods, the important results for modern work were in fact developed after their usefulness could be appreciated by an intelligent observer. There is very little honestly compelling evidence that pushing math for the sake of pushing math is likely to lead to practically important results more effectively than waiting until new math is needed and then developing it. Perhaps the most compelling case is number theory and its unexpected application to cryptography, which is still not nearly compelling enough to justify work on pure math (or even provide significant support).
Argument 2: Math is practically important today. The math I am working on is in a field that is practically important today, and not many people are qualified to work on it, so pushing the state of the art here is an excellent use of my time.
My friend's rebuttal: + Show Spoiler +Consider the actual marginal utility of advances in your field of choice, honestly. In the overwhelming majority of cases, the bulk of research effort is directed grotesquely inefficiently from a social perspective. In particular, a small number of largely artificial applications will typically support research programs which consume an incredible amount of intelligent mathematicians' time, compared to the time required to make fundamental progress on the actual problem that people care about. Here you have to make a different argument for every research program, which I would be happy to do if anyone offers a particular challenge.
My argument: + Show Spoiler +I agree with you. For the most part, applied souls dream up their advances and make them without relying on the mathematical machine. They invent the math they need to describe their ideas. Or perhaps they use a little of the pure mathematician's machine, but quickly develop it in ways that are more important to their work than the previous mathematical meanderings. I think you underestimate the role of mathematics as the grand expositor. It is the tortoise that trails forever beyond the hare of applied science. It takes the insights of applications, of calculus for example, and digests them. It reworks them, understands them, connects them, rigorizes them. The work of mathematics is not useful in your mind because a mathematician does not make a truly new applied advance. A mathematician invents and connects notations to ease the traversal, the learning, and most importantly the storage in working memory of past insights. What is the purpose of a category? An operad? A type theory? A vector bundle? The digit 0? When these languages were introduced, it could always be claimed they were worthless because the old languages could express the same content as these new languages. But somehow the new language makes it easier to conceptualize and think about the old ideas; it increases the working human RAM. And what of the poor student? He who must learn so many subjects is grateful when it is realized that many of those subjects are in fact the same: http://arxiv.org/abs/0903.0340 . Mathematics digests theories and rewrites them as branches of a common base. It makes it possible to learn more insights quickly and to communicate them to the next generation. So young applied scientists, perhaps generations later, benefit by more compactly and elegantly understanding the insights of their forebearers. Then, the mathematician dreams, they are freer to envision the next great ideas: http://arxiv.org/abs/1109.0955So why the mathematician's focus on solving specific problems? Why so much energy to characterize finite groups? It is not that these problems are important. It is that they serve as testbeds for new languages, for new characterizations of old insights. The problems of pure math are invented as challenges to understand an old applied language, not to invent a new one.
So TL why, honestly, do we fund theoretical math?
  
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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.
To not explore theoretical math is like not inventing new words in a language. It will stagnate and die or be replaced by another, more flexible language.
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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.
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Aside from the fact that you never know what people will discover and how it'll impact the world (which is brilliant justification in my eyes - we shouldn't ever abandon the desire to just learn more about how things work)...
Most maths can be found to have real world applications in some way. When I learned that complex numbers and quaternions had real life applications, I was convinced that maths was worth the time we put into it. Hell, if we can find uses for numbers that "don't exist" (quoted because of the arguable inaccuracies with saying they don't exist), we can find uses for other stuff some day.
Also, physics is only getting weirder 
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On January 19 2012 16:39 Cuddle wrote:
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.
To not explore theoretical math is like not inventing new words in a language. It will stagnate and die or be replaced by another, more flexible language.
I think this a very nice simplification of my final argument. Math is language and its value is entirely as a language!
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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. 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.
My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience.
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First of all, applied mathematics relies on pure/theoretical mathematics. That's how it works, right? People get stuff straight in an abstract setting, other people go ahead and adapt it to their given, (more or less) 'real world' applications, and use it. In fact, I don't think the current state of mathematics could have been achieved if everyone just looked at their own problem and just did some mathematical magic to make it work, for some theories, it is simply necessary, to see them in a bigger, broader context.
Second, mathematics needs to be swallowed, digested/processed, and spewn (? is this the correct word? :D) out again so that the next generation of mathematicians can do the same. Look at some older papers/books/whatever of famous results, it is remarkable how streamlined, elegant and fitting into the bigger context these results are formulated now, when one presents them to students, as opposed to how they were written down once. This is a necessary process, just to allow the broad mass of mathematicians to a) understand the theory and to b) put them in a greater context to, possibly, develop generalizations and (!) applications.
So, what I'm saying: Yes, it is of valuable time to study pure mathematics. Does it serve humanity and are there better ways to use one's time? Oh well, that's somewhat too deep and dodgy for me now. :-)
Edit: I just saw your last post. The fact that the foundation and/or first motivation to study a certain abstract, 'pure' problem, often comes from an application (be that physics or economy or whatever) does not change the need to consequently study the abstract problem.
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On January 19 2012 16:56 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. 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. My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience.
Ah, you replied before I had time to reupdate my post. I think my edit has the answer you're looking for. Reread and let me know. 
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On January 19 2012 16:56 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. 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. My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience.
I have a question:
What sort of math do we study and develop for the sake of it?
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On January 19 2012 17:00 RageOverdose wrote:Show nested quote +On January 19 2012 16:56 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. 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. My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience. I have a question: What sort of math do we study and develop for the sake of it?
The calculation of homotopy groups of spheres, for instance
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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.
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On January 19 2012 17:03 Muirhead wrote:Show nested quote +On January 19 2012 17:00 RageOverdose wrote:On January 19 2012 16:56 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. 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. My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience. I have a question: What sort of math do we study and develop for the sake of it? The calculation of homotopy groups of spheres, for instance
Such things can be useful in statistical physics, a subject with countless extremely important applications. Unless you have studied a lot of physics you don't understand how many nontrivial results from maths they actually use.
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Math, I feel is much like most other research which are only marginally significant on the fringes of the social world...but it seems invaluable as one progresses further in almost any field. Much like a language, especially a programming language Um... overall I suppose I agree with your view...although I find myself believing applications that arise like number theory's applications to cryptography are rather invaluable....they essentially secure all of our communications... >_>
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On January 19 2012 17:12 Klockan3 wrote:Show nested quote +On January 19 2012 17:03 Muirhead wrote:On January 19 2012 17:00 RageOverdose wrote:On January 19 2012 16:56 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. 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. My friend would claim that Newton invented calculus not as a mathematical exercise, but because he was trying to understand physics. The mathematics was invented to understand the motions of the world around him, aka applied physics. But nowadays we seem to develop math more "for its own sake", and less to understand things we see and experience. I have a question: What sort of math do we study and develop for the sake of it? The calculation of homotopy groups of spheres, for instance Such things can be useful in statistical physics, a subject with countless extremely important applications. Unless you have studied a lot of physics you don't understand how many nontrivial results from maths they actually use.
I would be interested in a source for this . Are the calculations of extremely large dimensional homotopy groups valuable, or is it mostly about low (say less than 50) dimensions?
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On January 19 2012 17:14 Aletheia27 wrote:
although I find myself believing applications that arise like number theory's applications to cryptography are rather invaluable....they essentially secure all of our communications... >_>
Nah, the only reason we need better and better encryptions is because due to number theory the methods for breaking these encryptions also have gotten stronger and stronger. So the existence of the number theory field is needed because the number theory field exists. If it just magically stopped existing 20 years ago we wouldn't have needed the better methods since we wouldn't be able to break the things we had 20 years ago.
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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.
*edit* The point of math being a language is a good one too, strong notation and/or methods are developed to solve purely academic problems. These tools however can then be used to simplify and generate other results in more applied problems.
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Don't forget about Riemann. We would never have had relativity if Einstein hadn't learned some Riemannian geometry.
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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.
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.
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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.
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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?
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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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Math is truth. It is the application of pure logic. To deny the importance of math is to deny the importance of truth.
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This is really an article about math education, but you should read it. It makes the same argument I would make for studying mathematics for its own sake, but more eloquently than I would have. This one too.
Most people, even many people who extol the virtues of mathematics up and down, seem to be under the delusion that mathematics is a science. As such, it seems only natural that attention to and funding for mathematics research be allocated according to how useful it is -- a return on an investment. But mathematics is not a science; it's an art. The occasional applications of math to practical matters are not the purpose of math, they are byproducts and happy accidents. People who question the usefulness of pure math don't really understand what mathematics is, and people who leap to the defense of pure math with "it builds the foundation for later applications" are sort of missing the point, good intentions aside.
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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?
You know the phrase "Dwarfs standing on the shoulders of giants ". It's because it is So MUCH easier to further progress something that was discovered earlier. Even if it was in completely different setting, ineffective. maybe not even correct. You don't really appreciate that fact till you do research at the frontier yourself (believe me, I also didn't appreciate till I started doing research). Einstein maybe could have developed Riemanian Geometry himself (though Riemann was one of the greatest mathematicians of all time and a genius himself) but I doubt he would be able to develop both Riemanian Geometry and Relativity Theory. He wasn't almighty and there is a limit (even for the greatest of minds) to how much one can think of.
Moreover, a lot of progress is not done by huge discoveries but is incremental. For that incremental process you need past work. It would be a huge impediment not being able to use advance mathematics in any field that applies mathematical techniques. And developing these techniques was non-trivial. As someone said it took years before they became applicable in practical progress. If you read history of math you will see how people struggled for decades to understand the notion of integral, define notion of continuity or understand the notion of cardinality. Tools from mathematics become wildly used with long delay.
Measure Theory and Functional Analysis also seemed too abstract even in eyes of mathematicians that witnessed their development and now are basic tools used everywhere.
And finally, your arguments against research in mathematics can be applied to anything. The rule of research is that bright minds pursue what they find interesting. And no one can tell if what they discover will or will not be useful in the future. These people have passion and talent for doing that.
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A recent example from finance. This is taken from the book by Oksendal "Malliavin Calculus for Levy Processes with Applications to Finance"
"Malliavin calculus were introduce by Paul Malliavin (..) as an infinite-dimensional integration by parts technique.For several years there was only one known application. Therefore, since the theory was considered quite complicated by many, Malliavin calculus stayed relative unknown theory also among mathematicians for some time. Many mathematicians simply considered the theory too complicated compared to the results it produced."
And now it is used all over sophisticated finance. So here we have a contemporary example were even mathematicians themselves and since mid 1990s it's been applied extensively.
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I work very much on the limit between where math can be said to be of practical use: particle physics phenomenology.
Will there ever be any practical applications from TeV scale particle collisions? Meh, not sure. Not out of the question.
Is the science connected to physical reality? Most definitely yes.
Will there be applications coming out from the technical constructions of the LHC? Probably yes.
Will there be applications on the existence/non existence of the Higgs? Not anytime soon for sure, maybe never.
When I saw the title I first open it with the intent to troll ("lol no use for math wtf srsly? i never use math and I do fine!!") but I found a kindof serious well thought through OP, so I though I could actually share some inside info, but then I read the replies and I see several people in the trade already has done so. Now there is not much for me to do here more than comment on the other comments.
The "art" argument is ridiculous. If the only reason to study math would be it's own beauty, then I'd strongly urge every government to cut all funding to mathematics... It'd be like any other art, only that you would have to spend years and years to understand it. 99.9% of the people would have no chance to understand it. Then please spend that money on sorting out the starving situation in Africa instead please.
Cutting funding completely for everything except engineering and trust the engineers to themselves find the mathematics. Well, as some people have said, some engineers would be more focused on actual use, and would not grasp the maths well enough to really advance maths. It would have to be the more math-type engineers to think about how math could be advanced to further engineering. Why not call those math-type engineers "mathematicians"?
There has to be a balance for how much funding is spent on applications, and on theory. Imo, there is a bit too much spent on theory atm. While I have not a good idea of how much is spent on non-application mathematics, I know that quite a lot is spent on particle physics, not least due to LHC. I have the feeling that there could have been better things done for that money. Then string theory...  now THAT is a talent sink. So many smart masters students that get fascinated by string theory and waste their career on this thing existing only in their minds...
I know, because I was very close to be one of those myself. I went to italy to do my master project, and essentially went there and picked the most abstract theoretical subject I could find for my master project (supergravity). I was young and enthusiastic about exploring and pushing the front of physics and understanding. I got offered to stay for PhD there, but decided to go back to sweden for personal reasons. In sweden I presented my master project and afterwards I got some questions from the audience, one of which was "Can you measure any of this?". I don't see myself as stupid or narrow minded or anything, but during the year working on this project in italy, not one time had that though crossed my mind. Measure? What? :o I ended up taking a PhD in particle physics phenomenology in sweden instead, doing things that you actually can measure, somehow exploring nature if you want, although I don't see how my work will ever be of public use.
So yeah, we should definitely keep funding research on theory, even without applications in sight, but in moderations. Theory with possible or even probably application, short or long term, deserve much more funding ofc. Not always easy to tell the difference though. Particle physics for example, and specially string theory for example is overfunded imo. There is a continuous scale
- from engineering that is of clear public use (combustion physics)
- through physics that is barely generating things of public use (advanced quantum mechanics, GR)
- through physics that may be of use (particle physics)
- through physics that is very far from any applications (strings)
- to pure mathematics without any current applications in physics.
these should be funded less the further down the list, but it's not easy to place a specific area on this scale, and not clear exactly how much decreasing it should be.
and OP: do you know that groups are used A LOT in particle physics? Not sure exactly what groups you work on, but I think any work on groups and their classification is of potential use in particle physics. E_8 is the master group in string theory for example, who could have guessed when they first classified the lie groups and found some stupid exceptional group of very high dimension that didn't really fit into their system.
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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?
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That and the whole assumption of this thread is wrong.
The OP thinks a enormous amount is spent on math research, which isn't true (compare it to other sciences or engineering at it's laughable). Also, he thinks mathematicians do pure research 100% of the time, and disregards consulting work, like providing tools/assistance to other sciences/research groups, and all that jazz. Also, don't forget computer science is largely pure math. Without modern CS work in what position do you think we would be?
Seriously, if you take the toll benefit of mathematical research on the long run and the toll that was spent on research, it is ridiculous to think, imo. I just agree that math shouldn't be overpopulated with students who lack real commitment to their research, but preventing that is the job of people who analyse research grants anyways.
Btw, I'm not a mathematician, I'm an undergraduate electrical engineer on my 2nd year in uni.
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On January 19 2012 19:45 Plexa wrote: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. Show nested quote +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.
You are absolutely correct and I framed my statement poorly.
I did not mean to say that a math person could not apply his intellect to a successful career elsewhere--I would not say that it is a matter of ability. Rather, my point is, if what a person wants to do more than anything else is to push the boundaries of theoretical or pure math, and that person was shamed or forced into giving this up to do something "practical," a part of their spirit is sacrificed and you won't be getting that person's best effort. Of course if a talented mathematician decides on his own to pursue something else, the ability to think systematically and logically (even creatively in a way?) will serve him well in almost any field.
By 'egghead' I meant "smart person pursuing an academic (rather than commercial) field." It is a term that carries a negative connotation though, and my point was hindered by using it. To be clear, I do not think of mathematicians as particularly frail or unadaptable. By 'wilt like flowers' I mean to the extent that any passionate person is lessened when they are deprived of the thing that drives them.
I admire math and mathematicians, as well as all those who pursue "pure science" type fields of study -- I can think of no more merit-based system than highly advanced math.
I got up to differential equations before I accepted that I lacked the attention to detail (and other important qualities) to progress further in math. Rather than be angry that I struggled through so much math that I don't use (a common response), I am more sad that I was not more successful, and insecure about what it says about me that I have performance issues with math. I then made a choice to go in another direction and have achieved a measure of success. But thankfully no one with power has ever said to me, "You know what, you are no longer allowed to do [insert things that I love] because it is not useful to society." I would wilt like a flower in my new, "practical" job.
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I think this topic comes up because of two things going on:
1) Mathematics has had tremendous success as a tool in the sciences
2) Many people who do mathematics finding it appealing on a intellectual or artistic level. The unexpected power of new formulations or approaches can be mind boggling (galois theory, complex analysis, cantor's theory of the infinite).
Non-mathematicians that use mathematics in their profession often see 1) but not much of 2). Therefore they have trouble understanding people who prioritize 2) to 1).
Moreover, they aren't completely distinct things. Value in 1) may contribute to 2) and vice-versa.
Mathematicians like to do math. Why make them unhappy by restricting the scope of their investigations to the applied?
I certainly do not think it is a waste of time for a mathematician to indulge in highly abstract subjects even if they have no clear application. What would your friend have them do instead? Go spend his life working in charitable organizations? In terms of social value, it is probably better, he may even help save lives of homeless people.
But it's foolish to have such a "utility" based short-sighted view. People aren't robots, and we have passions for various things. For some mathematicians it may be the search for truth and structure, chess players love to play chess, physicists want to learn more about how our physical world operates.
I know the american culture has very little appreciation or understanding of mathematics. One could attribute this to the lack of general aptitude, but I don't think that's the case.
Part of it is the sad curriculum and AP calculus based approaches to things like multivariable calculus, differential equations and linear algebra, which completely extract the philosophy that the subjects are based on. The other aspect is that mathematics requires patience and careful reflection, difficult for children and nowadays many adults as well.
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EPT
