Keeping Math Real

John von Neumann:

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I’art pour I’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. (from "The Mathematician")

This warning is rather similar to Gordon Tullock’s.   HT to Jeff Helzner.

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  • http://profile.typekey.com/RobinZ/ Robin Z

    Thanks for the quote – as an engineer, much of my exposure to maths has been through the lens of reality.

    (Hmm, I wonder if this is why postmodern texts get so abtruse…)

  • Paul

    So… Re(f(x)) is not sufficient?

  • Michael

    While there is some truth to what Von Neumann says (certainly one should hold his opinions in high regard), I am not convinced that these dangers are very real.

    For starters, almost all modern mathematics is at very least concerned with connections to other areas of mathematics if not “reality” (whatever that means). Even if one is pursuing a particular idea for its beauty, it it rare that one does not have other applications in mind.

    Second, there are always unintended consequences in life. Since mathematics objects, say numbers, describe the physical world, any true statement about numbers is necessarily a true statement about “reality.” As has been proven time and time again, many times when a mathematician explores the ensuing mathematical structures, profound consequences are learned about “reality.” I’m thinking here of the applications of non-euclidean geometry, higher dimensional manifolds, representation theory, etc. Furthermore, explorations into pure mathematics often find applications to “real-world” problems that were not at all envisioned by the explorers. I’m thinking here of the profound consequences on modern day society of cryptography, which is based on much deep number theory that originally began as an exploration of relationships between integers (cf. A Mathematician’s Apology).

    Third, there is certainly great value to humanity to understand deep truths that do not necessarily have immediate applications. Particle physics. Cosmology. Marine biology. Heck, just about any science has people working on understanding the physical world (I guess what the author considers “reality”) for the sake of understanding the physical world. Mathematicians can just do it on the cheap.

  • http://hanson.gmu.edu Robin Hanson

    Michael, do you think math has changed fundamentally in this regard since von Neumann, or do you think you understand the math enterprise better than he did?

  • http://antimeta.wordpress.com Kenny Easwaran

    von Neumann was a great mathematician, but it’s not obvious that this makes him an authority on a general criterion of value in mathematics. Also, he does leave outs for fields surrounded by correlated subjects, and for any “discipline is under the influence of men [sic] with an exceptionally well-developed taste”. I wonder which of these categories he would have put set theory in the 1920’s in, which is a field to which he made some great contributions. Apart from the discovery of the axiom of choice and the few decades when it was first applied to other fields of mathematics, set theory has always been relatively isolated from any sort of empirical sources.

  • http://yudkowsky.net/ Eliezer Yudkowsky

    Re: Robin:

    Ouch.

    But it is best to tread very carefully in applying Aumann reasoning to those who are dead, and who can no longer agree with you.

    Our era sustains us with more strength than we realize, for we tend to attribute our successes to our own wisdom.

  • Caledonian

    Michael, do you think math has changed fundamentally in this regard since von Neumann, or do you think you understand the math enterprise better than he did?

    We might ask quantum physicists a similar question about Einstein.

    When a senior, respected researcher in a field says that something is possible, he is probably right; when he says something is impossible, he is probably wrong.

    Von Neumann is wrong. Mathematics cannot get away from reality, because we only understand it by looking at reality, and there is no danger in exploring mathematics beyond known, practical application.

  • Recovering irrationalist

    Robin, the best genius is still fallable, leading experts are often wrong, and surely Michael is entitled to a differing opinion without being taunted as thinking he’s smarter than teacher.

    Having said that, I agree with von Neumann.

  • Michael

    Robin says:
    “Michael, do you think math has changed fundamentally in this regard since von Neumann, or do you think you understand the math enterprise better than he did?”

    I do not think math has changed fundamentally since the time of von Neumann–if anything, mathematics has become far more specialized than in his day, which leads to explorations that are more likely to further removed from (conceivably) “real-world” motivations.

    I also do not think I understand the math enterprise better than he does.

    But, SO WHAT? Should the discussion end there? Some brilliant mathematician states an opinion about the enterprise of mathematics and we all must agree with him? I do not feel like doing any research on the matter whatsoever, but without a doubt I can find an equally brilliant mathematician who directly contradicts von Neumann’s statement (again, cf. A Mathematician’s Apology).

    On the other hand, by definition I have more hindsight than von Neumann. I have seen extremely deep results from the purest of mathematics used in physics and cryptography, to name just a couple of examples. Take non-Euclidean geometry. Once upon a time, mathematicians removed an axiom from Euclid’s postulates just for kicks. Some beautiful mathematics was created. Then, over a century later, voila! Actual, real-world space turns out NOT to be euclidean–an application of all that beauty.

    If anything, as Hardy points out, there can be much more danger involved in “applied mathematics” (cf. Hiroshima and Nagasaki). And of course, even Hardy, a brilliant mathematician, also has anachronistic thoughts on the enterprise of mathematics, as much of the number theory he worked on has turned out to have very practical applications.

  • http://hanson.gmu.edu Robin Hanson

    Michael, I did not mean to end the conversation, but rather to explore the evidence that is von Neumann’s opinion. Your example from geometry is surely one of which he was well aware. Yes the opinion of brilliant folks on the other side would count as evidence too, but surely the mere fact that you might find such evidence if you looked is not by itself much evidence.

  • Michael

    Robin says:
    “…rather to explore the evidence that is von Neumann’s opinion.”

    I have already provided significant evidence to the contrary that is not merely one person’s opinion. I have not seen any evidence BEYOND his opinion. Even he does not provide any evidence. (I cannot find the reference “The Mathematician” to which you refer, so I do not know whether he provides any evidence outside this quote.)

    Take for another example the Langlands conjecture. A multitude of branches with VERY different sources: automorphic forms, algebraic varieties and Galois representations, beautifully related well after each was independently developed. There are a slew of theorems that relate disparate areas of mathematics or mathematics and the physical world that were discovered a posteriori. On the other hand, I cannot think of any area of mathematics (besides maybe set theory, but I am certainly no expert on that subject) that has been developed for a long time without applications and interconnections to other areas of mathematics.

    “…the mere fact that you might find such evidence if you looked is not by itself much evidence.”

    Fair enough. I do not intend to look for such evidence, so I will instead continue to submit my own.

  • Caledonian

    Von Neumann’s opinion is not evidence. It is data. There is a non-trivial difference.

    I am increasingly unsure what it would mean for mathematics to ‘be like art’. Art is subjective, defined by aesthetic perception – the artist is free to do as he pleases to express and invoke emotion. Mathematics is diametrically opposed: it is utterly objective and utterly constrained. Mathematicians are free to find the systems they explore elegant and beautiful, or not. It makes absolutely no difference, because they cannot change the least part of them.

    Perhaps von Neumann was concerned about the peril of exploring only the systems that appear to be aesthetic, ignoring ‘ugly’ math that the need to model reality might compel us to examine. But his opinion does not seem to be clear, nor well-founded.

  • komponisto

    There is something slightly ironic about posting this warning of von Neumann’s in the midst of Eliezer’s series on quantum mechanics.

    “What could something as abstract as operators on an infinite dimensional space possibly have to do with reality? Everyone knows that reality is composed of billiard balls bouncing around in three-dimensional space!”

    Oops.

    It keeps happening, time and time again: people want to condemn abstract mathematics for supposedly being out of touch with reality, and then the critics turn out not to have a clue about the latter.

    When will people learn?

    (And von Neumann, as one of the architects of the mathematical apparatus of quantum mechanics, should have known better — and probably would have, had he lived long enough.)

  • http://profile.typekey.com/RobinZ/ Robin Z

    “What could something as abstract as operators on an infinite dimensional space possibly have to do with reality?”

    Not to be overly glib, but let me change the question: what could something as abstract as Taylor series on infinitely differentiable functions possibly have to do with reality?

    On a fundamental level, probably not that terribly much. On the level of engineering analysis, a great deal.

    There are a terrible lot of concepts in mathematics. Focusing on the ‘hits’ doesn’t do anything for you unless it tells you what to expect next. And, as far as I can tell, mathematics does not do so in the way you imply.

    (I’m not denying that mathematics provides a lot of predictive benefit, of course – the formalization of finite element analysis is an obvious and valuable contribution of applied math to engineering, turning a shortcut in mechanics to a general scheme for solution of differential equations. This, I believe, hardly refutes John von Neumann’s claim.)

  • Douglas Knight

    The source of the quote.

    I would like to add some trivia:
    I think Gauss’s work on conformal and differential geometry, stemming from his work as a surveyor was a more influential predecessor to GR than axiomatic hyperbolic geometry; the surface of the Earth is manifestly non-euclidean.
    Langlands does statistical mechanics today.

  • http://geniusnz.blogspot.com GNZ

    In defense of Michael – I don’t think this is a mathematical question its a question for students of human behaviour*. And I think we should be more careful in how we attribute expert status.

    On the other hand I think both he and his detractors are right. Maths has a certain anchor in reality that maintains a sort of value in the discipline – but at the same time it could be crippled compared to its potential by social forces.

    * Or rather those with the sort of dual expertise that Elizer recently (and rightly) requested of philosophers who wanter into areas that overlap physics. (if von neumann had that then I retract my comment)

  • Michael

    “What could something as abstract as operators on an infinite dimensional space possibly have to do with reality? Everyone knows that reality is composed of billiard balls bouncing around in three-dimensional space!”

    Fourier series.

  • komponisto

    Michael, you did realize that the quotation marks were in the original, didn’t you? (I was “quoting” a hypothetical interlocutor, not poising that question myself, obviously!)

    Robin Z, I’m not sure where you’re coming from. One the one hand, you seem to be supporting my point by giving yet another example (what could something as abstract as Taylor series on infinitely differentiable functions possibly have to do with reality?) — but in the same sentence, you suggest that this might be regarded as a “glib” counterexample to my argument. (How in the world…?)

    There are a terrible lot of concepts in mathematics. Focusing on the ‘hits’ doesn’t do anything for you unless it tells you what to expect next. And, as far as I can tell, mathematics does not do so in the way you imply.

    I imply nothing like you seem to think. The point was not that anybody should have predicted in advance that functional analysis would have specific applications to fundamental physics. Rather, the point was that the critics have the wrong conception of what mathematics is for. According to them, apparently, mathematics should focus on “reality”, as defined by whatever model of the latter people currently have in their heads. But this is wrong. It would be closer to the truth to say that the purpose of mathematics is to expand our imaginative capabilities about possible models of reality — something that people clearly need to be doing more of, as the history of quantum mechanics demonstrates.

  • Caledonian

    It would be closer to the truth to say that the purpose of mathematics is to expand our imaginative capabilities about possible models of reality

    And it would be the truth to say that mathematics has no purpose beyond determining what the consequences of premises are. Anything we can later use the math for is just gravy.

  • http://fac-staff.seattleu.edu/dohertyd Davis

    Even if it is correct, von Neumann’s claim is not particularly useful, as he provides no method for identifying fields suffering from his claimed problem.

    To provide an example: for a little while, I was an algebraic geometer. Prior to my entry into the field, moduli spaces (specifically of curves) had become a major focus of study, though the subject had been around since at least Riemann. For some long time, the notion of a parameter space for curves (or surfaces, or higher-dimensional objects) was a very abstract idea, with no obvious connection to empirical reality at all — perhaps a candidate for von Neumann’s criticism. Somewhere along the way (I don’t know the exact history of this), the string theorists came in to inform us that moduli spaces have a significant role to play in their work.

    If the string theorists turn out to be correct, then what was once an extreme abstraction may have direct applications to reality after all. Even if they’re wrong, there’s no way of knowing whether the topic may be useful in the future.

    And there lies the problem: many initially-abstract areas of study do not find applications at first, and we have no way of determining which ones will end up being useful. I think Caledonian’s view is optimal; let the mathematicians keep doing math. Some of it will find applications, some of it will not, but asking them to focus on that will have negative results in the long run.

  • http://packbat.livejournal.com/ Robin Z

    I apologize, komponisto – I don’t know how I came up with what I said. What I would that I had said is: just because mathematicians derive new fields of study doesn’t mean they are relevant to anything, and just because they will become relevant to something doesn’t mean they will put any amount of effort into them beforehand. (Examples of the former are difficult to conceive, although I suspect the property analyzed in the Collatz conjecture is boring enough to be useless; examples of the latter include the Cooley-Tukey FFT algorithm, which was invented by Gauss around 1805 but whose utility was not truly valued until 1965.) I mentioned Taylor series because those don’t appear in “fundamental” physics, despite the careful and clever mathematics that created them.

    It would be closer to the truth to say that the purpose of mathematics is to expand our imaginative capabilities about possible models of reality — something that people clearly need to be doing more of, as the history of quantum mechanics demonstrates.

    This is an interesting historical question: to what extent does mathematical invention drive the advance of physics? Newton had a solid background in mathematics, but I’ve never heard of any mathematical development recent to his time being the impetus for his invention of calculus.

    Basically, I suspect Caledonian and Davis have the right of it, save that the empirical sciences can and do lead to the formulation of new mathematical constructs, and to the concentration of attention on neglected ones. Other than that, I don’t see any sign that commenting on math’s correlation with science is fruitful.

  • http://cob.jmu.edu/rosserjb Barkley Rosser

    I am going to defend von Neumann on this one. Most of the examples that people have put forward of deeply abstract mathematical concepts that have proven to be useful in practice predate von Neumann’s statement (that includes cryptanalysis, for which the deepest applied work was done during WW II, a fact von Neumann was almost certainly apprised of, even though it was deeply classified and remained so for decades afterwards). Mathematics has become more specialized, and while michael is right that there is a lot of cross-referencing between branches of math, and so forth, there have not been a whole lot of pure math ideas that have had major applications in reality since the death of von Neumann (I can think of a few though), certainly not very many compared with the volume of publication in pure math journals that goes on.

    There is also a bit of an oddity in some of the examples. So, yes, we now know that non-Euclidean geometry is useful for describing not just the earth’s surface but a generally relativistic universe. However, math allowed for both Euclidean and non-Euclidean geometry. It was reality that said which was “correct,” or if you prefer, “more useful.”

    Also, I would note that at its highest levels, there remains profound uncertainty regarding “mathematical truth.” Some of this is due to the problems arising from Godel’s work, well known to von Neumann, and some arises from the same sorts of arguments that showed up in the Euclidean versus non-Euclidean geometry debate: which axioms of which math are “true” or “more useful” (and can we distinguish those criteria?). The axiom of choice is one such matter, but there are others, and we find such constructs as constructivist and intuitionistic maths having comebacks in some quarters (and even spilling over into economics, although few economists read this stuff).

  • http://hanson.gmu.edu Robin Hanson

    I’d be surprised if anyone with a stature, seniority, and breadth in math remotely similar to von Neumann were to disagree much with him on this point. Here we mostly see “young punks” overconfidently dismissing the wisdom of such a master.

  • Caledonian

    When an esteemed professor says a thing is impossible…

  • http://profile.typekey.com/RobinZ/ Robin Z

    …someone will quote a distinguished but elderly science fiction author to refute them. And, in any event, John von Neumann made not the slightest reference to impossibility.

  • komponisto

    Robin H.:

    I’d be surprised if anyone with a stature, seniority, and breadth in math remotely similar to von Neumann were to disagree much with him on this point.

    G.H. Hardy has already been mentioned.

    But actually, I don’t necessarily disagree with the literal content of von Neumann’s remarks. It’s just that this is the kind of quotation that is very easily misinterpreted by people who are suspicious of pure mathematics in general.

  • http://yudkowsky.net/ Eliezer Yudkowsky

    P’rsnally, I’d put my young punk’s weight on the side of von Neumann here – the further you get from reality, the more exquisite the aesthetic taste you need to keep your math pretty. I am not disagreeing with him, at all –

    But I would advocate that, in general, being born more than fifty years after someone, gives you a much stronger prior for thinking that you might be able to disagree with them, than if it were only your strength against theirs – and not the weight of fifty years of combined human thought, standing behind you; possibly without your even beginning to realize how much your own era has influenced you, how different past times really were.

    The cheap example is, “Newton was a Christian – do you think the universe has changed since then, or do you think you’re smarter than he was?” “I think I was born more than twenty years later” is a perfectly legitimate response to that.

  • http://hanson.gmu.edu Robin Hanson

    Eliezer, yes on topics where human thought has greatly advanced we should feel more free to disagree with ancient wise ones. But this is a topic where I don’t think we’ve learned much in the last half century.

  • http://imaginarypolitics.wordpress.com Unit

    It seems though that this a problem for all human activity. One could replace math with econ, philosophy, basketball, paleontology, astrophysics, politics, etc…It seems that generally speaking our activities are getting further and further from reality, whatever that means, and trying to justify human activity is very hard without mentioning god or other incomprehensible things.

  • http://imaginarypolitics.wordpress.com Unit

    I maybe didn’t express myself correctly. Von Neumann was a giant in both “pure” and “applied” math. The title of the post says “keep math real”. My point is that both pure and applied math can fall prey of the dangers that von Neumann mentions.

  • http://yudkowsky.net/ Eliezer Yudkowsky

    But this is a topic where I don’t think we’ve learned much in the last half century.

    I’m proposing we’re biased to underestimate how much of our beliefs, and especially, our disagreements with the past, are the product of our era as compared to our personal intelligence and tendencies. I.e. people who say, “I don’t care if it was once a common entertainment, I would never set a cat on fire.”

    I mean, ultimately, I don’t disagree with you or von Neumann in this particular case. Maybe we didn’t learn all that much in the last half-century… or maybe we did, and this particular issue happens to turn out the same? But I think that Modesty applied to a major figure of the past is an extremely different thesis from Modesty applied to a major figure of the present.

  • Caledonian

    Or, instead of using someone’s status and achievements as a guide to guessing about the logical coherence and strength of their arguments for their positions, perhaps we should evaluation their positions on the arguments themselves.

    Smart people can still believe stupid things. Just because people are smarter (read: more processing power) than we are doesn’t mean that we have to blindly accept any position they happen to hold. Nor does it mean that we should.

    Newton was a genius. He was also a Christian. What arguments convinced him to become a Christian, and how compelling are they? How well do they hold up in the light of current knowledge, and were they both logically valid and sound?

  • http://fac-staff.seattleu.edu/dohertyd Davis

    But this is a topic where I don’t think we’ve learned much in the last half century.

    Are you sure about that?

    The specific example I provided is from the past half-century. Major parts of algebraic geometry would have seemed to be a perfect example for von Neumann’s criticism (I even thought so when I went into it), yet now appears to be vital in current physics research.

    It’s cute that you dismiss critics as “young punks”, but you’ll find a similar attitude throughout the entire field of mathematics — I’ve yet to meet a math researcher who shares von Neumann’s concern, and I’ve met many. Many, many mathematicians work with no concern for reality. They allow others to figure out whether their work can have applications. And sometimes, even the most abstract subjects do.

    So I’m claiming that (a) most mathematicians would disagree with von Neumann (and not just “young punks” like me), and (b) there’s no way of determining which abstract nonsense is potentially useful in describing reality, and which is not. Would you disagree with either of these claims? Because I claim either one is a solid reason to take issue with von Neumann’s argument.

  • Douglas Knight

    I agree with Davis:
    Even if it is correct, von Neumann’s claim is not particularly useful, as he provides no method for identifying fields suffering from his claimed problem.

    To elaborate, people who claim to be disagreeing with von Neumann are just saying where to draw the line. They’re being more precise (although they may be no better at offering forward-looking advice) and it’s not at all obvious that they’re disagreeing.

    One can interpret von Neumann as saying that most people draw the line in the wrong place. If he is saying that, it may be right to adjust one’s personal line, but I suspect that the people stating their assessments have adjusted them, just now, in response to von Neumann.

    But that may be an error, because opinion may have shifted; he is surely talking about the median opinion of his time, at least more strongly than about the future. Reading the rest of his article, I think it might be a condemnation of Bourbaki (although it might be a little early) and I think general opinion has turned against Bourbaki.

  • Douglas Knight

    Robin Hanson:
    I’d be surprised if anyone with a stature, seniority, and breadth in math remotely similar to von Neumann were to disagree much with him on this point. Here we mostly see “young punks” overconfidently dismissing the wisdom of such a master.

    OK, so now are you trying to “end the conversation”?

    Also, are you trying to imply that we should be concerned about the self-aggrandizing nature of the claim? (that mathematics would fall apart without exceptional taste-makers like vN)

  • http://cob.jmu.edu/rosserjb Barkley Rosser

    komponisto,

    Hardy’s work was done before von Neumann’s statement, although certainly the full applications of it were not completed by then.

    Davis,

    Good example, but one example does not disprove the general argument (I agreed in my comment that there were some). I do not think that either Robin (or von Neumann) is/was for shutting down pure math research (I certainly am not). I think the point was that von Neumann was forecasting that a) such research would become much more specialized, and b) that much less of it would lead to such fortuitous revelations/usefulness regarding reality, unrealized at the time of the work. Even though there have been a few examples such as yours, I think the weight of evidence is on the side of von Neumann on both points.

  • http://cob.jmu.edu/rosserjb Barkley Rosser

    Oh, and Eliezer is right, of course, that we have learned a lot in the last 50 years. Indeed, one of the things we have learned is that von Neumann was correct in his forecast, an increasing amount of increasingly specialized pure math research has been done and is being done, with substantially fewer serendipitous discoveries that have translated into serious real world applications unforeseen by their developers.

  • komponisto

    Hardy’s work was done before von Neumann’s statement, although certainly the full applications of it were not completed by then.

    It doesn’t matter when Hardy’s work was done; the point is that if you posed the question to Hardy, “Do you think mathematics derives its legitimacy from its connection to (physical) reality?”, he would have answered in the negative.

  • Caledonian

    We generate the reputation of a thinker by evaluating the quality of his work, not vice versa.

    If we judged the quality of work by the reputation of the person producing it, we’d still believe that heavier objects fall faster because Aristotle said so.

    This attitude of reverence towards ancestral icons is incompatible with scientific inquiry and, more generally, rationality itself.

  • http://cob.jmu.edu/rosserjb Barkley Rosser

    komponisto,

    Did you think I was disagreeing with you? The point was whether or not von Neumann’s forecast about the future was correct or not. I claim it mostly was, and citing the fact that Hardy’s purely theoretical work came to be used practically does not disprove that when Hardy’s work was done prior to von Neumann’s prediction, although the dates of application are also relevant.

    Of course, Hardy was certainly one of the most publicly ardent advocates ever of the idea of pure math being justified as its own end, pure math pour pure math, like l’art pour l’art.

  • Unknown

    Caledonian… heavier objects do, in fact, fall faster:

    1)due to differing air resistance to a heavy or light body (this is actually the reason explicitly mentioned by Aristotle).

    2)even in a vacuum, due to the gravitational attraction of the falling body upon the planet.

  • Caledonian

    1) actually concerns density and shape, not mass or weight.

    2) is far too small an effect for Aristotle to have detected – on the scale available to him, his claim was incorrect. Furthermore the claim is not that the more massive object falls faster, but that the Earth falls faster up to it.

    So you’re right in a trivial and limited sense, but completely wrong in the broader course of the argument.

  • Unknown

    Caledonian, for reason #3, see http://www.iop.org/EJ/abstract/0143-0807/8/2/006.

  • Caledonian

    Yes, Unknown, I’m sure those quantum mechanical arguments totally validate Aristotle’s claims.

    Next, will you demonstrate that the brain is actually a cooling apparatus and not the center of cognition, which is actually the heart?

  • Dihymo

    Mathematics being a generalized system of ideas tightly bound together does represent reality.

    Mathematicians being human do not represent reality.

    Mathematicians talk about math.

    Hence we’re playing telephone and calling it reality.

    Danger, Will Robinson!

  • Dihymo

    Unknown, it’s really simple. The force is different, the acceleration is the same. The mass which you are measuring the acceleration of cancels out.

    If I put a nickel (little mass) into a cup (big mass) and then I take the nickel out (cancel out little mass during the acceleration) what’s left? The cup.

    If I put a quarter (medium mass) and then take the quarter out (medium mass again) what’s left? The cup.

    In both cases it is the cup which determines the acceleration not the coin.

    You are correct though that the smaller acceleration of the Earth toward different masses is different. But that is not what Aristotle said. He said it depends on the mass of the object not the mass of the Earth. Aristotle is the grandfather of the big mass faster acceleration theory. This is exactly the theory that people still cling to because they make the same mistake he did.

    The only way you could get out of it is to say you meant the Earth.

  • gandalf

    I totally understand and support what von Neumann (and later V. I. Arnold) said. I am surprised that this is not screaming obvious to everyone here.

    Mathematics today has no resemblance to the real world, and never will have again. I sometimes feel there is almost a cartel at work, a massive clan of academics, whose sole purpose is to justify their existence by developing theory after more esoteric theory, none of which actually matters. They need to do this because people need their PhDs, academics need to keep the hundreds of millions of dollars of grants flowing, and they need to keep publishing to go from being assistant processors to associate professors to professors to emeritus.

    It’s their livelihood, fellas. Do you really expect them to rock the boat? What if the grants stop, or if the public start questioning the value of keeping up these behemoths that are pure maths departments?

    Just like the credit crunch and the financial services industry today, the higher mathematics community just another industry that is not interested in governing itself. it couldn’t care less.

    Sometimes I think it’s even worse than that. I almost think mathematicians actually enjoy living in their fairy-tale land, in their make-believe world that they have created because they can’t handle the real world.

    Here are some sample topics of recent papers taken from a randomly chosen journal:
    · “A Banach space without a basis which has the bounded approximation property”
    · “A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy”
    · “A class of idempotent measures on compact nilmanifolds”

    If you think any of these have any resemblance with the world we live in (or people writing these have the slightest interest about the real world), you are living in the same cloud-cuckoo land.