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")
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.
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.
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.
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.
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.
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.
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.
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.
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)
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.
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.
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?
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.
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.
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!
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?
Caledonian, for reason #3, see http://www.iop.org/EJ/abstr....
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.
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.
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.
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.
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.
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.
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.
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)
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.
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.
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?