The status of Perelman’s proof of the Poincare conjecture is still somewhat confusing. For some background see a previous posting and a later follow-up. Last week the ICTP in Trieste issued a press release entitled Poincare Conjecture Solved, which states that Perelman’s proof “has been confirmed by an international group of mathematicians whose findings were presented to participants at a conference” at the ICTP. The conference was a summer school, and the press release goes on to claim that “The 60 participants, more than half from the developing world, reaffirmed the approving judgement of the mathematicians.”
If you look at the write-ups of the talks, the only relevant thing in print is in the lecture notes of Carlo Sinestrati where he states “The details of the proof are still being checked by the experts in the field; however, the main ideas of the papers are by now widely understood.” So, it’s not clear who exactly is supposedly now willing to vouch for Perelman’s proof, and the comment that the students at the school “reaffirmed” that it is a proof is kind of silly, given the difficulties involved.
There’s a month-long summer school going on right now at MSRI in Berkeley, sponsored by the Clay Mathematics Institute. Many relevant materials are available at the web-site of the summer school. As far as I know, none of the experts there is yet quite willing to claim that a complete proof using Perelman’s techniques has been written down and checked, although they do seem to be getting close to this point.
Not Nobel laureate said,
“Aside from the various theoretical dead ends, a more fundamental problem with string theory is that it operates at an energy scale many orders of magnitude beyond what is measurable today. Thus it’s non-testable, hence meta-physical.”
This is one of popular miths of string M-theory.
It, like almost all of popular claims on string M-theory, is simply false.
String theory is perfectly testable. The first version of string theory was tested in the strong force regime and abandoned.
The introduction of 26D was forced because the unobserved tachions predicted by 4D version of the bosonic theory.
The next version (+ fermions) were also (un-)tested, for example so many times was claimed the inminent discover of supersimmetry and the first verification of supersimmetric string theory by crackpots!
In cosmological issues, string M-theory has been also tested. A complete failure in the words of Krauss, possibly the poor theory of history of physics according to Woit (discrepancy with experimental data is around 50 orders of magnitude).
The attemtp to explain inflaction and dark matter from string theory also has failed. Regarding cosmological brane theory, the recognized specialist Linde showed that “popular” (i.e. string theory is marvellous, string theory solves the most difficult open problems, etc. etc.) papers by string theorists were completely wrong and predicted the contrary to observed “inflationary” data.
Moreover, it is often ignored that before to explain NEW physics, string theory may explain known physics. In this simple, elementary, step, string M theory does not explain nothing already known. In fact, even the derivation of GR from string theory is based in heuristic reasoning and ad hoc hyphotesis.
It can be shown that string M-theory is incompatible with known experimental data.
The incompatibility with certain statistical mechanics data has forced to some string theorists to develop the so-called non critical approach that violate basic principles of usual “critical” string theory.
The incompatibility with thermal phenomena has forced to abandon all of standard Hilbert-Fock quantization of branes in favor of the new doubled space approach (so-called tilde operators) and the new thermal states of TFD-Dp-brane theory.
In the past, the “derivation” (not in the rigorous sense of term) of GR from 10D superstring action was one of main popular claims of string community. Curiously, now physicists and astronomers maintain doubts in the validity of GR at cosmological scales and well-known modifications like MOND, AQUAL, etc. are being studied and tested. Since that string theory claims just small-scale modifications for GR, one may see that also here string theory is a failure here. For example, string theory is incompatible with standard TF law.
From unitary Schwartz string action one cannot explain experimental data which IS explained by Lindblad semigroups axiomatic theory.
Etc, etc, etc.
All i am saying is standard, it is not speculation. For example, TF law is the basis for one of standard methods in gauging distance in spiral galaxies, Fourier thecniques in L-space (string theory and standard (e.g. Weinberg manual) QFT both work only with H-space) received Nobel Prize for chemistry in 1991, etc. specialists in quantum mechanics have said in many occasions that string theory is not fundamental (see me previous post on Witten for a quote extracted from the conference Quantum Future. It is well-known that true specialists (i.e. people that has really advanced the field) in quantum theory durely critiqued last Witten claims on the generalization of quantum mechanics from M-theory. In fath, it is aknowledged by many recognized specialists in the field that Witten (with no contribution to the field) misunderstands quantum mechanics.
The last year, a Nobel laureate (Freeman J. Dyson) did a similar criticism to Brian Greene in his review of book the Fabric of Cosmos. Dyson reacts atonished to Greene string interpretation of quantum mechanics saying “He rejects [standard view] without any serious discussion”. Dyson ignores that IS precisely string theory literature: no serious discussion, only speculation, conjetures, bad math, superfitial insight, etc.
This is the reason why i claim that string theory is wrong like a TOE. It is amazing that questions that i said some years ago (then i was ridiculized) begin to be supported by string theorists now.
In fact, i said in the past that string theory was rather standard (simple) and people ridiculized. In fact, one guy contacted with Lubos Motl and this “hiring” to me.
Fortunately, after of his Nobel Prize, David Gross has said a phrase very similar to i said two years ago: that string theory is not revolutionary.
The guy said to Lubos Motl đź™‚
If all i am saying about failure of string theory to explain known data is standard, why do string theorists ignore it?
Because are arrogant people and think that understand things when have only a superfitial knowledge of things.
For example some of them are very excited with TFD generalization of brane theory. TFD theory was know decades ago. Now we are working in more geernla stuff TFD II, NESOM-TFD, etc.
Other example, the most recent and radical modification of NC string theory posted in ArXiv by Nanopoulos uses mathematical tools in projected dynamics mathematics developed by Brushels School in the 60s but abandoned by the own School in favor of the recent LPS formalism in Gelfand triplets in the 90s!!!
It is more, not only again string ideas are outdated, even copying the interesting work done by others (e.g. Prigogine), they copy incorrectly!!
For example, the equation (10) of arXiv:hep-th/9403133 is simply wrong for anyone with a minimum insight in generalizations of quantum mechanics (e.e. Solvays conferences, etc.)
Note, i contacted with Nanopoulos for explaining it but he ignored to me, now i does not explain to him that almost of next sections of that and other papers are misleading. It is waste of time contact with string theorists.
My criticism to that preprint is correct, in fact it is supported by one of members of the School, prof. Gonzalo Ordońez from Ilya Prigogine institute on Texas that i contacted for verifying.
Batakis introduces electroweak structure more or less by hand without noticing that CP_2 spinor connection possesses naturally electroweak gauge group as holonomy group
I think this latter is Finkelstein and Jauch’s (and Speiser and Schiminovich’s) quaternion quantum mechanics. This theory also had a rationale for the Higgs mechanism other than expediency.
Aside from the various theoretical dead ends, a more fundamental problem with string theory is that it operates at an energy scale many orders of magnitude beyond what is measurable today. Thus it’s non-testable, hence meta-physical. Having said that, the same can be said about LQG, spin foams and any other theories whose predictions, if they even make any, lie outside the energies attainable at the Tevatron and the LHC or perhaps astrophysically observable.
“Pure reason” has never been a productive form of scientific enquiry.
Tony Smith mentioned in his posting the paper of Batakis about H=M^4×CP_2 Kaluza-Klein theory. We had a discussion about the paper with Tony for a couple of months ago.
Batakis introduces electroweak structure more or less by hand without noticing that CP_2 spinor connection possesses naturally electroweak gauge group as holonomy group. The problems of KK scenario become obvious when one looks for the spectrum of Dirac operator in H.
a) The two chiralities of H-spinors allow an identification as quark and lepton type spinors when one couples leptons/quarks to n=1/n=3 multiple of Kahler gauge potential of CP_2. The holonomy group U(2)_ew has a natural identification as electroweak gauge group. Separate conservation of lepton and baryon numbers is predicted.
b) The problems are that only right-handed covariantly constant neutrino is massless whereas other states have mass scale defined by CP_2 size. Also the correlation between color and ew quantum numbers for the spinor modes is wrong: only right-handed neutrino corresponds to color singlet. Thus Kaluza-Klein type theory as a limit of something more general is out of question.
Batakis does not notice that CP_2 already unifies color and electroweak symmetries. If one considers space-time as a 4-surface in H and induces spinor structure to the space-time surface (bundle induction is mentioned in the 20 first pages of any text book about bundles and means in recent case projecting of the gamma matrices of H to space-time surface), one obtains electroweak gauge field as classical gauge fields inheriting their dynamics from the dynamics of space-time as 4-surface. Color gauge potentials can be identified as projections of Killing vectors of color isometries.
From this it is a long way to a generalization of string model predicting correctly the massless sector of the theory and mass spectrum of elementary particles and hadrons. A profound generalization of conformal symmetries is needed and emerges naturally when one formulates quantum theory as a theory of free classical spinor fields in the “world of classical worlds” consisting of 3-surfaces in H and endowed with Kähler geometry. The spectrum of Dirac operator and mass calculations see the five chapters in the second part of p-Adic TGD.
Thank-you, Peter. I think I’ve got it.
I think I mostly agree with you, but a few comments.
The string theory community IS wasting time. They are sitting around waiting for someone else to come up with a new idea. Sure, if someone comes up with a wonderful, compelling new idea with lots of evidence for it, they’ll take it up. But as you say, they’ve set the bar very high. The present situation allows them to get jobs, awards, grants, etc, etc, based on worthless work on string theory, but not for work on other ideas (unless these ideas are quickly successful). I still think that if people start publicly acknowledging that string theory has failed, this will pull the plug on this unhealthy situation. If people can’t get a grant proposal to do string theory funded, but have to try and come up with something else, this would have a huge positive effect.
The leaders of the string theory community are well aware that if they start publicly talking about how badly things are going, they’re quickly going to get into this kind of trouble. I think that’s why many of them will privately agree that string theory is in bad shape, but say very different things publicly. I hope to have some role in not letting them get away with this.
I agree with your three ideas of things to do, and am trying to do them. I hope that reading this weblog will help students think independently. A lot of what I have to say is aimed at them. I also do try and identify and point to new work that might be promising. Unfortunately it’s rather discouraging how little of this there is from my point of view. I personally think that a deeper mathematical understanding of the standard model is what is needed for progress, but virtually no one is working on this. The interaction between particle theory and mathematics has narrowed down in recent years to virtually just topological string theory and enumerative problems involving Calabi-Yau 3-folds. This work is interesting, but it is mathematically very narrow.
Curious said “… I have had several occasions to discuss your blog with our (U Chicago) string theorists — and they read it and they agree with it, and then they ignore it. … The best alternative ideas around are not good enough to recruit string theorists. That is not the result of their obstinacy. These alternative ideas are simply not good enough, period. …”.
I disagree, and here is a concrete example of a model that unifies gravity and the standard model and is (afaik) ignored by the conventional string theory community.
N. A. Batakis, in Class. Quantum Grav. 3 (1986) L99-L105, wrote a paper entitled Extra gauge field structure uncovered in the Kaluza-Klein framework. In it Batakis said:
“… In a standard Kaluza-Klein framework, M4 x CP2 allows the classical unified description of an SU(3) gauge field with gravity. … the construction of an additional SU(2) x U(1) gauge field structure is uncovered. The construction involves a properly modified ‘gravitoweak connection’ and supplies a mechanism analogous but not redundant to the Kaluza-Klein ansatz … As a result, M4 x CP2 could conceivably accommodate the classical limit of a fully unified theory for the fundamental interactions and matter fields. …”.
If Curious is correct, then I would think that his U. Chicago string theorists would see the Batakis paper as a “spark to inflame” them into developing and completing the work of Batakis.
If, on the other hand, Peter is correct, then those string theorists would feel threatened by a competing non-string (not even LQG) theory and dismiss it without careful evaluation, perhaps even attacking anyone who is seriously interested in it.
I am curious to see what reaction the string theory colleagues of Curious might have to the Batakis paper.
I do not want to distract you with endless arguments going to and fro etc
inasmuch as I think we are on the same side of the issue. Thereof I’ll comment on only one point of
your answer; it is a very important point — to you. The question is how to stop the wastage of
time on ST; that’s what we both chiefly want.
You say that you deeply care for the field and you cannot just sit back and see how it disintegrates and how young people become demoralized. I know you do feel the pain. So do I. But critisizing ST in the way you do it – and you do a jolly good job of it – it just does not work. You are convincing the wrong people; those that do not waste time anyway.
I have had several
occasions to discuss your blog with our (U Chicago) string theorists — and they read it and they agree with it, and then they ignore it. It is too late in the game for your arguments to work on them.
One cannot turn a tide on a completely negative message. I do not say that YOU have to be positive about something. But someone better be; otherwise ST is unstoppable. You are trying to combat a belief system using rational arguments against it; that is as futile as Sean’s struggle against creationists. Painful as it is to watch the wastage of effort by first-class minds, the only way to stop it is to be creative; that’s where we all collectively failed.The best alternative ideas around are not good enough to recruit string theorists. That is not
the result of their obstinacy. These alternative
ideas are simply not good enough, period.
I do not believe in great men visions either but
even more am I sceptical of
the “fertile soil” model of which you and Smolin equally partake. The community of German physicists from which Heisenberg emerged was not at all like a healthy community. It was more like a viper’s nest,
with duelling schools, less then enthusiastic welcome of radical ideas, and a community riddled by mutual distrust, anti-semitism, and typically German professorial anti-everything. The community that you have in mind did exist but it was initially marginal and had
not more than 20-30 people at any time. It is always possible to
create such a mini-community of like-minded people inside a larger hostile community.
The power of Heisenberg was not that he had roots
in some healthy worldwide community; initially, there was tremendous opposition to QM. It was the truth of his ideas; it is the truth that won the people against their worst selves. ST does not strangle new thought; this thought is simply not there.
We can only do three things to stop ST:
(i) generate this new all-winning idea by ourselves (ii) foster independent thinking in our students and
(iii) spot this new idea and champion it when it emerges. I cannot think of anything else that might work towards the goal.
You look at the ST community and see people wasting their time on a dud. I see it as a barrier this new idea has to jump over;
it is a mighty high barrier and it works as an excellent
obstacle w/o which the race is meaningless and a perfect deterrent to bad ideas. The ST community is also a potential pool of people that will be able
to develop and complete this new idea when its time finally comes. This community is not wasting time,
it is waiting for a spark to inflame.
Perhaps, all of the above is not really news to you and you have heard such arguments many a time. By no means I suggest that you should stop what you are doing; you would not listen anyway. All that I am asking from you is to think once in a while: Are the means that I chose helping to bring about the goal that I chose? If the means are not helpful, another tactics is needed.
I enjoy reading your honest, principled, and intellectually challenging blog but it mainly
serves to vent the frustration; it does not go to the root of
the problem and it does not suggest a winning strategy. Nobody ever became healthy by
a realization of one’s sickness, using your analogy. We need a diagnosis, a treatment plan and medicine — if not a cure. You offer a death sentence to a terminally ill patient. Is that enough?
Curious asked “… Could it be that people rage against the ST because the dominance of this admittedly failed programme painfully reminds them their own failure to produce a great new idea? …”.
I don’t think so. I think that the rage is directed at the massive publicity campaign to present conventional superstring theory as the only possible program for unification of gravity and the standard model.
For example, the 1 July 2005 issue of Science listed the 25 most important questions in Science today. About the number 5 question, Can the Laws of Physics Be Unified?, Charles Seife wrote:
“… Gravity clashes with quantum theory so badly that nobody has come up with a convincing way to build a single theory that includes all the particles, the strong and electroweak forces, and gravity all in one big bundle. But physicists do have some leads. Perhaps the most promising is superstring theory. Superstring theory has a large following because it provides a way to unify everything into one large theory with a single symmetry—
SO(32) for one branch of superstring theory, for example
—but it requires a universe with 10 or 11 dimensions, scads of undetected particles, and a lot of intellectual baggage that might never be verifiable. It may be that there are dozens of unified theories, only one of which is correct, but scientists may never have the means to determine which. Or it may be that the struggle to unify all the forces and particles is a fool’s quest. …”.
It seems to me that Seife and Science are saying that conventional superstring theory, their “most promising” approach, may or may not produce a unique unified theory, but if it fails to do so, then “the struggle to unify all the forces and particles is a fool’s quest”.
In other words, anyone who pursues any approach other than conventional superstring theory is characterized as a “fool”.
That attitude, which is prevalent not only in the scientific media such as Science, but also in the popular media and in the culture of conventional superstring theorists themselves, is where my rage is directed.
PS – Just for the record, the number 1 question listed by Science was What Is the Universe Made Of?, as to which Charles Seife wrote:
“… Ordinary matter and exotic, unknown particles together make up only about 30% of the stuff in the universe; the rest is this mysterious antigravity force known as dark energy … at the moment, the nature of dark energy is arguably the murkiest question in physics—and the one that, when answered, may shed the most light. …”.
Thankfully, Seife’s article did not present conventional superstring theory as the “most promising” answer to that question, but I have little doubt that members of the conventional superstring community will declare that their theory is the “most promising” approach to an answer.
I’ve got no idea by what you mean in claiming that Hilbert’s research program dominates mathematics. Hilbert was a very broad mathematician, and did a wide variety of things. Some of the areas he worked in that were most identifiable with him are relatively unpopular now (e.g. the formalist approach to foundations of mathematics). Mathematicians work on a wide variety of problems, from a variety of points of view, and they’ve been making real progress (e.g. the Wiles proof of Fermat, the possible proof of Poincare using ideas of Hamilton and Perelman). A lot of what they do is unproductive, but the field is healthy. Theoretical physicists could learn a lot from this, and not just possibly useful mathematical ideas.
I don’t think I’m criticizing string theory in order to find out what’s wrong with it. In the end, that’s very simple: it abandons the successful core mathematical concepts at the foundation of the standard model and adopts an extremely speculative and not very promising alternative, one that has now conclusively failed. From the beginning of the fad in 1984, it was clear to me and to many other people that string theory wasn’t an obviously promising idea. It predicted nothing and there were clear reasons for this.
In 1984 it wasn’t unreasonable to work on string theory. The theory was not very well understood and one could hope that further work on it would show a way around its problems. In 2005 the situation is very different. It is completely unreasonable to now believe that these problems can be overcome. Current attempts to create a unified theory out of string theory are both mind-boggling ugly and utter failures.
I won’t disagree with you or anyone else who tries to argue that I should devote more of my time to working on positive alternatives and less to criticizing the current situation. Maybe this is right. But I find what has happened to the field I care deeply about extremely disturbing. Each year it has gone from bad to worse, and is increasingly dominated by pseudo-scientific garbage, heavily promoted to the public. I don’t find myself able to ignore this.
I’m not a big fan of the “great man” view of science, the idea that progress depends on one brilliant person. More commonly, scientific progress comes from a community of people working on promising ideas. When Heisenberg came up with quantum mechanics he didn’t do this all by himself. He was part of a healthy community in which many people were trying many different things. The theoretical physics community is not healthy. It is sick and has been getting sicker. When you’re sick, the first thing you need to do is acknowledge this, then figure out what you need to do to get better.
Thank-you for your thoughtful remarks. I’ve been reading
this blog for some time and I still cannot determine what exactly is your problem with the ST:
you give so many different reasons and none of these various reasons look entirely
convincing, hence your need to repeat yourself time and time again, in order to convince yourself and others. I have a hunch that you have started this blog to find out precisely that — what is the problem with the ST as a programme, at the deepest level — something that you intuitively feel — and you write this blog for yourself — not for the benefit of the others. You suggest various answers to this one question, but these answers do not fully satisfy you. And so is the case with your readers, many of whom are string theorists. They read your blog, they agree, and then keep on doing what they are doing. If the answer were right, they would stop doing ST and start doing something else.
I do not know what that right answer is. You have to find it out for yourself and, please, keep on trying; it is important.
One thing I am sure of is that the problem with the ST is not that it is a fad or that it dominates HEP or that it is unprecedented in physics or mathematics or that it is unpredictive or plain wrong. The problem is US.
Hilbert’s research program had dominated and still
dominates mathematics to a much greater extent than
string theory dominates high energy physics. It was viewed as a passing fad by many leading
mathematicians at the turn of the 20th century. It was
enormously sucessful, several excesses aside. Its
success is not and was not self-evident, as it does not logically follow from any rational argument; it is one man’s vision. By all criteria it was a fad. It could’ve been a misguided fad as well. Fad or no
fad, tangible results started to stream out immediately
and still do. There was no time lag between the
promise of a wonderland and the result.
The news about the string theory is not that a certain fad dominates a certain field but how unproductive that fad turned out to be. Perhaps, one’s man intuition
is not enough for physics. Or maybe we listened to the wrong man.
I believe that it is entirely appropriate for a fad to dominate the field, and it happended repeatedly, in both physics and mathematics. But most of these previous fads were productive and that’s why we do not call these fads “fads” anymore. For lack of better ideas and guiding experimental results,
ST manages to dominate the field without the benefit of being productive.
And yet still, if every string theorist
will read this blog and agree that ST is not even wrong that would not change the situation one bit, because it is unclear what’s right. The dominance of ST is not the result of groupthink or worldwide conspirancy as some people believe. It is the looming, annoying testimony to
OUR failure to come up with better ideas
and decisive experiments. The problem is US not THEM. Never before physicists were unable to come up with a good idea for such a long time — and that is certainly not the fault of the ST community.
Could it be that people rage against the ST because the dominance of this admittedly failed programme painfully reminds them their own failure to produce a great new idea? My feeling is that at the bottom, that’s the right answer, and it is very sad and disconcerting.
While physics and math, like all human endeavor, have always experienced a certain number of misguided fads, I think the superstring theory story is without parallel. In physics the closest analog I can think of is the craze for S-matrix theory during the 60s, but that only lasted a decade or so.
No one research program has ever dominated mathematics the way string theory dominates particle theory. The excessively formal style of Bourbaki is more an issue of bad pedagogy and exposition than bad research. While the members of Bourbaki were writing the very formal Bourbaki textbooks, they also were doing a wide range of different wonderful mathematics research. Take a look at the Bourbaki seminar writeups during the fifties and sixties. There’s all sorts of exciting new mathematics there, being written up often in an accessible way, with very little evidence of bad effects of too much formalism.
Curious: Your comments about axiomization are not true. Axiomization is not as big a deal now as it was in the early twentieth century mainly because it was so successful: the current foundations are adequate for most purposes. Foundational matters still arise, though. For example, a long-open problem in abelian group theory (the Whitehead problem) has turned out to be independent of the usual axioms of set theory (ZFC).
Foundational techniques are beginning to be used to settle purely mathematical questions. There are some results in analysis whose only known proof requires “nonstandard analysis”. Model theory was used to prove the Mordell conjecture in the function field case.
Curious: “…axiomatization of mathematics peaked in
popularity in the first half of the century and
then faded out.”
Actually, this isn’t true. The field simply evolved into something qualitatively new. Foundations of Mathematics is a big subject today. And as much as String theory bugs me, I have a terrible feeling that history will look at Strings in a similar light. A lot of the maths of M-theory has to be important to physics, as Witten says. The fact that what wins out might not have anything to do with M-theory from certain points of view doesn’t alter the fact that detailed arguments connecting the old ideas to the new will inevitably turn up. And all history will see is another turning tide in the ocean – not the swirl of water washing over into the tiny lagoon at the end of the beach.
Peter then Tony asks an interesting question:
has it ever been an analog of superstring theory in mathematics? Perhaps, they mean:
was there a school of thought that captured a lot of attention, hype and effort and produced very or relatively little after many years of persistent study? The answer is surely, yes,
though we may disagree on details. Say,
axiomatization of mathematics peaked in
popularity in the first half of the century and
then faded out. It is crushingly boring and
not a very good way to come with interesting
math. Another program, N. Bourbaki used to be as influential as Witten, but few people still care. One can come with more examples.
Observe also that in mathematics the logic
proof stands on its own; there is no emprical
input. By contrast no physical theory is self-
consistent or logically complete. We accept
these theories not because these theories are
rigorous (they may be actually highly dubious)
but because they work. Such a situation is
acceptable for any physical theories because
their difficulties are delegated to a yet
unknown more general theory. However,
such a situation is not acceptable for a TOE;
it should be as consistent as a mathematical
proof. I think that the problem with the string
theory is exactly that: it aspires to be what
no physical theory had ever been: a domain
of pure logic and mathematics, with acceptance criteria peculiar to these fields.
It cannot live to that self-imposed standard.
There could be in this sence no analog of the string
theory in mathematics: the complete analog would
be a mathematical theory that needs a
physical experiment to decide the truth of
the theorem. So far, even computer experiments are not exactly welcome.
The huge groups and large sums of money necessary to do some kinds of experiments have certainly had a bureaucratic effect on some areas of experimental physics, but this doesn’t really explain what has been going on in particle theory, where large groups aren’t needed.
Particle theory is much, much, much more faddish than mathematics, and attitudes are very different. To oversimplify, in mathematics someone who quickly publishes a not really great paper on what seems to be the hottest topic would generally be thought of as someone shallow, unwilling to take the time to become a real expert in something and do some serious work. In particle theory, someone who doesn’t jump on the latest fad is often thought of as an intellectual lightweight who isn’t smart enough to quickly absorb something new and work on it.
The roots of this attitude in particle theory come from the days when experiments were producing unexpected new results. The most ambitious people would then jump into trying to explain them. The problem now is that there haven’t been any especially new unexpected experimental results in particle physics for a long time, so this kind of attitude has become dysfunctional. Mathematicians have never had experiments to feed them new clues as to what to think about, so they have a tradition of people spreading out and digging in for much longer term research projects.
Peter, you say “… The way the particle theory community has refused to acknowledge …[that]… over the last 20 years a huge amount of evidence has accumulated that, as an idea for unification … Superstring theory … doesn’t work … is something that has no analog I can think of in mathematics.
What is the relevant difference between the math and physics communities?
First, it seems to me that superstring theory is not the only example in which the physics community refuses to recognize obvious failure.
For example, huge amounts of money and manpower have been spent over several decades on magnetic confinement fusion machines. At first, it was worth exploring (like superstrings in 1984), but now, decades later, even though it is clear to any reasonable person that magnetic confinement fusion is not going to be a significant source of energy, the ITER project is getting under way.
For another example, it was clear to any reasonable scientist that the International Space Station would not produce results that would come close to justifying its costs, but it was built anyway.
Those projects (superstring theory, ITER, ISS) all have something in common: they are in fact promoted by growth-seeking bureaucracies.
It seems to me that a distinguishing factor between physics and math is that
the physics community has evolved into a group of bureaucracies that use committee/consensus to enforce uniformity of thought because they feel that any independent thought threatens the fundamental bureaucratic goal of growth,
the math community is a lot of individuals, with uniformity of thought being enforced primarily by relatively objective standards of logical proof.
In my opinion, it is no accident that the last big success in theoretical particle physics was the Standard Model of the 1970s, which roughly coincided in time with the rise of large collaborative physics institutions like Fermilab and SLAC.
It also may be no accident that the last big experimental particle physics success was Fermilab’s taking of data about the T-quark, and that the most significant data was taken over a decade ago, during its first run.
On the other hand, the math community has continued to make substantial advances, such as Wiles’s proof of Fermat and Perelman’s possible proof of 3-dim Poincare. Both of those advances were made by individuals, not large collaborations. It may be that collaborations are necessary to validate their work, but the initial work was by individual initiative.
Maybe the problem with physics is its current sociological structure, something that Burton Richter (former SLAC director) tried to point out in his paper at http://xxx.lanl.gov/abs/hep-ex/0001012 in which he said “… In the 500-strong collaborations of today, we … have a bureaucratic overlay to the science with committees that decide on … speakers, paper publications, etc. The participating scientists are imprisoned by golden bars of consensus …”.
Perhaps particle physics might advance more if it loosens the golden bars of consensus, and allows 1000 flowers to bloom (and perhaps lets committees be evaluators rather than enforcers of consensus thought).
It seems to me that astrophysics has come closer to following that path, as it has been successful by taking interesting data such as WMAP and not attempting to enforce any consensus interpretation of the data (being tolerant of interpretations of dark energy and dark matter ranging from conventional cosmological constant dark energy to MOND and many others).
Perelman is kind of a special case since the problem is that he hasn’t actually written down a proof, just an outline. Turning his outline into a real proof is a lot of work, especially since he is using some techniques few people are familiar with. The classification of finite groups proof is very long, but at least supposedly all the details are there.
I know people have been working on shorter versions of the four-color and finite group classification theorems, not sure what the status is. For another related story, there’s the claimed proof by Hales of the Kepler conjecture. This one involves both issues of computer calculations and such a complex argument that there have been problems getting it refereed.
The classification theorem for finite simple groups is likely to remain impossible for any one individual to verify. Perelman’s proof should be ‘easier’ since it requires verification of new mathematical tools.
Also, is there a proof of the Four-Colour theorem that does not need computers??
Seems like proof in the strict traditional sense is not always possible.
What do you think?
First of all, stop submitting large numbers of comments, this is not a forum set up for you to go on and on. I’ve deleted all except the first one.
About your points: there’s never any absolute logical certainty of the validity of a proof of the complexity of the one Perelman has outlined. But if several people who really have a lot of experience dealing with the techniques involved put in the time necessary to go over the proof with a skeptical eye and announce that they are convinced by it, it’s extremely likely that the proof is valid. But you don’t have to take their word for it, that’s the whole point of the culture of mathematics. The mathematics community historically won’t recognize a proof until it is written down in sufficient detail to allow anyone who wants to to check it.
How well a proof is checked is a function of how much people care about the result. There are lots of proofs in the literature that are probably invalid if looked at carefully, but no one does this because hardly anyone cares about the result. For things like the Wiles proof of Fermat and this proof, the result is so important that you can be sure the proofs will be carefully checked. The whole process is also to some degree self-correcting. If a mistaken argument gets into the literature, and if it is an important one, other people will sooner or later try and use it to do other things, and if it is wrong it will give them wrong results which will ultimately be noticed.
The smooth 4d Poincare conjecture is still open, no significant progress that I know of.
I think the way mathematics deals with conjectures like 3d Poincare and the way physics has dealt with superstring theory are completely different. Over the years lots of good evidence accumulated for 3d Poincare: all attempts to construct spaces that violated it failed, it was shown to be equivalent to other conjectures about topology for which there was significant evidence, and, as Morgan points out, it fit into a larger web of conjectures for which there was a lot of evidence.
So, the general assumption has been that the conjecture is true, but that new mathematical ideas were needed to prove it. It seems that, together with Hamilton’s work, Perelman has come up with these ideas. But still, until there is a complete proof written down and experts have gone over it and vetted it (under normal circumstances this would happen in a formal refereeing process before publication), mathematicians are not going to be comfortable saying there is a proof. From experience they know that a plausible sounding outline of a proof and a proof with the details worked out are two different things. Until you work out the details you can’t be sure a subtle problem has been missed (e.g. for instance recall the problem with the initial proof of Fermat by Wiles).
Superstring theory in 1984 was just a somewhat wild conjecture with little evidence to back it up. At the time you could argue it was an idea worth working on, but over the last 20 years a huge amount of evidence has accumulated that, as an idea for unification it doesn’t work. The way the particle theory community has refused to acknowledge this is something that has no analog I can think of in mathematics.
“and the press release goes on to claim that “The 60 participants, more than half from the developing world, reaffirmed the approving judgement of the mathematicians.”…”
Peter raised a good point on how these 60 guys supposed to “reaffirm” the judgements of the mathematicians? By what? Maybe by their faith or confidence in the authority and credibility represented by the hats these mathematicians wear? If they have not spent 5 years trying to rigorously check every single step of the logic in Perelman’s proof, they probably have run it through a spell checker to make sure it contains no English grammar error. Maybe that’s what they mean “reaffirm”.
Suppose Perelman is already a well known and well established guy, He is the No.1 guy in math, and he made this unusual claim that he proved the Poincare Conjesture just today. And the proof is 1000 pages long and extremely hard to read and go through? How does the No.2 guy in math supposed to believe in this No.1 guy? He has two choices:
1.He believes it because he trusts the No.1 guy and because of the authority power that the No.1 guy is smarter than him and whatever No.1 says must be true, and he is not going to waste time questioning the No.1 guy.
2.He question it until he can convince himself that the logic really works. So he need to spend time, tremendous amount of time, maybe several years, 5, 10 years? Rigurously going through every step. After exhaustive search, he could not find a single logical fraud that could render the original proof invalid. Thus the No.2 guy claim to the world that he has gone through the No.1 guy’s proof, and he discovered nothing wrong. Thus the proof is valid.
But certainly there is always the possibility that despite of the honest effort No.2 put in, he might still have missed something. Or even the remote possibility that No.2 gets a little bit lazy after spend one year going through 2/3 of the paper. How are we supposed to know that is the case or not. How are we even supposed to know that No.1 wasn’t being dishonest and he took the first approach since he was lazy.
So that is the question the next guy in line, No.3, has to ask and answer. Does he believe in No.1 and No.2? And just like No.1, he has two choices, either he believes in No.1 and No.2 by faith and confidence, and save himself some time. Or the No.3 has to do the hard work and going through the whole rigorously thing himself. Being less smart than No.1 and No.2, it would be even harder and take him more time if he wishes to go through the whole thing himself. So there is an even better chance No.3 gets lazy and become dishonest, and just make the “re-affirmation” claim before he finish reading 1/2 of the paper.
And what about the No. N guy, after all the previous N-1 guys claimed the proof was valid? Would he choose to nod his head with the rest of the gang. Or would he want to become the first child who point out the fact that the emperor could actually be naked, and all previous N-2 guys are all just blindly follow their trust in the judgement of the “smart guys” before them?
That’s an interesting philosophy question. And one that really worth thinking about, in the light that 60 physicist all collectively choose the first approach, i.e., “re-affirm” purely based on their faith in their math colleagues, instead of actually spend time proof reading the actual proof. And I bet 99.9% of mathematicians, 99.999% of the physicists community, and 99.999999% of the human population, would all very likely put their faith in the top 3 math guys and “re-affirm”, instead of wasting their own time in trying to figure out the thing themselves. And I guess Peter, me myself, all readers of this blog, are all probably belong to this majority group like the 60 physicists.
Now are we really so sure the emperor is not naked?
There are two kinds of believe systems, One, faith based, you believe therefore you believe. That we call religion. Another, you believe based on evidences and based on rigorous logic and reasoning, that’s science.
Unfortunately when it comes to extremely difficult math or physics problems, virtually in-accessible to the majority of people even in the field, evidence based religions system could be easily replaced by faith based believe system.
What happens a couple hundred years down the road. When the most difficult math problems become so difficult that no human being can resolve them and we have to just trust the proof spitted out of the printer from a super computer. Will we then become a civilization with a religion that worship machines? And science cease to exist? Will that happen eventually?
Peter, you cited in an earlier post Morgan’s paper at http://www.ams.org/bull/2005-42-01/S0273-0979-04-01045-6/S0273-0979-04-01045-6.pdf
In that paper, Morgan said: “… After Thurston’s work, notwithstanding the fact that it has no direct bearing on the Poincare Conjecture, a consensus developed that the Poincare Conjecture (and the Geometrization Conjecture) were true. Paradoxically, subsuming the Poincare Conjecture into a broader conjecture and then giving evidence, independent from the Poincare Conjecture, for the broader conjecture led to a firmer belief in the Poincare Conjecture. …”.
That is a fascinating commentary on the process of acceptance of conjectures within the mathematics community.
Can you comment on comparing that process of acceptance of the 3-dim Poincare Conjecture by the math community with the process of acceptance of superstring theory as the only possible Theory of Everything by the physics community?
Also, what is the status of the smooth Poincare conjecture in 4 dimensions?
I may be out of touch, but if I recall correctly as of a few years ago it was still unsolved, and Donaldson and Kronheimer say in their book The Geometry of Four-Manifolds (Oxford 1990): “… Smale’s h-cobordism theorem: if X and Y are h-cobordant then they are diffeomorphic … breaks down in four dimensions … “. Also, Milnor’s paper at http://www.math.sunysb.edu/~jack/PREPRINTS/poi-04a.pdf (updated June 2004) says “… In particular, if M4 is a homotopy sphere, then … M4 is homeomorphic to S4 . It should be noted that the piecewise linear or differentiable theories in dimension 4 are much more difficult. It is not known whether every smooth homotopy 4-sphere is diffeomorphic to S4 … As one indication of the complications, Freedman showed, using Donaldson’s work, that R4 admits uncountably many inequivalent differentiable structures. …”.
Has the smooth Poincare conjecture in 4 dimensions been solved, or has significant progress been made?