Witten Interview

Graham Farmelo has posted a very interesting interview he did with Witten last year, as part of his promotion of his forthcoming book The Universe Speaks in Numbers.

One surprising thing I learned from the interview is that Witten learned Calculus when he was 11 (this would have been 1962). He quite liked that, but then lost interest in math for many years, since no one gave him more advanced material to study. After years of studying non math/physics subjects and doing things like working on the 1972 McGovern campaign, he finally realized physics and math were where his talents lay. He ended up doing a Ph.D. at Princeton with David Gross, starting work with him just months after the huge breakthrough of asymptotic freedom, which put in place the final main piece of the Standard Model.

If only back in 1962 someone had told Witten about linear algebra and quantum mechanics, the entire history of the subject could have been quite different. It seems quite possible that within 5 years he would have picked up quantum field theory and maybe started thinking about Yang-Mills generalizations of QED, perhaps, at 16, beating Weinberg and Salam to the electroweak theory. Surely he could have figured out how to do one loop calculations in gauge theory, beating Gross/Wilczek/Politzer to asymptotic freedom and a Nobel prize, possibly a few years early. If he had done this at Princeton, he would have overlapped with John Schwarz, who surely would have then been more interested in pursuing gauge theory than string theory. So, no superstring theory or 1984 “revolution”, and who knows what different sort of path the history of the field would have taken.

A lesson for all parents: if your child is an off-the-scale genius, learning Calculus at age 11, don’t even think about trying to give them a normal childhood. Push them, hard, to skip grades, get to college/grad school early. Do whatever it takes.

I did though find some of the later parts of the interview quite depressing. While acknowledging that neither he nor anyone else has been able to figure out what string theory actually is, this hasn’t shaken Witten’s faith that it’s the only viable path towards a unified theory. Most disturbing, on the topic of the landscape he says that he has gone from finding it upsetting to reconciling himself to the idea. For years, whenever asked about how evidence could be found for string theory, he would point to the naturalness arguments indicating that something like SUSY had to happen at the electroweak scale. Now that the LHC has falsified this and there’s nothing to point to as any sort of “test of string theory”, he shows no signs that this falsification has in any way shaken his faith.

Looking to the near future, he’s most optimistic about the “It from Qubit” business. Maybe he’s right and something will come of this, but I’ve seen no indication of a path to a unified theory in this direction (how do you get the Standard Model? Or has he just completely given up on that?).

I don’t have time right now to transcribe the most relevant portions of the interview, might find time later, or maybe Farmelo will make available a transcription.

Update: As explained in the comments, the advice to parents was not meant to be taken seriously. No, your child is not going to grow up to be Edward Witten, and they do not need to hurry up to revolutionize physics before it is too late.

Sabine Hossenfelder had posted a transcript of the interview here. I’ll add some extracts and some more comments about the interview.

About the landscape:

These two puzzles although primarily the one about gravity which was discovered first are perhaps the main motivation for discussions of a cosmic landscape of vacua. Which is an idea that used to make me extremely uncomfortable and unhappy. I guess because of the challenge it poses to trying to understand the universe and the possibly unfortunate implications for our distant descendants tens of billions of years from now. I guess I ultimately made my peace with it recognizing that the universe hadn’t been created for our convenience.

GF [00:20:43] So you come to terms with it.

EW [00:20:45] I’ve come to terms with the landscape idea and the sense of not being upset about it. As I was for many years.

GF [00:20:49] Really upset?

EW [00:20:50] I still would prefer to have a different explanation but it doesn’t upset me personally to the extent it used to.

GF [00:20:56] So just to conclude what would you say the principal challenge is all down to people looking at fundamental physics.

EW [00:21:01] I think it’s quite possible that new observations either in astronomy or accelerators will turn up new and more down to earth challenges. But with what we have now and also with my own personal inclinations it’s hard to avoid answering new terms of cosmic challenges. I actually believe that string slash M theory is on the right track toward a more deeper explanation. But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is missing or where to find it.

If you theory is not well understood, you don’t even know what sort of thing is missing, and a multiverse is being invoked to explain away why it can’t be tested, the situation seems clear: you have a failed theory. Yes, failure may be personally upsetting to you, but, that’s science.

GF [00:23:20] There’s a famous book about night thoughts of a quantum physics. are there night thoughts of a string theorists is where you have a wonderful theory list developing you know unable to test it. Does that ever bother you.

EW [00:23:31] Of course it bothers us but we have to live with our existential condition. But let’s backtrack 34 years. So in the early 80s there were a lot of hints that something important was happening in string theory but once Green and Schwarz discovered the anomaly cancellation and it became possible to make models of elementary particle physics unified with gravity. From then I thought the direction was clear. But some senior physicists rejected it completely on the grounds that it would supposedly be untestable. Or even have cracked it would be too hard to understand. My view at the time was that when we reached the energies of the W, Z and the Higgs particle we’d get all kinds of fantastic new clues.

EW [00:24:11] So. I found it very very surprising that any colleagues would be so convinced that you wouldn’t be able to get important clues that would shed light on the validity of a fundamental new theory that might in fact be valid. Now if you analyze that 34 years later I’m tempted to say we were both a little bit wrong. So the scale of clues that I thought would materialize from accelerators has not come. In fact the most important clue possibly is that we’ve confirmed the standard model without getting what we fully expected would come with him. And as I told you earlier that might be a clue concerning the landscape. I think the flaw in the thinking of the critics though is that while it’s a shame that the period of incredible turmoil and constant experiment and discovery that existed until roughly when I started graduate school hasn’t continued. I think that the progress which has been made in physics since 1984 is much greater than it would have been if the naysayers had been heeded and string theory hadn’t been done in that period.

“34 years later I’m tempted to say we were both a little bit wrong”??? No, others had good arguments and were right about this (string theory is untestable and has nothing to do with LHC-scale physics), and you had bad arguments and were quite wrong. That this clear result is not being acknowledged and is having no effect on faith in string theory is disturbing.

Update: For another interview with an influential theorist, Sean Carroll has an interview with Leonard Susskind. I don’t think this is a good thing, but Susskind has been very influential in blazing the path that Witten now seems headed down (invoke the multiverse to justify giving up on unifying particle physics, hope very general “it from qubit” considerations will explain gravity). The interview explains in detail Susskind’s point of view.

Update: Farmelo has another interview with a string theorist up, this time it’s Michael Green. When asked if he’s troubled by string theory not being experimentally testable, Green says (19:20):

I don’t think at the moment there’s anything directly to test, because we don’t know what its predictions are.

and says that string theory should really be called “string (not yet a) theory”. Earlier (16:40), he explains

The ingredients of something are there, but it’s clearly not formulated in the right language, and because it’s not formulated in the right language, we don’t really know how to even make sense of its predictions. It doesn’t have any really genuine rigorously derived predictions yet.

Green has been working on string theory for forty years, an entire professional lifetime during which string theory has gone from a relatively simple “(not yet a) theory”, with a true theory seeming not far away, to a much more complicated “(not yet a) theory”, with no progress towards an actual theory in sight. Farmelo doesn’t ask the obvious question of why people shouldn’t interpret this story straightforwardly as the story of a failed speculative idea that never worked out.

This entry was posted in Multiverse Mania, Uncategorized. Bookmark the permalink.

38 Responses to Witten Interview

  1. Schrodinger's Rat says:

    Rather bizarre that after learning calculus at 11 nobody bothered to give him any interesting math for years. If his parents were both illiterate farm workers, then I could understand their not appreciating his talent. But his father (Louis Witten) was a theoretical physicist who’d done a post-doc at Princeton. He must surely have known just a little bit of post-calculus math he could have shared at the dinner table.

  2. Richard says:

    … if your child is an off-the-scale genius, learning Calculus at age 11 …

    I think your exposure to US undergraduates (ie to the products of miserable US high school education) may have coloured your perceptions a little here.

    To me this seems a little precocious (in particular a little more precocious than my never-to-amount-to-anything I was), but nothing truly extraordinary.

  3. kim8 says:

    Anecdotally my understanding is being a child prodigy is not a good indication if one wishes for their success as an adult (what with the added pressure, etc.), although of course there are always unexpected exceptions.

  4. Olly Johnson says:

    Please tell me that the “lesson for all parents” is a joke that I’m missing?

    I tried to explain why at https://twitter.com/BristOliver/status/1124192117125939201

  5. Anonymous says:

    Hi Peter,

    I don’t think that your speculations about what Witten might have done in his teens had he stayed on the math/physics path are any kind of justification for your “lesson to parents”. Who knows what might have happened had he been pushed hard? Your instincts may be different, but mine tell me that pushing kids hard like that is not the way to success, especially not in creative fields. And expecting groundbreaking work from a 16-year old, no matter how talented, seems unrealistic.

    With respect,
    Anonymous

  6. Richard Gaylord says:

    bee hossenfelder just posted a transcript of the interview

    http://backreaction.blogspot.com/2019/05/graham-farmelos-interview-of-edward.html

  7. Shaltut says:

    That plot in the third paragraph 🙂

    I am not surprised that Witten won’t leave string theory. For in this religion, he is the high priest.

  8. Matt Grayson says:

    Peter,

    Pushing a genius is great for everybody else, but destroys the genius as a human being.

    Producing Mozart was child abuse. I know more than a few music prodigies, and their personalities are largely frozen at the age that they were forced to practice 100% of their waking hours. Only the ones so extremely talented that they could achieve world recognition without sacrificing their childhoods have anything like a normal life. I’d put Witten in that last category.

  9. Peter Woit says:

    All,
    The suggestion to parents was tongue-in-cheek. I felt the need to add some humor to this since I found Witten’s announcement he has reconciled himself to multiverse pseudo-science profoundly depressing. Imagining a better parallel universe was the only way to cheer myself up. Witten is a very special case, not a good argument for what to do with other children.

  10. Matt Grayson says:

    BTW, when I was 11, I went to the town library, picked up Hocking and Young’s Topology book, saw the Alexander Horned Sphere, screamed, and ran away. (Their illustration is far the creepiest picture in the literature.)

    Apologies. I’m expecting a “Please no more about childhood” directive any second now…

  11. Lars says:

    As far as I can see, even in Witten’s “special” case, his parents made an excellent decision NOT to push him into math and physics at age 11.

    Witten seems not to have “suffered” too much (if at all) from his wanderings.

    He may not have received a Nobel Prize (yet), but he did get the Fields medal and perhaps not coincidentally, Witten also seems to be a very content, socially well adapted human being .

    And I suspect that like his own parents, he is probably also a very good parent — wisely disregarding popular songs like ” Momma’s , don’t let your babies grow up to be string theorists.”

    Ha ha ha.

  12. Peter Woit says:

    Matt and others,
    Yes, enough about 11 year olds.

  13. Amitabh Lath says:

    Alternative histories are fun. There is an experimental version centered on your own august institution. In 1968 the Columbia-BNL experiment was looking at dimuon production mapping out the recently theorized Drell-Yan spectrum. They saw a “feature” at 3 GeV (you can google “Lederman shoulder 3 GeV”) . It wouldn’t go away no matter how hard they polished the data.

    https://www.mediatheque.lindau-nobel.org/research-profile/laureate-lederman

    I heard this story from Leon Lederman himself. They considered exploring this 3 GeV anomaly in more detail but it didn’t pan out. All the PIs went on to do other less interesting things.
    At this point in the story Leon pauses:
    “If we had followed up we would have scooped the parton discovery and the charmonium discovery. We could have called it the Lee-on”.
    (Pauses for effect)
    “After T.D. Lee, of course.”

    A lesson for all scientists: if your data is showing interesting features, get a better spectrometer. Push the lab directorate, the funding agencies, skip other experiments. Do whatever it takes.

    In this alternate history the top quark and higgs are discovered at ISABELLE in Long Island.

  14. David Appell says:

    When Witten is discussing the cosmic landscape, he says “…the possibly unfortunate implications for our distant descendants tens of billions of years from now.”

    What does he mean by this?

  15. student says:

    from the transcript

    EW [00:17:48] But you’ve never been tempted down the other route. The other options are not.

    EW [00:17:52] I’m not even sure what you would mean by other routes.

    GF [00:17:54] Loop quantum gravity?

    EW [00:17:56] Those are just words. There aren’t any other routes.

    Peter,

    I’ve always wondered what Witten thinks of LQG and now I know.

    I’ve always found it surprising such a towering figure in physics has zero interest and curiosity in one branch of physics, LQG.

    Do you have any idea why he’s so dogmatic on strings as the only way to QG?

  16. Peter Woit says:

    David Appell,

    I’m guessing he’s referring to the fact that the vacua of the string landscape are just metastable, so could in principle tunnel to a different lower energy vacuum.

  17. Peter Woit says:

    student,
    I do think Witten has paid attention to LQG, just believes that it has problems that are not likely to get solved. What’s unclear is why he sees this as so different than the situation with string theory, which also has long-standing problems that have not gotten solved. That he’s more optimistic about the prospects for his own creation is not really surprising.

    What I do find really surprising is what seems to me like a change in tone towards more dogmatism about string theory. Witten in the past has always been pretty cautious in what he says, and I would have expected the negative LHC results to make him even more cautious about making claims about string theory being the only way to go. Instead the change in his tone seems to be in the exact opposite direction. Odd.

  18. Milkshake says:

    I think “faith in string theory” is a bit unfair. He doesn’t have faith in the sense that a religious person has faith. Instead, it seems, there have been so many positive developments in string theory, and the theory is so rich, that he may use induction to conclude that it is on the right track.

  19. Peter Woit says:

    Milkshake,
    There’s no fully objective way to measure “positive” vs. “negative” developments in string theory (I’d argue that the negative now far outweighs the positive, Witten undoubtedly sees things differently). Similarly, where one person sees a “rich” structure, another may see a “mess”.

    Witten is a genius and surely has many valid technical things he can point to, but I think his evaluation of the state of string theory is highly colored by his emotional and intellectual investment in and commitment to a certain vision he came under the spell of in 1984. His reaction to the null LHC results and the landscape shows that it will be extremely hard to sway him from his belief in that vision. “Faith” isn’t an inappropriate word to describe what is going on.

    There is an historical analog of this situation: Witten’s IAS predecessor Einstein, who until the end of his life remained convinced that the way forward to a unified theory lay in a classical geometric framework extending GR, rather than through quantum theory and quantum field theory. I think Einstein was wrong, misled by faith in a vision he first came to in his mid-thirties, and Witten’s situation is not completely different.

  20. A says:

    “What I do find really surprising is what seems to me like a change in tone towards more dogmatism about string theory.”

    Not really surprising. It’s called the backfire effect and shows that not even brilliant geniuses are immune to cognitive bias.

  21. Low Math, Meekly Interacting says:

    One thing I found odd: Witten moved on from QFT because the problems he was working on (non-perturbative QCD and related things, I guess) were “intractable”.

    The Landscape, with its effectively infinite vacua and lacking any sort of selection principle to narrow down the options to something that looks like our corner of the multiverse, seems rather intractable. Same goes for gravity in the transplanckian regime, about which string/M theory has apparently nothing to say because, again, the methods needed are non-perturbative.

    At least while banging ones head against the QCD wall one can compare fruitless attempts with experiment. I’m not sure what the appeal of working fruitlessly on something that can’t be compared with observation might be. “Beauty”?

  22. Peter Woit says:

    LMMI,
    Witten has always kept changing what he actually works on, looking for places he can make real progress. I don’t think he’s actually worked on the Landscape, or ever will, since that can’t go anywhere.

    An accurate way to characterize the current situation I fear is that Witten and most leading string theorists have simply given up on unification. But instead of admitting that string theory was a dead end for this, they have decided to argue that the multiverse solves the problem so they don’t need to think about it any more. From the beginning David Gross argued that this was the big danger: the multiverse would get used as an excuse for giving up (at Strings 2003 he was quoting Churchill as “Never, never, never, never, never give up.”) I had thought Witten had a similar point of view to this, but that seems to be no longer true, as he has moved from finding “giving up” upsetting to making his peace with it.

  23. Anonyrat says:

    So, what is Witten working on these days? Yes, I can look at arxiv — but I can’t fathom the significance or “bigger picture”.

    Thanks in advance!

  24. Peter Woit says:

    Anonyrat,
    I have no idea if Witten now has some “bigger picture” he is pursuing. From the Farmelo and other interviews, it seems that he’s interested in various currently popular ideas about how to get quantum gravity and spacetime as emergent out of some sort of non-gravity quantum theory (“it from qubit”). Much of this is now being pursued at the level of trying to solve low-dimensional toy models. For the absolute latest from him, tomorrow he’ll be talking at the Yau birthday conference in Cambridge (I was hoping to get up there this weekend, but recovering from a cold, stayed home). His title and abstract is at
    http://www.math.harvard.edu/conferences/calabi19/poster/program.pdf
    and says

    Title: “Unorientable Two-Manifolds, Super Riemann Surfaces, And Random Matrices”
    Abstract: Recently, P.Saad, S. Shenker, and D. Stanford showed that what is arguably the simplest model of quantum gravity, which is the Jackiw-Teitelboim (JT) model in two spacetime dimensions, can be understood as a random matrix theory. This result depends on the facts that JT gravity computes volumes of moduli spaces,and on the fact that those volumes have a random matrix interpretation. In this talk (reporting on work with D. Stanford), I will explain how to extend these results to the case that an ordinary Riemann surface is replaced by an unorientable two-manifold and/or a super Riemann surface.

  25. Justin says:

    Hi Woit,

    I understand that you were joking about pushing children to do physics. But, were you joking when you said that Witten would have discovered the electroweak theory and asymptotic freedom at the age of 16?

  26. Peter Woit says:

    Justin,
    The stories of Witten’s ability to learn and absorb very difficult material unusually quickly are many. If he had started studying higher level math and physics seriously at age 12 (1963), by the time of the Weinberg-Salam model (1967), it’s possible he would have understood the ideas behind the model, much less likely he would have been in the right place to do something with them. However, for asymptotic freedom, which wasn’t discovered until 1973 (ignoring ‘t Hooft’s unpublished calculation, I’m counting discovery as including understanding the significance for QCD of the result), he would have ten years to get this done before Gross/Wilczek/Politzer. That he could have discovered asymptotic freedom before age 22 seems not implausible at all. Actually, if he had headed directly to graduate school at Princeton after his 1971 undergrad degree, it seems quite possible he rather than Wilczek would have been the one doing the asymptotic freedom work.

  27. Blake Stacey says:

    Looking to the near future, he’s most optimistic about the “It from Qubit” business. Maybe he’s right and something will come of this, but I’ve seen no indication of a path to a unified theory in this direction (how do you get the Standard Model? Or has he just completely given up on that?).

    My sense of the “It from Qubit” crowd is that they want to get spacetime, or at least some kind of processed spacetime-like product, out of quantum entanglement. The priority is not so much to get the Standard Model from more basic principles, but to find a setting that can include both it and gravity.

    Now, I haven’t so far been blown away by their writings on this. To me, it sounds plausible that they’ll get a geometric way of talking about many-body quantum states, but I’m far less persuaded that one can really go in the reverse direction. In addition, the features cited in their models don’t really seem to do more than scratch the surface of what quantum mechanics has to offer, being features that can be emulated classically (per the Gottesman–Knill theorem) and/or found in theories that have an underlying layer of local hidden variables (like the Spekkens toy model). “It from Barely Nonclassical Bits” just doesn’t have a revolutionary ring to it.

  28. Brett Smolenski says:

    A rather unpleasant question, but one worth considering here: what paths should one take if our universe truly was described by a particular choice of compactification parameters, yet no way to determine what that choice should be? Now Witten and other string theorist believe, or have faith, in at least the first premise of this question, so perhaps the paths they are choosing to take are just their depressing answers to an inherently depressing question. I hope the premise of this question is false, but the universe does not seem to care much for my hopes.

  29. Peter Woit says:

    Brett Smolenski,
    It’s of course logically possible that all parameters of the SM are enviromental, determined by some unknown pre-big bang physics in an essentially random way. If we knew this to be true, of course we should give up thinking about the origin of the SM and do something else, and I guess that’s what Witten and others think they are doing.

    The problem is that we have absolutely no evidence at all for this. There is zero evidence for unification via a string theory compactification. If you’re going to do science, you are supposed to have scientific evidence for your claims, and there is none here.

    For another related interview, Sean Carroll has posted an interview with Leonard Susskind here

    https://www.preposterousuniverse.com/podcast/2019/05/06/episode-45-leonard-susskind-on-quantum-information-quantum-gravity-and-holography/

    where Susskind responds to a question about this not by claiming any positive evidence for it, but by just saying no one else has a better answer:

    “My answer is always, “Yeah, what do you have that’s better?”

    This method of argument works universally: for any unsolved problem you can argue that it can’t be solved (and, if you have tried and failed, you may strongly believe this). But if you want want your “it can’t be solved” argument to be “better” than “it can be solved, we just don’t know how yet”, you have to have some evidence for your argument, and there isn’t any here.

  30. Marko says:

    “Yeah, what do you have that’s better?”

    But there indeed *are* better, or at least equally promising, approaches. For example, higher gauge theory looks promising to revive the idea of unification of everything (gravity included), based on higher category structures instead of Lie groups (my recent draft shows how to construct the SM and gravity using 3-groups). With a little more research and a clever choice of a 3-group, this framework could potentially be able to reduce the number of free parameters in the SM. There are also other alternatives.

    So string theory is really not the only game in town anymore, even for unification of matter, let alone gravity. But the vast majority of people just listens to Witten and Susskind, and ignores everyone else. That’s the problem.

    🙂
    Marko

  31. Peter Woit says:

    Marko,

    I’ve never understood the argument “we’ve devoted 35 years and the efforts of thousands of people to see if our speculative idea works, and by now we understand in detail why it can’t work. So, our speculative idea is better than your much more poorly understood speculative idea, which seems to also have some problems.”

    Or, more briefly: “yes, our idea doesn’t work, but at least we understand in detail why it doesn’t work, so that makes it better.”

  32. shantanu says:

    I would like to see a debate between Witten and some LQG person on LQG. Incidentally some of his own colleagues at Princeton such as Frans Pretorius are working on LQG.

  33. Peter Woit says:

    shantanu,
    I think there is zero chance of ever seeing that. Back at the height of arguments over LQG vs. string theory in 2006-7 Witten showed little evidence of willingness to engage in that argument, even less now.

  34. IM says:

    Dear shantanu,
    I believe that the only time Witten indirectly criticized some of the ideas underlying the research program of LQG in public was his paper “A Note On The Chern-Simons And Kodama Wavefunctions,” see https://arxiv.org/abs/gr-qc/0306083
    To understand the relevance of that paper to LQG and its response (by Smolin and others) you can read the Wikipedia page: https://en.wikipedia.org/wiki/Kodama_state
    Interesting to note that in Witten’s gr-qc/0306083 he never mentions the name “l.q.g.” but only cites Smolin’s hep-th/0209079
    Regards,

  35. Michael Harney says:

    I can’t believe Ed Witten has bought into the landscape nonsense. This is one of the brightest physicists in the last 30 years accepting unfalsifiable predictions. Truly bizzare.

  36. tulpoeid says:

    (@ Peter and Justin,)

    “That he could have discovered asymptotic freedom before age 22 seems not implausible at all. Actually, if he had headed directly to graduate school at Princeton after his 1971 undergrad degree, it seems quite possible he rather than Wilczek would have been the one doing the asymptotic freedom work.”

    When I read this in the original post I took it as part of a great joke. Apologies, but I still do. Mathematical genius is one thing and good sense is another; if a lack of the latter is demonstrated so rigorously by an individual, then s/he cannot be expected to hands down discern study topics that will lead to concrete advancement.

  37. Peter Woit says:

    tulpoeid,

    If you look at the history of the discovery of asymptotic freedom, it was done independently (although there are questions about leakage of info via Sidney Coleman) at Harvard by one graduate student (David Politzer) and at Princeton by David Gross working with his student Frank Wilczek. Witten quickly ended up working with David Gross at Princeton after he arrived there. If he had arrived a couple years earlier he still likely would have started working with Gross, and it’s not at all implausible that he would have been the one Gross set on working on the beta function calculation, not Wilczek. Knowing Witten, it’s also not at all implausible that he would have completed the calculation and understood its significance faster than the Wilczek/Gross pair did. Similarly, if he had gone to Harvard, he quite possibly would have worked with Coleman, who would have sent him down the same path Politzer started down, with the same result, possibly quicker.

    Witten has great mathematical talent, but also great talents as a physicist, and it is the physics that more strongly motivates him than the math. Over the years his greatest work has often brought new mathematics to bear on physics, but this more often has been in the service of solving physics problems, not mathematics problems. I think blaming the string theory debacle on theorists being too “mathematical” and lacking “physical intuition” is a huge mistake. String theory unification is a very “physical” idea, just happens to be a bad one. The mistakes in judgement that have caused generations of physicists to pursue this failed idea are not ones that have their origin in too much exposure to mathematical culture.

Comments are closed.