Book Reviews: The Trouble with Physics, Not Even Wrong

Two recent popular science books provide a startling peek into the deep scientific and sociological troubles in the world of superstring theory. Not Even Wrong by Peter Woit and The Trouble with Physics by Lee Smolin together triangulate the core of the trouble. If you, like me, have been distracted from the foundational problems of physics by the ongoing two-decade fascination with chaos and complexity in the popular literature, now is the time to get back to observing the “deep” stuff. It is starting to get seriously interesting again.

I suspect if you’ve been following physics through casual browsing and media reports like I have, you have probably formed an opinion along these lines: superstring theory is the only game in town, and is basically in good shape except that it cannot be experimentally verified because the energies involved are “too high.” That is, you probably thought it is a sound scientific theory in principle, just not verifiable in practice. What I learned from these two books is that first, it is not sound even in principle, and second, it is not even a theory at all — just a collection of mathematical clues to the existence of a potential theory.

Taken together, the books make for perhaps 3-4 weekends worth of serious reading effort. Since the authors’ metaphoric/natural language explanations of various technicalities are actually more confusing than the math itself would be, you might want to read the books with wikipedia handy, to look up technical bits and pieces that you think you might understand better if you saw the actual equations. But the reading is absolutely worth it. I have rarely been this seriously challenged when reading popular science, and I came out of the experience feeling exhausted, but with a sense that for once, I’d earned that warm glow of wisdom, instead of having been deluded into a false sense of comprehension by a clever writer.

The Two Viewpoints

Peter Woit, now an adjunct at Columbia and author of the Not Even Wrong blog is a physicist by training who shifted to mathematics early in his career. This is the less polished of the two books, but paradoxically, more useful and interesting as a result. Had Woit carefully judged his potential audience and picked the appropriate mix of exposition, metaphor and mathematics, the book would have been easier to read, but not been as revealing as it is. Unlike most popular science authors, Woit doesn’t attempt to make the reader feel like a genius. If you are not a physicist, but have training in some other mathematically-oriented discipline, you’ll probably get about a third of the way through the book before you start to drown. But the overall sociological and philosophy-of-science argument is straightforward enough to follow. Technically, Woit focuses his critique primarily on the structure of the quantum physics aspect of the field.

Smolin’s book is different. First, it is better structured and you’ll get a lot deeper into it before floundering (if you are a non-physicist and you actually think you understand non-mathematical popular physics books, you are either deluded or a closet prodigy). Smolin himself is a respected physicist (now at Canada’s unique Perimeter institute) who has published extensively in both superstring theory and one of the alternatives you don’t hear about — general-relativity based quantum gravity. A deep love of physics permeates every page of the book, and Smolin tries his best to base his conclusions on a sophisticated (and classical) understanding of what legitimate science is about. By contrast, despite his best efforts, Woit’s writing reads like a thinly-veiled polemic in parts. Each book, by itself, provides a critically incomplete look at the situation, but taken together, they provide a reasonably complete assessment.

The Technical Critique

The technical critique hinges on two major elements and several minor ones. The first is the lack of experimental evidence. Superstring theory has dominated fundamental physics since around 1980. Despite more than 25 years of development, it still has no experimental support or any realistic hope of experimental support in the near future. This is apparently historically unprecedented. By itself, this wouldn’t be so worrisome if it weren’t for the fact that superstring theories don’t look verifiable or falsifiable even in principle. This forms the basis for much of the philosophical critique.

The second element has to do with general relativity, and is particularly emphasized by Smolin’s book. You have probably heard some claim to the effect that superstring theory naturally requires and predicts gravity. Not quite. Smolin argues carefully that the most mathematically and philosophically compelling way to think about gravity is through background independent theories. Theories, like Einstein’s general relativity, where the laws do not depend on a particular structure of space-time (like Euclidean or Riemannian), but which actually tell you how space-time behaves based on the distribution of mass and energy. Superstring theories, by contrast, are background-dependent. You have to assume the structure of space-time to get started. That gravitons (hypothetical particles that carry the gravity field) pop out of superstring theory doesn’t help. You want background independence.

Since superstring theory grew out of quantum field theories and the Standard Model rather than out of general relativity, there are a great many other specific technical critiques, most of which are inherited from the the concepts and techniques of the Standard Model (which itself is at least experimentally verified to unprecedented levels within its explanatory scope). Woit’s book gets into some of these at a fairly detailed level, that I am not competent to summarize. One common thread to all these critiques though, is that the successful conceptual advances and technical solutions that do exist should be interpreted as advances in mathematics, not physics. In fact the influence of superstring theory, particularly that of the field’s leading figure, Ed Witten, has led to a renaissance of sorts in mathematics, and earned Witten a Fields medal.

The Philosophical/Aesthetic Critique

The main philosophical critique of superstring/M theory is that it appears to be fundamentally unfalsifiable due to its intrinsic nature: it has enough mathematical flexibility that its parameters can be set to match any observed reality. In fact there are apparently trillions of possible “theories” that form an entire landscape of possible instance theories which could fit any observed reality. This suggests that what we are seeing is not an actual theory, but instances of the trajectories of a theory we don’t yet know (a rough, but limited analogy: if you observed various pendulums and documented each damped oscillation trajectory as a “theory” you would probably be looking for the familiar simple-harmonic-oscillator equation).

But what are we to make of a (hypothesized to exist) theory that can fit any reality? You could either toss out your methodological dogmas about falsifiability and appeal to ideas like the anthropic principle, or you could stick to your guns and decide that this is not real science. Both Smolin and Woit stick to their guns. Going in, I personally was fairly receptive to the postmodern philosophies of “ironic” science that are being used to bolster superstring theory, but I came away convinced that Smolin and Woit are probably right. It is not that postmodern philosophies, such as those of Paul Feyerabend, are bad. It is that they are being employed in bad faith in the service of flawed concepts.

The aesthetic critique is actually more interesting. Superstring/M theory proponents claim that the fundamental beauty of their mathematics is an argument in its favor. Smolin challenges this framing and cites a number of examples in history where apparent beauty turned out to be flawed and gave way to a more correct (and perhaps more beautiful) idea. Woit, on the other hand, calls the “beauty” bluff and pokes at it. He concludes that the theory is not very beautiful at all, but is extraordinarily full of technical ugliness as well as a big failure by that one metric of beauty that scientists tend to apply, Occam’s razor. Superstring theory has vastly more primitive parameters than the Standard Model and therefore, along the Occam-razor dimension, is much uglier.

The Sociological Critique

Both authors lay out the evidence to show that the entire field of physics has been taken over by a culture of intellectual bad faith that dominates discourse, prevents alternative viewpoints from getting traction (or funding), and somehow manages to perpetuate itself. Some calibration might help. Just how bad is this situation, viewed from the perspective of other fields? To a certain extent, these criticisms are usually being leveled all the time at the dominant paradigms in every field. I have heard versions of these critiques in the three academic fields I am personally most familiar with (my own home discipline of control theory, and the closely related disciplines of artificial intelligence and operations research). Calibrating against these examples and others, I have to conclude that the picture painted by Woit and Smolin does look pathologically toxic and unhealthy.

Among the things Smolin in particular points out, which I have never seen in such egregious form in my field, are the deification of a few priestly figures like Witten, the open suppression and exclusion of non-string approaches, the striking homogeneity of physics departments and the insularity of conferences in the field. Reading the anecdotes illustrating the pathologies at work, I felt for the first time that my own field is relatively healthy and open.

The Bogdanov Affair is a particularly interesting illustration of the sociological problems in superstring theory. The short version of the story is this: a prestigious journal apparently accepted clear mathematical nonsense for publication, despite peer review by competent reviewers. Woit examines this case in detail and concludes that the lack of sound foundational concepts in the field has created a world full of technicians who can’t even understand each other’s dense calculations as garbage.

So where might all this have come from. Both Woit and Smolin are careful not to blame individuals, and this seems like a sincere attempt to be fair. The one suggestion that comes through, regarding a possible cause, is the very structure of American science. After the great conceptual advances before World War II, American physics (which, by its dominance, has meant world physics until recently) somehow slid into an era where people asking foundational questions were marginalized, and a “Shut up and calculate!” technician ethic took hold, leading to a vast number of technically brilliant physicists taking over the field, leaving little room for philosophical introspection and alternate conceptual frameworks.

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About Venkatesh Rao

Venkat is the founder and editor-in-chief of ribbonfarm. Follow him on Twitter

Comments

  1. ‘… American physics (which, by its dominance, has meant world physics until recently) somehow slid into an era where people asking foundational questions were marginalized, and a “Shut up and calculate!” technician ethic took hold, leading to a vast number of technically brilliant physicists taking over the field, leaving little room for philosophical introspection and alternate conceptual frameworks.’

    I think your review is good up to this ending. But the “shut up and calculate” philosophy is due to Feynman, and is the precisely criticism he had of string theory:

    ‘… I do feel strongly that this is nonsense! … I think all this superstring stuff is crazy and is in the wrong direction. … I don’t like it that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation … All these numbers [particle masses, etc.] … have no explanations in these string theories – absolutely none!’ – Richard P. Feynman, in Davies & Brown, ‘Superstrings’ 1988, at pages 194-195. (Quotation courtesy of Tony Smith.)

    The problem with superstrings is that the 6 extra dimensions have to be postulated as rolled up and invisible, which prevents their 100 or so parameters (size and shape) being known. It’s impossible to experimentally find these parameters because the only way to reveal the exact structure of the Calabi-Yau manifold would be by Planck-scale scattering experiments in a particle accelerator the size of the galaxy. Without these 100 inputs to the theory being known, there are 10^500 possible outputs, and the theory is so vague it’s not falsifiable. That’s why it’s not science but deluded groupthing maintained by paranoia, arrogance, censorship and bigoted charlatanism towards genuine ‘alternatives’.

    I don’t think there is anything wrong with calculating falsifiable predictions really, just with the kind of non-calculating, arm-waving, priestly, dictatorial, wishful-thinking approach to physics that many hotshot string theorists prefer.

    Best wishes,
    nigel

  2. Thanks Nigel. I think the “shut up and calculate” phrase was in Smolin’s book, in reference to the way 70s era physicists were trained, but the Feynman quote is an interesting take on ‘calculate’ as well. You are probably right that the distinction lies in what you are calculating. In “A beat of a different drum” on Feynman’s work, the author characterizes the difference between Feynman and Julian Schwinger in roughly this way: Schwinger was a brute-force calculator while Feynman was a conceptualizer.

    It is interesting that you call string theorists non-calculating. I got the impression that they are viewed as doing dense, incomprehensible mathematics that explore the mathematical structure of the mystery-theory without getting to falsifiable tests.

    In my own field there is a relatively healthy mix of conceptualizers who love abstractions, pragmatic folks who have no patience for abstractions, “beauty” driven calculators and “data” driven calculators. I’ve never felt oppressed by one kind.

  3. Hi Venkat,

    String stuff makes me very angry because it’s declared to be the mainstream theory, when in fact it’s half baked speculation that can’t calculate anything solid and checkable; not a single falsifiable calculation has come from it. It’s not possible to ever calculate anything seeing that there are so many parameters in the theory whose values are arbitrarily adjustable and cannot be observed.

    I think that Lee Smolin and Peter Woit are deliberately avoiding a confrontation with people like Edward Witten over the deeper issue that string theory is actually based on incorrect speculations that spin-2 attractive gravitons mediate gravity and that standard model forces unify at very high energy. There is experimental evidence against spin-2 attractive gravitons and against supersymmetric (or any other) unification at very high energy.

    Gravity can be predicted accurately (within experimental error) from a quantum gravity model utilising spin-1 gravitons which push masses together. This makes accurate calculations of other phenomena too but was censored off arxiv in 2002. It puts gravity into the standard model very simply. This blows apart most of the requirements for string theory. String theorists ignore work such as this, or loudly oppose it without having first bothered to read it. They’re prejudiced in favor of the particular speculations they work on even though they have not a shred of objective evidence, and that’s very dangerous for them and harmful to others. Other theorists may have wrong ideas, but at least they can test them.

  4. It really needs to be pointed out that whereas the lack of experimental support for Superstrings is universally regarded as a major problem, the lack of background independence is not, and building theories that have such is more a personal quest on Smolin’s part than a universally-acknowledged essential ingredient of any “ultimate” theory. Thanks to bubble chamber data, the experimental support for Special Relativity is now completely overwhelming, whereas the experimental support for General Relativity is not. Background independence as a requirement not only requires one to accept G.R., but also a certain philosophical interpretation, and I for one do not, especially as no-one seems to have been able to do anything much with it anyway. In any case, some of us, including myself, object to the notion that the least understood part of the theory (quantum (?) gravity) should have the most impact in building models.

  5. “The Bogdanov Affair is a particularly interesting illustration of the sociological problems in superstring theory. The short version of the story is this: a prestigious journal apparently accepted clear mathematical nonsense for publication, despite peer review by competent reviewers. Woit examines this case in detail and concludes that the lack of sound foundational concepts in the field has created a world full of technicians who can’t even understand each other’s dense calculations as garbage.”

    It’s worth pointing out that the Bogdanov papers weren’t about string theory, so I’m not sure how this is a criticism of string theory. I, personally, find it extremely unlikely that this mess had anything to do with HEP theory being “full of technicians who can’t even understand each other’s dense calculations as garbage” and MUCH more to do with a couple of lazy, shoddy referees. I’ve had a wide rang of experiences with referees, some of them clearly gave the paper a cusory reading at best. I think that there is a lesson to be learned from this affair but it’s not about string theory, it’s about the serious flaws in the peer review process.

    The anecdotes which Woit tells of prominent Harvard theorists being unable to tell if the papers were nonesense are, I suspect, exaggerations at best (perhaps even a flat out lie). I’m quite certain that any of my grad students could tell that those papers were nonesense.

  6. Wow, looks like I’ve stirred up quite a hornet’s nest here. I thought after so many months, the controversy would have died down and you physicists would have tired of the debate! Thanks for helping me understand the books better.

    Bog, you are partly right, Woit says the Bogdanov papers were actually about quantum algebra and topological quantum field theory, but also says, (page 216): “The discussion section at the end of their three identical papers is all about relations of their work to superstring theory and the problem of supersymmetry breaking” … later on the same page, he cites a referee report that begins “motivated by string theory results…” Without being a physicist myself, I guess I am guilty of not appreciating these finer distinctions.

    Chris, thanks for the clarification. I personally found the philosophical argument for background independence quite alluring, but I understand it is not quite as compelling a critique as the lack of experimental evidence.

    And to re-iterate, I am NOT a physicist, so thank you all for pointing out some of my errors. Am just launching into Penrose’s “Road to Reality” seriously, so I hope to improve the quality of my outsider-commentary on physics as I write more. I want to emphasize that the language in the review (“concludes that the lack of sound foundational concepts in the field has created a world full of technicians who can’t even understand each other’s dense calculations as garbage.”) is my paraphrase of the overall sentiment that I got from Woit’s writing, so any inaccuracy in representing his opinion are my fault and Woit probably meant to convey a more precise critique.

    Rgds,

    Venkat

  7. ‘The anecdotes which Woit tells of prominent Harvard theorists being unable to tell if the papers were nonesense are, I suspect, exaggerations at best (perhaps even a flat out lie). I’m quite certain that any of my grad students could tell that those papers were nonesense.’ – bog

    Bog, making personal statements about other people being liars and your own graduate students being brilliant is ineffectual if you retain anonymity, so why not tell us your name and bathe in the glory of your own making?

    BTW, you’ve missed Woit’s point entirely:

    ‘The one thing the journals do provide which the preprint database does not is the peer-review process. The main thing the journals are selling is the fact that what they publish has supposedly been carefully vetted by experts. The Bogdanov story shows that, at least for papers in quantum gravity in some journals [including the U.K. Institute of Physics journal Classical and Quantum Gravity], this vetting is no longer worth much. … Why did referees in this case accept for publication such obviously incoherent nonsense? One reason is undoubtedly that many physicists do not willingly admit that they don’t understand things.’

    – Woit, Not Even Wrong, Jonathan Cape, London, 2006, p. 223.

    I personally submitted a paper to the editor of Classical and Quantum Gravity (a U.K. Institute of Physics journal) and it was refereed and returned to me by the editor with an anonymous referee’s report stating that it was since it was fact-based and made falsifiable predictions, it was incompatible with string theory speculations and should be censored out. There was no comment on my calculations and no fault found in them whatsoever. The entire reason for rejection was incompatibility with M-theory. So I can personally tell you, these bigoted stringers aren’t scientists, they don’t use science for any purpose whatsoever, in fact, they are all anti-science as Feynman explained:

    ‘Science is the organized skepticism in the reliability of expert opinion.’ – R. P. Feynman (quoted by Smolin, TTWP, 2006, p. 307).

  8. nc,

    Have you actually looked at the Bogdanov papers? You don’t need to be a genius to tell that they’re rubbish, you just need to actually read them. I’m arguing that the referee didn’t even read them (this wouldn’t be shocking – it happens all the time). It’s almost inconceivable that any professor working in HEP theory couldn’t tell that they were nonsense (at Harvard or anywhere else) assuming that said professor ACTUALLY READ the papers. Since these papers were published in obscure journals (also not put in the arxiv) and written by unknown authors it seems to me very likely that nobody at all ever looked at them.

    It’s probably true that many physicists don’t like to admit that they don’t know things but, again, I don’t see what this has to do with string theory. It’s worth mentioning that I’m not a string theorist. I just think that this line of attack is very misleading, in fact it’s probably a staw man. The criticism that is being made here applies quite generally to physics as a whole and I think it’s unfair to put all the blame on string theorists.

    As for my anonymity, I’m preserving it for fear that Witten will have me “wacked”.

  9. Venkat,

    Thanks for the well-done review of my book and Lee Smolin’s. I think you give an accurate and reasonable take on what is in the books.

    bog,

    What’s in my book about the Bogdanovs and Harvard is just a direct quote from an e-mail sent by someone who was visiting there at the time. Undoubtedly a bit of an exaggeration, but if you look at Lubos Motl’s blog you can find an extensive defense of the claim that the Bogdanov papers are not nonsense. I don’t know this for a fact, but I’d suspect that one of the people the e-mail writer may have been observing was Motl (at the time a Junior Fellow, on his way to an assistant professorship). The referee reports are available, and at least one of them indicates that the referee read the paper since he/she gave a long list of things that needed to be changed to make the paper acceptable for publication (it was published after this rewriting to address the referee’s comments).

    I don’t think the Bogdanov affair is specifically an indictment of string theory, but it is an indictment of quantum gravity/cosmology in general, a field which is now heavily dominated by string theory. At the time I initially wrote the chapter of the book about the Bogdanovs I had some qualms about whether I was making too strong a claim about how bad the situation was in this field. The increasingly large amount of absurdity being published as “string cosmology” and the behavior of people like Motl has convinced me over the past few years that the problem I pointed to in that chapter is a very real and increasingly serious one.

  10. Peter,

    Again, one might argue that this sort of shoddy refereeing is quite a widespread problem and not specific to cosmology/quantum gravity/string theory. Remember the story of the computer science grad students who wrote a computer to randomly generate scientific-sounding papers and then got one such paper published in a journal? I quite agree with you that this is a serious problem, however, I personally don’t find that conduct of cosmologists/quantum graviters/string theorists differs significantly from the rest of physics. (There’s a lot more speculation, of course, as compared to something like condensed matter theory but that’s to be expected because those fields are inherently more speculative.) Some sring theorists have not taken the high road in responding to your criticisms and this is lamentable, however, I suspect you’d find the same thing if you had published a lengthy critique of any other research programme.

    It’s worth being clear that I do appreciate your point of view and think that you do make some good points (though the stuff which surfaces frequently on your blog about string theory being a religious cult with Witten as head guru is, I think, a little silly). Differences of opinion in science are a good thing and this kind of controversy has a positive aspect to it. However, I do think that some of the criticisms being launched against string theory are misleading or unfair (of course many of the claims made by string theorists are overstated, but two wrongs don’t make a right). Anyway, I have enjoyed all this lively debate!

  11. Thanks all for an interesting conversation. I couldn’t have hoped for a more stimulating start in the first week of this blog’s existence :)

    I am just starting to explore the world of physics blogs as I cue up more interested-onlooker articles on physics. Pointers to other blogs besides Peter’s welcome!

    Venkat

  12. tubelite says:

    My understanding of physics is at Dave Barry levels (E=H2O), so I can kind of figure out how cars run on water, but string theory is beyond me.

    However, I find the idea of a theory which can has enough knobs that you can tweak it to explain anything, but which does not make predictions which can be experimentally verified, to be dangerously close to theology.

    If you really want something like that, why not go with “It’s God’s will…” For every possible question, you have a simple, unfalsifiable answer “It’s God’s will that the proton decayed to a beta carotene”. Explains everything, but tells you nothing. Saves you a lot of headaches trying to comprehend the math, at any rate.

    The sociological critique reminds me of Pratchett’s “Special and Inevitable Anthropic Principle”, put forward by Unseen University’s Professor of Anthropics, which claims that the entire reason for the universe to exist was for the eventual evolution of the Professor of Anthropics.

  13. Hadn’t heard of that Pratchett (Terry, I assume?) quote. Right on :)

    Quasi-teleological arguments always lend themselves to parody that way. Daniel Quinn’s Ishmael pokes at linear tellings of the Darwinian evolution story that end with “And then finally homo sapiens evolved” by having a starfish tell the story, which ends, “And then finally, the starfish evolved”

    But its use in supporting string theory aside, the anthropic principle does have a certain appeal to it when applied to the multiple-universes scenario (the weaker version which argues that earth has life-supporting properties because we wouldn’t have evolved to remark about it otherwise, is uncontroversial I think).

  14. tubelite says:

    If you want to undermine any serious and educated point of view with booger jokes, I’m your man. I have two more: one is the precursor to the SIAP, again by Pterry:

    Many people are aware of the Weak and Strong Anthropic Principles. The Weak One says, basically, that it was jolly amazing of the universe to be constructed in such a way that humans could evolve to a point where they make a living in, for example, universities, while the Strong One says that, on the contrary, the whole point of the universe was that humans should not only work in universities but also write for huge sums books with words like “Cosmic” and “Chaos” in the titles.

    …and the second from Vonnegut:
    “Beware of the man who works hard to learn something, learns it, and
    finds himself no wiser than before,” Bokonon tells us. “He is full of
    murderous resentment of people who are ignorant without having come by
    their ignorance the hard way.”

    Thank you, I’ll shut up now :)

  15. “[He] had read much, and although he generally forgot what he read, there were left with him from his reading certain nebulous lights, begotten by other men’s thinking, which enabled him to talk on most subjects. It cannot be said of him that he did much thinking for himself — but he thought that he thought.” — Anthony Trollope in some book.

  16. Ah, falsifiability. A good scientific theory must make testable predictions. But let me ask a basic question. Why exactly must the universe be so predictable as to allow us to make testable predictions (as opposed to explaining available data)? What constrains the universe thus?

    — Viraje

  17. I used to think “Of Course ! The Universe is explicable and predictable” until a friend, philosopher and guide quoted someone saying “Mother Nature does not owe it to us to be explicable”..when put that way, I did loosen up on “Her” a little bit after that. She does tempt the reasoning mind into “believing” otherwise though.

    Questions: Is the statement “The Universe is predictable / consistent enough to allow us to make testable predictions based on simple and elegant models” a belief / axiom that all science is based upon? Or can we say that this statement need not be believed for scientific pursuit? Can we drop the “based on simple and elegant models” part of the statement and still find satisfaction in trying to explain the Universe?

  18. Yes, this is a fundamental philosophical problem that has not been adequately addressed that I know of, and here is a thought experiment that helps. What if the sun were suddenly replaced by a giant apple of the same size? In an instant. Before and after, regular physics applies. But there is, let us say, the ‘big apple event.’ This is actually entirely possible if you buy the Popper model of science which says basically that no scientific theory can every be verified, only falsified. Would you call regular physics false, true almost everywhere (in a measure-theoretic sense of only violating point on the time axis), or what? Your theory of the universe would then be your favorite grand unified theory plus the axiom that “On such and such a time, an a giant apple replaced the sun.”

    Point on the ‘simple and elegant’: that’s a deep issue right there, starting from Occam’s razor down to very modern ideas of Kolmogorov-Chaitin complexity, that I’ll talk about in my review of Gregory Chaitin’s “Meta Math: the search for omega,” a book whose bottom line is that ‘simple, elegant and predictable’ is the exception rather the rule if you believe that the continuum (as in 1 d time or 3d space) has any real meaning. You can either drop the idea of simple/elegant/predictable or drop your most basic sense of reality, the sense of “continuousness” that forms the fabric of our perception.

    Most scientists today are Popper-ians, so they DON’T take the simple/elegant as an axiom, but merely an operational truth about anything they can usefully work on.

    There are a few more subtleties here that I’ll think about and maybe tease out later.