Time after time

by on March 11, 2009

Rev. of Sean Carroll, From Eternity to Here:  The Origin of the Universe and the Arrow of Time.  Forthcoming from Dutton (Penguin), 2009.

Time just isn’t what it used to be.  And space has gotten to be a bit of a problem, as well.  When I was a lad, physicists told me that they had these things pretty well figured out: they had discovered material evidence of the Big Bang, they had adjusted their conception of the age and evolution of the universe accordingly, and, having recalculated the universe’s rate of expansion (after Hubble’s disastrous miscalculations threw the field into disarray), they were working on the problem of trying to figure out whether the whole thing would keep expanding forever or would eventually slow down and snap back in a Big Crunch.  The key, they said, lay in finding all the “missing mass” that would enable a Big Crunch to occur, because at the time it looked as if we only had two or three percent of the stuff it would take to bring it all back home.  When I asked them why a Big Crunch, and a cyclical universe, should be preferable to a universe that just keeps going and going, they told me that the idea of a cyclical eternity was more pleasing and comfortable than the idea of a one-off event; and when I asked them what came before the Big Bang, they patted my head and told me that because the Big Bang initiated all space and time, there was no such thing as “before the Big Bang.”

But now they tell me that most of that account of the world is wrong.  For one thing, the expansion of the universe seems to be accelerating, which puts a crimp in the plans of everyone who’d been counting on its eventual collapse; worse still, no one can explain why it is that the universe is different now than it was, say, 14 billion years ago, or why it will be different 14 billion years from now.  For the simple and stupefying fact remains that the laws of physics are reversible; nothing in those laws prevents time from running backwards, and it’s entirely possible to have universes in which conscious entities remember the future and remark offhandedly to each other that you can’t get some eggs without breaking an omelet.  And yet, our universe obeys those reversible laws of physics even though effects follow causes, old age follows youth, and systems move from states of low entropy to states of high entropy.  How can this be?  How might it be otherwise?

It’s above my pay grade, this much I know.  But thanks in part to local fluctuations in my corner of the universe that allow me to read books before they are written (these are known technically as Borges-Boltzmann Waveforms, or more colloquially, “wrinkles in time”), I can reveal that Caltech physicist Sean Carroll will have addressed—if not quite “answered”—these questions in his new book, From Eternity to Here: The Origin of the Universe and the Arrow of Time. (Not to be confused with this superficially similar book, which has been published in parallel universe XGH0046, where Frank Viola gave up a promising baseball career in order to become a Christian writer.)

Carroll will have set himself a difficult task: on the one hand, the questions before him are so fundamental and vexing that they are taken seriously only by cosmologists, sages, and stoners.  How did I get here?  Where does that universe go to?  Why isn’t it the same as it ever was?  On the other hand, From Eternity to Here will take its place in a genre that has emerged into prominence over the past few decades, the Popular Explanation of Incomprehensible Physics.  From Steven Weinberg’s The First Three Minutes to Stephen Hawking’s A Brief History of Time to Brian Greene’s The Elegant Universe, such books have appeared roughly once every sunspot cycle, and they usually try to speak to a readership that can’t follow the math but is willing to try to understand why the Second Law of Thermodynamics isn’t just a metaphor for how things fall apart and why the Uncertainty Principle isn’t just a metaphor for how you change things by looking at them.  In other words, the difficulty in writing such books lies in figuring out (1) how much popular misconception needs to be cleared away, (2) how familiar your readers are with things like the light-bulb-in-the-moving-train example, and (3) how much detail you need in order to explain the truly recondite stuff.

For example: one important aspect of Carroll’s argument will have involved the question of how to think about small-scale, anomalous conditions in the universe.  Like us: if indeed the universe is proceeding apace to its eventual heat death, then where do humans come in?  Perhaps, following a provocative suggestion from 19th-century Austrian physicist Ludwig Boltzmann, we might argue that there are potentially vast differences between the macrostate of the universe and a tiny microstate thereof, just as there might conceivably be rooms in which all the fast-moving molecules have congregated in one corner.  The problem with that argument, Carroll will have noted, is that the current state of our microstate is far too complex to be explained by such random fluctuations: all you would really need to make the point is a “Boltzmann Brain” to develop from random molecules somewhere in the universe, form the thought “I think therefore I am, and hey, what’s all this then,” and return to dust again.  Developing an entire biosphere just to spite the forces of entropy seems a bit . . . well, excessive.

Furthermore, Carroll will have rejected the notion that the universe is homogeneous and isotropic.  Instead, he will argue that thanks to something called “spontaneous inflation,” local exceptions to the general rule happen all the time, and that consequently, the universe can be clumpy rather than smooth, and our little corner of it might not look like all the rest.  Carroll may write, “We should certainly entertain the possibility that our observable patch is dramatically unrepresentative of the entire universe, and see where that leads us.”  This is deeply counterintuitive; it goes against the Copernican principle, according to which it is a bad idea to think that our immediate surroundings differ appreciably from the rest of the universe, and it relies for its plausibility on some very advanced math.  To put this another way, you know you’re in difficult straits when you hear a physicist say that our standard conception of the Big Bang relies on “classical general relativity,” because when physicists say “classical,” they mean “quaint.”  As Carroll may argue:

Most of us suffer under the vague impression—with our intuitions trained by classical general relativity and the innocent-sounding assumption that our local uniformity can be straightforwardly extrapolated across infinity—that the Big Bang singularity is a past boundary to the entire universe, one that must somehow be smoothed out to make sense of the pre-Bang universe.   But the Bang isn’t all that different from future singularities, of the type we’re familiar with from black holes. We don’t really know what’s going on at black-hole singularities, either, but that doesn’t stop us from making sense of what happens from the outside. A black hole forms, settles down, Hawking-radiates, and eventually disappears entirely. Something quasi-singular goes on inside, but it’s just a passing phase, with the outside world going on its merry way.

The Big Bang could have very well been like that, but backwards in time. In other words, our observable patch of expanding universe could be some local region that has a singularity (or whatever quantum effects may resolve it) in the past, but is part of a larger space in which many past-going paths don’t hit that singularity.

Carroll’s larger idea is that ours is one of many not-merely-possible but actually existing universes, that the Big Bang is not the origin of them all, and that in some of them, time may run backwards, forwards, sideways, or not at all.  It is not an utterly alien idea, and Philip Pullman has some fun and games with some aspects of it in the popular book series His Dark Materials.  The passage above, though, seems to dramatize Carroll’s problem quite nicely: the readership imagined here is one that suffers under a vague impression of the Big Bang because its intuitions have been trained by classical general relativity.  How big is this readership, exactly?  And is it expanding?  Carroll’s challenge here lies in disabusing some of his readers of concepts they haven’t gotten to yet, such that he will have had to say, “here’s general relativity, and here are its implications, in layperson’s terms.  OK, well, when it comes to the Big Bang it turns out to be wrong.  So now let me explain quantum gravity, which we don’t quite understand yet.”

It will be a remarkable testimony to Carroll’s skills as a writer and public intellectual that he will have helped to accelerate the expansion of a readership for such things.  Along the way, he will have offered a cogent and compelling explanation for why our universe has such rigid rules about time; he will have suggested that even empty space isn’t empty space; and he will have sketched a picture of a cosmos populated by “baby universes” of all descriptions.  Where are we in that picture?  I won’t say, because I don’t want to give away the beginning.  But I can say that From Eternity to Here will have been a richly rewarding reading experience, even by the exacting standards of the genre, for everyone willing to give it the time.

I didn’t say, because I didn’t want to give away the beginning. Where are we in that picture?  Along the way, he offers a cogent and compelling explanation for why our universe has such rigid rules about time; he suggests that even empty space isn’t empty space; and he sketches a picture of a cosmos populated by “baby universes” of all descriptions. It is a remarkable testimony to Carroll’s skills as a writer and public intellectual that he has helped to accelerate the expansion of a readership for such things.

Carroll’s challenge here lies in disabusing some of his readers of concepts they haven’t gotten to yet, such that he has to say, “here’s general relativity, and here are its implications, in layperson’s terms.  OK, well, when it comes to the Big Bang it turns out to be wrong.  So now let me explain quantum gravity, which we don’t quite understand yet.”  And is it expanding?  How big is this readership, exactly?  The passage below, though, seems to dramatize Carroll’s problem quite nicely: the readership imagined here is one that suffers under a vague impression of the Big Bang because its intuitions have been trained by classical general relativity. It is not an utterly alien idea, and Philip Pullman has some fun and games with some aspects of it in the popular book series His Dark Materials.  Carroll’s larger idea is that ours is one of many not-merely-possible but actually existing universes, that the Big Bang is not the origin of them all, and that in some of them, time may run backwards, forwards, sideways, or not at all.

In other words, our observable patch of expanding universe could be some local region that has a singularity (or whatever quantum effects may resolve it) in the past, but is part of a larger space in which many past-going paths don’t hit that singularity. The Big Bang could have very well been like that, but backwards in time.

Something quasi-singular goes on inside, but it’s just a passing phase, with the outside world going on its merry way.  A black hole forms, settles down, Hawking-radiates, and eventually disappears entirely. We don’t really know what’s going on at black-hole singularities, either, but that doesn’t stop us from making sense of what happens from the outside.  But the Bang isn’t all that different from future singularities, of the type we’re familiar with from black holes.  Most of us suffer under the vague impression—with our intuitions trained by classical general relativity and the innocent-sounding assumption that our local uniformity can be straightforwardly extrapolated across infinity—that the Big Bang singularity is a past boundary to the entire universe, one that must somehow be smoothed out to make sense of the pre-Bang universe.

As Carroll argues:  To put this another way, you know you’re in difficult straits when you hear a physicist say that our standard conception of the Big Bang relies on “classical general relativity,” because when physicists say “classical,” they mean “quaint.” This is deeply counterintuitive; it goes against the Copernican principle, according to which it is a bad idea to think that our immediate surroundings differ appreciably from the rest of the universe, and it relies for its plausibility on some very advanced math. Carroll writes, “We should certainly entertain the possibility that our observable patch is dramatically unrepresentative of the entire universe, and see where that leads us.” Instead, he argues that thanks to something called “spontaneous inflation,” local exceptions to the general rule happen all the time, and that consequently, the universe can be clumpy rather than smooth, and our little corner of it might not look like all the rest. Furthermore, Carroll rejects the notion that the universe is homogeneous and isotropic.

Developing an entire biosphere just to spite the forces of entropy seems a bit . . . well, excessive.  The problem with that argument, Carroll notes, is that the current state of our microstate is far too complex to be explained by such random fluctuations: all you would really need to make the point is a “Boltzmann Brain” to develop from random molecules somewhere in the universe, form the thought “I think therefore I am, and hey, what’s all this then,” and return to dust again. Perhaps, following a provocative suggestion from 19th-century Austrian physicist Ludwig Boltzmann, we might argue that there are potentially vast differences between the macrostate of the universe and a tiny microstate thereof, just as there might conceivably be rooms in which all the fast-moving molecules have congregated in one corner. Like us: if indeed the universe is proceeding apace to its eventual heat death, then where do humans come in?  For example: one important aspect of Carroll’s argument involves the question of how to think about small-scale, anomalous conditions in the universe.

In other words, the difficulty in writing such books lies in figuring out (1) how much popular misconception needs to be cleared away, (2) how familiar your readers are with things like the light-bulb-in-the-moving-train example, and (3) how much detail you need in order to explain the truly recondite stuff.  From Steven Weinberg’s The First Three Minutes to Stephen Hawking’s A Brief History of Time to Brian Greene’s The Elegant Universe, such books have appeared roughly once every sunspot cycle, and they usually try to speak to a readership that can’t follow the math but is willing to try to understand why the Second Law of Thermodynamics isn’t just a metaphor for how things fall apart and why the Uncertainty Principle isn’t just a metaphor for how you change things by looking at them.  On the other hand, From Eternity to Here takes its place in a genre that has emerged into prominence over the past few decades, the Popular Explanation of Incomprehensible Physics. Why isn’t it the same as it ever was? Where does that universe go to? How did I get here?  Carroll has set himself a difficult task: on the one hand, the questions before him are so fundamental and vexing that they are taken seriously only by cosmologists, sages, and stoners.

(Not to be confused with this superficially similar book, which has been published in parallel universe XGH0046, where Frank Viola gave up a promising baseball career in order to become a Christian writer.)  But thanks in part to local fluctuations in my corner of the universe that allow me to read books after they are written (these are known technically as Borges-Boltzmann Waveforms, or more colloquially, “wrinkles in time”), I can reveal that Caltech physicist Sean Carroll has addressed—if not quite “answered”—these questions in his new book, From Eternity to Here: The Origin of the Universe and the Arrow of Time.  It’s above my pay grade, this much I know.

How might it be otherwise?  How can this be?  And yet, our universe obeys those reversible laws of physics even though effects follow causes, old age follows youth, and systems move from states of low entropy to states of high entropy. For the simple and stupefying fact remains that the laws of physics are reversible; nothing in those laws prevents time from running backwards, and it’s entirely possible to have universes in which conscious entities remember the future and remark offhandedly to each other that you can’t get some eggs without breaking an omelet. For one thing, the expansion of the universe seems to be accelerating, which puts a crimp in the plans of everyone who’d been counting on its eventual collapse; worse still, no one can explain why it is that the universe is different now than it was, say, 14 billion years ago, or why it will be different 14 billion years from now. But now they tell me that most of that account of the world is wrong.

When I asked them why a Big Crunch, and a cyclical universe, should be preferable to a universe that just keeps going and going, they told me that the idea of a cyclical eternity was more pleasing and comfortable than the idea of a one-off event; and when I asked them what came before the Big Bang, they patted my head and told me that because the Big Bang initiated all space and time, there was no such thing as “before the Big Bang.” The key, they said, lay in finding all the “missing mass” that would enable a Big Crunch to occur, because at the time it looked as if we only had two or three percent of the stuff it would take to bring it all back home.  When I was a lad, physicists told me that they had these things pretty well figured out: they had discovered material evidence of the Big Bang, they had adjusted their conception of the age and evolution of the universe accordingly, and, having recalculated the universe’s rate of expansion (after Hubble’s disastrous miscalculations threw the field into disarray), they were working on the problem of trying to figure out whether the whole thing would keep expanding forever or would eventually slow down and snap back in a Big Crunch.  And space has gotten to be a bit of a problem, as well.  Time just isn’t what it used to be.

1

Paul 03.11.09 at 2:02 pm

Time must have a stop !!

2

Michael Bérubé 03.11.09 at 2:06 pm

Time isn’t holding up. Time isn’t after us.

3

Henry 03.11.09 at 2:13 pm

He flexes like a whore. Falls wanking to the floor. His trick is you and me, boy.

4

Michael Bérubé 03.11.09 at 2:23 pm

And then one day you find ten years have got behind you.

Oh, and it’s probably worth noting that Paul’s comment was up three minutes before I posted this.

5

dsquared 03.11.09 at 2:27 pm

Time isn’t what it used to be, yet.

6

Michael Bérubé 03.11.09 at 2:34 pm

We’re gonna need more tenses.

7

steven 03.11.09 at 2:44 pm

Developing an entire biosphere just to spite the forces of entropy seems a bit . . . well, excessive.

The notion that the evolution of life on Earth somehow violates or constitutes an exception to the second law of thermodynamics is also sometimes proposed by creationists, but it’s not true.

8

Michael Bérubé 03.11.09 at 2:53 pm

Good thing that no one here said that life violates the Second Law! I was just paraphrasing the “Boltzmann’s brain” problem.

9

JLR 03.11.09 at 3:04 pm

Good thing that no one here will say that life violates the Second Law! You will just be paraphrasing the “Boltzmann’s brain” problem.

10

Michael, if you search for the words “quaint” or “future singularities” in here I bet you’ll wind up deciding to dock some of your copy editor’s pay. Also when I buy a copy of this book and discover that it does not refer generously to Ilya Prigogine, it will make me angry and I’ll blame it all (everything up to the Big Crunch at least) on you. Just letting you know ahead of time.

The notion that the evolution of life on Earth somehow violates or constitutes an exception to the second law of thermodynamics is also sometimes proposed by creationists, but it’s not true.

Life doesn’t violate the law — it mocks, spites, and humiliates the spirit of the law while following the letter without exception. As a sometime student of the thermodynamics of living systems I think this is one of those situations where it’s appropriate to anthropomorphize, because, well look, that’s *us* we’re talking about, among others.

11

Michael Bérubé 03.11.09 at 4:19 pm

if you search for the words “quaint” or “future singularities” in here I bet you’ll wind up deciding to dock some of your copy editor’s pay

Searching for “nothing in those laws prevents time from running backwards” is even worse! What am I was I paying that guy for?

12

Sean Carroll 03.11.09 at 4:20 pm

radish — Prigogine will not be generously referred to, so you will be angry. But at least you will have bought the book, so my purposes will have been served. And you’ll be blaming Michael, so good news for everyone, really.

We’re definitely going to need some more tenses. At least, if we want to make the future perfect.

13

ffrancis 03.11.09 at 4:27 pm

Uhh, Michael, are the repeated passages in slightly different wording insufficient editing or are you illustrating the workings of multiple not quite parallel universes?

14

Ben Alpers 03.11.09 at 4:34 pm

Correct me if I’m wrong, Michael, but as a literary scholar writing about physics, have you not become All That is Wrong With the Academy and, hence, All That is Wrong With Our Culture, and, hence, the epicenter of the Crisis of the West?

Dangeral indeed!

15

steven 03.11.09 at 4:39 pm

Good thing that no one here said that life violates the Second Law! I was just paraphrasing the “Boltzmann’s brain” problem.

Sorry, I will not have realized that when you will have said that our biosphere “spite[s] the forces of entropy” you will have been engaging in anthropomorphism rather than claiming a rule-violation, as radish will be on the point of explaining, for which misunderstanding I will then apologize.

16

Prigogine will not be generously referred to, so you will be angry.

Oh well. What’s past is epilogue… Seriously though, I’ve never been able to figure out why it is that my guy tends to get short shrift in the literature. Is it just that his own writing is practically incomprehensible (at least in English), or is it some sort of conspiracy to deny Belgium its rightful place in scientific history, or what? It’s not like anybody actually disagrees with him on merits as far as I can tell.

17

Ginger Yellow 03.11.09 at 4:50 pm

As a non-physicist who is very interested in cosmology, the “baby universes” idea has always had a lot of appeal to me, but I can never get a handle on how cosmologists conceptualise it without reference to something akin to a branescape. It’s hard enough for us mortals to conceptualise one universe expanding, while not expanding into anything. It’s considerably harder to imagine two or indeed an infinite number of baby universes doing the same thing. Surely these universes have some sort of spatio-temporal relationship to each other, but how? Yet there seem to be plenty of people who endorse some variant of the bubble universes idea without endorsing M-theory.

Anyway, are these the sorts of things Sean addresses?

18

Sean Carroll 03.11.09 at 4:55 pm

Lots of people disagree with Prigogine on the merits. He basically turned against the entire standard picture of explaining the Second Law in terms of the statistics of microstates a la Boltzmann. Cosma Shalizi lays it out.

I do touch briefly on issues of organization and complexity, which are clearly related to the arrow of time but much less well-understood. So basically I throw them out there as mysteries to be contemplated, rather than attempting any explanations.

19

Sean Carroll 03.11.09 at 4:57 pm

20

dan 03.11.09 at 5:11 pm

We’re gonna need more tenses.

back to basics: viva infinitives!

21

Michael Bérubé 03.11.09 at 5:42 pm

Ben @ 13: surely you’re aware that I teach literature in order to bring about physical transformations in the structure of the universe?

Sorry, I will not have realized that when you will have said that our biosphere “spite[s] the forces of entropy” you will have been engaging in anthropomorphism rather than claiming a rule-violation

Yeah, Steven, I will have had the thought that the phrase “developing an entire biosphere just to spite the forces of entropy” sounds all-too-self-consciously anthropomorphic, to the point of self-parody. I think.

But right now, honestly, I don’t care what the creationists seize on, because they’ll seize on anything. Remember that Robert Wright essay in the New Yorker ten years ago, an excerpt from Nonzero in which he basically accused Steven Jay Gould of aiding and abetting the creationist menace by stressing the discontinuities in the evolutionary record and insisting, in Wonderful Life, that the biosphere we inhabit could have turned out radically differently? Well, I thought that was a very terrible argument, unworthy of a bright guy like Wright. And guess what? Apparently, the creationists really liked Wright’s attack on Gould. So it seems to me that trying to think about physics and biology in ways that creationists can’t misuse is something of a mug’s game.

And I see no reason to take the claims of this “Sean Carroll” seriously, as if he has any idea what the book will or won’t contain.

22

cs 03.11.09 at 5:45 pm

I’m not much of a scientist, but I followed the Wikipedia link to “Boltzman’s Brain”, and I’m pretty sure that the whole “paradox” is based on a false assumption, which is that the relative likelihood of something existing is equal to the likelihood of that thing spontaeously arising from random fluctuations in a chaotic system. We know the history of how our world (including the people in it) came to be, and it was a much less unlikely process than the random formation of a Bolzman brain.

23

Ginger Yellow 03.11.09 at 5:48 pm

“So it seems to me that trying to think about physics and biology in ways that creationists can’t misuse is something of a mug’s game.”

True, but at the same time you do so without unnecessarily handing them tools to mislead the incurious and uninformed. The recent kerfuffle over New Scientist’s “Darwin Was Wrong” cover story being a perfect example. Indeed, New Scientist, for all its merits, has a long track record of needless sensationalism.

24

Ginger Yellow 03.11.09 at 5:48 pm

Perhaps I should say “counterproductive” rather than “needless”. Clearly they feel there is a commercial need for it.

25

Ah, interesting link, which definitely explains the short shrift. So us Prigogine fans are “New Age twinks?” I’ll interpret that as meaning “his philosophical views are irritating and stupid and demean his previous scientific contributions, and his contributions weren’t all that novel.” I’m more sympathetic to the idea that Onsager (who I don’t recall ever hearing of, not being a chemist or physicist) got there first, and will look into it.

As a quick and dirty response though I would point out that to biologists “Prigogine’s work on statistical mechanics and the origins of irreversibility … [being] in fact part of orthodox statistical mechanics” is a feature, not a bug. It’s the very reason that dissipative structure is a useful way to look at how living systems manage to “spite” the second law. In physics and chemistry there’s no need to puzzle over how, exactly, the system you’re observing embodies less and less entropy over time.

Physical scientists don’t run into systems like that unless they go looking for them, but in biology those systems are the entire universe of discourse. Darwin described a mechanism by which systems that started out spiting the second law could remain (or become increasingly) spiteful over time, but we still didn’t have an empirical description of the difference between “life” and “non-life.” We had to settle for “knowing it when we saw it.”

Prigogine is considered seminal by this New Age twink because, with dissipative structure, he provided a way to tie Darwin down, using ropes provided by Boltzmann and Shannon. All of his anti-determinist stuff is incomprehensible anyway IMO. Or at least I don’t care about it. Now maybe Onsager did get there first, and maybe y’all in physics don’t care about having the option of measuring “life” in thermodynamic terms, but to some of us it’s a very big deal indeed.

Whether “reversible small-scale dynamics can lead to large-scale effects which are irreversible on any reasonable time-scale” isn’t a problem for us. Assuming our understanding of the fine structure of the Universe is more or less correct, they do that all the time. So what? The problem for us is how do we explain those *particular* systems that are not just irreversibility-emerging-from-reversible-substrate, but which insist on going, so to speak, in the “wrong” direction. Toward increasing complexity. Darwin showed us *how* they did it, but left us asking, well what is it that’s special and different about those systems? How do we identify them? Why doesn’t it happen more often? Why does it happen at all? If there’s a better description out there than dissipative structure I’d love to hear it.

Anyway now that I’ve bought your book the arrow of time prevents me from unbuying it :)

NB: my spell checker flags Prigogine, but not Onsager, thus supporting the conspiracy theory.

26

John Emerson 03.11.09 at 6:49 pm

I’m 100% with Radish. Also, Berube has given me a chance to unleash one of my more obscure rants. Thank you, Michael! (The management may want to have a word with you in the office, though.)

The odds are that Carroll’s book is, with all due respect, crap. I collect this genre of crap and will certainly buy and read this particular load. (Some reviews.)

I have read Shalizi on Prigogine, have followed his links, and have tracked down some of the references in in the library, as well as two or three related books he recommended in correspondence. (Not to say that he agrees with me.)

Based on that reading, there are two problems with Prigogine: first, he unsuccessfully tried to extend insights from his own work into cosmology; and second, he overestimated the value of his own work in biology. I’ll stipulate to these points.

What I find valuable in Prigogine is his history of thermodynamics from the beginning through Boltz, Poincare, Kolmogorov, Onsager, et al, all the way to Prigogine and (not mentioned) Mandelbrot. The gist is that the concepts of time you get from fundamental physics and socmology are useless and harmful when discussing anything as large as a speck of dust or as small as a solar system. In particular, it’s harmful when discussing any kind of life form and any kind of intelligent being.

Shalizi has pointed out that most of what Prigogine said about these topics was well known to physicists decades ago. None of those figures are in the least obscure. My point is that none of the difficulties rising from thermodynamics ever entered scientific common sense, scientistic ideology, or (by and large) philosophy of science. For decades I’ve been asking people about the three body problem, which is a century old and severely damages classical cosmology (the motherland of scientific ideology), and almost no one knows what it is.

These points are here-and-now relevant. Georgescu-Roegen, a mathematical economist, tried to bring entropy (irreversability) to the attention of economists a generation ago, and Samuelson sneered at him. Mandelbrot did work on economics almost that long ago, and he was pigeonholed in a closed alcove separated from the rest of the profession. Mirowski explained all this stuff a decade or more ago, and Durlauf said

The heterodox ideas found here and elsewhere have had no impact on economics as a whole, just as the body of science studies research has had no effect on the natural sciences. Mirowski would attribute this to the “vested interest” of neoclassical economics. An alternative explanation is that research programs such as those found here—which fail to provide either new empirical insights or criticisms of existing practice that are intellectually compelling, let alone constructive ways to proceed—do not have enough substance to warrant a claim on intellectual resources. In my judgment, in this instance the marketplace of ideas is working efficiently.

.

For various reasons (venality, corruption, powerful friends, laziness, disciplinary nativism, the radical evil in human nature) economics has been invulnerable until very recently. “You can’t argue with success” is America’s Rule One. But now that the economists have destroyed five trillion dollars, millions of jobs, millions of retirement plans, and well over a decade of supposed economic growth, people might be listening to the criticisms for the first time. Hopefully, a decade from now the economists will all be out on the street like the rest of us, dumpster diving and digging up roots, with only a few of them still employed pontificating on Fox News and the WSJ opinion page.

And hopefully by that time it won’t only be crazy people who are talking about these things.

27

John Emerson 03.11.09 at 7:05 pm

I do touch briefly on issues of organization and complexity, which are clearly related to the arrow of time but much less well-understood. So basically I throw them out there as mysteries to be contemplated, rather than attempting any explanations.

You might have mentioned that every single one of the entities in out experience is in this mystery zone, and that for this reason your book has less real impact that you might wish.

Once you decide that “real” = “scientifically intelligible”, since fundamental physics is scientically more intelligible than organization and complexity, and call it “a mystery”, then anyone trying to speak reasonably about actuality is vulnerable to accusations of anti-science, Luddism, anti-intellectualism, irrationalism, etc. That’s exactly how economists gained their dominance — by seeming like physicists and accusing everyone else of obscurantism, stupidity, etc.

All anyone has to do is explain that none of the entities of our experience are describable by fundamental physics, and for that reason the truths of fundamental physics are not very helpful for anyone studying these entities. But for many fundamental physics is an object of devotion, like the cow, Kaaba or Jesus Fucking Christ, and its no use talking to those people.

Eddington explained all this almost a century ago, but if people paid attention to him the temple would have been destroyed by earthquakes and mere anarchy would have been loosed on the world, physics porn would have become harder to sell, the suspension of disbelief would have become much more difficult, and many rat-orgasm-units of utility would have been destroyed utterly.

28

John Emerson 03.11.09 at 7:08 pm

29

John Emerson 03.11.09 at 7:12 pm

“Boltzmann” and “cosmology”, if anyone cares.

30

Chris S 03.11.09 at 7:34 pm

“‘time flies like an arrow, fruit flies like a banana” – G. Marx

31

chris y 03.11.09 at 7:38 pm

Sorry to go off topic, but why, when I navigate normally to CT in FF3, does it show nothing later than 6th March? If I hadn’t found a link to this on Carroll’s blog, I’d still believe you’d all gone on strike here.

32

tom bach 03.11.09 at 7:50 pm

chris y: me too, a real puzzle.

33

Righteous Bubba 03.11.09 at 8:03 pm

Sorry to go off topic, but why, when I navigate normally to CT in FF3, does it show nothing later than 6th March?

Recent posts show for me, same browser.

34

John Quiggin 03.11.09 at 8:12 pm

Somewhere between the cache and the browser, time has frozen.

You can move to a universe where time is arrowlike by clicking on comments in FF3, then choosing Home.

I was going to write a post explaining how to use this workaround.
Then I realised I should do another explaining how to use the workaround to
read the post explaining the workaround …

35

des von bladet 03.11.09 at 8:38 pm

I was going to write a post explaining how to use this workaround.
Then I realised I should do another explaining how to use the workaround to
read the post explaining the workaround …

You just need to temporarily reverse the polarity of the causality flux (assuming your Spiterium Biospheres are fully charged; otherwise, I am reliably informed, they may blow).

36

Nick 03.11.09 at 8:42 pm

We’re gonna need more tenses.

Tenses? Where we’re going, we’re not going to need tenses . . .

37

des von bladet 03.11.09 at 8:49 pm

For decades I’ve been asking people about the three body problem, which is a century old and severely damages classical cosmology (the motherland of scientific ideology), and almost no one knows what it is.

My previous boss had a set of Poincare’s collected works (in French, which I don’t believe he could read) on his window-sill, so it slightly depends who you ask. But while these days I hang out with people who hang out with real cosmologistes, I was under the impression it took more than three(3) bodies to form a cosmos, classical or otherwise?

And not withstanding the work of Sussman and his collaborators, I always feel a strong urge to say “tidal friction; the end”. (Not being qualified to actually say it I usually don’t, but you’re never underqualified for a blog-comment thread.)

38

Donald A. Coffin 03.11.09 at 9:01 pm

Yes, but where, in all of this, is the infield fly rule explained?

39

Elliot Tarabour 03.11.09 at 9:33 pm

40

John Emerson 03.11.09 at 9:43 pm

The problem with three bodies is that while explaining three bodies is impossible, and each succeeding body makes things worse.

A Nobelist biologist (possibly Wald or Szent-Gyorgy) who got a fellowship at the Princeton Institute for Advanced Study was eager to find out what the big physicists could teach him. One of them said that he had nothing to say about anything including more than one electron, and that was the end of the interaction.

heisenberg said that quantum theory was difficult, but turbulence in fluids was impossible. But turbulence is very simple physics; you wouldn’t even need molecules, much less life forms. (That is, you could get turbulence with liquid argon flowing over properly-shaped channels made of elemental copper or iron). Fundamental physics sucks as soon as it needs to take it shoes off to count the eleventh particle. (Actually, long before that, but I write vividly.)

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onymous 03.11.09 at 10:49 pm

Everyone knows what the three-body problem is. But it influences job choices much more rarely than the two-body problem, so it gets less attention.

42

JP Stormcrow 03.12.09 at 12:56 am

Tenses? Where we’re going, we’re not going to need tenses . . .

We’re gonna need a bigger universe of discourse.

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Jake 03.12.09 at 1:17 am

The problem with three bodies is that while explaining three bodies is impossible, and each succeeding body makes things worse.

Are you talking about the three-body problem as in the orbital motions of the Sun, Earth, and Moon?

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John Emerson 03.12.09 at 1:29 am

Any three bodies. E-M-S is a single case.

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Jake 03.12.09 at 2:45 am

And you met cosmologists who had never heard of this?

Or do you practice a religion that rejects the validity of numerical integration?

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John Emerson 03.12.09 at 3:02 am

They all know about it (see 26.5), but when writing physics porn they ignore the consequences. It’s the entering wedge of chaos, etc., and the reason why Heisenberg thought turbulence would be impossible to study.

How they deal with it is with sentences like:

I do touch briefly on issues of organization and complexity, which are clearly related to the arrow of time but much less well-understood. So basically I throw them out there as mysteries to be contemplated, rather than attempting any explanations.

.

Otherwise there’d be no porn.

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John Emerson 03.12.09 at 3:07 am

Numerical Integration and the Three Body Problem. From this 2007 book it seems that the problem remains.

48

Walt 03.12.09 at 3:31 am

Read further. It says that whatever technique it’s talking about doesn’t work, so you have to use a higher-order Runge-Kutta technique, which provides “reasonably accurate solutions”.

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John Emerson 03.12.09 at 3:43 am

Aren’t “reasonably accurate solutions” what Science is all about, after all?

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John Emerson 03.12.09 at 3:57 am

My son learned a mathematical technique called “guess-and-check” in 5th grade. Numerical integration seems like a much more sophisticated mathematical technique than that, of course.

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Walt 03.12.09 at 4:28 am

Numerical integration is probably less sophisticated than guess-and-check. But that doesn’t prevent NASA from putting a probe in orbit around Saturn.

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roy belmont 03.12.09 at 5:27 am

We experience the passing of the present moment into what we call the future. We can mark it, record it, separate part of it and describe and name that part, but it’s a seamless, smooth passage as experienced, before and after.
The solidity of the present seems to be already there as we arrive to it, and it doesn’t seem that fantastic to imagine the past as still containing that solidity we experienced when it was our present. Just that we can’t reinhabit it, which makes it easy to believe it’s gone.
The seamlessness because it’s part of the same larger thing, not chopped into measurable pieces.
In order for it to have some kind of beginning it has to be happening in something that’s outside of itself. Time within eternity is one way to look at it.
From an eternal p.o.v. even what we think of as time wouldn’t appear as it does to us, moving, particalized, vanishing and becoming. It would just all be there.
The connection between me writing this in the subjective present and a reader reading it in another subjective present is solid, in that it connects those two solid moments and is made up of them.
This is real, and for purposes of argument, solid, as I’m writing it.
It’s still real as someone reads it, it isn’t disappearing and reappearing.
And it’s just an artifact, not even the totality of the two moments and the solid connection between them.
We liken it to the fluid solidity of water, “the river of time”, but that’s more metaphor than description. And it’s a tautology, because time is what makes the river distinct. Water moving downhill. Without the movement it’s something else. Without time there’s no movement.
Our view is too incomplete, too subjective and isolated, what it is we’re calling time is a subjective experience of something much different. Just as the human subjective experience of matter is much different than “what’s really there”.
Just as with matter without some kind of fixed perspective, human subjectivity being as good as any other, there’s no handle for the image.
We’re looking at something that holds all moments when we talk about “what time is”.
Even if time really does have a beginning and end point, it will have just as much validity, or solidity, anywhere along the line. Simultaneity of the present all the way. From a non-temporal p.o.v.
Which may or not be possible. Maybe even to look at time you have to be in it.
It’s hard to imagine that eternal simultaneity because it requires fitting things into the same imaginative space as other things occupying precisely the same space, because we can’t think about individual moments without giving them solidity, distinct from other moments..
This does add weight to Deutsch’s multiverse idea, though there’s an anthropocentric parallelism in his theory as I understand it, a kind of spatial sidestep, that makes it less than complete. It’s infinity, or near infinity, from the human perspective, the human-sized universe having other parallel human-sized universes in near-endless profusion.
But why not multiple universes infinitely smaller, infinitely larger, as well as infinitely lateral. Multiple everywheres in all directions. And time like that.

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iain 03.12.09 at 10:09 am

“We’re gonna need more tenses”

The major problem is simply one of grammar, and the main work to consult in this matter is Dr. Dan Streetmentioner’s Time Traveler’s Handbook of 1001 Tense Formations. It will tell you, for instance, how to describe something that was about to happen to you in the past before you avoided it by time-jumping forward two days in order to avoid it. The event will be descibed differently according to whether you are talking about it from the standpoint of your own natural time, from a time in the further future, or a time in the further past and is futher complicated by the possibility of conducting conversations while you are actually traveling from one time to another with the intention of becoming your own mother or father.

Most readers get as far as the Future Semiconditionally Modified Subinverted Plagal Past Subjunctive Intentional before giving up; and in fact in later aditions of the book all pages beyond this point have been left blank to save on printing costs.

The Hitchhiker’s Guide to the Galaxy skips lightly over this tangle of academic abstraction, pausing only to note that the term “Future Perfect” has been abandoned since it was discovered not to be.

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engels 03.12.09 at 12:05 pm

John, try comparing http://www.crookedtimber.org with crookedtimber.org. I think when you go to the first address you only see the older posts and by doing what you say you are getting to the second address, which shows the newer ones. I don’t think it is a caching issue because the effect is the same after clearing the cache.

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Sock Puppet of the Great Satan 03.12.09 at 3:10 pm

‘I’m not much of a scientist, but I followed the Wikipedia link to “Boltzman’s Brain”, and I’m pretty sure that the whole “paradox” is based on a false assumption, which is that the relative likelihood of something existing is equal to the likelihood of that thing spontaeously arising from random fluctuations in a chaotic system.’

Y’know, I thought of this as a possible justification for belief in an afterlife: after all, if there is an infinite universe (beyond the limits of the visible universe) or number of parallel universes, then in all that randomness there must be an entity which is a facsimile of me waking up with a replica of my memories I will have when I die, and with similar responses to stimuli, in a pleasant environment. Hooray!

Then I realized that there’s also worlds in which a facsimile that thinks its me is a genocidal dictator, another one where that facsimile is in a prison being farmed for food and body parts, another where the facsimile is a member of a thirty-foot long froglike species being treated for that species’ equivalent of schizophrenia, or another where the facsimile is about to suffer long enduring intolerable pain. So the universe being an infinite improbability generator didn’t seem so comforting then.

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Miracle Max 03.12.09 at 3:12 pm

I miss the Flyin Spaghetti Monster.

57

Perezoso 03.12.09 at 3:24 pm

The trash used as fuel for the winos’ bonfire will not be returning, Emo-san. The arrow of time rides a Harley fatboy towards Futurity, or at least our idea of Futurity. For that matter, stochastics-fetish, or uncertainty fetish, should not be mistaken for applied physics itself, which is more about, yeah, Harleys, or H-bombs (serio, read Bricmont on some of the misapplications of chaos and entropy).

58

Bruce Baugh 03.12.09 at 4:17 pm

This may be late to the party, but the three-body problem isn’t a killer for NASA because they get to keep re-doing the calculations. Dad designed ranging systems, so this was something I grew up hearing about. There are very sophisticated complications, but it boils down to the fact that you can keep measuring and getting a rolling average kind of data, rather than being stuck with once-forever calculations.

As with a lot of life, really.

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salient 03.12.09 at 5:08 pm

From this 2007 book it seems that the problem remains.

*wincing* As an (apprentice) mathematician, specializing in analysis which is the field of study in which the three-body problem is a triviality, am I in any way obligated to explain to John Emerson the difference between computation and characterization, a distinction which completely invalidates the complaint he’s been going on about?

Canonical example: The function f(x) = \sqrt{x^2 + 1} does not have an antiderivative. So, if arc length for some curve is given by \int_a^b \sqrt{x^2 + 1} dx, it is not possible to compute arc length directly (so we find a reasonable approximation procedure). However, it is possible to characterize the arc length precisely: I just did.

Aren’t “reasonably accurate solutions” what Science is all about, after all?

For a precise and careful definition of what we mean by “reasonably accurate” : yes. An undergraduate mathematician usually spends their U.S. junior or senior year, or their freshman year at a more rigorous university elsewhere, doing coursework in which “reasonably accurate” is characterized and its characterization is thoroughly justified.

At its most fundamental level, approximation to within error range, and finding appropriate characterizations of these approximations, is what analysts do (and mathematical physics is essentially a subset of mathematical analysis, if we lump in partial differential equations and discretization under the banner of “analysis”).

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John Emerson 03.12.09 at 5:36 pm

As a pragmatist I’m absolutely fine with “reasonably accurate” calculations. But that isn’t the kind of thing from which fundamental physicists get their conviction that their science is the really true science of really real foundational reality, and that the other sciences are lesser. And it’s not where the economists got the idea that they were smarter than anyone else because they seem like physicists and use a lot of difficult math.

The reversibility of time in fundamental physics does not us what time really is. It’s what time really is for the entities studied by fundamental physics, but not what time really is for the biosphere or for human life. Our commonsense notion of time is useless and harmful if we’re studying fundamental physics, but the fundamental physicists notion of time (and historicity and contingency) is useless and harmful for someone studying biology or human society.

And the reversibility of the time of fundamental physics is not the great new WOW discovery it always is claimed to be in physics porn. It’s been around for at least 50 years. It comes up in pop physics as much as it does for the same reason that lady detectives in crime shows always seem to be slender and busty with great hair and great complexions. No one would care otherwise.

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Jake 03.12.09 at 5:57 pm

The real question is: why are cosmologists and physicists so much more resistant to trolling than philosophers and historians?

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Lee A. Arnold 03.12.09 at 6:02 pm

There is a larger question raised by the 3-body problem. It is whether the universe is essentially mathematical, or whether it may be something else, besides.

The point about the 3-body, or n-body, problem is that there is no closed-form deterministic solution — it has to be done by workarounds and approximations. As Stephen Toulmin wrote, the result is that, after failing to meet the challenge of Leibnitz that Newton’s system was not a completely successful science, in the 19th century “working physicists in Britain took the empiricist line: it was enough to balance the books by improving the match between calculations and observations. …the Three-Body Problem tended to fade into the background, being treated as a metaphysical, not a scientific issue.”

Poincare’s exhaustive examination of the n-body problem late in the 19th century showed that even small variations in initial conditions can end in enormously different results, which has developed into the idea called “deterministic chaos,” sometimes called the “butterfly effect.” This “chaos” is categorized as “well-understood,” although as Gell-Mann pointed out, it is a permanent source of uncertainty because it enhances the effects of ignorance of “fine-grained” prior outcomes, including the effects of measurement errors.

There n-bodies remain, in practice. Philosophers of science have lots of questions, though, because it seems to be combined with the mathematical-cognitive issues that Poincare and Brouwer touched upon and which are reappearing at this time. (Brouwer believed that mathematics originated in a primal temporal split in the field of attention, which he called the “twoity,” and which appears to be similar in conception to the Platonic and Neoplatonic idea of the audacious Dyad. In other words, Time would be necessarily bound-up with mathematics.) Physicists may scoff at this, but they should be reminded that good science may issue from several different originations, not merely observational and/or rationalistic, but also purely philosophical. Einstein’s relativity is a case which was triggered by purely epistemological musing.

And this is only the start of the issue, for complex living systems. Because if you bring in the biosphere you are in further trouble using mathematical predictions. In fact you can approach complex systems in two ways, with varying degrees of partial success based upon the kinds of system. There is the global view, which tries overall equations. And there is the view of a complex system as a compartmentalized model, with connections or flows between all the little boxes. Then you have a sort of n-body computational problem, only now with different equations between each, and at different times of effectiveness. No one supposes that we can successfully predict the number of individuals in the species in a wildlife food web, for example. It hardly matters whether probabilities or agent-based mathematics are used, instead of a strictly deterministic approach.

After that, we would need a full discussion of evolutionary emergence. Here we shall forego the monograph, only to say that biological evolution is “weakly predictive” and that one major sticking point, as Stuart Kauffmann has argued extensively, is that the “space” of possible mutations is not finitely pre-statable, in order to begin the calculating.

That takes us by a giant leap to non-computability and non-halting problems, and then we shall take another leap beyond that, to: “What else is there, if mathematics isn’t the only thing in the universe?”

The answer appears to be something toward which both the later Wittgenstein and Gödel were headed: we can make a tree of higher semantic categories, and the concepts involved in this are NOT algorithmic. It is notable in this regard that, as Hao Wang reports, Gödel was a theist: he came to the conclusion that although the brain would have to be a Turing machine, the mind is not — and that the world is both rational, and a Leibnitzian monadology with God as the central monad!

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James Joyce 03.12.09 at 6:15 pm

Then’s now with now’s then in tense continuant. Heard. Who having has he shall have had. Hear! …

Signifying, if [tongues may talk] tungs may tolkan, that, primeval conditions having gradually receded but nevertheless the emplacement of solid and fluid having to a great extent persisted through intermittences of sullemn fulminance, sollemn nuptialism, sallemn sepulture and providential divining, making possible AND EVEN INEVITABLE, after his a time has a tense haves and havenots hesitency, at the place and period under consideration a socially organic entity of a millenary military maritory monetary morphological circumformation in a more or less settled state of equonomic ecolube equalobe equilab equilibbrium.

Gam on, George! Nomo-morphemy for me!

p. 599

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salient 03.12.09 at 6:22 pm

“What else is there, if mathematics isn’t the only thing in the universe?”

That is a very odd question, because it presupposes anyone in the discussion believes mathematics ‘is’ the only thing in the universe.

Mathematics is a study of the consequences of adopted axioms. The more interesting mathematics, in my determinedly biased opinion, provide us with the ability to characterize and model real-world phenomena, and (ideally) make testable predictions with our models.

The idea that we’d be able to characterize every phenomenon that occurs in the universe with a computational-friendly mathematical model is at least as wildly implausible as the idea that we’d be able to give every thing in the universe a name.

And I’m not even going to touch the premise that maybe there’s a distinction between a mathematical model and a thing, which you are accepting without justification only to dismiss. At best it’s counter-intuitive.

both the later Wittgenstein and Gödel were headed

You exhaust your credibility in this last paragraph (if not long before). Gödel demonstrated that, with finitely many axioms of sufficient sophistication to produce an arithmetic structure, it won’t be possible to prove every true statement that can be made within the system. So, mathematics restricts itself to those statements which we can prove. Gödel demonstrated that mathematics must restrict itself to proving a proper subset of the statements that are consistent with the axioms we adopt. You can cup your hands and go WoooOOOOOoooo about central monads if you like, but at least take credit for your own propositions: don’t pawn them off on name-checks.

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John Emerson 03.12.09 at 6:36 pm

61: I roused 2 or 3 physicists over at Unfogged.

Gödel was also an admirer of Eisenhower and was suspicious of Kennedy because of Kennedy’s militarism. Though he was Viennese, preferred American pop music to Austriann. When he wasn’t being a genius or mentally ill he seems to have been a fairly mundane guy. Everyone should read Hao Wang on Gödel and other topics.

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John Emerson 03.12.09 at 6:41 pm

I think that salient is making more noise than he can back up. Some mathematicians accept his view of math, many don’t, and as far as I can tell what Lee Arnold says about Goedel squares with that Goedel’s friend and editor Hao Wang says about him.

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Righteous Bubba 03.12.09 at 6:50 pm

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Daniel 03.12.09 at 6:59 pm

I have to agree with JE here, Salient. The biographical fact that Goedel was a dualist and believed that a form of dualism (that “the human mind […] infinitely surpasses the powers of any finite machine”) was implied by his theorem, is really quite well established. It might or might not be right, but that’s what Goedel believed.

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Perezoso 03.12.09 at 7:03 pm

Undecidability matters only when there’s some argument/program/algorithm that’s undecidable. AS realized by Church and Turing undecidability may present a problem (not so much when self-referentiality is verboten a priori). Goedel ‘s numbering system counterargument (really to Whitehead/Russell’s logicism) itself questionable. Goedel probably had an ax to grind against Russell too, perhaps due to religion/politics . Goedel’s mostly negligible , like platonic mathematics, and mystical readings of quantum mechanics and stochastics itself. (the class of “not negligible” would be like depleted oil reserves, or a drought, plagues, etc. A stanford physicist may have a great insight into the Big Bang, or some parallel worlds BS interpretation of copenhagen, etc: where’s his solution to the CA drought? ).

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onymous 03.12.09 at 7:08 pm

The reversibility of time in fundamental physics does not us what time really is. It’s what time really is for the entities studied by fundamental physics, but not what time really is for the biosphere or for human life.

“The reversibility of time” is a strange phrase to use, anyway. Time keeps going in one direction. The microscopic processes of fundamental physics are reversible, and lead to irreversible macroscopic processes. To say the process is reversible is to say if you can have state A turning into state B, you can also have state B turning into state A. But this doesn’t mean time is running backwards. Earlier today I walked into my office, took off my coat, and sat in this chair, and in a little while I will stand up, put on my coat, and leave my office. This doesn’t mean time will be running backwards in a little while.

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salient 03.12.09 at 7:32 pm

I think that salient is making more noise than he can back up.

Maybe. Probably. Oh, definitely. And then I blame my squeaky office chair when the noise irritates people.

Some mathematicians accept his view of math, many don’t

I’m in the position of both readily agreeing with you, and wondering what alternative “view of math” you have in mind.

what Lee Arnold says about Goedel squares with that Goedel’s friend and editor Hao Wang says about him.

That’s fair. But this doesn’t carry any more weight (in terms of what Goedel rigorously demonstrated versus dreamily hypothesized) than Einstein’s hypotheses about a God who eschews dice. Since Lee was not suggesting the hypothesis carried very much weight (as I see in a re-read, there’s a lot of hedging), I walk back that criticism: it was inaccurate and unfair. My apologies.

One of my noisome errors, actually: Taking issue with the final paragraph, which I should have ignored, while I didn’t take issue with passages that were clearly problematic. I was intending to call Lee on the fact that Lee doesn’t seem to know what a “closed-form deterministic” model is, or what it means to have a deterministic model at all, and Lee’s subsequent paragraphs demonstrate a problematic lack of understanding: Lee sets up a very strange interpretation of the universe-as-mathematical-model only to knock it down.

I suspect very few mathematicians would look at a chair and say, “ah, that is a mathematical object!” Is its behavior when pushed across the room, i.e. its motion, accurately predicted with mathematical models? Yes, it seems so. Is the chair itself a piece of mathematics? Only if you adopt a very weird and counter-intuitive notion of what ‘mathematics’ is. I imagine that even the most devoted of Grand Unified Theorists would go no further than to hypothesize that all interactions between matter can be characteristically modeled (such a model may not be will not be computational and certainly is not the same thing as the entities whose behavior it models).

Anyone who finds themselves asking, What else is there, if mathematics isn’t the only thing in the universe? needs to seek out further clarification about what mathematics is / is not / can be / can’t reasonably be.

I suppose Lee’s comment is an example of what problematic characterizations of mathematics and physics in the popular literature can do to an unsuspecting layperson. Call it the Gary Zukav Problem. If you’re proposing anything in your book that your audience will feel they only really understand while experiencing an induced hallucination, rethink your pedagogical strategy.

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John Emerson 03.12.09 at 7:45 pm

Earlier today I walked into my office, took off my coat, and sat in this chair, and in a little while I will stand up, put on my coat, and leave my office.

Physics reversibility means that “earlier” states actually can be returned to exactly. You just rearrange the quarks to the way that were before. Thermodynamics says that observable reversion to an earlier state is only formally possible the way that it’s formally possible for monkeys to type out the works of Shakespeare, except much, much less so. In the example you give, there isn’t reversal, because your digestion, respiration, etc. are still working.

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onymous 03.12.09 at 7:53 pm

Yes, John, thank you, I am a physicist and understand all of that, I was just trying to oversimplify to explain why it’s processes that are reversible, not time itself, as you seemed to imply. (Posting under a different pseud here than the one I use on Unfogged, as I mentioned there recently.)

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John Emerson 03.12.09 at 7:58 pm

The things I’m reading are saying that time itself is reversible in the world of fundamental physics, or that there is no time, or that time is a dimension of space and there’s no present moment, etc. This seems to be what physics porn, which is a whole pop genre and which also is found in AP (“Philosophy of Time”, ed. Poidevin) is always keying on. But the level of complexity/ number of particles above which it is no longer true is very low.

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salient 03.12.09 at 8:04 pm

A trouble with many worlds theory, even before we get to Max Tegmark: it hypothesizes there is a universe in which extraordinarily weird things happen regularly. In particular, it seems unbelievably odd that we just happen to inhabit a universe in which rudimentary probabilistic models, e.g. quantum electrodynamics, are such reliable predictors of the specific phenomena they are intended to model. What kind of physics is being developed by scientists in a world, among these many worlds, in which phenomena that we envision as extraordinarily improbable occur fairly often? What kind of models are they developing?

Fun to think about, but I suspect Max Tegmark is being coy when he talks with his interviewer about “structure” in that linked interview.

Well, Galileo and Wigner and lots of other scientists would argue that abstract mathematics “describes” reality.

I love the scare quotes around “describes” in this sentence… also, Schellnhuber and Pierrehumbert lots of other scientists would argue that carbon emissions due to humans burning fossil fuels “has had an effect on” climate change.

Well, the hypothesis predicts a lot more to reality than we thought, since every mathematical structure is another universe. Just as our sun is not the center of the galaxy but just another star, so too our universe is just another mathematical structure in a cosmos full of mathematical structures. From that we can make all kinds of predictions.

I guess the interviewer didn’t know to ask, specifically, for “testable hypotheses.”

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onymous 03.12.09 at 8:06 pm

time itself is reversible in the world of fundamental physics

There is no sense in which that is true. Fundamental processes are reversible. Time, to quote a crappier song than the ones quoted above, keeps on ticking.

that there is no time

It’s possible that time is an emergent property of something timeless, but so far as I know there’s really not even a good toy model of how that would work. Space emerging from something spaceless is a much more robust and plausible concept. (The well-informed can insert mumblings here about the timelessness of the Wheeler-DeWitt equation, mumble mumble Hamiltonian constraints in quantum gravity, maybe even timelike Liouville modes, but it’s hard to defend them as more than mumblings.)

time is a dimension of space

Time is a dimension of spacetime. But it’s not at all the same thing as space, and anyone who tries to tell you otherwise needs more remedial lectures in relativity.

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onymous 03.12.09 at 8:08 pm

A trouble with many worlds theory, even before we get to Max Tegmark: it hypothesizes there is a universe in which extraordinarily weird things happen regularly.

No no no no no. The wavefunction is still an amplitude with a probability interpretation. Extraordinarily weird things are parts of the wave function with tiny, tiny amplitude. All “many worlds” means is taking seriously that there is no non-unitary collapse. In other words, it’s just quantum mechanics, with no unnecessary pieces tacked on.

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onymous 03.12.09 at 8:11 pm

Clearly Emerson is right about one thing: popular books about physics fill people’s heads with all kinds of nonsense.

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John Emerson 03.12.09 at 8:16 pm

Onymous, I can tell you that the misinterpretations I’m citing are not rare and don’t seem to come from Zukov-style idiots. I disagree with them as much as you do, but am now wondering whether you arent jumping from your own conclusions on the question to the claim that no one at all says or believes the things that I’m saying they say. Maybe you’re interpretation of these things is right, but is it as universally held by top people as you say it is?

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salient 03.12.09 at 8:33 pm

In other words, it’s just quantum mechanics, with no unnecessary pieces tacked on.

That’s not exactly John Wheeler’s interpretation, is it? Last I heard, and I may have misheard, the conventional interpretation is that cases of universes in which sufficiently habitable conditions for life to develop must satisfy conditions which are sufficiently close to ours to eliminate all the unreasonable cases from consideration. In other words, in the universes in which observations would deviate substantially from our own, there’s no scientists in existence to observe the phenomena. In other words (to ape Schrödinger), people don’t live in worlds in which a particle of material with predicted half-life of five minutes regularly decays in less than a minute’s time, because those universes don’t adequately support life. Please do correct me where I’m wrong.

(And I should mention that by “extraordinarily weird things” occurring I don’t intend “nonsensical things”, e.g. walking through walls or whatnot.)

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salient 03.12.09 at 8:45 pm

Here’s a thought experiment: suppose I go ahead and execute Schrödinger’s experiement, with a particle of material that has five-minute half-life, but I raise the curtain after two seconds. By accomplishing a measurement, I have “split” the universe into two universes, A and B. A is far more likely, but B is nontrivial. Let us assume our model of the radiation predicts a 1% chance of B. No matter which world I end up inhabiting, I repeat this doctored experiment for as long as I can, as frequently as I can, let’s suppose several times a minute, or concurrently if you prefer. So, the me that occupies B may end up in BA or BB, the me that occupies BB may occupy BBA or BBB, and after n many performances there exists a me that occupies B^n. Which seems to be a universe in which extraordinary things are happening fairly regularly, for sufficiently large n. As a scientist in the world B^n, for what value of n do I start considering abandoning the model of particle radiation that I have?

Perhaps, even, (unseriously,) at what point have I performed the experiment sufficiently many times so as to preclude my own existence? And, wouldn’t you much rather live in a world in which I wasn’t being a pest about this? (But I sincerely appreciate replies.)

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John Emerson 03.12.09 at 9:44 pm

Off in the porn zone again. I try to help people.

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notsneaky 03.12.09 at 9:58 pm

What is this concept called “time” and why should I care?

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James Joyce 03.12.09 at 11:31 pm

salient, Why not just read Kurt Gödel? Quotations from Hao Wang, A Logical Journey [bracketed clarifications by Wang]:

0.2.1 My theory is a monadology with a central monad [namely, God]. It is like the monadogy by Leibnitz in its central structure.

0.2.2 My theory is rationalistic, idealistic, optimistic, and theological.

9.1.8 It is an idea of Leibnitz that monads are spiritual in the sense that they have consciousness, experience, and drive on the active side, and contain representations (Vorstellungen) on the passive side. Matter is also composed of such monads. We have the emotional idea that we should avoid inflicting pain on living things, but an electron or a piece of rock also has experiences. We experience drives, pains, and so on ourselves. The task is to discover the universal laws of the interactions of monads, including people, electrons, and so forth. For example, attraction and repulsion are the drives of electrons, and they contain representations of other elementary particles.

9.1.9 Monads (bions, etc.) are not another kind of material particle; they are not in fixed parts of space; they are nowhere and, therefore, not material objects. Matter will be spiritualized when the true theory of physics is found. Monads only act into space; they are not in space. They have an inner life or consciousness; in addition to relations to other particles (clear in Newtonian physics, where we know the relations between the particles,) they also have something inside.

9.1.17 One needs some Arbeitshypothese or working hypothesis in considering the question whether one should pursue certain metaphysical projects now. My working hypothesis is that the project under consideration has not yet been studied from the right perspective. Specifically, previous attempts have been hampered by one combination or another of three factors: (1) lack of an exact development of science; (2) theological prejudices; and (3) a materialistic bias. The pursuit, unhampered by any one of these three negative factors, hasn’t been tried before.

6.1.2 Either the human mind surpasses all machines (to be more precise: it can decide more number-theoretical questions than any machine) or else there exist number-theoretical questions undecidable for the human mind. [It is not excluded that both alternatives may be true.]

6.3.1 Note that the question of whether there exist finite non-mechanical procedures (such as those involving the use of abstract terms on the basis of their meaning,) not equivalent with any algorithm, has nothing whatsoever to do with the adequacy of the definition of “formal system” and of “mechanical procedure.” …Note that the results mentioned in this postscript do not establish any bounds for the powers of human reason, but rather for the potentiality of pure formalism in mathematics.

7.3.6 Philosophy as an exact theory should do to physics as much as Newton did to physics. I think it is perfectly possible that the development of such a philosophical theory will take place within the next hundred years or even sooner.

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Lee A. Arnold 03.12.09 at 11:47 pm

I’ll leave the Wittgenstein for an after-school exercise.

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Elliot Tarabour 03.13.09 at 12:33 am

The many world hypothesis is not particularly economical. I think there are more effective interpretations of QM that address the “simple-minded” approach of MWI. For example the information theoretic interpretation posed by Jerome Rothstein seems like it might be just a tweak or two away from the truth. IMHO.

e.

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Lee A. Arnold 03.13.09 at 1:28 am

(1) Sorry, “James Joyce” was me. I copped the moniker to quote from page 599 of Finnegans Wake.

(2) I thought a “closed-form” is an equation where you plug in the numbers to get immediate results, e.g. not iterative or recursive.

(3) The reason why the failure of closed-form solutions to the n-body problem matters to the philosophy of science is because throughout the 18th and 19th centuries scientists very precisely DID hope that we would be able to “characterize every phenomenon that occurs in the universe with a computational-friendly mathematical model.”

(4) Before accusing Gödel of dreamily hypothesizing, we may want to run it through, again:

I would guess his originating point is here, footnote 48 of “On formally undecidable propositions of Principia Mathematica and related systems” (1931):

“The true source of the incompleteness attaching to all formal systems of mathematics, is to be found — as will be shown in Part II of this essay — in the fact that the formation of ever higher types can be continued into the transfinite (cf. D. Hilbert, “Uber das Unendliche,” Math. Ann. 95, p. 84), whereas in every formal system at most denumerably many types occur. It can be shown, that is, that the undecidable propositions here presented always become decidable by the adjunction of suitable higher types. A similar result also holds for the axiom system of set theory.”

It appears that he started applying this reasoning to all concepts, i.e. semantic content in general. Wang quotes Gödel many times looking for a “concept theory.” I think “concept” here means the necessary semantic component of the relata of formal logical propositions. I think that what he didn’t conceive is that “concept” always includes the one-step hierarchy of the higher logical context it is under — in other words, a step up OUT of the formal proposition — something which the later Wittgenstein approached by showing that natural language sentences may come from different intentions, (or by the same token, that the same intention can be expressed in different sentences.)

Gödel in Wang, A Logical Journey (MIT, 1996):

7.3.20 The analysis of concepts is central to philosophy. Science only combines concepts and does not analyze concepts. It contributes to the analysis of concepts by being stimulating for real analysis. Einstein’s theory is itself not an analysis of concepts (and does not penetrate into the last analysis;) its metaphysics (with its four-dimensional frame) deals with observations which are the given for science. Physical theories change quickly or slowly; they are stimulating to investigate but are not the correct metaphysics. Exact reasoning, positive integers, and real numbers all occur in metaphysics. (It is not sure that topology also does.) For example, natural objects differ more or less, and metric space is concerned with how much they differ. Abstract structures are naturally chosen. Analysis is to arrive at what thinking is based on: the inborn intuitions.

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Matt McIrvin 03.13.09 at 2:12 am

But that isn’t the kind of thing from which fundamental physicists get their conviction that their science is the really true science of really real foundational reality, and that the other sciences are lesser.

Ohhh, so that’s all that’s bothering you. I thought you were saying that radios shouldn’t work because electromagnetic theory doesn’t apply to macroscopic objects.

I recommend reading up on effective field theory. I tend to think it has some philosophical significance as a kind of application of humility to physics.

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John Emerson 03.13.09 at 3:55 am

It’s useless, Matt. Physics porn will be with us forever. I have a quote somewhere from Philip Anderson, a solid state physics Nobelist, and he thinks that fundamental physicists are assholes too. But nothing will changes. Sure, there are good economists, good philosophers, and good physicists out there. But they’re lost in the noise.

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onymous 03.13.09 at 5:08 am

I have a quote somewhere from Philip Anderson, a solid state physics Nobelist, and he thinks that fundamental physicists are assholes too.

Whereas Philip Anderson is universally thought to be a perfectly nice and ego-free guy? Christ, dude, you have no idea what you’re talking about.

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onymous 03.13.09 at 5:11 am

McIrvin is right, by the way, about effective field theory; a significant chunk of what fundamental physicists do involves quantifying precisely how little, and in what ways, the things we study can influence large-scale phenomena.

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John Emerson 03.13.09 at 6:33 am

Yes, I know, Onymous. Surprisingly, economists and philosophers are also above all criticism and are good all the time, except for the odd Philip Anderson or Ilya Prigogine you can find in any field.

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Rob G 03.13.09 at 3:20 pm

he thinks that fundamental physicists are assholes too.

I think this is the weak form of the Misanthropic Principle.

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Lee A. Arnold 03.13.09 at 3:55 pm

John, it’s very odd indeed. People automatically think they know what they know. Everything I wrote in the comment that Salient got foolishly angry about is straight from the seriously reasoned viewpoints written by the people I named. My only own speculation in that comment is that both Wittgenstein and Gödel (and I would add Poincare, Brouwer, several living philosophers, and last but not least Gregory Bateson) appear to be heading in the same direction as to WHAT TO DO ABOUT the limits of logic in both mind and nature. (I would speculate further that will end-up being understood as one of the most significant things about philosophy in the 20th century.) And I’m not a theist, but it is fascinating that the greatest logician since Aristotle was led to reintroduce theism. In every other respect I think agree with his list:

(Wang writes, p.316) “On another page [written around 1960], under the rubric ‘My philosophical viewpoint,’ Gödel lists fourteen items which appear to be an attempt to outline his fundamental philosphical beliefs:

1. The world is rational.
2. Human reason can, in principle, be developed more highly (through certain techniques.)
3. There are systematic methods for the solution of all problems (also art, etc.)
4. There are other worlds and rational beings of a different and higher kind.
5. The world in which we live is not the only one in which we shall live or have lived.
6. There is incomparably more knowledge a priori than is currently known.
7. The development of human thought since the Renaissance is thoroughly intelligible (durchaus einsichtige).
8. Reason in mankind will be developed in every direction.
9. Formal rights comprise a real science.
10. Materialism is false.
11. The higher beings are connected ot the others by analogy, not by composition.
12. Concepts have an objective existence.
13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
14. Religions are, for the most part, bad — but religion is not.

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John Emerson 03.13.09 at 4:31 pm

Well, another CT thread ruined by me. Don’t try to figure out my motivations. I’m just a CT plot function, like Satan or Iago, driven on by an unexplainable, irrational need to devastate civilization and wreak havoc on my innocent victims.

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salient 03.13.09 at 4:46 pm

I try to help people.

It’s probably hopeless (but thanks).

Matter will be spiritualized when the true theory of physics is found.

Shucks. At the very least this (again) confirms I was wrong to accuse you of projection. The WoooOOOooo is indeed coming from inside the house.

The reason why the failure of closed-form solutions to the n-body problem matters to the philosophy of science is because throughout the 18th and 19th centuries scientists very precisely DID hope that we would be able to “characterize every phenomenon that occurs in the universe with a computational-friendly mathematical model.”

Okay, this explanation of why the $n$-body problem is historically important makes sense. Specifically, it makes much more sense than what you said earlier: “It is whether the universe is essentially mathematical, or whether it may be something else, besides.”

Also, it’s the 21st century, now. We’ve grown up about what we can and cannot do (I hope I can trust onymous that conventional in-field interpretations of “many worlds” doesn’t extend beyond the constraints of onymous’ own characterization), and are much more sensible about recognizing how very little we can indeed characterize with mathematical models. (Apparently, a la Emerson, with some exceptions.) Your previous quote presented the problem as, if I may summarize/characterize, a kind of universal mystical contemplation, rather than a resolved problem which historically has helped to shape the philosophy of mathematics (and bring that philosophy within proper constraints, thereby resolving the problem). Like Russell’s paradox, which helped shape our understanding of the constraints on set theory.

To your credit, it’s apparently Goedel’s dreamy contemplation in response to the far more blatantly absurd proposition that mathematics can characterize all phenomena that have occurred or could occur with computational-friendly models. So, alright, you were speaking on behalf of Goedel. The idea is itself a mischaracterization of what has been demonstrated uncontroversially.

I know most of the advances in analysis occurred post-1870 and were controversial, but again, these are historical problems that illustrated the constraints on mathematics, rather than mystical problems that require or induce this central monad conjecture. Goedel was free to hypothesize about a God through which the truth of logical propositions exists as consequence, exactly as Einstein was free to hypothesize about a God who wouldn’t leave everything to chance. It’s equally arbitrary relative to the well-established theory, equally controversial, equally unproven, equally dreamy.

Before accusing Gödel of dreamily hypothesizing, we may want to run it through, again

Sure. So, to be fair and precise, referencing only what you’ve quoted: Philosophy as an exact theory should do to physics as much as Newton did to physics. I think it is perfectly possible that the development of such a philosophical theory will take place within the next hundred years or even sooner.

If that’s not dreamily hypothesizing, what is? (While some subset of ‘ideas envisioned by very bright people while dreamily hypothesizing’ turn out to be exceptionally accurate characterizations, that doesn’t entice me to accept all such hypotheses wholesale.)

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Lee A. Arnold 03.13.09 at 6:13 pm

Salient, don’t try and dance your way out of this. They are not mystical problems, they are questions about the “law” that would allow the invention of the incompleteness theorem, since algorithms in the formal language that it is about, won’t invent that theorem of themselves.

I will be happy to acknowledge that there is a mathematician or physicist who doesn’t think the universe is essentially mathematical, should you be one. (You’re of a breed that was rare until recently.) But THEN the question is, “What else is there?” You can ridicule Gödel for being anti-materialist and a pure idealist, but you ought to come up with something more that the mere intellectual prejudices of our time, in doing so. In the words of C.S. Peirce, “People who think they don’t have a metaphysics usually just have a really bad one.”

In other words, we certainly have NOT “grown up about what we can and cannot do.” We have barely scratched that surface. In cosmology, as you well know, we have far more questions than answers. Science and mathematics are likely to go on forever. Gödel’s theorem virtually guarantees that, about mathematics. And there will be changes in how we do science and mathematics. The history of the n-body problem demonstrates it.

So: the question posed at the top is Time. We are not much further along than Augustine on the issue, and he lived before the invention of clocks! Now, as Gödel said about time, again from Wang, “The real idea behind time is causation… Causation in mathematics, in the sense of, say, a fundamental theorem causing its consequences, is not in time, but we take it as a scheme in time.”

Okay then, in regards to the question of time and mathematics, let’s look at the resolution of the search, starting with Leibnitz and ending in Poincare, for a closed-form solution to the n-body problem.

We stopped looking, and we approximately solve for n-bodies by recursion, i.e. repeated calculations in time. Yet the equations themselves have time IN them. Let me ask you, since you seem to know, is there anyone in physics who looks at the fact that time is used at two DIFFERENT logical levels (again: in the equations, and BY the repeated steps of the necessary computations) to solve this thing?

(For clearly, this the sort of question that led Gödel to study Husserl. If we don’t know what time is, we’d better not dismiss any percepts or concepts involving it. I also assume that Einstein considered it; he talked extensively to Gödel for many years and Einstein was an epistemologist after all, but his observations if written down are not published.)

P.S. Far from making a proof that Gödel was given to dreamily hypothesize, you avoid his chain of thought. That, again, appears to be: how can it be that “the formation of ever higher types can be continued into the transfinite” or anywhere else for that matter — and whether formalism is always stuck at one type level (or more accurately, between two adjacent levels) and why the semantical content in categorial hierarchies across multiple levels can only be empirically determined, i.e. can’t be computed. Oh — and whether that is related to time.

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John Emerson 03.13.09 at 7:14 pm

Salient and onymous:

Probably I agree with you about most of the substantive questions here, though I can’t be sure of that. What I’ve been talking about is popular writing about physics, particularly what I call physics porn, which takes abstruse points from physics and makes it seem more exciting, more relevant to everyday life, and usually more new than it really is. I’m especially thinking about the genre of popular writing grounded on fundamental physics which makes lurid statements about “the reality of time”, etc., and I suspect that Carroll’s book is part of this genre. Nobody has said I’m wrong, but if I am, of course I owe Carroll an apology, not that I think he cares much.

The error of this kind of writing is often or usually an error of writing — i.e., not an error of science. My claim is that when Carroll whizzed by a key point as briefly as he apparently did (#46), he was misleading his readers without actually saying anything false.

Contrary to your opinion, I think that this kind of misleading writing is very common and comes from very smart people, and at worst ends up with misunderstandings like those in Wittgenstein’s example of a man who put on galoshes so he wouldn’t fall into the empty space between atomic nuclei. And I think that the authors could easily avoid that, but their sales would suffer.

Prigogine seems to be in bad odor, rightly or wrongly, but his history of thermodynamics was the first time I heard of Onsager and Kolmogorov and the first lucid narration of that story I’d ever heard. It’s old hat to physicists, but they haven’t shared it with the world the way they’ve shared relativity, quantum theory, cosmology, etc., etc., in part, I think, because it conflicts with part of scientific mythology. These ideas have only recently started to enter into educated common sense, and too much energy has gone to minimizing its consequences, wrongly in my opinion. After all, most of this stuff was around 50 years ago. But let me just refer people to Eddington on entropy and the arrow of time, which is the main thing left out or slipped past in these titillating presentations.

And yes, I think that this is relevant to the weaknesses of economics. I’m not just picking up handy pieces of shit to fling at my favorite enemy, I’ve thought this all along but found few chances to speak about this particular topic. The authors in 26 explain this further, and I should add Geoffrey Hodgson on Economics forgetting history.

And last, I don’t think that I’ve been trolling. I think that what I’ve written has been both relevant and useful. Obviously I don’t accept the bureaucratic “not my department and above my pay grade” way of evading with these kinds of questions. Maybe that does make me a troll.

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onymous 03.13.09 at 7:27 pm

Surprisingly, economists and philosophers are also above all criticism and are good all the time, except for the odd Philip Anderson or Ilya Prigogine you can find in any field.

Look, I don’t know enough about Prigogine to say much, really (though I trust that Cosma does), but Anderson is a damned good physicist whose ideas permeate not only his own area of condensed matter physics but also have found their way into particle physics and cosmology. This still doesn’t mean one should take him seriously when he starts wildly accusing other entire fields of being misguided, but nowhere did I imply that he is not a great physicist. (Nor, for that matter, did I imply that anyone is “above all criticism and good all the time”.)

[Prigogine’s] history of thermodynamics was the first time I heard of Onsager and Kolmogorov and the first lucid narration of that story I’d ever heard. It’s old hat to physicists, but they haven’t shared it with the world the way they’ve shared relativity, quantum theory, cosmology, etc., etc., in part, I think, because it conflicts with part of scientific mythology.

This is unfortunate, because Onsager and Kolmogorov are indeed among the giants of 20th century physics and mathematics. The trouble here isn’t so much “conflicting with scientific mythology” as a lack either of willingness of people to popularize their work, or in popular interest in their work. There are far more physicists studying condensed matter systems than studying cosmology, for instance, but the former don’t tend to write books for the general public. Whether it’s because they don’t want to, or because the audience isn’t there, or because publishers think the audience isn’t there, I don’t know. But it is unfortunate.

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onymous 03.13.09 at 7:34 pm

Salient, to finally try to answer your questions (briefly), when you say “the conventional interpretation is that cases of universes in which sufficiently habitable conditions for life to develop must satisfy conditions which are sufficiently close to ours to eliminate all the unreasonable cases from consideration”, you’re describing some form of the “anthropic principle”. This is a separate issue from many worlds — it’s about why the parameters of our universe have the values they do, and it’s been most plausibly argued by Weinberg to be the reason for the smallness of the cosmological constant. Some people might tie up these issues with many worlds, but it’s misguided, and doesn’t get at the basic point that the probabilities predicted by quantum mechanics precisely fit our observations. Many worlds doesn’t mean giving up the usual probability calculations; it’s a mistake to think every possibility counts with equal weight, which would be in blatant contradiction with zillions of experiments, most of which could happily have different results without threatening the existence of human life (so the anthropic principle can’t save this misinterpretation of many worlds).

As for “…after n many performances there exists a me that occupies Bn. Which seems to be a universe in which extraordinary things are happening fairly regularly, for sufficiently large n”, you can ask this question of any theory that predicts things happening probabilistically, including rolling dice, independent of quantum mechanics. So I don’t see how it can speak to any issues involved in interpreting quantum mechanics.

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John Emerson 03.13.09 at 7:37 pm

Onymous, why don’t you just use my phrase, which is “physics porn”. You seem to agree with me entirely.

Prigogine had lots of audience when he started writing this stuff, but that seems to have pissed physicists off enormously.

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John Emerson 03.13.09 at 7:45 pm

Thermodynamics isn’t sexy for the same reasons that organic chemistry isn’t. No enemies, no struggle, no revolution, just a lot of science. Evolution and classical mechanics and cosmology are sexy because of the opposition.

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Rob G 03.14.09 at 12:56 am

Classical mechanics has enemies and is sexy? Much as I loved my third year course, knowing this would have given it that extra oomph.

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salient 03.14.09 at 1:18 am

I don’t think that I’ve been trolling.

Agreed, John, for what it’s worth (but don’t you run Trollblog?). I’m certainly learning from this discussion (especially from onymous), and I can’t be the only human being reading CT who has some misconceptions or incomplete conceptions about these philosophical issues that have since been cleared up, or at least, brought into the light and acknowledged.

Let me ask you, since you seem to know, is there anyone in physics who looks at the fact that time is used at two DIFFERENT logical levels (again: in the equations, and BY the repeated steps of the necessary computations) to solve this thing?

I wouldn’t know an answer to most “does there exist a physicist who” questions. And for evidence of that, see Righteous Bubba’s reference to Max Tegmark (whose name, granted, I should have memorized under the “physicists who are known to the general public and colleagues mostly for their sensational conjectures rather than their justified and verified theorems” Rolodex — but I’m still at the stage where I’m building my familiarity with these folks’ work).

However: from my perspective, you are confusing representations with entities. You are presupposing that “time is used at two DIFFERENT logical levels.” This is like asking, do you know of “any philosophers who are taking a look at the fact that, because every entity in the universe is made up of words, we can view any phenomenon as a sentence?”

In third-semester calculus, you saw (or would’ve seen) the chain rule for functions of multiple variables. You should be comfortable by now with the idea that a formula represents a a relation between sets, or let’s say a relationship between quantities which are allowed to vary within specified sets, e.g. an interval. Sure, sometimes the interval [0,1] gets called “the time from 0 to 1 hour” as a kind of concrete memory aid, and sometimes we interpret a variable as a representation of time. But the variable ‘t’ is not time itself; it’s better to think of it as a representation of a variable quantity. At which point your comment about the philosophy of time doesn’t apply to the example.

But I suppose it would be sensible to conflate the two if you believe mathematics is a universe. At that point, I don’t know how we could communicate successfully. In particular, I don’t know how to translate an interpretation of a model of a phenomenon, into a system where the model is the phenomenon. It just strikes me as equivalent, roughly, to saying “the universe is made out of words.” And, to me, equally absurd.

Onymous, why don’t you just use my phrase

…Banking off of this, let me try a crude analogy. (1) I (and my colleagues, advisers, mentors, etc) work within a bubble, gradually expanding its boundaries. I find appropriate mathematical characterizations of specific relationships between bodies of matter that, when used to predict the results of an event, conjecture results that seem regularly accurate to observed results, to within error tolerance. This kind of numerical work isn’t done by everyone in (1); some of us develop tools, theorems, characterizations that while not computational improve our understanding of mathematical structures. This work, also, extends the boundary of the bubble.

(2) Apparently there are any number of physicists who make their living writing conjectures about the possible shapes this “bubble of well-characterized phenomena” could take. When these conjectures imply falsifiable or at least testable hypotheses, or when these conjectures provide us with insight into the constraints or boundaries our work may run up against, this kind of work is useful and meaningful.

(3) I suspect there are any number of people who rely on their professional authority from good work done in (1) to sell books that engage (2). “No harm, no foul” only comes into play here when there is clearly no harm done.

(4) Those who devote themselves entirely to rigorously philosophizing on these matters attempt to position themselves outside the bubble and characterize the limitations on how far I (and my colleagues) will potentially get. That’s fine and commendable; at the very least, it’s definitely good to have a great deal of skepticism about what we will, and will not, ever accomplish.

My problem (with Goedel, apparently, not with you): Goedel seems to abandon that skepticism in his metaphysics. One of his premises (#3 in the list above) is that every ‘problem’ can be characterized and solved, essentially that any phenomenon has a predictive characterization. Why would anyone assume that, or begin with that foundation? Goedel is brilliant enough to excuse his hubris, I guess.

But the only circumstances in which Goedel’s theory implies the existence of transfinite axioms is one in which you hang on to his #3 premise: that we can develop a complete logical/mathematical characterization of every phenomena that occurs in the universe.

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Matt McIrvin 03.14.09 at 2:11 am

1. physicists in general

2. particle physicists/cosmologists/etc.

3. physicists who write popular books

4. particle physicists/cosmologists/etc. who write popular books

5. whoever it was who supposedly kept people from taking Ilya Prigogine seriously

Because I don’t think these groups of people are all the same. I suspect that groups 2 and 5 are almost completely disjoint; particle physicists probably never knew Ilya Prigogine from a hole in the ground in the first place. (I have no strong opinion about him personally; I read his book when I was a kid and thought it was interesting but hard to understand, but don’t remember much about it.)

I had some experience back in the day with group 2 and found that the percentage of assholes there was pretty comparable to the general population.

Academic physicists like to bitch and moan about what gets in the popular books all the time–looking at the science shelves at Barnes & Noble can be a really depressing experience–though from time to time you see a good one. Personally, I suspect that the reason they lean so heavily toward relativity, cosmology, weird quantum physics phenomena, string theory, etc. is not that the scientific establishment is guarding a mythos but that publishers find these topics sexy and will sometimes buy books on those subjects. I agree that some more good pop books about statistical mechanics and condensed-matter physics would be a grand thing (there were some good ones in the Scientific American Library series, if I recall correctly).

Finally, I find it really hard to believe that Cosma Shalizi, of all people, would be working within any sort of triumphalist narrative of particle physics as ultimate key to ultimate reality. In fact I wish the mayor would light the Shalizi-signal to get him to come over here, but I suspect he’s busy.

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Michael Bérubé 03.14.09 at 2:33 am

Obviously I don’t accept the bureaucratic “not my department and above my pay grade” way of evading with these kinds of questions.

Obviously. And obviously, I blame Obama.

But yes, if Sean’s book turns out to be in the genre of The Dancing Wu Li Masters instead of the genre of The First Three Minutes, then I will apologize profusely to John for having led him to write all these comments about physics porn. Mea proleptica culpa.

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John Emerson 03.14.09 at 3:22 am

105:

I am above all antagonistic to anyone who writes a book about the philosophy of time from the physicist’s point of view which slips past Eddington’s point, which basically tells us that in terms of the world of our experience –the human world, the biosphere and the world of entropy — the commonsense notion of the arrow of time and the open, undecided future is about right. So basically this means that the answer is #5, with a few revisions and footnotes. Not especially #1-#4. But also and above all #6: people from any area who privilege some kind of restricted or customized fundamental-physics model as the most real description of reality, above all those who extrapolate from it. EG, economists, as shown by Mirowski and Georgescu-Roegen. Also, many analytic philosophers, as exemplified by Poidevin’s Oxfor U.P “Philosophy of Time” anthology.

Secondarily, I think that the anti-Prigogine pileon is excessive. People should have begun by ignoring and not reading the Toffler intro as I did. Prigogine isn’t really New Age, or at least much of what he said wasn’t, but more commonsensical: something like “In our [experienced] world we move from the past to an unpredictable future, and the unpredictability is radical and major. Our past-to-future time is not a subjective illusion; without irreversible time there can be no life forms.” I think that Bricmont was all wrong on this.

There are criticisms of Prigogine which I don’t understand, so I don’t know how thoroughgoing my defense of him should be, but certainly I learned a lot from him, and I think that a lot of other people need to learn some of that. I vastly prefer his popular works to the competing popular works.

I’ve paid a lot of attention to Shalizi on these questions and have no idea how he would respond to these questions. I may have wronged Carroll, in which case he can add himself to the list of people who have been wronged by myself and other trolls, but his #18 set me off.

Note that Radish is an apparently completely sensible , non-trollish giologist, and I suspect that he speaks for a high proportion of biologists. I mentioned Philip Anderson not as a perfectly nice person, but as a physicist who felt that the privileging of fundamental physics as some sort of Truth of Everything is really wrong. Unfortunately I haven’t had time to dig up the quote yet; It’s somewhere in a recent book on emergence edited by Bedau and others.

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John Emerson 03.14.09 at 3:31 am

S/B Secondarily, I think that the anti-Prigogine pileon is excessive. People should have begun, as I did, by ignoring and not even reading the Toffler intro.

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Lee A. Arnold 03.14.09 at 6:06 am

Salient: “In particular, I don’t know how to translate an interpretation of a model of a phenomenon, into a system where the model is the phenomenon.”

My pure conjecture is, that’s one reason why Gödel turned to Husserl.

Gödel mentions in Wang that Husserl spent over twenty years studying time, and that the work was lost from Husserl’s manuscripts. (This wouldn’t have been mere gibberish, since Husserl had studied under Weierstrass.) I think Gödel may have wondered whether Husserl had come upon a useful discrimination in intentionality (the psychological kind) that would make such differences in logical type to be scientifically tractable. Not “conflating” representation and entity, but somehow combining them into a new sort of study or operation, a “theory of concepts.” For a physics that includes the observer, why not?

(By the way, an entire movement tried to analyze every phenomenon as a sentence: the logical positivists. So one could in fact name a number of philosophers who tried to do that!)

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Elliot Tarabour 03.14.09 at 3:13 pm

John,

Two words: more fiber

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John Emerson 03.14.09 at 4:45 pm

Fuck off, Elliot.

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Walt 03.14.09 at 5:36 pm

Lee, this is debateable:

Salient, don’t try and dance your way out of this. They are not mystical problems, they are questions about the “law” that would allow the invention of the incompleteness theorem, since algorithms in the formal language that it is about, won’t invent that theorem of themselves.

Set theory is incomplete, and yet the incompleteness of set theory is expressible in set theory. An algorithm that starts from the axioms of set theory w0uld eventually discover it.

Also, the fact that the three-body problem doesn’t have closed solutions isn’t that important. The definition of closed solution is a social fact, not a mathematical one. (What makes “sin x” a closed solution?) Problems can fail to have closed solutions, and yet not exhibit chaos.

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Lee A. Arnold 03.14.09 at 10:13 pm

Thanks for that. I am trying to find the proof of the incompleteness of set theory. Do you have a reference?

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salient 03.15.09 at 12:56 am

Bolded here and there for thread-skimmers’ convenience.

I am trying to find the proof of the incompleteness of set theory. Do you have a reference?

Irony Police may annihilate me for this, but… Goedel’s On Formally Undecidable Propositions of Principia Mathematica is widely considered a proof that any axiomatic system with finitely many axioms (like set theory) is automatically either incomplete or inconsistent, if it’s a sophisticated enough system to support arithmetic (which the standard set theory axioms are). So, it’s exactly what you’re asking for: by proving that any axiomatic system is incomplete or inconsistent, Goedel himself proved the incompleteness of set theory by proving that the system derived from the standard set theory axioms is incomplete, as a special case of his general theorem. Goedel further demonstrated that in any such case, one can express the incompleteness within the structure of that system, as Walt said; he proved this by producing an arithmetic equation that is true under the laws of the system but whose truth cannot be verified using the system’s laws alone. I’m getting all of this from V.A. Uspensky’s presentation of Goedel’s theorems and their proofs, which I thought to be uncontroversial (but which doesn’t include Goedel’s philosophizing about metaphysics). Also: in light of this discussion thread, I plan to read this – linking it in case anyone else is interested.

Now, apparently Goedel (in his original work?) (or elsewhere?) went on to philosophize about the potential metaphysical interpretations of this theorem (file this under “things Salient should have known but didn’t,” if there’s still room in that bulging file), but his proof of the above ‘incomplete or inconsistent’ proposition is nowadays accepted and uncontroversial.

As for specific propositions that are independent of set theoretic axioms, which might be what you’re looking for depending on what you mean by “the incompleteness of set theory,” check out the Axiom of Choice and the Axiom of Determinacy. I think you’d get a kick out of reading some of the writing about their mutual inconsistency in the context of set theory: it’s very interesting, from a meta-mathematical perspective. Each of the above axioms seems fairly reasonable and even intuitive, but it is impossible for a logically consistent set theoretic system to assume that both of them hold. (For what little it’s worth, to my knowledge I’ve never met a mathematician who would be unwilling adopt the Axiom of Choice, or who has ever adopted the Axiom of Determinacy, while working on a problem.)

I think it’s also worth thinking about what “incompleteness” means. Part of my (admittedly unjustifiably strong) reaction to Goedel’s dreamy philosophizing is that he, out of hubris, seems to have missed the point of his own proof: we can’t characterize everything. We just can’t. And out of what we can characterize, a lot of the characterizations won’t be computational. So we have to approximate. Everything that I do, day to day, can be summed up under the banner of “approximation to within error tolerance” (which one generally seeks to prove can be chosen arbitrarily small).

Goedel does acknowledge this limitation, of course: it would take a “higher being” to do what humans so clearly cannot.* But there’s no a priori reason to believe that such solutions are possible or to believe that there exists a higher being capable of such solution-finding. At the point where he posits such an existence, Goedel dons the garb of a mathematician to lecture as a theologist, and his credentials as a mathematician shouldn’t automatically carry any more authority in theology than someone who does brilliant work in, say, geology. His ideas are interesting, but they don’t carry the force or credibility of proof.

*I guess this idea of “so clearly cannot” would have been controversial up through at least 1870 or so, but it seems to be the mainstream perspective in the physical sciences, at least applied physical sciences… We do what we can, and as onymous so vividly stated with regard to field theory, it is becoming increasingly obvious just how little about our universe we can characterize, from fundamentals, with models that are arbitrarily precise and yet computable/falsifiable. (From another point of view, the achievements that have resulted from what we can do within the constraints of an incomplete axiomatic mathematical system, are quite amazing.)

(Oh, and regarding logical positivism: it was an intentional reference. I just finished reading Broom of the System a few days ago, so Wittgenstein et al has been on my mind.)

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salient 03.15.09 at 1:00 am

Oh, and Lee: for a specific reference, you might try V. A. Uspensky’s book on the theorem. I tried to find a link on Amazon for you but couldn’t find a current edition. You might be able to find it in the library; I don’t know.

The definition of closed solution is a social fact, not a mathematical one.

I don’t know… the definition of “closed-form expression” is mathematical, insofar as any phrase that mathematicians define and use is “mathematical.” I’m not sure what you intend when you say “social fact” though, so maybe we agree on this?

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Walt 03.15.09 at 3:18 am

Lee: It follows directly from Gödel’s theorem for Peano Arithmetic (which is the version its usually given in these days, since the Russell-Whitehead system is something of a museum piece). The key is that all of the concepts necessary are directly definable as sets. So the “set of all theorems derivable from the axioms of set theory” is a definable set. Gödel’s theorem is a theorem of set theory, and the fact that set theory can express Peano Arithmetic is a theorem of set theory. Therefore the incompleteness of set theory is a theorem of set theory. As Salient mentioned, there are also concrete statements that are known to be not provable in set theory, such as the Continuum Hypothesis. (So “the Continuum Hypothesis is unprovable” is a theorem of set theory.)

Set theory is sufficiently expressible that you should be able to give a short direct proof that it’s incomplete, but I don’t know one off the top of my head.

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Lee A. Arnold 03.15.09 at 4:11 am

Well you’ll have to walk me through this. The question I am getting at, though I may not be expressing it very well, is whether if you set up a Turing machine with ZFC algorithms, and programmed it to make combinations, it would eventually turn up Gödel’s proof. I thought the answer was “no.”

I am not studied in set theory, but I thought that “set theory” proper is just ZFC, or the like. Gödel demonstrated the incompleteness of the axioms of ZFC, only if his same recursive extensions are introduced. In other words you still need Peano arithmetic, i.e. with multiplication. Arithmetic without multiplication (Pressburger arithmetic) on the other hand is complete: it is also, I would gather, expressible as set theory. Is this correct?

I thought ZFC set theory is more or less a set of rules for thinking, for framing the methods of proof of mathematics, and that it has its own theorems — but it doesn’t have the primitives of your subject, which you still must introduce, whether it’s arithmetic with or without multiplication, or the order of the lepidoptera, or all the rules of a nation’s judicial system.

Now I have read that set theory by itself has unresolved paradoxes, and that ZFC is consistent with or without the continuum theorem, and that Wittgenstein derided parts of set theory as being as close to mysticism as you can get. (Not that I don’t like actual mysticism.) Certainly set theory allows hierarchies constructible to the absolute, which looks like the source of some of its paradoxes.

But to have it be about something, you still need to add the semantic primitives, and you still need to judge, as an outside observer, whether what you are doing with set theory follows the meaning of those primitives. (Which is where the later Wittgenstein jumped in to show that natural language doesn’t follow a scheme like that.) I have never heard of set theory algorithms that are able, of themselves, to logically lead to Gödel’s proof, without being preprogrammed to do so. I would think that if formal languages could do so, we’d have general, self-discovering artificial mathematical intelligence by now.

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Walt 03.15.09 at 5:06 am

The answer to your first question is “yes”. If you wrote a computer program to systematically generate theorems in ZFC, it would eventually generate the proof to Gödel’s theorem, and the proof that ZFC was incomplete. Addition and multiplication is definable within ZFC, so you get it for free. You don’t need to add it.

The semantic primitives are all expressible in ZFC. Of course, it still requires an outsider to say, “yes, that’s what I really mean by that semantic primitive”, but that’s true of the semantics of 2+2 just as well.

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Lee A. Arnold 03.15.09 at 5:15 am

How does it generate multiplication? Is it done by power sets?

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Lee A. Arnold 03.15.09 at 5:17 am

And how would it generate Cantor’s diagonal arguments? Is that done in the same way?

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Lee A. Arnold 03.15.09 at 5:20 am

Salient, Gödel’s beliefs #1 “The world is rational” and #3 “There are systematic methods for the solution of all problems” are High Baconism, major tent-poles of early modernism, although now they may be just a methodological direction. By the 1950’s Gödel was an oddball outlier among philosophers of science. But as I understand it, his incompleteness theorem doesn’t contradict either belief, because he thought you could continue build new things outside older formal limitations.

And there were, and still are, at least two compelling, a priori reasons to hold onto belief in the rationality and discoverability of the world:

The first reason is sometimes called Leibnitz’s Law of Sufficient Reason (although Leibnitz certainly wasn’t the first to notice it) — i.e., that everything that happens, happens for some prior reason. There are counterexamples such as emissions of subatomic particles, but because they are repeating and regular in form, we are led to suppose that there may be reasons. (And there are things like extreme and absolute counterexamples, such as the beginning of the universe, or why is there anything at all, instead of nothing.) Some events have reasons but might not be predictable (such as the moment an apple falls from a tree,) or else we have to use probability, but these still fall under the law.

The second reason for belief in a rational world is that we are able to construct finite typologies of things — there are only so many subatomic particles, only so many natural elements, only so many species of apple tree, etc. Nobody has an explanation for this either, but of course it’s related to Occam’s razor and it propels reductionism.

These two reasons are put together and react powerfully with a third: the millennial Great Chain of Being was turned upside down. That is, we no longer believe (most of us, anyway) that the universe emanates downward from a Creator, with no gaps in the creation. This was rather precisely inverted in the modern period to the evolutionary view that things are constructed upwards from tiny bits, atoms or whatever, and that there are lacunae in the picture.

Gödel very clearly decided that the world is rational, that math can stay, and so science has the wrong metaphysics and needs a rewrite.

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Walt 03.15.09 at 5:40 am

Multiplication doesn’t need power sets. You can define a set that consists of the natural numbers, and then you show that there exists a function on that set that fits the definition of multiplication.

Cantor’s diagonalization argument is straightforward to express in ZFC, and is how you prove that the power set of a set is bigger than the original set.

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Lee A. Arnold 03.15.09 at 3:10 pm

Thanks for this. I want to find the way in which it is understood that knowledge cannot be reliably computed by formal language up and down many hierarchical levels of semantic categories. In other words, that they really don’t do semantics well at all. I thought that was true of set theory, but it looks like my mistake is in not understanding that set theory automatically expresses at least one semantics, that of arithmetic.

So then it seems to me then if you were to run the algorithms of set theory on a computer to find the incompleteness proof and other new, undiscovered proofs of mathematics and metamathematics, then the real problems are that (1) it takes an enormous amount of computation time, (2) the computer wouldn’t know when it has run past a completed proof, in order to alert you, i.e. it doesn’t know where to insert “endpoints” (which would be different than the halting problem,) and (3) you might not be able to reduce it to another language, in order to comprehend it yourself. Are these correct?

I may have to leave it here in order to do other stuff, but thanks again.

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Walt 03.15.09 at 6:30 pm

1 is true. 2 is false (the computer can always tell when it has a complete proof). 3 is probably true (I’m not 100% sure I understand what you mean), in that computer-generated proofs are usually hard for people to understand.

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Lee A. Arnold 03.15.09 at 8:08 pm

I don’t understand #2. If we ran the algorithms on a computer and Gödel had never lived and the rest of us maybe never suspected the incompleteness theorem, how would the computer trigger itself to alert us that it had discovered the incompleteness theorem? Or any other theorem in the future that we don’t know about now? What are its criteria for having discovered a complete theorem? Wouldn’t it just continue combinatorically reapplying set theory algorithms beyond that point?

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Elliot Tarabour 03.15.09 at 9:07 pm

fuck off Elliot
his shortest ever

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salient 03.15.09 at 9:41 pm

Lee, I think maybe you’re talking about theoretical Turing reduction models, whereas I think Walt is talking about existing applications, e.g. to solve the four-color mapping problem. Opposite sides of the theoretical-applied spectrum. But I may be misunderstanding one or the other of you.

If you are asking about Turing reducibility, i.e. an “oracle” computer that can solve problems, then the answer is: the computer would “solve” a given problem by reducing it to an elementary logical combination of problems that are known to have solutions. Once the problem has been so reduced, the computer would recognize the reduction is complete, and report that reduction to us in a logic-symbolic language which we’re presuming we would understand.

However, I noticed you asked about “Gödel had never lived and the rest of us maybe never suspected the incompleteness theorem” — It is important to note that (1) a universal oracle that would be capable of such a task has not been realized in real life, and probably couldn’t be, and (2) even if realized, there’s no guarantee such a computer would pursue proving the kind of theorems we’re interested in, without guidance as to what answer it’s trying to find (I believe all theoretical Turing models assume that a specific and falsifiable question is asked of the oracle to prompt its operation).

In other words, if none of us suspected the incompleteness theorem, how would a computer know to characterize it and pursue its proof, instead of one of the infinitely many other theorems that can be characterized within the system? If we wanted a computer to determine what proofs are important and then prove them, instead of mucking about arbitrarily, we’d need to induce in it some intuitive guidance, which gets us into neural networking and artificial intelligence and ideas I’d rather not speculate about; my thoughts on the matter would be even more useless than average.

Relatedly, there exists a canonical story, possibly mythical, that the nowadays-common classical trigonometry proof that \angle ABC = \angle ACB implies that AC = AB (completed via AA similarity of \triangle ABC and \triangle ACB using CB = BC) was first discovered by an oracle computer designed to find geometry proofs. So oracles designed to perform specific tasks, moving from theoretical to applied territory, do exist and have been used. The standard examples are the various proofs of the four-color map theorem. I don’t think these practical-purpose oracles start from any kind of set-theoretic axioms, though; I think at the very least computation on the real numbers or at least the integers (as a field) is naively assumed in any such computer model.

Ok, some potentially interesting resources for you (and anyone so inclined). All PDF links:

A very readable explanation of Turing reducibility can be found this paper by Martin Davis.

On the other end of the theoretical-practical spectrum, if the controversy surrounding applied computational proofs interests you, check out this paper E. R. Swart.

If the technical aspects of how such a proof is constructed interest you, here’s a very readable summary of the combinatorial reducibility of the four-color map problem, written by Robin Thomas, who offers this caveat with surely no small amount of sly humor: “[V]erifying all of this without a computer would require an amount of persistence and determination my coauthors and I do not possess.” :-)

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Walt 03.16.09 at 12:27 am

A theorem is just a statement that follows from the premises. The computer will just churn out theorem after theorem, and 99.99998% of them will be completely boring. But if you wait long enough, it will churn out the incompleteness theorem, and all other interesting theorems.

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Walt 03.16.09 at 12:42 am

Salient: The computer rediscovered the proof you mention, but the proof itself goes back to the Greeks. I think the people who wrote the program hadn’t seen the proof before.

A fact that deserves to be better known is that Euclidean geometry is complete: there is algorithm to determine whether or not any statement in Euclidean geometry is true.

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salient 03.16.09 at 12:05 pm

The computer rediscovered the proof you mention, but the proof itself goes back to the Greeks. I think the people who wrote the program hadn’t seen the proof before.

Okay, good to know – I’ve always suspected the version of the “the computer discovered it” story I’d heard was apocryphal / misleading. It’s too basic a proof for it to be plausible that it’d been overlooked for so long.

But if you wait long enough, it will churn out the incompleteness theorem, and all other interesting theorems.

In some sense, but given that there are infinitely many logical constructions that could be proven or disproven, it’s also possible the computer would never get around to proving something you didn’t already know (assuming finite computational speed). Some of this depends on programming intuition into the computer about what propositions to pursue first, and what corollaries to ignore.

A fact that deserves to be better known is that Euclidean geometry is complete: there is algorithm to determine whether or not any statement in Euclidean geometry is true.

Do you happen to have a link or source to recommend, for a development of this? I’d be interested in its history, the implementations, etc. With such a resource and with regard to “deserves to be better known,” as a teacher, I can probably do my part :-)

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Walt 03.17.09 at 3:08 am

Do you mean finite time? Given a fixed computational speed, the computer would eventually prove every theorem. Interesting theorems may require 10^22 years, though.

The geometry theorem has two steps. The first step is to show that statements in Euclidean geometry can all be rephrased as statements about systems of multivariate polynomials equalities or inequalities with real coefficients. I don’t know a reference for the first step. The second step is to prove that there is an algorithm for solving such systems. This theorem is by Tarski, and goes under the name “quantifier elimination” for real closed fields. A book on model theory would be a good starting place.

I know there’s a book devoted specifically to this subject, but I don’t remember the name. Searching Amazon, it might be Mechanical Geometry Theorem Proving by Shang-Ching Chou.

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