Archive for the ‘Metaphysical Spouting’ Category

Why Philosophers Should Care About Computational Complexity

Monday, August 8th, 2011

Update (August 11, 2011): Thanks to everyone who offered useful feedback!  I uploaded a slightly-revised version, adding a “note of humility” to the introduction, correcting the footnote about Cramer’s Conjecture, incorporating Gil Kalai’s point that an efficient program to pass the Turing Test could exist but be computationally intractable to find, adding some more references, and starting the statement of Valiant’s sample-size theorem with the word “Consider…” instead of “Fix…”


I just posted a 53-page essay of that name to ECCC; it’s what I was writing pretty much nonstop for the last two months.  The essay will appear in a volume entitled “Computability: Gödel, Turing, Church, and beyond,” which MIT Press will be publishing next year (to coincide with Alan T.’s hundredth birthday).

Note that, to explain why philosophers should care about computational complexity, I also had to touch on the related questions of why anyone should care about computational complexity, and why computational complexity theorists should care about philosophy.  Anyway, here’s the abstract:

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance.  In this essay, I offer a detailed case that one would be wrong.  In particular, I argue that computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction and Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest.  I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

Weighing in with 70 footnotes and 126 references, the essay is basically a huge, sprawling mess; I hope that at least some of you will enjoy getting lost in it.  I’d like to thank my editor, Oron Shagrir, for kicking me for more than a year until I finally wrote this thing.

My diavlog with Anthony Aguirre

Saturday, July 24th, 2010

Bloggingheads has just posted an hour-long diavlog between the cosmologist Anthony Aguirre and your humble blogger.  Topics discussed include: the anthropic principle; how to do quantum mechanics if the universe is so large that there could be multiple copies of you; Nick Bostrom’s “God’s Coin Toss” thought experiment; the cosmological constant; the total amount of computation in the observable universe; whether it’s reasonable to restrict cosmology to our observable region and ignore everything beyond that; whether the universe “is” a computer; whether, when we ask the preceding question, we’re no better than those Renaissance folks who asked whether the universe “is” a clockwork mechanism; and other questions that neither Anthony, myself, nor anyone else is really qualified to address.

There was one point that sort of implicit in the discussion, but I noticed afterward that I never said explicitly, so let me do it now.  The question of whether the universe “is” a computer, I see as almost too meaningless to deserve discussion.  The reason is that the notion of “computation” is so broad that pretty much any system, following any sort of rules whatsoever (yes, even non-Turing-computable rules) could be regarded as some sort of computation.  So the right question to ask is not whether the universe is a computer, but rather what kind of computer it is.  How many bits can it store?  How many operations can it perform?  What’s the class of problems that it can solve in polynomial time?

My diavlog with Eliezer Yudkowsky

Monday, August 17th, 2009

Here it is.  It’s mostly about the Singularity and the Many-Worlds Interpretation.

(I apologize if Eliezer and I agreed too much, and also apologize for not quite realizing that the sun was going to set while I was speaking.)

And here’s the discussion that already took place over at Eliezer’s blogging-grounds, Less Wrong.

The complement of Atlas Shrugged

Monday, March 23rd, 2009

A few months ago I read Atlas Shrugged, the 1,069-page Ayn Rand opus that was recently praised by Stephen Colbert (for its newfound popularity with beleaguered CEOs).  As I mentioned in the comments of a previous post, like many other nerds I went through a brief Aynfatuation around the age of 14.  Rand’s portrayal of an anti-mind, anti-reason cabal of collectivist rulers, who spout oleaginous platitudes about love and self-sacrifice even as they mercilessly repress any spark of individuality, happens to be extremely relevant to at least two cases I’m aware of:

  1. Soviet Russia.
  2. The average American high school.

But it didn’t last long.  Even in the midst of it, I could see problems: I wrote a term paper analyzing the rape scene in The Fountainhead as immoral and irreconcilable with the rest of an otherwise supremely-rational novel.  And ironically, once I went to college and started doing more-or-less what Rand extols as life’s highest purposes—pursuing my ambitions, tackling math and science problems, trying to create something original—her philosophy itself seemed more and more quaint and irrelevant.  I snapped out of it before I reached Atlas.  (Or did I subconsciously fear that, if I did read Atlas, I’d be brainwashed forever?  Or did I just figure that, having read the 752-page Fountainhead and dozens of essays, I already got the basic idea?)

So, having now returned to Atlas out of curiosity, what can I say?  Numerous readers have already listed the reasons why, judged as a conventional novel, it’s pretty bad: wooden dialogue, over-the-top melodrama, characters barely recognizable as human.  But of course, Atlas doesn’t ask to be judged as a conventional novel.  Rand and her followers clearly saw it as a secular Bible: a Book of Books that lays out for all eternity, through parables and explicit exhortation, what you should value and how you should live your life.  This presents an obvious problem for me: how does one review a book that seeks, among other things, to define the standards by which all books should be reviewed?

Mulling over this question, I hit on an answer: I should look not at what’s in the book—whose every word is perfect by definition, to true believers who define ‘perfect’ as ‘that exemplified by Atlas Shrugged‘—but at what’s not in it.  In other words, I should review the complement of the book.  By approaching the donut through the hole, I will try to explain how, even considering it on its own terms, Atlas Shrugged fails to provide an account of human life that I found comprehensive or satisfying.

(Though on the positive side, it still makes much more sense than my 11th-grade English teacher.)

Without further ado, here are the ten most striking things I noticed in the complement of Atlas Shrugged.

  1. Recent technologies.  For a novel set in the future, whose whole point is to defend capitalism, technology, innovation, and industry, Atlas is startlingly uninterested in any technologies being developed at the time it was written (the fifties).  For Rand, the ultimate symbol of technological progress is the railroad—though she’s also impressed by steel mills, copper mines, skyscrapers, factories, and bridges.  Transistors, computers, space travel, and even plastic and interstate highways seem entirely absent from her universe, while nuclear energy (which no one could ignore at the time) enters only metaphorically, through the sinister “Project X.”  Airplanes, which were starting to overtake trains as a form of passenger travel even as Atlas was written, do play a tiny role, though it’s never explained where the busy protagonists learned to pilot.  Overall, I got the impression that Rand didn’t really care for technology as such—only for what certain specific, 19th-century technologies symbolized to her about Man’s dominance over Nature.
  2. Curiosity about the physical universe.  This, of course, is related to point 1.  For Rand, the physical world seems to be of interest only as a medium to be bent to human will.  When I read The Fountainhead as a teenager, I found myself wondering what Rand would’ve made of academic scientists: people who generally share her respect for reason, reality, and creative achievement, but not her metaphysical certainty or her hatred of all government planning.  (Also, while most male scientists resemble a cross between Howard Roark and John Galt, it must be admitted that a tiny minority of them are awkward nerds.)
    In Atlas, Rand finally supplies an answer to this question, in the form of Dr. Robert Stadler.  It turns out that in Rand’s eschatology, academic scientists are the worst evil imaginable: people smart enough to see the truth of her philosophy, but who nevertheless choose to reject it.  Science, as a whole, does not come off well in Atlas: the country starves while Stadler’s State Science Institute builds a new cyclotron; and Dr. Floyd Ferris, the author of obscurantist popular physics books, later turns into a cold-blooded torturer.  (That last bit, actually, has a ring of truth to it.)
    More important, in a book with hundreds of pages of philosophizing about human nature, there’s no mention of evolution; in a book obsessed with “physics,” there’s no evidence of any acquaintance with relativity, quantum mechanics, or pretty much anything else about physics.  (When Stadler starts talking about particles approaching the speed of light, Dagny impatiently changes the subject.)  It’s an interesting question whether Rand outright rejected the content of modern science; maybe we’ll pick up that debate in the comments section.  But another possibility—that Rand was simply indifferent to the sorts of things an Einstein, Darwin, or Robert Stadler might discover, that she didn’t care whether they were true or not—is, to my mind, hardly more defensible for a “philosopher of reason.”
  3. Family.  Whittaker Chambers (of pumpkin patch fame) pointed out this startling omission in his review of 1957.  The characters in Atlas mate often enough, but they never reproduce, or even discuss the possibility of reproduction (if only to take precautions against it).  Also, the only family relationships portrayed at length are entirely negative in character: Rearden’s mother, brother, and wife are all contemptible collectivists who mooch off the great man even as they despise him, while Dagny’s brother Jim is the wretched prince of looters.  Any Republicans seeking solace in Atlas should be warned: Ayn Rand is not your go-to philosopher for family values (much less “Judeo-Christian” ones).
  4. “Angular,” attractive people who also happen to be collectivists, or “shapeless” people who happen to be rational individualists.  In the universe of Atlas, physical appearance is destiny—always, without exception, from John Galt down to the last minor villain.  Whenever Rand introduces a new character, you learn immediately, after a one-paragraph physical description, everything she wants you to know about that character’s moral essence: “angular” equals good, “limp,” “petulant,” and so on equal bad.  Admittedly, most movies also save the audience from unwanted thought by making similar identifications.  But Rand’s harping on this theme is so insistent, so vitriolic, that it leaves little doubt she really did accept the eugenic notion that a person’s character is visible on his or her face.
  5. Personalities.  In Atlas, as in The Fountainhead, each character has (to put it mildly) a philosophy, but no personality independent of that philosophy, no Objectively-neutral character traits.  What, for example, do we know about Howard Roark?  Well, he has orange hair, likes to smoke cigarettes, and is a brilliant architect and defender of individualism.  What do we know about John Galt?  He has gold hair, likes to smoke cigarettes, and is a brilliant inventor and defender of individualism.  Besides occupation and hair color, they’re pretty much identical.  Neither is suffered to have any family, culture, backstory, weaknesses, quirks, or even hobbies or favorite foods (not counting cigarettes, of course).  Yes, I know this is by explicit authorial design.  But it also seems to undermine Rand’s basic thesis: that Galt and Roark are not gods or robots, but ordinary mortals.
  6. Positive portrayal of uncertainty.  In Atlas, “rationality” is equated over and over with being certain one is right.  The only topic the good guys, like Hank and Dagny, ever change their minds about is whether the collectivists are (a) evil or (b) really, really evil.  (Spoiler alert: after 800 pages, they opt for (b).)  The idea that rationality might have anything to do with being uncertain—with admitting you’re wrong, changing your mind, withholding judgment—simply does not exist in Rand’s universe.  For me, this is the single most troubling aspect of her thought.
  7. Honest disagreements.  Atlas might be the closest thing ever written to a novelization of Aumann’s Agreement Theorem.  In RandLand, whenever two rational people meet, they discover to their delight that they agree about everything—not merely the basics like capitalism and individualism, but also the usefulness of Rearden Metal, the beauty of Halley’s Fifth Concerto, and so on.  (Again, the one exception is the disagreement between those who’ve already accepted the full evil of the collectivists, and those still willing to give them a chance.)  In “Galt’s Gulch” (the book’s utopia), there’s one judge to resolve disputes, but he’s never had to do anything since no disputes have ever arisen.
  8. History.  When I read The Fountainhead as a teenager, there was one detail that kept bothering me: the fact that it was published in 1943.  At such a time, how could Rand possibly imagine the ultimate human evil to be a left-wing newspaper critic?  Atlas continues the willful obliviousness to real events, like (say) World War II or the Cold War.  And yet—just like when she removes family, personality, culture, evolution, and so on from the picture—Rand clearly wants us to apply the lessons from her pared-down, stylized world to this world.  Which raises an obvious question: if her philosophy is rich enough to deal with all these elephants in the room, then why does she have to avoid mentioning the elephants while writing thousands of pages about the room’s contents?
  9. Efficient evil people.  In Atlas, there’s not a single competent industrialist who isn’t also an exemplar of virtue.  The heroine, Dagny, is a railroad executive who makes trains run on time—who knows in her heart that reliable train service is its own justification, and that what the trains are transporting and why is morally irrelevant.  Granted, after 900 pages, Dagny finally admits to herself that she’s been serving an evil cause, and should probably stop.  But even then, her earlier “don’t ask why” policy is understood to have been entirely forgivable: a consequence of too much virtue rather than too little.  I found it odd that Rand, who (for all her faults) was normally a razor-sharp debater, could write this way so soon after the Holocaust without thinking through the obvious implications.
  10. Ethnicity.  Seriously: to write two sprawling novels set in the US, with hundreds of characters between them, and not a single non-Aryan?  Even in the 40s and 50s?  For me, the issue here is not political correctness, but something much more basic: for all Rand’s praise of “reality,” how much interest did she have in its contents?  On a related note, somehow Rand seems to have gotten the idea that “the East,” and India in particular, were entirely populated by mystical savages sitting cross-legged on mats, eating soybeans as they condemned reason and reality.  To which I can only reply: what did she have against soybeans?  Edamame is pretty tasty.

Murray Rothbard and Eliezer Yudkowsky take different routes to some of the same conclusions.

Quantum Computing Since Democritus Lecture 21: Ask Me Anything

Monday, September 1st, 2008

In the final Democritus installment, I entertain students’ questions about everything from derandomization to the “complexity class for creativity” to the future of religion.  (In this edited version, I omitted questions that seemed too technical, which surprisingly was almost half of them.)  Thanks to all the readers who’ve stuck with me to this point, to the students for a fantastic semester (if they still remember it) as well as their scribing help, to Chris Granade for further scribing, and to Waterloo’s Institute for Quantum Computing for letting me get away with this.  I hope you’ve enjoyed it, and only wish I’d kept my end of the bargain by getting these notes done a year earlier.

A question for the floor: some publishers have expressed interest in adapting the Democritus material into book form.  Would any of you actually shell out money for that?

Quantum Computing Since Democritus Lecture 18: Free Will

Monday, July 28th, 2008

If you don’t like this latest lecture, please don’t blame me: I had no choice! (Yeah, yeah, I know. You presumably have no choice in criticizing it either. But it can’t hurt to ask!)

Those of you who’ve been reading this blog since the dark days of 2005 might recognize some of the content from this post about Newcomb’s Paradox, and this one about the Free Will Theorem.

Quantum Computing Since Democritus Lecture 17: Fun With The Anthropic Principle

Thursday, July 24th, 2008

Here it is. There was already a big anthropic debate in the Lecture 16 comments — spurred by a “homework exercise” at the end of that lecture — so I feel absolutely certain that there’s nothing more to argue about. On the off chance I’m wrong, though, you’re welcome to restart the debate; maybe you’ll even tempt me to join in eventually.

The past couple weeks, I was at Foo Camp in Sebastopol, CA, where I had the opportunity to meet some wealthy venture capitalists, and tell them all about quantum computing and why not to invest in it hoping for any short-term payoff other than interesting science. Then I went to Reed College in Portland, OR, to teach a weeklong course on “The Complexity of Boolean Functions” at MathCamp’2008. MathCamp is (as the name might suggest) a math camp for high school students. I myself attended it way back in 1996, where some guy named Karp gave a talk about P and NP that may have changed the course of my life.

Alas, neither camp is the reason I haven’t posted anything for two weeks; for that I can only blame my inherent procrastination and laziness, as well as my steadily-increasing, eminently-justified fear of saying something stupid or needlessly offensive (i.e., the same fear that leads wiser colleagues not to start blogs in the first place).

The array size of the universe

Friday, May 23rd, 2008

I’ve been increasingly tempted to make this blog into a forum solely for responding to the posts at Overcoming Bias. (Possible new name: “Wallowing in Bias.”)

Two days ago, Robin Hanson pointed to a fascinating paper by Bousso, Harnik, Kribs, and Perez, on predicting the cosmological constant from an “entropic” version of the anthropic principle. Say what you like about whether anthropicology is science or not, for me there’s something delightfully non-intimidating about any physics paper with “anthropic” in the abstract. Sure, you know it’s going to have metric tensors, etc. (after all, it’s a physics paper) — but you also know that in the end, it’s going to turn on some core set of assumptions about the number of sentient observers, the prior probability of the universe being one way rather than another, etc., which will be comprehensible (if not necessarily plausible) to anyone familiar with Bayes’ Theorem and how to think formally.

So in this post, I’m going to try to extract an “anthropic core” of Bousso et al.’s argument — one that doesn’t depend on detailed calculations of entropy production (or anything else) — trusting my expert readers to correct me where I’m mistaken. In defense of this task, I can hardly do better than to quote the authors themselves. In explaining why they make what will seem to many like a decidedly dubious assumption — namely, that the “number of observations” in a given universe should be proportional to the increase in non-gravitational entropy, which is dominated (or so the authors calculate) by starlight hitting dust — they write:

We could have … continued to estimate the number of observers by more explicit anthropic criteria. This would not have changed our final result significantly. But why make a strong assumption if a more conservative one suffices? [p. 14]

In this post I’ll freely make strong assumptions, since my goal is to understand and explain the argument rather than to defend it.

The basic question the authors want to answer is this: why does our causally-connected patch of the universe have the size it does? Or more accurately: taking everything else we know about physics and cosmology as given, why shouldn’t we be surprised that it has the size it does?

From the standpoint of post-1998 cosmology, this is more-or-less equivalent to asking why the cosmological constant Λ ~ 10-122 should have the value it has. For the radius of our causal patch scales like

1/√Λ ~ 1061 Planck lengths ~ 1010 light-years,

while (if you believe the holographic principle) its maximum information content scales like 1/Λ ~ 10122 qubits. To put it differently, there might be stars and galaxies and computers that are more than ~1010 light-years away from us, and they might require more than ~10122 qubits to describe. But if so, they’re receding from us so quickly that we’ll never be able to observe or interact with them.

Of course, to ask why Λ has the value it does is really to ask two questions:

1. Why isn’t Λ smaller than it is, or even zero? (In this post, I’ll ignore the possibility of its being negative.)
2. Why isn’t Λ bigger than it is?

Presumably, any story that answers both questions simultaneously will have to bring in some actual facts about the universe. Let’s face it: 10-122 is just not the sort of answer you expect to get from armchair philosophizing (not that it wouldn’t be great if you did). It’s a number.

As a first remark, it’s easy to understand why Λ isn’t much bigger than it is. If it were really big, then matter in the early universe would’ve flown apart so quickly that stars and galaxies wouldn’t have formed, and hence we wouldn’t be here to blog about it. But this upper bound is far from tight. Bousso et al. write that, based on current estimates, Λ could be about 2000 times bigger than it is without preventing galaxy formation.

As for why Λ isn’t smaller, there’s a “naturalness” argument due originally (I think) to Weinberg, before the astronomers even discovered that Λ>0. One can think of Λ as the energy of empty space; as such, it’s a sum of positive and negative contributions from all possible “scalar fields” (or whatever else) that contribute to that energy. That all of these admittedly-unknown contributions would happen to cancel out exactly, yielding Λ=0, seems fantastically “unnatural” if you choose to think of the contributions as more-or-less random. (Attempts to calculate the likely values of Λ, with no “anthropic correction,” notoriously give values that are off by 120 orders of magnitude!) From this perspective, the smaller you want Λ to be, the higher the price you have to pay in the unlikelihood of your hypothesis.

Based on the above reasoning, Weinberg predicted that Λ would have close to the largest possible value it could have, consistent with the formation of galaxies. As mentioned before, this gives a prediction that’s too big by a factor of 2000 — a vast improvement over the other approaches, which gave predictions that were off by factors of 10120 or infinity!

Still, can’t we do better? One obvious approach to pushing Λ down would be to extend the relatively-uncontroversial argument explaining why Λ can’t be enormous. After all, the tinier we make Λ, the bigger the universe (or at least our causal patch of it) will be. And hence, one might argue, the more observers there will be, hence the more likely we’ll be to exist in the first place! This form of anthropicizing — that we’re twice as likely to exist in a universe with twice as many observers — is what philosopher Nick Bostrom calls the Self-Indication Assumption.

However, two problems with this idea are evident. First, why should it be our causal patch of the universe that matters, rather than the universe as a whole? For anthropic purposes, who cares if the various civilizations that arise in some universe are in causal contact with each other or not, provided they exist? Bousso et al.’s response is basically just to stress that, from what we know about quantum gravity (in particular, black-hole complementarity), it probably doesn’t even make sense to assign a Hilbert space to the entire universe, as opposed to some causal patch of it. Their “Causal-Patch Self-Indication Assumption” still strikes me as profoundly questionable — but let’s be good sports, assume it, and see what the consequences are.

If we do this, we immediately encounter a second problem with the anthropic argument for a low value of Λ: namely, it seems to work too well! On its face, the Self-Indication Assumption wants the number of observers in our causal patch to be infinite, hence the patch itself to be infinite in size, hence Λ=0, in direct conflict with observation.

But wait: what exactly is our prior over the possible values of Λ? Well, it appears Landscapeologists typically just assume a uniform prior over Λ within some range. (Can someone enlighten me on the reasons for this, if there are any? E.g., is it just that the middle part of a Gaussian is roughly uniform?) In that case, the probability that Λ is between ε and 2ε will be of order ε — and such an event, we might guess, would lead to a universe of “size” 1/ε, with order 1/ε observers. In other words, it seems like the tiny prior probability of a small cosmological constant should precisely cancel out the huge number of observers that such a constant leads to — Λ(1/Λ)=1 — leaving us with no prediction whatsoever about the value of Λ. (When I tried to think about this issue years ago, that’s about as far as I got.)

So to summarize: Bousso et al. need to explain to us on the one hand why Λ isn’t 2000 times bigger than it is, and on the other hand why it’s not arbitrarily smaller or 0. Alright, so are you ready for the argument?

The key, which maybe isn’t so surprising in retrospect, turns out to be other stuff that’s known about physics and astronomy (independent of Λ), together with the assumption that that other stuff stays the same (i.e., that all we’re varying is Λ). Sure, say Bousso et al.: in principle a universe with positive cosmological constant Λ could contain up to ~1/Λ bits of information, which corresponds — or so a computer scientist might estimate! — to ~1/Λ observers, like maybe ~1/√Λ observers in each of ~1/√Λ time periods. (The 1/√Λ comes from the Schwarzschild bound on the amount of matter and energy within a given radius, which is linear in the radius and therefore scales like 1/√Λ.)

But in reality, that 1/Λ upper bound on the number of observers won’t be anywhere close to saturated. In reality, what will happen is that after a billion or so years stars will begin to form, radiating light and quickly increasing the universe’s entropy, and then after a couple tens of billions more years, those stars will fizzle out and the universe will return to darkness. And this means that, even though you pay a Λ price in prior probability for a universe with 1/Λ information content, as Λ goes to zero what you get for your money is not ~1/√Λ observers in each of ~1/√Λ time periods (hence ~1/Λ observers in total), but rather just ~1/√Λ observers over a length of time independent of Λ (hence ~1/√Λ observers in total). In other words, you get diminishing returns for postulating a bigger and bigger causal patch, once your causal patch exceeds a few tens of billions of light-years in radius.

So that’s one direction. In the other direction, why shouldn’t we expect Λ to be 2000 times bigger than it is (i.e. the radius of our causal patch to be ~45 times smaller)? Well, Λ could be that big, say the authors, but in that case the galaxies would fly apart from each other before starlight really started to heat things up. So once again you lose out: during the very period when the stars are shining the brightest, entropy production is at its peak, civilizations are presumably arising and killing each other off, etc., the number of galaxies per causal patch is minuscule, and that more than cancels out the larger prior probability that comes with a larger value of Λ.

Putting it all together, then, what you get is a posterior distribution for Λ that’s peaked right around 10-122 or so, corresponding to a causal patch a couple tens of light-years across. This, of course, is exactly what’s observed. You also get the prediction that we should be living in the era when Λ is “just taking over” from gravity, which again is borne out by observation. According to another paper, which I haven’t yet read, several other predictions of cosmological parameters come out right as well.

On the other hand, it seems to me that there are still few enough data points that physicists’ ability to cook up some anthropic explanation to fit them all isn’t sufficiently surprising to compel belief. (In learning theory terms, the measurable cosmological parameters still seem shattered by the concept class of possible anthropic stories.) For those of us who, unlike Eliezer Yudkowsky, still hew to the plodding, non-Bayesian, laughably human norms of traditional science, it seems like what’s needed is a successful prediction of a not-yet-observed cosmological parameter.

Until then, I’m happy to adopt a bullet-dodging attitude toward this and all other proposed anthropic explanations. I assent to none, but wish to understand them all — the more so if they have a novel conceptual twist that I personally failed to think of.

Floating in Platonic heaven

Thursday, May 15th, 2008

In the comments section of my last post, Jack in Danville writes:

I may have misunderstood [an offhand comment about the "irrelevance" of the Continuum Hypothesis] … Intuitively I’ve thought the Continuum Hypothesis describes an aspect of the real world.

I know we’ve touched on similar topics before, but something tells me many of you are hungerin’ for a metamathematical foodfight, and Jack’s perplexity seemed as good a pretext as any for starting a new thread.

So, Jack: this is a Deep Question, but let me try to summarize my view in a few paragraphs.

It’s easy to imagine a “physical process” whose outcome could depend on whether Goldbach’s Conjecture is true or false. (For example, a computer program that tests even numbers successively and halts if it finds one that’s not a sum of two primes.) Likewise for P versus NP, the Riemann Hypothesis, and even considerably more abstract questions.

But can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like?

I submit that the key distinction is between

  1. questions that are ultimately about Turing machines and finite sets of integers (even if they’re not phrased that way), and
  2. questions that aren’t.

We need to assume that we have a “direct intuition” about integers and finite processes, which precedes formal reasoning — since without such an intuition, we couldn’t even do formal reasoning in the first place. By contrast, for me the great lesson of Gödel and Cohen’s independence results is that we don’t have a similar intuition about transfinite sets, even if we sometimes fool ourselves into thinking we do. Sure, we might say we’re talking about arbitrary subsets of real numbers, but on closer inspection, it turns out we’re just talking about consequences of the ZFC axioms, and those axioms will happily admit models with intermediate cardinalities and other models without them, the same way the axioms of group theory admit both abelian and non-abelian groups. (Incidentally, Gödel’s models of ZFC+CH and Cohen’s models of ZFC+not(CH) both involve only countably many elements, which makes the notion that they’re telling us about some external reality even harder to understand.)

Of course, everything I’ve said is consistent with the possibility that there’s a “truth” about CH floating in Platonic heaven, or even that a plausible axiom system other than ZFC could prove or disprove CH (which was Gödel’s hope). But the “truth” of CH is not going to have consequences for human beings or the physical universe independent of its provability, in the same way that the truth of P=NP could conceivably have consequences for us even if we weren’t able to prove or disprove it.

For mathematicians, this distinction between “CH-like questions” and “Goldbach/Riemann/Pvs.NP-like questions” is a cringingly obvious one, probably even too obvious to point out. But I’ve seen so many people argue about Platonism versus formalism as if this distinction didn’t exist — as if one can’t be a Platonist about integers but a formalist about transfinite sets — that I think it’s worth hammering home.

To summarize, Kronecker had it backwards. Man and Woman deal with the integers; all else is the province of God.

The bullet-swallowers

Tuesday, May 13th, 2008

Question for the day: what do libertarianism and the Many-Worlds Interpretation of quantum mechanics have in common? Interest in the two worldviews seems to be positively correlated: think of quantum computing pioneer David Deutsch, or several prominent posters over at Overcoming Bias, or … oh, alright, my sample size is admittedly pretty small.

Some connections are obvious: libertarianism and MWI are both grand philosophical theories that start from premises that almost all educated people accept (quantum mechanics in the one case, Econ 101 in the other), and claim to reach conclusions that most educated people reject, or are at least puzzled by (the existence of parallel universes / the desirability of eliminating fire departments). Both theories seem to have a strong following with nerds who read science fiction and post to Internet discussion groups, but a relatively poorer following with both John Q. Public and Alistair K. Intellectual. (Needless to say, these stereotypes tell us almost nothing about the theories’ validity.)

My own hypothesis has to do with bullet-dodgers versus bullet-swallowers. A bullet-dodger is a person who says things like:

Sure, obviously if you pursued that particular line of reasoning to an extreme, then you’d get such-and-such an absurd-seeming conclusion. But that very fact suggests that other forces might come into play that we don’t understand yet or haven’t accounted for. So let’s just make a mental note of it and move on.

Faced with exactly the same situation, a bullet-swallower will exclaim:

The entire world should follow the line of reasoning to precisely this extreme, and this is the conclusion, and if a ‘consensus of educated opinion’ finds it disagreeable or absurd, then so much the worse for educated opinion! Those who accept this are intellectual heroes; those who don’t are cowards.

In a lifetime of websurfing, I don’t think I’ve ever read an argument by a libertarian or a Many-Worlds proponent that didn’t sound like the latter.

We know plenty of historical examples where the bullet-swallowers were gloriously right: Moore’s Law, Darwinism, the abolition of slavery, women’s rights. On the other hand, at various points within the last 150 years, extremely smart people also reasoned themselves to the inescapable conclusions that aether had to exist for light to be a wave in, that capitalism was reaching its final crisis, that only a world government could prevent imminent nuclear war, and that space colonies would surely exist by 2000. In those cases, even if you couldn’t spot any flaws in the arguments, you still would’ve been wise to doubt their conclusions. (Or are you sure you would have spotted the flaws where Maxwell and Kelvin, Russell and Einstein did not?)

Here’s a favorite analogy. The world is a real-valued function that’s almost completely unknown to us, and that we only observe in the vicinity of a single point x0. To our surprise, we find that, within that tiny vicinity, we can approximate the function extremely well by a Taylor series.

“Aha!” exclaim the bullet-swallowers. “So then the function must be the infinite series, neither more nor less.”

“Not so fast,” reply the bullet-dodgers. “All we know is that we can approximate the function in a small open interval around x0. Who knows what unsuspected phenomena might be lurking beyond it?”

“Intellectual cowardice!” the first group snorts. “You’re just like the Jesuit schoolmen, who dismissed the Copernican system as a mere calculational device! Why can’t you accept what our best theory is clearly telling us?”

So who’s right: the bullet-swallowing libertarian Many-Worlders, or the bullet-dodging intellectual kibitzers? Well, that depends on whether the function is sin(x) or log(x).