What I Believe But Cannot Prove

Each year, John Brockman’s Edge asks a collection of deep thinkers a profound question, and gives them a couple of hundred words to answer: The World Question Center. The question for 2005 was What Do You Believe Is True Even Though You Cannot Prove It? Plenty of entertaining answers, offered by people like Bruce Sterling, Ray Kurzweil, Lenny Susskind, Philip Anderson, Alison Gopnik, Paul Steinhardt, Maria Spiropulu, Simon Baron-Cohen, Alex Vilenkin, Martin Rees, Esther Dyson, Margaret Wertheim, Daniel Dennett, and a bunch more. They’ve even been collected into a book for your convenient perusal. Happily, these questions are more or less timeless, so nobody should be upset that I’m a couple of years late in offering my wisdom on this pressing issue.

Most of the participants were polite enough to play along and answer the question in the spirit in which it was asked, although their answers often came down to “I believe the thing I’m working on right now will turn out to be correct and interesting.” But to me, there was a perfectly obvious response that almost nobody gave, although Janna Levin and Seth Lloyd came pretty close. Namely: there isn’t anything that I believe that I can prove, aside from a limited set of ultimately sterile logical tautologies. Not that there’s anything wrong with tautologies; they include, for example, all of mathematics. But they describe necessary truths; given the axioms, the conclusions follow, and we can’t imagine it being any other way. The more interesting truths, it seems to me, are the contingent ones, the features of our world that didn’t have to be that way. And I can’t prove any of them.

The very phrasing of the question, and the way most of the participants answered it, irks me a bit, as it seems to buy into a very wrong way of thinking about science and understanding: the idea that true and reliable knowledge derives from rigorous proof, and anything less than that is dangerously uncertain. But the reality couldn’t be more different. I can’t prove that the Sun will rise tomorrow, that radioactive decays obey an exponential probability law, or that the Earth is more than 6,000 years old. But I’m as sure as I am about any empirical statement that these are true. And, most importantly, there’s nothing incomplete or unsatisfying about that. It’s the basic way in which we understand the world.

Here is a mathematical theorem: There is no largest prime number. And here is a proof:

Consider the list of all primes, pi, starting with p1 = 2. Suppose that there is a largest prime, p*. Then there are only a finite number of primes. Now consider the number X that we obtain by multiplying together all of the primes pi (exactly once each) from 2 to p* and adding 1 to the result. Then X is clearly larger than any of the primes pi. But it is not divisible by any of them, since dividing by any of them yields a remainder 1. Therefore X, since it has no prime factors, is prime. We have thus constructed a prime larger than p*, which is a contradiction. Therefore there is no largest prime.

Here is a scientific belief: General relativity accurately describes gravity within the solar system. And here is the argument for it:

GR incorporates both the relativity of locally inertial frames and the principle of equivalence, both of which have been tested to many decimal places. Einstein’s equation is the simplest possible non-trivial dynamical equation for the curvature of spacetime. GR explained a pre-existing anomaly — the precession of Mercury — and made several new predictions, from the deflection of light to gravitational redshift and time delay, which have successfully been measured. Higher-precision tests from satellites continue to constrain any possible deviations from GR. Without taking GR effects into account, the Global Positioning System would rapidly go out of whack, and by including GR it works like a charm. All of the known alternatives are more complicated than GR, or introduce new free parameters that must be finely-tuned to agree with experiment. Furthermore, we can start from the idea of massless spin-two gravitons coupled to energy and momentum, and show that the nonlinear completion of such a theory leads to Einstein’s equation. Although the theory is not successfully incorporated into a quantum-mechanical framework, quantum effects are expected to be unobservably small in present-day experiments. In particular, higher-order corrections to Einstein’s equation should naturally be suppressed by powers of the Planck scale.

You see the difference, I hope. The mathematical proof is airtight; it’s just a matter of following the rules of logic. It is impossible for us to conceive of a world in which we grant the underlying assumptions, and yet the conclusion doesn’t hold.

The argument in favor of believing general relativity — a scientific one, not a mathematical one — is of an utterly different character. It’s all about hypothesis testing, and accumulating better and better pieces of evidence. We throw an hypothesis out there — gravity is the curvature of spacetime, governed by Einstein’s equation — and then we try to test it or shoot it down, while simultaneously searching for alternative hypotheses. If the tests get better and better, and the search for alternatives doesn’t turn up any reasonable competitors, we gradually come to the conclusion that the hypothesis is “right.” There is no sharp bright line that we cross, at which the idea goes from being “just a theory” to being “proven correct.” Rather, maintaining skepticism about the theory goes from being “prudent caution” to being “crackpottery.”

It is a intrinsic part of this process that the conclusion didn’t have to turn out that way, in any a priori sense. I could certainly imagine a world in which some more complicated theory like Brans-Dicke was the empirically correct theory of gravity, or perhaps even one in which Newtonian gravity was correct. Deciding between the alternatives is not a matter of proving or disproving; its a matter of accumulating evidence past the point where doubt is reasonable.

Furthermore, even when we do believe the conclusion beyond any reasonable doubt, we still understand that it’s an approximation, likely (or certain) to break down somewhere. There could very well be some very weakly-coupled field that we haven’t yet detected, that acts to slightly alter the true behavior of gravity from what Einstein predicted. And there is certainly something going on when we get down to quantum scales; nobody believes that GR is really the final word on gravity. But none of that changes the essential truth that GR is “right” in a certain well-defined regime. When we do hit upon an even better understanding, the current one will be understood as a limiting case of the more comprehensive picture.

“Proof” has an interesting and useful meaning, in the context of logical demonstration. But it only gives us access to an infinitesimal fraction of the things we can reasonably believe. Philosophers have gone over this ground pretty thoroughly, and arrived at a sensible solution. The young Wittgenstein would not admit to Bertrand Russell that there was not a rhinoceros in the room, because he couldn’t be absolutely sure (in the sense of logical proof) that his senses weren’t tricking him. But the later Wittgenstein understood that taking such a purist stance renders the notion of “to know” (or “to believe”) completely useless. If logical proof were required, we would only believe logical truths — and even then the proofs might contain errors. But in the real world it makes perfect sense to believe much more than that. So we take “I believe x” to mean, not “I can prove x is the case,” but “it would be unreasonable to doubt x.”

The search for certainty in empirical knowledge is a chimera. I could always be a brain in a vat, or teased by an evil demon, or simply an AI program running on somebody else’s computer — fed consistently misleading “sense data” that led me to incorrect conclusions about the true nature of reality. Or, to put a more modern spin on things, I could be a Boltzmann Brain — a thermal fluctuation, born spontaneously out of a thermal bath with convincing (but thoroughly incorrect) memories of the past. But — here is the punchline — it makes no sense to act as if any of those is the case. By “makes no sense” we don’t mean “can’t possibly be true,” because any one of those certainly could be true. Instead, we mean that it’s a cognitive dead end. Maybe you are a brain in a vat. What are you going to do about it? You could try to live your life in a state of rigorous epistemological skepticism, but I guarantee that you will fail. You have to believe something, and you have to act in some way, even if your belief is that we have no reliable empirical knowledge about the world and your action is to never climb out of bed. On the other hand, putting aside the various solipsistic scenarios and deciding to take the evidence of our senses (more or less) at face value does lead somewhere; we can make sense of the world, act within it and see it respond in accordance with our understanding. That’s both the best we can hope for, and what the world does as a matter of fact grant us; that’s why science works!

It can sound a little fuzzy, with this notion of “reasonable” having sneaked into our definition of belief, where we might prefer to stand on some rock-solid metaphysical foundations. But the world is a fuzzy place. Although I cannot prove that I am not a brain in a vat, it is unreasonable for me to take the possibility seriously — I don’t gain anything by it, and it doesn’t help me make sense of the world. Similarly, I can’t prove that the early universe was in a hot, dense state billions of years ago, nor that human beings evolved from precursor species under the pressures of natural selection. But it would be unreasonable for me to doubt it; those beliefs add significantly to my understanding of the universe, accord with massive piles of evidence, and contribute substantially to the coherence of my overall worldview.

At least, that’s what I believe, although I can’t prove it.

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  2. One of the essays dealt with the mathematical definition of proof — similar to your post. I can’t tell you which one or who it was by as my copy of the book is sitting next to my toilet at home, where it made for some okay bathroom reading. And a lot of the biology in the book is total crap, in case you care.

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  4. Logic is a very binary system – true, false and that’s it.

    Is it possible to extend logic into the realm of “almost” in any kind of rigorous way? I’m tempted to say that the answer is statistics, or some form thereof, but maybe not.

    Point being that the world of “almost” is where science lives, obviously, so do you think that making that distinction is enough encapsulate the above?

  5. couldn’t X be composite, which still provides a contradiction by requiring the existence of larger primes? e.g. 2*3*5*7*11*13 + 1 = 30031 = 59*509…

    but i’m just being anal, great post!

  6. — anon

    59 and 509 are both prime numbers GREATER THAN 13. This contradicts your assumption that 13 is the LARGEST prime number.

    The proof is assuming there are only a finite number of primes, 13 in your case. The number p = 2*3*5*7*11*13 + 1 is not divisable by any of the primes between 2 – 13. Therefore, there exists a prime greater than 13… Either the number p, or its composite factors.

  7. hi shaun, right, i think we are making the same point…? my point was X is not always prime, but it always implies the existence of larger primes.

  8. Logic is a very binary system – true, false and that’s it.

    From my admittedly amateur experience, formal logic can be a hell of a lot weirder than that.

  9. Kieran: What do you have in mind? In classical logic, statements are in principle either true or false (though in practice this can be unknowable).

  10. Sean, lovely post, it is precisely what I was trying to explain to one of the students I tutor as well as the woman who waxes my legs at the spa. The woman was extremely unimpressed with scientists, who have nothing but flimsy theories, and data that fits the theory, but may also fit another yet to be discovered theory. Theories which cannot state “this is eternally true!” with certitude. I told her but don’t you understand, this is the strength in science, the ability to be flexible, to realize that our understanding is partial and we may have to discard a theory tomorrow because of new data. But that 2 of the main scientific theories of how the world work have been experimentally verified over and over (verified, not proven). She was not impressed and a change of subject.

    It is strange, it reminds me of people who think that the strongest people in the world are those who never waver, never doubt, when really the opposite is true, the people who can withstand all sorts of challenges, to intellect/motives/knowledge/opinions, and come around to a new way of thinking if the challenges make sense and are sound, these are the strong ones.

    It is not politically correct to say so, I suppose.

  11. Wonderful post, Sean. This is why I love science. I’m so moved I not even going to quibble about anything today.

  12. Nice post, Sean — but I’m surprised no one’s questioned your stated belief that contingent truths are more interesting than necessary truths by reason of their contingency. On behalf of the Mathematician Awareness League, I wish to remind Cosmic Variance readers that some people find necessary truths more interesting by reason of their necessity! (Are those people posting to Cosmic Constancy instead?)

    Oh, as for me? I’m partial to those necessary truths that no one would have thought to prove if not for contingent truths.

  13. Perhaps Kieran speaks of one amongst the following:

    Paraconsistent logics or relevance logics or perhaps any many valued logic system. Modal logics are not so simple either. Even the constructivist logics, which remove the law of excluded middle has that Not P does not necessitate P.

  14. It strikes me that the question was poorly poised.

    “What do I believe that I cannot test?” would have been far better,
    and getting to the point of reasonableness.

    “What do I believe that I cannot test properly?”.

    For Example, Cosmologists, are confronted with Inflation everyday.
    So some aspects of inflation have been tested…but I would say, certainly not properly. So I am interested in this theory – but don’t yet believe it.

    I am attending a “String Theory” conference next month where I expect to be innundated with the “Landscape theory” to explain why we live in a Universe that appears to be accelerating. This idea seems almost hopeless to ever test, and so I am not interested in it at all.

    So Sean…we know your views on God. But what do you believe in that you cannot test (properly)…

  15. Scott, interestingness is very much in the brain of the beholder, not inherent in the object. It’s a good thing that not everyone thinks the same way as me (on this as on so many other things).

    For example, I’m very interested in the possibility of an unobservable multiverse, which Beep is not. I wouldn’t go so far as to say that I believe in it (at this point), but I hope that someday we’ll have a theory, which we can test, and which makes unambiguous predictions about what happens outside our parochial little patch of spacetime.

  16. Not to be a quibbler or anything, but in order to prove that there are an infinite number of primes, you must treat N = 2*3*…*p + 1 in 2 separate cases: N is prime and N is composite. And you might’ve given Euclid credit for this (I personally consider this to be the defn of the “perfect proof” – neat, short, elegant, easy to understand, and far reaching.

  17. yes kavik…that was my original point! perhaps a cleaner way to state the proof:

    consider the list of primes up to a prime p. then consider X = 2*3*…*p + 1. by the fundamental theorem of arithmetic, there exists a prime q that divides X. q cannot be in the original list, as shown by sean. so p

  18. Anon, Kavik are you two objecting to proof by contradiction? I know that there are some hard core mathematical fundamentalists who won’t accept proof by contradiction 🙂

    Also, some mathematical facts are not as unambiguous as one might think. Take e.g. the concept of uncountable infinities, like the cardinality of the set of real numbers,
    see e.g. here

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  20. Interesting post, Sean. But it seems like you’re really talking about two separate (albeit related) issues:

    1) Science works inductively rather than deductively.
    That is, rather than concluding that a proposition must be true because it follows logically from an accepted axiom, we conclude that a proposition is probably true because it agrees with observation, and “probably” converges to “almost certainly” with the continued accumulation of relevant observations.

    2) Science is based on untestable assumptions.
    For instance, if we’re claiming such-and-such experiment is evidence of the way the world really is, then we’re assuming that there isn’t an omnipotent God who deliberately disrupts the results of all science experiments in order to deceive scientists. (I’d say there’s no good reason to think that such a God exists, but of course I can’t prove he doesn’t.) Or, to use your example, we’re assuming that the scientist really did the experiments at all, rather than just being a brain in a jar receiving data that doesn’t have anything to do with his surroundings. Or for that matter having sprung into being five minutes ago with the false memory of doing the experiment already recorded in his brain.

    But I think it’s worth noting that any attempt to make claims about the world we live in has the same problems. A mathematician can say with certainty that given the axioms of Euclidean geometry the interior angles of a triangle must add to 180 degrees. But a mathematician cannot prove that the angles of a triangle add to 180 degrees in the real world, because he can’t prove that the axioms of Euclidean geometry hold in the real world. (Of course you can convince yourself by observation that Euclidean geometry holds reasonably well in the world around you, but then you’ve invoked the same assumptions you use every time you reason from observation — I’m not a brain in a non-euclidean jar, etc.)

    So the only claims about reality that can be proven (in the deductive sense) are negative claims: “it’s not simultaneously true that these two contradictory statements hold”. But there’s no way to prove that one particular set of logically consistent statements describes the real world better than any other — not without making assumptions.

    But the nice thing about science is that it makes the minimal set of assumptions. The assumptions of science are, roughly:
    (1) My observations (and memories of past observations) correspond to the actual world.
    (2) The world can be described by consistent, universal laws, which can be articulated in a fairly simple form.
    If we want to include what I called a “separate issue” above, we can add a third assumption:
    (3) It is reasonably to draw conclusions about these laws by reasoning inductively from my observations.

    But my point is, every method for acquiring knowledge about the world makes at least these same assumptions. If you believe, say “Every word of the King James Bible is literal truth”, you still have to assume that you can trust your own powers of observation when it comes to reading the words of the Bible, and that you can accurately remember what it said. And furthermore in trying to use it to draw conclusions about the world around you, you have to assume you are accurately observing the world around you in order to make any comparison. If I can tell you “all men are descended from Adam”, but I can’t tell you reliably whether any particular entity I encounter is a man or, say, a toaster oven, then that piece of “knowledge” is devoid of meaning.

    Likewise, if I believe the laws of the universe are constantly changing, or only apply in some cases but not others, or are too complex to ever be understood by humans, etc., then there’s no way to know how they apply to the world around me. A law that may or may not apply in any given case isn’t really a law at all.

    I’ve kind of rambled on, so I’m going to cut myself off here, but my point is:
    When it comes to deciding what we should believe about the world, you can either believe science which basically says “Trust your powers of observation and go from there”, or you can believe some other belief system, which says “Believe X, and also trust your powers of observation except perhaps where they conflict with X.” So science can reasonably said to make the minimal number of assumptions about the world (except perhaps for some completely useless belief systems like “nothing exists”, or “I can’t assume anything.”)

    I guess that was Sean’s point: not trusting your observations gets you nowhere. I’m just adding that trusting your observations and very little else gets you science.

  21. Perhaps “believe” and “prove” are not the right words. How about

    What I think seems likely, though there isn’t any evidence.


    What I think seems likely, though the evidence is currently incomplete.


    What the evidence suggests, and I think soon the evidence will strongly suggest.