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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67 Responses to What I Believe But Cannot Prove

  1. anon says:

    whoops! looks like my post got cut off because i used &lt incorrectly:

    “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 &lt q &lt= X. therefore for any p we have shown the existence of a larger prime q (which is not necessarily X!), implying the infinitude of primes…

    but again, just nitpicking!”

    count iblis, i don’t think we are objecting to proof by contradiction…we are objecting to sean’s statement “therefore X, since it has no prime factors, is prime.” if he starts with the assumption that he can list all of the primes (i.e., there is a finite number of them), he must construct a prime not in the list to provide a contradiction. X is not always that prime, but it does imply the existence of one by the fundamental theorem of arithmetic – which is still a contradiction!

  2. anon says:

    whoops again!!!! &lt = <!

  3. Count Iblis says:

    Anon, I see your point. However, this sort of ambiguity is typical of proof by contradiction. At a certain point you know that the assumption you started out with is false. But must you then stop using that assumption or can you continue and find even more obvious contradictions?

    Sean was just continuing with the original assumption and concluded that since X is not divisible by any primes in the list (which are assumed to be all the primes that exists) it must be prime itself.

    What Sean did is not wrong. Note that a sentence like “if Y is true then Z” is always true if Y is false. So, you’ll never find erroneous results by continuing with the original assumption even when you can already conclude that it is false. Therefore why not continue with it if that makes the proof simpler? 🙂

  4. Elliot says:

    And now for something completely different.

    “I Believe In Love” sung by Don Williams

    I don’t believe in superstars,
    Organic food and foreign cars.
    I don’t believe the price of gold;
    The certainty of growing old.
    That right is right and left is wrong,
    That north and south can’t get along.
    That east is east and west is west.
    And being first is always best.

    But I believe in love.
    I believe in babies.
    I believe in Mom and Dad.
    And I believe in you.

    Well, I don’t believe that heaven waits,
    For only those who congregate.
    I like to think of God as love:
    He’s down below, He’s up above.
    He’s watching people everywhere.
    He knows who does and doesn’t care.
    And I’m an ordinary man,
    Sometimes I wonder who I am.

    But I believe in love.
    I believe in music.
    I believe in magic.
    And I believe in you.

    Well, I know with all my certainty,
    What’s going on with you and me,
    Is a good thing.
    It’s true, I believe in you.

    I don’t believe virginity,
    Is as common as it used to be.
    In working days and sleeping nights,
    That black is black and white is white.
    That Superman and Robin Hood,
    Are still alive in Hollywood.
    That gasoline’s in short supply,
    The rising cost of getting by.

    But I believe in love.
    I believe in old folks.
    I believe in children.
    I believe in you.

    But I believe in love.
    I believe in babies.
    I believe in Mom and Dad.
    And I believe in you.


    Sorry every once in a while I just feel compelled to disrupt the micro-granular intellectual processes I see.


  5. Neil B. says:

    C’mon, make it something simple and basic, like, “I believe that things continue to exist even while not being observed.” Or try, “I believe that the past really existed, not just in the present and its currently-existing apparent evidence of the past.” Any honest logical positivists here that want to say they don’t really even believe that much?

  6. spaceman says:

    As long as there are people, there will always be ATM (Against the Mainstream) theorists pushing the boundaries of thought. Just look at the fabulous examples we have today. Certain people still maintain the Earth is flat (e.g. the Flat Earth Society). Some believe, including certain South African politicians, that HIV does not lead to AIDS; although I am not so sure how many of those HIV-skeptics would be willing to stick themselves with an HIV containing needle. Others think that evolution is completely incorrect and the Earth is 6000 years old. A non-neglible number of educated folks, as well as 2/3 of Americans, think the Big Bang theory is all wrong and there is a concerted conspiracy afoot among cosmologists to suppress alternatives. Still others believe that CO2, the bubbly gas found in your sparkling water or root beer, is not a greenhouse gas.

    These MNPs (Maverick Natural Philosophers) will come up with all kinds of convoluted reasons, often untestable and almost without exception unassailable, for why these views are correct. If the evidence turns out to be overwhelmingly powerful against their views, then the MNPs turn to universal skepticism and/or philosophical fallibilism (which is actually self-defeating if you think about it, but who said the majority of these people are reasonable); I call this the “if I am not right, then nobody can be” reaction. Nevertheless, some ATM theorizing can be productive (save the examples provided in first paragraph! esp. the HIV/AIDS one) because it forces us to think differently about our Universe.

  7. Allyson says:

    This post took me to a very Pirsig place.

    spaceman, I believe that what people say they believe, and what they actually believe, are often completely different things.

    I can’t prove it, but if anyone wants to hand me some grant money…

  8. Mark H. says:

    I say leave proof to math and whiskey.

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  10. Aaron F. says:

    Just wanted to let you know I love this essay! ^_^

  11. Neil B. says:

    I posted something similar to the quantum lottery thread, but it belongs here too:

    If all possible universes exist, even restricting to “having the same laws of physics,” we get the following conundrum for an empirical approach to verifying/confirming truth: Of the 10^10^(1,000, …) universes (not literally infinite if we imagine any kind of discontinuity of space), there will be a subset in which the least probable things happen, as already noted. But instead of peculiar events per se, consider the implications for defining “the laws of physics”: in some universes, for example Co-60 decayed throughout their history on an average of about a few hours instead of five years. (I am one of those who would have been quickly fried in such a universe, or at least where it started during part of my career… .) In such a case, what is the meaning of “half-life” etc? Would physicists and thinkers in such worlds be allowed to say from theory that the half-life was “really” about 5 years, and their whole world is a statistical fluke? Or, is empirical verification part of the very meaning of probability of occurance, in which case the subset of universes “really have different laws of physics.”, and many seem not to have any at all about such matters. Well?

  12. Yvette says:

    Lovely essay. However, I will note that when I first heard the question my immediate answer to it was more to say that I believe that people are fundamentally good at heart. Now you could argue that obviously I’ve had personal experiences leading me to this conclusion and we could make up some test to conclude things either way. But then we’d have to argue over what was meant by “goodness,” we’d start swapping stories of all the terrible things we’ve experienced during our lives, and no one would fully agree.

    So I dunno, I guess the question’s alright for me so long as you’re careful what you apply it to.

  13. Osama Ben says:

    The preponderance of evidence currently is that Hitler was not fundamentally good at heart. Anybody have a definition of “goodness” that changes that evidence?

    Here in New Mexico we have an individual in prison for raping and murdering a 12-year old girl–while he was on parole from another rape charge. A couple of years ago he attempted to hire another convict to kill the father of the 12-year old girl because the father was pushing for the death penalty. I suggest this person is not fundamentally good at heart. Is there a definition of “goodness” that includes this individual?

    Just askin’, that’s all.

  14. Osama Ben says:

    What I started to write was that the essay is superb; one that should be assigned reading for every high school AP science student–or every AP student.

  15. Jane says:

    OK, what to you believe without very good evidence (however you define that)? That’s obviously what the question was trying to get at.

  16. Janet Leslie Blumberg says:

    Sean’s wonderful post on how science works would seem to lead toward respect for those who live within or work within any area of inquiry that exercises a similarly high critical awareness of its own tradition and a commitment to open discussion and a willingness to revise even the most revered (or hoary) principles in order to get closer to the subject matter. So…

    Well, here’s an amazing example from within the Christian community of how to respectfully carry on genuine discussion with those we may deeply disagree with, without dismissing an entire institution and way of life. It’s from “Soulforce,” a group that’s been traveling to Christian campuses to affirm the full humanity of gay, lesbian, bisexual, and transgendered students on those campuses. You’ll find the following at http://www.soulforce.com/blogs/. Reflecting on visits last week to Seattle universities, one soulforce member writes:

    “Ironically, the assumptions that lead us to accept the Bible as inerrant and perfect are the same assumptions that stop us from fully including lesbian, gay, bisexual, and transgender individuals into our churches and schools. At its base is the assumption that our worldview is shared, unchanging and unwavering, throughout all time and every category we would claim for our own. It is a form of prejudice. And the foundation of it all is fear: fear of our ability to cope with change, fear of having to wrestle with new ideas and situations, fear of losing our Self, fear of being alone, fear of being wrong.

    “Faith cannot grow in concrete ground. It needs good, tilled earth. So we must wrestle with the earth we are blessed with, to sift it and question it, to tug at its roots and examine them, to prepare its branches for the grafting of new truths and revelations, to water it with thought, and nourish it with fervent study. Uncertainty cannot scare us, and—like Scripture asserts—we must prepared to submit our deepest truths to the ways of God. If we are to become the new creature, transformed, we cannot fear. There is no fear in love; that is the lesson—Northwest University included—must learn.”

    This is now Janet again: here’s the part I would emphasize for scientists to consider in their own consciences, just as Dawkins weighs on the conscience of thoughtful Christians:

    “At its base is the assumption that our worldview is shared, unchanging and unwavering, throughout all time and every category we would claim for our own. It is a form of prejudice. And the foundation of it all is fear….”

    Now Sean’s blog on how science operates (“What I Believe that I Cannot Prove”) dispells these kinds of assumptions about science. So how can someone like Dawkins in good faith use science as an exclusivist “rationality” in order to dismiss an entire group of people, who vary greatly from one another and many of whom operate (within their faith) with the openness and humilty evidenced by soulforce? Think of the religious people who led the fight against slavery, or the Civil Rights Movement in the South, or the Truth and Reconciliation process in South Africa…. What Dawkins is doing is illiberal, ungenerous, and deeply divisive, but worst of all, intellectually insupportable because it clothes prejudice and bigotry in the garb of scientific openness and honesty, and science doesn’t deserve to be so misused in the public arena, any more than religion deserves to be misused as a misguided weapon against science in the public arena.

    [This is a follow-up to a conversation over on “God Flights” but I didn’t know how to use pingback! By the way, I wonder if those flights over Ohio were a response to the devastating snowstorms in Ohio at that time, asking for comfort and safety for Ohioans at a time of state-wide emergency?]

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  20. ragtag says:

    anon writes,

    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!

    No, your’re not being anal, just wrong. There’s absolutely no way X can be composite, given the assumptions in the proof.

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  23. Jean-Denis says:


    You are absolutely right, but this may deserve laying out the details.

    anon writes,

    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!

    No, your’re not being anal, just wrong. There’s absolutely no way X can be composite, given the assumptions in the proof.

    The reason that X cannot be composite, is:

    if X was composite, there would exist a prime number Pj that divides X. Therefore the remainder of the division of X by Pj would be 0. But we have shown that the remainder of the division of X by *any* prime (remember: there are a finite number of primes in the assumption) is 1. The conclusion is that none of the primes divides X, and therefore the only divisors X has are 1 and X. Therefore X is prime. Since X is larger than the largest prime Pn, this contradicts the assumption that Pn is the largest prime.

    Therefore there is no largest prime: there is an infinite number of prime numbers.

    Also, all of this assumes the following definition of a prime number: a positive whole number is prime if and only if it has exactly two divisors (1 and itself).

    (forgive my awkward English. English is not my native language).


  24. brian says:

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

    Have you ever made a mistake in a proof? How do you know for sure that you didn’t make one this time? It may be improbable that you, and everyone else has missed a glaring error every time you’ve checked it, but its not, strictly speaking, impossible. For that matter, have you conclusively proven that your brain wasn’t being tampered with by aliens when you were working out the proof causing you to jump to the wrong result? Even something as simple as calculating “2+2=4” is subject to the same problems.

    Ultimately, even a mathematical proof isn’t rock-hard certain, because our reasoning about it still has to be done with brains that work in the messy real world, bringing us back to the level of empirical evidence and degrees of likelihood, but no certainties.

  25. Mark says:

    Sean Lake wrote:

    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.

    Yep, you can very naturally extend logic to take into account uncertainty thanks to Bayes’ identity. Furthermore, if you want your rules for reasoning about uncertainty to be sensible (in some weakly defined sense) it’s pretty much the only way.

    E. T. Jaynes spells it all out in his book Probability Theory: The Logic of Science. It’s a really inspiring read – equal parts maths, philosophy, pragmatics and opinion.