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. Thinker says:

    I would be interested to know the following simple questions:

    1. If some outside observer is preventing us from statistically testing paranormal events (by forcing them to follow random distributions), can this be tested for? This may provide more evidence for the non-existance of paranormal events – i.e. even if some outside observer was hiding the existance of paranormality, we were unable to detect their influence.
    2. If our DNA extended from one family (noahs) can we test for this using standard genetics? Can we estimate the number of generations, and does this hold? This may show that the timeline is correct, or otherwise 🙂
    3. Why more people think science is so awesome when in fact up to 50% of papers may be wrong.

  2. Michael Pohoreski says:

    Sean, Logic is NOT just binary. Why do you think there is such a thing as tertiary logic?
    True, False, Indeterminate

  3. Jean-Denis says:

    My take:

    1- If paranormal phenomenon are undistinguishable from random events, then their reality is totally irrelevant.

    2- yes, it can be tested. Using DNA in remains of Cro Magnon and Neanderthal bones, it was assessed that the initial population of Neanderthal was composed of about 8000 individuals, while the initial population of Cro Magnon was composed of about 15000 individuals. I have strictly no idea how that was done by the way… If really pressed, I might be able to retrieve where I read that.

    3- the fact that xx% of scientific papers might be wrong is probably not a reason to think either that science is awesome or that it’s not (I will not tackle the question of what it means for a scientific paper to be right or wrong). If anything, it shows that scientists are really “re-searchers”, who look hard and long before they can find a tiny bit of advancement. This might be considered awesome.

    There are many reasons why science can be seen as awesome.

    Of course, there are countless examples of things we use everyday that would not have been possible without science, such as the GPS system or vaccines.

    A nice reason in a short sentence? Science is awesome because of its predictive power (well, that sentence is too short and could be read in several wrong ways).

    For me personally, science (and the scientific method) is awesome because it allows us (mankind) to improve our collective knowledge of how “things” work. It allows us to reach beauties and marvels unreachable otherwise. I will stop right now, lest I write a full essay.


  4. Thinker says:

    If paranormal events are being influenced, that IS an important result

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  8. Thomas says:

    Is it a cognitive dead end to believe as Plato does when he has Socrates say: I think that Euripides may have been right in saying, who knows if life be not death and death life; and that we are very likely dead; (Plato Gorgias, 493a)?

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  10. TwisMinion says:

    Maybe it could be phrased like this…

    What is something you believe but haven’t been able to convince anybody else of?


    What is something you believe that you aren’t quite sure is correct?

    If they asked this question to free thinkers they can get all sorts of good answers to the question because true free thinking allows you to stake mental ground, much like staking a claim in the days of the gold rush, without actually assuming that there’s gold in them thar hills. It’s not just that you want or expect to find the gold there, it’s that you believe it could be there, but have yet to dig…

    The issue Sean seems to have is with the word believe… But Sean’s a scientist, and so we forgive and even encourage his dedication to skeptical analysis.

    Words mean different things to different people. The word believe means one thing to a cosmologist, and quite another to a preacher, gambler or five year old on Christmas Eve. The word theory in science means a concept backed up by many, many individual facts, while in layman’s terms it is equivalent to a best guess. Math is less fuzzy than words, but only because advertisers haven’t figured out how to exploit them properly yet.

    Still, even Sean must have reacted to many conversations regarding the difference between human intuition and quantum physics with something to the tune of “I agree with you completely, that makes sense, here’s why we’re wrong…”


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  12. TP says:

    “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”. That sort of spoiled a very good piece.

    I think a better way to counter such an argument is if I am a brain in vat then the vat is the world for me and it is no less a world than the “true” world. I took this whole brain in a vat idea quite seriously some years ago. Now it just seems a bit ridiculous.

  13. Neil B. ? says:

    No TP, that attitude is “anti-realist”. If you think “things really exist” in some actual configuration, then a brain in a vat is just that, however confused it may be about it’s materially true status. Otherwise you might as well be a post-modernist etc, but you aren’t a physical realist (being able to measure as e.g for quantum issues is a matter of physical principle, and not practical “handicap” as for example the tricked brain in a vat or otherwise.

    But the real “brain in a vat” problem may be even worse than the “physical” issue of there being real worlds with actual properties independent of whether any given brain can accurately assess its status or not therein. The big problem comes from the AI notion that all thought (and even experience!) is really a formal calculation process, doable with a Turing machine etc. But if that’s so, then there’s no way to know if you’re even a “real brain” at all (in a vat, or not; regardless in this case) versus just the platonically real existence of the thought process itself as a mathematical model. You can’t explain what the difference would consist of. Any difference by definition would have to be something beyond the computations, such that strong AI is wrong (which I think it is, so this is not a problem for *me*, just for such believers like spurious great-modern philosopher Dan Dennett.)

    There is no way to logically model and validate thoughts of, “Wow, I really exist as/through a brain in a material world. I’m not just the process either on a computer or even merely a concept” because physical embodiment itself cannot be represented by logical processes. Only the thoughts themselves can be, and anything that can be thought of “computationally” is just as real as pure mathematical description. All strong AI type thought is just the information itself, it is a given already as the math describing it (you’ll have to reflect on that to get the point.)

    Note that the computations happening “in time” is no way out or can show “realness”, since processes can be mathematically represented as a “block” (look up “block universe.”) IOW, this forces the acceptance of modal realism for minds, even if the fullness of physical nature cannot be entirely modeled in math (pace Max Tegmark et al.)


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  15. Inquiry says:

    This is something every Atheist should read so that they can realize how much they profoundly misunderstand science as a whole and that they’ve been convicting hypocrisy at best.

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