The Big Picture
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idea, but if that idea predicts that certain outcomes happen only 1 percent
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of the time, and those outcomes keep happening, an honest Bayesian up-
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dating will eventually lead you to assign a very low posterior credence. You
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might assign a high prior credence to “Drinking coffee will give me the
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ability to accurately predict the future.” Then you drink some coffee, make
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predictions, find that your predictions didn’t come true, and update ap-
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propriately. If you do that enough, the data will wipe out your original
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prior. That’s called “changing your mind,” and it’s a good thing. Further-
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more, since the likelihoods are meant to be objective, gathering more and
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more data nudges everyone in the direction of the same set of ultimate be-
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liefs about the world.
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That’s how it’s supposed to work anyway. It’s up to each of us to honestly
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carry out the process in good faith.
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Evidence that favors one alternative automatically disfavors oth-
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ers. Imagine we are comparing two propositions, X and Y, and we observe 18
an outcome that has a 90 percent chance of happening under X and a 99
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percent chance of happening under Y. According to Bayes’s Theorem, after
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collecting that information, the credence we assign to X will go down.
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That can seem counterintuitive. After all, if X were true, we would have
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a 90 percent chance of obtaining that outcome— how can observing it
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count as evidence against this theory? The answer is just that it’s even more
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likely under the other theory. The shift in credences might not be large, but
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it will always be there. As a result, the fact that you can come up with an
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explanation for some event within some theory doesn’t mean that event
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doesn’t lower the credence you have for the theory. The converse is also true:
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if some observation would have favored one theory, but we obtained the
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opposite of that observation, that result necessarily decreases our credence
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for the theory.
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Consider two theories: theism (God exists) and atheism (God doesn’t
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exist). And imagine we lived in a world where the religious texts from dif-
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ferent societies across the globe and throughout history were all perfectly
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compatible with one another— they all told essentially the same stories and
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promulgated consistent doctrine, even though there was no way for the
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authors of those texts to have ever communicated.
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Everyone would, sensibly, count that as evidence in favor of theism. You
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could cook up some convoluted explanation for the widespread consistency
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even under atheism: maybe there is a universal drive toward telling certain
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kinds of stories, implanted in us by our evolutionary history. But we can’t
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deny that theism provides a more straightforward explanation: God spread
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his word to many different sets of people.
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If that’s true, it follows as a matter of inescapable logic that the ab-
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sence of consistency across sacred texts counts as evidence against theism.
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If data D would increase our credence in theory X, then not- D necessar-12
ily decreases it. It might not be hard to explain such inconsistency, even if
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theism is true: maybe God plays favorites, or not everyone was listening
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very carefully. That is part of estimating our likelihoods, but it doesn’t
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change the qualitative result. In an honest accounting, the credence we
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assign to a theory should go down every time we make observations that
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are more probable in competing theories. The shift might be small, but it
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is there.
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All evidence matters. It’s not hard to pretend we’re being good Bayes-
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ians while we’re actually cooking the books by looking at some evidence but
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not all of it.
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Let’s say a friend tells you that they believe in the Loch Ness Monster.
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There are pictures, they say, and they provide good evidence. Surely, you
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must admit, the likelihood of such pictures being taken is larger under the
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theory that Nessie is real than under the theory that she isn’t.
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True, but that’s far from the whole story. First, your prior for a monster
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living in a remote Scottish lake should be pretty small. Even then, if the
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evidence were sufficiently compelling, you should change your mind. But a
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few grainy pictures aren’t all the evidence. We should also take into account
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all of the searches in the loch that tried to find a monster and came up
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empty. Not to mention the evidence that the original famous photograph
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of Nessie was eventually admitted to be a hoax. We can’t pick and choose
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which evidence we want to consider; everything relevant should be brought
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to bear.
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Bayes’s Theorem is one of those insights that can change the way we go
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through life. Each of us comes equipped with a rich variety of beliefs, for or
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against all sorts of propositions. Bayes teaches us (1) never to assign perfect
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certainty to any such belief; (2) always to be prepared to update our cre-
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dences when new evidence comes along; and (3) how exactly such evidence
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alters the credences we assign. It’s a road map for coming closer and closer
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to the truth.
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Is It Okay to Doubt Everything?
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ludwig Wittgenstein, one of the greatest philosophers of the twenti-
eth century, began his doctoral studies at Cambridge as a student of
Bertrand Russell, a massively influential thinker in his own right.
Russell liked to tell the story of how a young Wittgenstein would deny that
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anything empirical— an assertion about the real world, rather than a logical
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provable statement— was truly knowable. In his relatively small quarters at
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Cambridge, Russell challenged Wittgenstein to admit that there was not a
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rhinoceros in the room. Wittgenstein refused. “My German engineer, I
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think, is a fool,” Russell wrote in a letter, though he later changed his mind.
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(Wittgenstein was Austrian, not German, and certainly no fool.)
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It’s an old parlor game among philosophers, seeing who can be the best
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at doubting seemingly obvious truths about the world. Skepticism, in the
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sense of doubting anything, was a popular school of thought in ancient
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Greece. The champions were the Pyrrhonists, followers of Pyrrho of Elis,
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who insisted that we couldn’t even be sure about the fact that we can never
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be sure about anything.
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A more recent contestant in the game was the seventeenth- century
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thinker René Descartes. He was not only a philosopher but also a mathe-
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matician and scientist, laying the foundations for analytic geometry and
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contributing to early work in mechanics and optics. If you have ever drawn
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x and y axes on a piece of graph paper, your life has been affected by René 35S
Descartes; he invented that little trick, which we now call “Cartesian coor-
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dinates.” In his philosophizing, Descartes was very influenced by the
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practice of mathematics. In particular, he was enchanted by the fact that in
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math we can prove statements beyond any doubt— at least, once we accept
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the relevant postulates.
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René Descartes, philosopher, mathematician, and
doubter of many things other than his own existence,
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1596– 1650. (Painting after Frans Hals)
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In 1641, Descartes published his celebrated Meditations on First Philos-
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ophy. To this day it is one of the books most likely to be assigned to college 26
students taking their first philosophy course. In Meditations, Descartes at-
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tempts to be as skeptical as possible about our knowledge of the world. You
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might think, for example, that you are sitting on a chair, and that the exis-
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tence of that chair is beyond dispute. But is it really? After all, you’ve un-
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doubtedly been quite sure about this or that belief in the past, and turned
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out to be wrong. When we are dreaming or hallucinating, there’s no ques-
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tion that we are “experiencing” things that aren’t actually happening. It’s
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possible, Descartes suggests, that we are dreaming even now, or that our
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senses are being tricked by an evil demon, one who (for whatever inscruta-
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But not to lose hope. Descartes concludes that there is one belief about
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which skepticism is impossible: his own existence. Sure, he reasons, we can
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doubt the existence of the sky and the Earth— our senses could be fooled.
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But he can’t be skeptical about himself; if he didn’t exist, who was it who was
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being skeptical? Descartes summarized this view in his famous cogito ergo
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sum: I think, therefore I am. (He first wrote that Latin phrase in the later
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work Principles of Philosophy, but the French formulation je pense, donc je suis 08
appears in the earlier Discourse on Method, aimed at a broader audience.)
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It would be an unsatisfying, solipsistic existence if each person could be
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convinced only that they themselves existed, and had to reserve judgment
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about everyone else. Descartes wants to build a foundation for justified
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belief about the whole world, not just himself. But he’s not allowed to ap-
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peal to anything he sees or experiences— after all, even if he himself exists,
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that evil demon could still be tricking him when it comes to the evidence
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of his senses.
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So as Descartes’s meditations continue, he realizes that he can salvage
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the reality of the world without ever leaving the comfort of his armchair.
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Not only do I think, he says to himself, but I can hold in my mind an idea
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of perfection— a clear and distinct idea, as a matter of fact. This idea, as well
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as my own existence, must have some cause, and the only possible cause is
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God. Indeed, God is himself perfect, and the property of “existing” is a
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necessary aspect of perfection— it is more perfect to exist than to not exist.
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Therefore, God exists.
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And then we are off to the races. If we are confident not only in our own
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existence but also in God’s, then we can be confident in much more than
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that. After all, God is perfect, and a perfect being wouldn’t allow me to be
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utterly deceived in everything I see and hear. God can overrule any tricky
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demons that might be trying to mislead me. So the evidence of my senses,
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and the objective reality of the world,
can largely be trusted. Now we can
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start doing science, secure in the knowledge that we are discovering truths
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about the universe.
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Descartes was a Catholic, and thought of himself as defending his reli-
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gious beliefs against the nagging doubts of skepticism. Not everyone else
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saw it that way. His proofs for the existence of God were perceived as blood-
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less and philosophical, divorced from the intense spiritual experience of
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lived faith. He was accused of atheism, which for most of recorded history
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was a way of saying “You don’t believe in God the way you are supposed to.”
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(Atheism was one of the crimes for which Socrates was sentenced to death,
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even though he talked about gods all the time. Meletus, one of his adversar-
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ies, ended up accusing him both of atheism and of belief in demigods.)
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Eventually, in 1663, Pope Alexander VII would place all of Descartes’s
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works on the Church’s Index Librorum Prohibitorum, the list of officially
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prohibited writings, where it joined books by Copernicus, Kepler, Bruno,
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and Galileo, among others.
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One of my college professors once told me that nobody could get a PhD in
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philosophy without writing a refutation of Descartes. It remained unclear
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which part of Descartes was supposed to be refuted— his initial skepticism
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and ability to doubt everything, or his laying foundations for secure belief
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through his conviction that both he and God certainly existed?
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Opinions on the existence of God, and in particular on Descartes’s pur-
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ported proofs, vary widely. But before even getting to that part of the argu-
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ment, most people feel a visceral reaction against “Cartesian doubt.” It
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strikes us as ridiculous and irritating to imagine that we can’t be sure of
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anything at all, not even the existence of the chair on which we are sitting.
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But in that part of his method, Descartes was completely correct. We
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may be quite convinced that the world around us is real, but we can’t be
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absolutely certain, beyond any conceivable doubt. We can even come up
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with a number of scenarios under which we could be fooled, beyond Des-
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cartes’s suggestions that we might be dreaming or being fooled by an evil
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demon. We could be a brain in a vat, receiving false impulses from wires