Marginalia in Revd Bayes' copy of Dr Wittgenstein's On Certainty
It is rare enough to find an important manuscript from the eighteenth century of which we have had no inkling before. It is rarer still to find one that provides a commentary on a work from two hundred years later. Yet this indeed is what Sotheby’s listed as Lot 6 in this week’s auction of rare books from ‘a distinguished private library’.
Ludwig Wittgenstein, 1889-1951 | On Certainty (ed. G. E. M. Anscombe and G. H. von Wright; trans. Denis Paul and G. E. M. Anscombe), 1951, with marginal notes by the Reverend Thomas Bayes, 1701-1761.
So reads the title of the listing in Sotheby’s catalogue. The marginal notes in question centre around §§337-341, though there is also material about §§95-97.
Reverend Bayes is not the first to read Dr Wittgenstein’s work through the lens of the epistemological theory that he founded. Luca Moretti & Crispin Wright and Olav Benjamin Vassend have also tried to wrest Wittgenstein’s unpolished notes into the Bayesian framework. Yet Bayes is more interested in correcting Wittgenstein than are these earlier authors; he would like to persuade the philosopher of his way of thinking about matters of belief.
Here follows a transcript of the original text (LW; block quote) with the marginal notes (TB; main text).
337. One cannot make experiments if there are not some things that one does not doubt. But that does not mean that one takes certain propositions on trust. When I write a letter and post it, I take it for granted that it will arrive—I expect this.
If I make an experiment I do not doubt the existence of the apparatus before my eyes. I have plenty of doubts, but not that. If I do a calculation I believe, without any doubts, that the figures on the paper aren’t switching of their own accord, and I also trust my memory the whole time, and trust it without any reservation. The certainty here is the same as that of my never having been on the moon.
What is this state of doubting of which he writes? A lack of certainty, a failure to assign maximal credence, so that to doubt something is merely not to be completely certain of it? Or more like a middling credence, so that to doubt something is to think it as likely as not, or something close to that? And yet neither seems to support the first sentence, nor his expansion on it in the second paragraph. I might have a middling credence that my apparatus exists and still perform an experiment using it, though I will likely end up learning less than if I were more confident that my apparatus exists. Why? Let H be a hypothesis I wish to test, E the proposition that it seems to me that my apparatus gives a particular reading, and K the claim that my apparatus exists. Then perhaps P(H | E & K) is quite high: supposing the apparatus exists, its seeming to give the reading in question lends much credence to the hypothesis. And yet P(H | E & not-K) = P(H): supposing the apparatus does not exist, its seeming to give a particular reading is irrelevant to the hypothesis. Then, by the sorts of laws of probability I took my generation to bequeath to posterity,
And, by the rule that governs how I should update my credences upon receipt of evidence, and that happily bears my name, Bayes’ Rule, my credence in H having learned E should be P(H | E). And so, putting these together, my new credence in H should be the weighted average of the high credence in H conditional on E and K, weighted by the prior credence in K, and the prior credence in H, weighted by the prior credence in not-K. Thus, the higher my prior credence is that the apparatus exists, the greater weight is given to the high credence in H conditional on E & K. So the experiment shifts my credences further towards that high credence the more confident I am that my apparatus exists; but, however it is set, it shifts my confidence somewhat; and so does not preclude experimentation in the way W. suggests.
And what goes for experimentation goes equally for the everyday evidence we receive that we take to tell us about the world. W. might imagine that, in order to do anything with the evidence of my senses concerning the location my quill, the capacity of my inkwell, the coarseness of the paper on which I write, there must be something I do not doubt; some proposition of which I must be doubtless that provides the basis for my response to this evidence. And yet the same point holds as held above: this time, let K be the proposition that the external world exists and my senses provide largely reliable information about medium-sized dry goods in my vicinity; let E say that it seems to me my quill is in front of me; and let H say my quill is indeed in front of me. Then, as before, it is most likely the evidence E will alter my credence in H, though how much will depend on how strongly I believe K. But I surely need not be certain of K.
338. But imagine people who were never quite certain of these things, but said that they were very probably so, and that it did not pay to doubt them. Such a person, then, would say in my situation: “It is extremely unlikely that I have ever been on the moon”, etc., etc. How would the life of these people differ from ours? For there are people who say that it is merely extremely probable that water over a fire will boil and not freeze, and that therefore strictly speaking what we consider impossible is only improbable. What difference does this make in their lives? Isn’t it just that they talk rather more about certain things than the rest of us.
Ah, so he is aware of my view of belief! And yet he thinks it makes no difference to his point? Yet I argued previously it does.
341. That is to say, the questions that we raise and our doubts depend on the fact that some propositions are exempt from doubt, are as it were hinges on which those turn.
342. That is to say, it belongs to the logic of our scientific investigations that certain things are in deed not doubted.
So it is not only that we cannot experiment without placing some things beyond the reach of doubt, but we cannot question either, nor even entertain doubt itself. I imagine he must think that to question is automatically to say what evidence one would take to bear on the question and how much, and that leads us back to the question of whether we must put something beyond doubt in order to say whether evidence bears on a question. But, as we saw above, we do not need to do that.
343. But it isn’t that the situation is like this: We just can’t investigate everything, and for that reason we are forced to rest content with assumption. If I want the door to turn, the hinges must stay put.
344. My life consists in my being content to accept many things.
With this last sentence, perhaps I start to glimpse his meaning, for I too think that we are content to accept many things, and that such sustained acceptance, at least over reasonable stretches of time, is part of what it is to live my life. But these things we accept are not propositions, and accepting them is not believing—that is the source of W.’s mistake; he is in the grip of a picture of belief that leads him astray. Rather these things we accept are what we might call ur-priors. They are the credences with which we begin our epistemic lives—or, better, since there is no sensible time to mark as the beginning of my epistemic life, they are built up gradually as I become aware of propositions and possiblities, and start to assign degrees of confidence to them. I do not base these initial credences on prior attitudes I have, for there are none there to guide me. An aside: This is a mistake I fear W. would have made even if he had considered more carefully my epistemology of credences and their evolution. I predict he would have imagined that I set these ur-priors by assigning certainty to some very general proposition—The external world exists and my senses are largely reliable guides to its nature, for instance—and then deriving the other unconditional and conditional credences from that via the laws of probability and other constraints that rationality places on them—I set a high conditional credence in a proposition conditional on it seeming to my senses that the proposition is true, for instance. But that’s not the case. Of course, I might assign high credence to such a general proposition and base my other credences on that; but equally I might not entertain such a proposition until a much later stage of my epistemic development, and in the meantime, I must still have ur-priors, conditional and unconditional, that set the course for the evolution of my credences in response to the evidence I receive. There are many different ways that rationality permits us to set these ur-prior credences, and I must select one amongst them to inhabit. And, once they are set, they provide the credences that evolve into other credences later, once I start to accumulate evidence; and they evolve in line with the rule that bears my name.
But what does it mean to accept these ur-priors? It is not a further, higher-order attitude I take to those credences. It is rather that those ur-priors encode my view of the world prior to obtaining evidence; I inhabit those ur-priors and use them to guide my new credences as evidence comes in. Indeed, it is misleading to say that I hold any particular attitude like acceptance to them, for I do not stand back and consider them and judge them positively in any way. Rather, I simply have them, inhabit them, and approach the world using them—my life consists in accepting these ur-priors.
Once we see it is these ur-priors that are the hinges, not propositions, we can understand more of W.’s claims. It is true that ur-priors are not questioned in an experiment, just as he takes hinges to remain unquestioned. Indeed, they provide the basis for your new degrees of confidence after the evidence from the experiment comes in.
This helps me to make sense of an earlier passage, based on a metaphor I didn’t understand at the time. I will annotate it there.
95. The propositions describing this world-picture might be part of a kind of mythology. And their role is like that of rules of a game; and the game can be learned purely practically, without learning any explicit rules.
96. It might be imagined that some propositions, of the form of empirical propositions, were hardened and functioned as channels for such empirical propositions as were not hardened but fluid; and that this relation altered with time, in that fluid propositions hardened, and hard ones became fluid.
97. The mythology may change back into a state of flux, the river-bed of thoughts may shift. But I distinguish between the movement of the waters on the river-bed and the shift of the bed itself; though there is not a sharp division of the one from the other.
The river-bed is my ur-prior, I would say, and, via my eponymous rule, it functions as a channel that directs our credences about the empirical propositions, just as W. says. That is, it sets the way that the credences in those propositions will change as I encounter new evidence. And yet, as W. says, “the river-bed of thoughts may shift”. That is, the ur-prior itself can change.
But how? Not in response to any evidence, surely, for my ur-prior includes my commitments to respond to future evidence in a particular way; receiving evidence cannot undermine it, for it anticipates each possible piece of evidence I might receive and provides a plan for how I should respond if I do. And yet other events can undermine it. We might call such events doxastic crises. They lead me to step back from the ur-priors that I inhabit and ask whether I would like to pick other ones, which I would then update on the evidence I’ve acquired so far in order to set my new credences.
What might be such doxastic crises? Havi Carel points out that becoming ill can have such an effect, leading the ill person to step outside the view of the world they’ve previously inhabited, and asking whether they would wish to retain it. A more mundane example occurs when I become aware of some new propositions to which I haven’t hitherto assigned a credence. Since the way we assign ur-priors is often sensitive to the propositions we entertain at the time we set them, becoming aware of some new propositions can lead me to realise I would have set the ur-priors differently had I been aware of these new propositions earlier. This seems to be what happens when the sceptic to which W.’s friend Moore is responding raises their sceptical concerns. I become aware of the proposition that the external world exists and my senses reliably detect its feature, and thereby become aware of the possibility that it might not exist, and the possibility that it does exist but my senses are systematically mistaken about it, and that growth in my awareness leads me to question whether I would have set my ur-priors differently had I been aware of that possibility when I set those ur-priors. More generally, realising that there are many ways I might have set my ur-priors that are different from the way I did can call the ur-priors into question, and my response might be to pick different ones, leading the river-bed of my thoughts to shift.
Nonetheless, it is odd to say that these doxastic crises ‘call the ur-priors into question’, as I have just done. Of course, it might be that we realise our ur-priors are irrational in some way: too extreme, if we think there is such a thing; incoherent; some other flaw. But, equally, we might call them into question even when they are fully rational. When we do so, we don’t come to think they are wrong in any sense; rather, we come to realise that there are many others that we might have had that, from the point of view of rationality, are equally acceptable.
But why is this a crisis? Usually, it is concerning to learn that I’m doing something wrong, but not worrying to learn that there are many other ways I might be doing things right. But the crisis arises because, before it comes, I inhabit these ur-priors; and that involves committing to them; and that in turn involves an implicit rejection of alternatives. Such rejection is reasonable from the point of view I inhabit, bequeathed to me by my ur-prior and my evidence, and so it is stable while I inhabit it. If you start with a different ur-prior, receives the same evidence that I do, and thereby come to have a low credence in a proposition to which I assign a high credence, I can say why you’re wrong from the point of view of my credences: I expect your credence to be less accurate than mine; I expect it to guide your decisions less well than my credences would. The crisis comes when I am shaken into no longer inhabiting my credences. From that point of view, I can no longer say that my old credences were right and yours wrong.