The following conversation is fictional, but I attempt to represent each speaker’s views according to points made in John Maynard Keynes, A Treatise on Probability (USA: Rough Draft Printers, 2008). In Keynes’s closing soliloquy, I put a lot of words in his mouth, but I find my interpretation defensible from ibid. pages 254-260. I submit this post to solicit feedback about the accuracy of my interpretation of Keynes in that closing soliloquy.
Hume [♠A, ♣A]: I always win with this hand.
Laplace [♥4, ♠7]: “Always” is an eternal word; he’s lying. Raise ten.
Hume [♠A, ♣A]: No, I’m serious. Table-talk aside, I wish I could account for my repeated success with just this hand, when the table has just this many players. I even win on video poker, so I cannot blame my dark shades and camo. Tonight is my 7th drawing of the hand this month, and I won the last 6 of them . . . and yet [sips sour blue vodka through a clear straw] I lack justification for my expectation that I will also win this 7th hand.
Keynes [♠K, ♣K]: Raise twenty. But hold on, David. If they’re good cards, then a high correlation of draws to wins should not surprise anyone; for opponents can play strong cards against you, commit no mistakes, and still lose to your hand of higher value.
Venn [♥9, ♦9]: But Maynard, you know as well as I that for the size of tables Hume plays, only a few combinations of pocket cards boast anywhere near a coin-flip’s chance of victory when abstracted from the psychology of betting. The difference in numerical odds between strong and mediocre pocket draws, moreover, never renders a win-streak of 6 likely. David identifies a deeper problem than that of numerical odds: he is trying to justify his recourse to past experience as the ground of his assumption that he will repeatedly beat these kinds of amateurs, at this size table, with his favored draw. The victories incumbent on his favored draw seem now as predictable to him as the observation of another sunrise or another white swan. He sounds like a braggart for it, but who can blame him? He’s asking what formula underwrites his confidence. We could try to construct a formula a posteriori from frequency sampling, but to justify induction for Hume we would have to follow him around the fair-dealing casinos, which would take years. I call.
Cartesian Demon [♣3, ♣10]: [Dips a broiled rabbit’s foot into a bowl of mustard] Hume’s problem has nothing to do with counting, guys. Have you considered how you will justify induction before you justify the reliability of your own memories? I didn’t think so. I call.
Keynes [♠K, ♣K]: I’ll answer your point in a minute, Demon, but assuming that we do possess justified memory, the rest of this table is barking up the wrong tree. From the details that Hume provides, his appeal to induction fails to convince if his 6 draws win 6 times, or his 6 million draws win 6 million times; for Hume emphasizes only what is similar about all the draws. We find ourselves unmoved that the same hand, at the same sized table, in the same casinos, continues to win. What motivate induction are the details that Hume omits: the number and variety of differences between the victory scenarios involving his favored draw. He seeks an inductive law because the trend persists despite the number and variety of opposing hands, of amateur plots, and of phases of the moon.
Laplace [♥4, ♠7]: What are you getting at, Maynard? [Glances at the demon] It sounds occult. The deal determines Hume’s hand, and all the deals are fair, and Hume’s expertise normalizes the psychological dynamics of the betting. He’s a veritable mind-reader, so he scares opponents into folding and reinforces his bravado with what are likely high-value cards. So is the inductive question: Will I continue to be a clever better from hand-to-hand? Will amateurs continue to insult the table by the size of their counter-raises? Will the dealer continue to deal fairly? You seem to target a broader induction, i. e. winning from the particular draw, and I don’t see how your emphasis on situational differences proves superior to a frequency analysis of Hume’s specific outcomes.
Hume [♠A, ♣A]: We don’t really have minds to read, by the way.
Venn [♥9, ♦9]: I completely agree with you, Pierre-Simon.
Keynes [♠K, ♣K]: Pierre-Simon, your last question provides an outstanding example to support my claim. In a case of ignorance about the fairness of the deal, do you opine that Mr. Hume enjoys an equal probability, from deal to deal, of drawing his favored pocket cards?
Laplace [♥4, ♠7]: You weary, me, Maynard. We clearly agree that a run of Hume’s favorite draws would cast suspicion on the fairness of the cards or the dealer.
Keynes [♠K, ♣K]: But what about the first draw of what turns out to be a run of draws?
Laplace [♥4, ♠7]: The likelihood of that draw, too, is obviously skewed.
Keynes [♠K, ♣K]: But you lack any justification to bias the probability of the first draw that way, because at the moment of the first draw, you possess no knowledge to offset the probability of a fair draw.
Demon [♣3, ♣10]: What does this have to do with the price of meat in China? Are you going to answer my memory challenge? Or have you already forgotten it? [Chortles, wheezing]
Keynes [♠K, ♣K]: Probability hinges on what we know, Demon, and in a similar vein, we justify induction by what we directly know. That some of our bodily movements proceed under the influence of our mind, for example, is indubitable. We see directly the probable relation between mind and action. Whether we attain such insight without memory is an interesting bugbear, but if we do, arguments that justify induction remain plausible. For our conclusion (C) that other people have minds is a priori probable, if we assume (H) that objects acquire their properties from a finite system of generators. Our direct observation (E) of our own mind-body interaction, moreover, increases the probability of C beyond its a priori value, via the relevance theorem. But in addition to this, and for the same reason, E also imparts an a priori probability to H. If H is a priori probable, then inductive inferences are valid. Validity here means not infallibility, but probability, and so generalizations probably indicate the properties to be exhibited by future instances of their objects.
Hume [♠A, ♣A]: I think your generalization may be a hasty one, Maynard, but we have to return to the game.[Hume pushes all-in. Everyone folds but Keynes. The flop reads: ♦K, ♠9, ♣9]
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