Some Mathematical Facts about Peirce's Game

Stephen Pollard Some Mathematical Facts about Peirce's Game Carolyn Eisele observed that, "Mathematics was so basic an ingredient of Peirce's tighdy integrated intellectual life that every area of his thought reflected his awareness of mathematical accomplishments in ages past as well as in the then current literature" (Eisele 1979, pp. 295-296). Peirce himself once complained to E. H. Moore that, "To attempt to make myself understood by anybody but a mathematician would be futile."1 And he offered the following advice to wouldbe philosophers: "Mathematicians always have been the very best reasoners in the world; while metaphysicians always have been the very worst. Therein is reason enough why students of philosophy should not neglect mathematics."2 I take it for granted that Peirce's philosophical work was mathematically informed and that his intellectual gifts naturally disposed him to think like a mathematician. So when he offers an example that cries out to be analyzed mathematically, I assume that this example was chosen, at least in part, because of its mathematical tractability. I also assume that Peirce recognized and considered significant at least some of its mathematical properties. Such an example is the game of chance that Peirce uses to model certain features of biological evolution. Wherever there are large numbers of objects having a tendency to retain certain characters unaltered, this tendency, however, not being absolute but giving room for chance variations, then, if the amount of variation is absolutely limited in certain directions by the destruction of everything which reaches those limits, there will be a gradual tendency to change in directions of departure from them. Thus, if a million players sit down to bet at an even game, since one after another will get ruined, the average wealth of those who remain will perpetually increase. Here is indubitably a genuine formula of possible evolution, whether its operation accounts for much or litde in the development of animal and vegetable species.3 Transactions of the Charles S. Peirce Society Winter, 2005, Vol. XLI, No. 1 190 Stephen Pollard I devote most of the following pages to an exposition of some key mathematical properties of Peirce's game. I do so because it seems so highly probable that an improved grasp of these mathematical features will help us better understand the philosophical significance of the game. Though I do offer a few suggestions about how the mathematics might relate to some of Peirce's philosophical hypotheses, I leave most of the philosophical work for scholars whose insight into Peirce goes much deeper than my own. I. Some Combinatorics Suppose you are standing on a line of stepping stones extending infinitely far behind and in front of you. And suppose you decide to take a journey of 4 steps: 2 forward, 2 back (not necessarily in that order). Since you will be going backward exactiy as often as you go forward, you are sure to end up where you started. But how many ways are there of doing this? The answer is 6: Stepl Step 2 Step 3 Step 4 Journey 1 Forward Forward Back Back Journey 2 Forward Back Forward Back Journey 3 Forward Back Back Forward Journey 4 Back Forward Forward Back Journey 5 Back Forward Back Forward Journey 6 Back Back Forward Forward We characterize a journey by labeling each of its steps either "forward" or "back". Indeed, since forward and back are the only options here, we only need to identify the forward steps. So the number of journeys with 2 forward steps and 4 total steps is just the number of ways of picking 2 of 4 things. This number is given by the binomial coefficient function "n choose k": ( ν, λ η In this case, we have: J k\-(n - k)\ f4l 4! 4-3-2 2!-(4 - 2) ! 2-2 = 6 The same technique applies to every case in the situation we are now imagining: the number of journeys with k forward steps and η total steps is just the number of ways of picking kofn things; that is, (fif~ U Some Mathematical Facts about Peirce's Game 191 We now consider a somewhat different situation. Suppose you are standing at the beginning of a...