The Discovery of Mathematical Probability Theory: A Case Study in the Logic of Mathematical Inquiry

ing from their actual experience with games of chance, a collection of possible outcomes of aleatory trials. This does not mean that the collection or complete enumeration of possible outcomes was directly perceived; it rather means that these inquirers were able to imagine the aleatory situation of dice-throwing, for example, in such a careful, attentive way so as to seize upon the complete enumeration of possible outcomes as the ‘mathematical form’ worthy of mathematical study, while disregarding those aspects of the ‘matter’ of the situation that were not of mathematical interest. 204 Among such mathematically unessential aspects of the ‘matter’ of dice-throwing to be disregarded we might include, say, the material of which the dice are made, the color of the dice, who the players are (so long as the game does not involve skill or skill can be assumed equal), whether the throw is rightor left-handed, and so on. I am suggesting, then, that the early mathematical probabilists discovered the fundamental probability set not by abstraction from a direct percept but rather by abstraction from an image. That is, they first imagined an aleatory situation under ideal conditions: perfect dice, equally skilled players, and so on. Then they abstracted from this image the conception of an enumerated collection of possible outcomes. This means that they transformed the possible individual imaginary outcomes of the idealized game into an abstract object, an enumeration or collection which, later in the history of mathematics, came to be conceived as a set. It is not important right now that the initial enumerations may have been incorrect for the purposes of estimating chances insofar as they did not distinguish between partitions and permutations. This distinction would be the later result of a more precise determination of the fundamental probability set in the context of the ‘analysis’ of problems in the calculus of chances, as we can see in Galileo’s ‘analytic’ work. The crucial point now is that the fundamental probability set was initially conceived, even if imprecisely, by way of abstraction from an idealized situation. Admittedly, this original image of a game under ideal conditions was the result of abstracting from actual experience with games. I would like to call this operation of abstracting an idea from actual perceptual experience an ‘idealizing abstraction’. Clearlying from actual experience with games. I would like to call this operation of abstracting an idea from actual perceptual experience an ‘idealizing abstraction’. Clearly the Peircean stance admits that ideas may arise by abstraction from perceptual 205 experience. But this does not mean that all ideas arise by idealizing abstraction from perceptual experience. Our first ideas, once conceived on the basis of experience, spur a continuous train of thought in the mind that, animated by the imagination, has a life of its own. Ideas engender ideas through the work of the imagination. Sometimes this imaginative work proceeds by way of abstraction. In the conception of a fundamental probability set, the imaginative work—literally, the making of a mathematical image or ‘sign’—consisted in abstracting the mathematical form from the image of idealized aleatory conditions. Once the concept of a fundamental probability set was abstracted, the course of mathematical thought upon this idea took a life of its own in the minds of a community of inquirers. 5.1. 2 Framing Analogy Analogy is an important method for framing mathematical hypotheses. In a very general sense, analogy is the inference that if two things agree with each other in one or more respects, they will agree in other respects as well. The nature of the agreement between the analogous things—whether individual objects, an object and its sign, relations between objects, systems of relations, and so on—may be one of strict correspondence or of some degree of resemblance. If the agreement is a strict one-to-one correspondence between (i) all the elements and (ii) the patters of relations among the elements that constitute two different things, the analogy is an isomorphism. If the number of elements is not the same, but the structural patterns of relation between the 206 elements of the two things are the same, the analogy is a homomorphism. Less strictly, an analogy between two things may be a relation of stronger or weaker resemblance. Peirce considered analogy to be a fourth type of inference, a mixed case between induction and abduction. In a relatively early effort at discussing the various forms of ampliative inference, Peirce writes that “among probable inferences of mixed character, there are many forms of great importance. The most interesting, perhaps, is the argument from Analogy in which, from a few instances of objects agreeing in a few well-defined respects, inference is made that another object, known to agree with the others in all but one of those respects, agrees in that respect also” (CP 2.787). In this case, we might understand the analogy to be between a sample of completely examined objects and another partially examined object. I think that the Peircean view that analogy is a mixed case between induction and hypothesis or abduction may be elaborated as follows. In an analogy, there is an enumeration of the aspects in which two things correspond to, or at least resemble, each other. This sort of enumerative evidence is usually inductive—it consists in a sample of observations about particulars. However, the inference is not from particular instances to a probable general rule; it is rather a conclusion to another particular aspect in which the analogous things plausibly agree. No definite probability is attached to the conclusion of the inference by analogy. The inference only suggests what may plausibly be the case about one of the things under examination on the basis of its 120 However, later in the development of Peirce’s doctrines of logic, he seems to have come to classify analogy as a species of abduction. The question of what is the form of an analogy according to Peirce and how it ought to be classified is a subject that still requires a thorough investigation of its own. Here I will only try to discuss it in sufficient detail for my present purposes, though I openly admit that my interpretation of analogy may be considered as an incipient attempt at an elaboration of the Peircean model rather than as a faithful account of Peirce’s own explicit views. 207 partial agreement with the other, and this is similar to a ‘habitual’ abductive conclusion. In analogy, therefore, no general rule is inferred, as it is in ‘creative’ abduction. Analogy, remains at the level of knowledge of particulars, and so it is not an inference that leads to generalization. It does however tend to extend our knowledge—in the Peircean sense of ‘extension’ versus ‘generalization’—by concluding that an object may plausibly belong to a known class of objects. With specific regard to necessary diagrammatic reasoning, Peirce writes that “deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts” (CP 3.363; emphasis mine). Therefore, here Peirce claims that the mathematician constructs an icon or diagram for study by way analogy, where we might take a “complete analogy” to be an isomorphism, or at least a homomorphism, between the thing represented and the diagrammatic representation of the thing. In the larger context of his thought, I think we should take Peirce’s position to be that mathematical ‘diagrams’ are often constructed by analogy, though mathematicians have other methods of construction depending on what the specific reasoning context requires. Be that as it may, it is precisely as an extensive inference—that is, as an inference that adds “breadth” rather than “depth” to our knowledge—that Bernoulli uses an analogy to justify, in one of a series of arguments, the application of mathematical probability to the study of natural phenomena. As I discussed in section 4.1. 1 , in his 1703-1704 correspondence with Leibniz, Bernoulli used an analogy between an urn filled with 208 pebbles and a human body containing sicknesses or diseased parts to warrant the application of the probability calculus to the study of natural events, such as diseases and storms. Bernoulli offers the analogy in response to his correspondent’s objection that the possible outcomes of natural events, unlike those of games of chance, are infinite—as Hacking points out, an objection that in contemporary terms amounts to claiming that there is no fundamental probability set for natural events (Hacking 1975, p. 163-164). Let me now expound Bernoulli’s inference by analogy. The initial step is to enumerate several characteristics with respect to which the urn and the human body “agree” or are analogous. Bernoulli lists two such characteristics in this case. Accordingly, the first premise of the analogy is that just as the urn contains white or black balls in a ratio that is unknown to us, so also the human body contains “the tinder of sicknesses within itself” (Bernoulli 1966, p. 76). For simplicity, we might interpret Bernoulli to mean that the human body contains healthy—analogous to white—and sick—analogous to black—parts in a ratio that is unknown to us. Bernoulli assumes that the ratio of sick to healthy parts provides an indication of the probability of death within a given time period. It is likely that Bernoulli conceived of this tinder box of sicknesses as containing various types of sicknesses, such as dropsy and plague, which he uses in other examples and which were commonly listed in mortality tables such as those gathered by John Graunt’s 1662 Bills of Mortality. This