of the bat” and the “blaze of day” are the phoros because they represent a concrete relationship between knowable entities that the writer supposes the reader understands (Metaphysics, II: 933b, 10–11).11 This concrete relationship is used to guide the reader in comprehending the abstract relationship in the theme between the “reason in the soul” and “things which are by nature most evident of all” (Perelman Olbrechts-Tyteca 373).
Based on Whewell’s description of the process, the creation of a formula to express a particular change in a phenomenon can be construed as an argument for an analogy between reason and experience. The phoros—the suggested description of the change, warranted by the well-known axioms of mathematics—comes from the domain of reason. The theme—the perceived but vaguely understood change in the natural phenomenon12—is derived from a domain of experience that has been “translated” into a quantitative description to permit comparison. The final formula is the epitomized analogy, the proposed conclusion that the mathematical arrangement is a legitimate descriptor of a change, or the relationship between changes in a group of phenomena.
The benefit of making an analogy between quantified observations of nature and a mathematical formula whose components are related by strictly defined operations is that the result allows experience to be cast into a form that could be reasoned about clearly and rigorously. Because mathematical argument was governed by the established principles of logic at this time, conclusions reached through its use were considered credible if supported by sufficient evidence. Once verified, these conclusions could be used as axioms for making deductive arguments. In A Preliminary Discourse on the Study of Natural Philosophy, Herschel recognizes the rigor that mathematical form brings to arguments about nature:
Acquaintance with abstract [mathematical] science may be regarded as highly desirable in general education, if not indispensably necessary, to impress on us the distinction between strict and vague reasoning, to show us what demonstration really is, and to give us thereby a full and intimate sense of the nature and strength of the evidence on which our knowledge of the actual system of nature, and the laws of natural phenomena, rests. (22)
According to Herschel’s admonition here, for an argument to be considered sufficiently robust to be “scientific,” it had to be made mathematically. This position reflects a consensus in nineteenth-century science that mathematical argument was the gold standard for making claims about natural phenomena. Given this sentiment, and the new self-consciousness engendered by works like Herschel’s and Whewell’s, there was a drive in all areas of natural investigation—even those in which there was no tradition of mathematization—to develop or use existing mathematical arguments to describe the changes and relationships between changes in natural phenomena (Cannon, Science in Culture 234–35).
Step Three: Verification
Once a formula is proposed, the next step in induction is to test the validity and limitations of the analogy by increasing the number of observations, and varying the conditions under which the data is gathered. This step is crucial when using mathematical arguments because it ensures that the necessary balance between the conceptual and the empirical is maintained.
The connection between the strength of conclusions and the number of trials/observations made to verify those conclusions was articulated at the beginning of the eighteenth century by Jakob Bernoulli in Ars Conjectandi (The Art of Conjecture) (1713). In the book, Bernoulli describes his famous “limit theorem,” which states that the calculated a posteriori probability of an event (p) gets closer to the true a priori probability of an event (P) the greater the number of trials (n) that are conducted (Chatterjee 168).
Both Herschel and Whewell were generally acquainted with mathematical probability, as evidenced in their discussions of the method of curves as a way of identifying the “true value” in a set of observations (Discourse 130, 217–19; Philosophy 2: 398–400). They also seem to have been aware of Bernoulli’s limit theorem for certifying the verity of the quantitative data and thereby the validity of the laws describing the relationship in the data. Whewell, for example, appeals to Bernoulli’s principle when he writes: “In order to obtain very great accuracy, very large masses of observations are often employed by philosophers, the accuracy of the results increases with the multitude of observations” (Philosophy 2: 406).
While expanding the number of observations tests the verity of a mathematical analogy, increasing the variety of conditions under which trials are conducted helps determine its scope. Herschel advocates for both methods of verification, explaining that precise testing of quantitative hypotheses across a variety of circumstances can expose deviations in the data that might limit the analogy’s scope or challenge its credibility:
In the verification of a law whose expression is quantitative, not only must its generality be established by the trial of it in as various circumstances as possible, but every trial must be one of precise measurement. And in such cases the means taken for subjecting it to trial ought to be so devised as to repeat and multiply a great number of times any deviation (if any exists); so that, let it be ever so small, it shall at least become sensible. (Discourse168)
Whewell’s and Herschel’s discussion of the process of testing empirical laws reveals two obvious objections that might be brought against nineteenth-century researchers trying to establish conclusions using mathematical analogies. The objections of not doing a sufficient number of experiments, and not doing them under a sufficiently wide range of conditions, though not necessarily fatal to a particular argument, could force the arguer-scientist back into the field or laboratory to make further observations and experiments, or could require him to defend the breadth and depth of his empirical work. To support their claims, natural investigators could either remind readers about the scope or number of observations they undertook or limit their claims to the extent that they matched the level of proof their audience believed could be verified by the extent of the empirical evidence supplied.
Step Four: Extrapolation
Once a mathematical formula has been sufficiently tested to be considered a reliable analogy within a specific set of parameters, its argument status is changed. Instead of being the ends of the argument, it becomes the means. Herschel describes this transformation when he writes,
These [empirical laws of nature], once discovered, place in our power the explanation of all particular facts, and become grounds of reasoning, independent of particular trial: thus playing the same part in natural philosophy that axioms do in geometry; containing . . . all that our reason has occasion to draw from experience to enable it to follow out the truths of physics by the mere application of logical argument. (Discourse 95)
The transformation from an argument for an analogy to an argument from an analogy is the result of the collapse of the phoros and the theme. This process is described by Perelman and Olbrechts-Tyteca, who write:
Analogy finds a place in science, where it serves rather as a means of invention than as a means of proof. If the analogy is a fruitful one, theme and phoros are transformed into examples or illustrations of a more general law, and by their relation to this law there is a unification of the fields of the theme and the phoros. This unification of fields leads to the inclusion of the relation uniting the terms of the phoros and of the relation uniting the terms of the theme in a single category, and, with respect to this category, the two relations become interchangeable. There is no longer an asymmetry between theme and phoros. (396)
According to Perelman and Olbrechts-Tyteca, the process of validation, when successful, pushes the phoros and the theme, and the reason and experience, together to the point where any asymmetry between the two is lost. With this transformation, however, the question remains: “Is the formula still the epitome of an analogy?” Although Perelman and Olbrechts-Tyteca comprehensively describe the decomposition of analogy, they offer no comment on whether analogy, once it has gone through this process of decomposition, is still an analogy or something altogether different. If the
