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 hallmark of an analogy is an asymmetry between its theme and phoros, then empirical laws in which the phoros and theme are conflated seems to be something different altogether.
Once a mathematical formula has made the transition from an analogy to a law or principle of nature, it can be used as a warrant for making further arguments about phenomena both related and unrelated to the original subject of the induction. In “Of the Application of Inductive Truths,” in Philosophy of the Inductive Sciences, Whewell offers astronomical tables as an example of how quantitative laws, once established inductively, are extended deductively to draw conclusions about subjects considered in the original induction, but not specifically used in the calculation of the laws: “Tables of great extent have been calculated, with immense labor, from each theory, showing the place which the theory assigned to the heavenly body at successive times; and thus, as it were, challenging nature to deny the truth of discovery” (2: 426). In Whewell’s example, he cites the laws of planetary motion, arrived at by observing a few heavenly bodies, and then extrapolated in tables to describe the motion of other like objects, as instances where deduction is applied to the same class of subjects that were considered in the original induction.
In other cases, an empirical law can be used to predict phenomena which were not the original subjects of the induction by which the law was established. Herschel cites Newton’s and others’ applications of the theory of gravitational attraction to deduce the anomalies in the motions of the planets as an example where inductively established laws lead, via deductive extrapolation, to arguments about phenomena not considered under the original laws:
We must set out by assuming this law [of gravitational attraction] . . . we then, for the first time, perceive a train of modifying circumstances which had not occurred to us when reasoning upwards from particulars to obtain the fundamental law; we perceive that all the planets attract each other . . . and as this was never contemplated in the inductive process. (Discourse 201)
By developing further mathematical calculations from the law of gravity to describe the amount of influence planets have on one another, and then using the results to predict eccentricities in their orbital paths, Newton and others seeking to verify or extend his theory of gravitation proved that the law of gravity accounted for anomalies in planetary motion that had previously puzzled researchers. If the theory of gravity had not been able to suitably account for these effects, for which it had been deduced to be the cause, then the credibility of the law would have been in jeopardy (202).
In the final stage of induction, mathematical analogies make an important transition from tentative conclusions to generally accepted warrants for further arguments. Although the laws established from these analogies are still open to emendation and clarification, they have passed an important threshold after which they are generally considered accepted principles of nature. As a result of their new status, they can serve as axioms from which extrapolations can be made about subjects that fall under their jurisdiction, or about phenomena not originally considered. In this capacity, mathematical warrants serve as engines of invention, suggesting new pathways for expanding natural investigation.
Conclusion
By examining in tandem the works of two of the most influential, nineteenth-century philosophers/methodologists of science, this chapter has endeavored to provide the background for assessing what constitutes the usual or commonly accepted criteria for making mathematical argument in science in the nineteenth century, and the appropriate stages by which mathematical warrants were thought to develop. Though fundamental disagreement existed between Herschel and Whewell on the ultimate source of natural knowledge, they both agreed that without quantitative laws, nature’s intricate and sometimes impossible-to-observe operations could never be brought to light. They also believed that the strength of mathematical arguments resided in their capacity to illuminate these operations in a precise and rigorous manner, which spared natural researchers from the weakness of memory and the illusions of experience. These obvious benefits of