If we accept the proposition that Herschel’s and Whewell’s opinions about the importance of quantification to the foundation of credible, scientific argument reflects and/or has influence on the opinions of other Victorian natural researchers, then we can assume that researchers making arguments about natural phenomena would aspire to use precise, quantified data to make their arguments compelling for their audiences. We can also assume that audiences assessing scientific arguments might praise or criticize them based on whether or not they were made using precise quantified data.
Step Two: Formulization
Once a standard for measurement is established and quantitative data is collected, the next step in quantitative induction is to describe the relationship in the data using a mathematical formula. With this step, the researcher proposes an analogy, wherein a particular relationship described by or derived from existing mathematical axioms is hypothesized to be analogous to the change in the natural phenomenon observed.
In the Philosophy of the Inductive Sciences, Whewell writes in detail about this process, suggesting that it has three steps: selection of the independent variable, construction of the formula, and determination of the coefficients (1: 382). He provides an example of the process using a hypothetical case in which astronomers attempt to discover the quantitative law describing how a particular star’s position changes in the heavens. In the scenario, the researcher begins with observational data on the star, which shows that, after three successive years, the star has moved by 3, 8, and 15 minutes from its original place.
After consulting the existing data, he casts about for the appropriate category of change, or Idea under which a law might be constructed to describe the star’s change in position.7 If the investigation is to be quantitative, the Idea must come from one of four possible categories: space, time, number, or resemblance.8 The researchers following the star settle on “time,” which becomes the independent variable (t) for the formula with which they will express their law describing the star’s movement.
After selecting an appropriate category of change, the scientist’s next duty is to determine exactly how the measured phenomenon changes with respect to that category. If the category selected is “time,” the researcher would ask, “How does the star’s position change with respect to time?”; “Are the changes in time and position uniform? Are they linear? Are they cyclical?”
The change in the star’s location of 3, 8, 15 minutes suggests that the alteration of its position with respect to the change of time is not regular. With the aid of his mathematical training, the researcher would quickly recognize that the series “can be obtained by means of two terms, one of which is proportional to time, and the other to the square of the time . . . expressed by the formula at + btt” (Philosophy 2: 383).
Once the apparent manner of change has been described in the formula, the magnitude of the coefficients—the fixed numerical constants by which the independent variables are multiplied—needs to be established. In the formula, at + btt, a and b are the coefficients. As Whewell explains, the magnitude of a and b could be established by figuring out what values were required to get the results described in the observations. To generate the series 3, 8, 15 from the equation at + btt if time increases 1, 2, 3, etc., a must equal 2 and b must equal 1.9
For Whewell, the creation of a formula, which at this stage was considered a hypothetical representation of the change in a particular phenomenon, was an attempt to colligate (or collect) the instances of change under a single mathematical description. This move can be understood as an effort to make the case for a particular analogy between experience and reason (i.e., between observed data and known mathematical principles).
Analogy can be defined broadly as an argument for or from the resemblance between dissimilar constituents.10 For example, Benjamin Franklin argued for accepting the resemblance between electricity and fluids in his efforts to explain the operation of the Leyden jar. Once this analogy was accepted, researchers such as Henry Cavendish used it as a basis from which to develop mathematical and mechanical explanations about the behavior of electricity (Jungnickel and McCormick 174–81). In the New Rhetoric, theorists Chiam Perelman and Lucie Olbrechts-Tyteca explain that analogies have two constituent parts, the phoros and the theme (373). The phoros is the part of the analogy with which the audience is familiar. It provides a structure, value, and/or meaning by which the unknown or unvalued theme can be understood or characterized. For example, in the analogy from Aristotle, “For as the eyes of bats are to the blaze of day, so is the reason in our soul to the things which are by nature most evident of all,” the “eyes 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