the door for the participation of mathematical argument in the development of natural knowledge. However, experience—the data from observation and experiment—always acts as a limiting and shaping force on its contribution. These shared beliefs about the necessary balance between reason and experience represent the fundamental principles guiding Whewell’s and Herschel’s opinions about the possible strengths and potential weaknesses of mathematical argument as well as the manner in which robust, mathematical arguments about nature could be developed.
Mathematical Arguments and the Inductive Process
The delicate balance between experience and rationality plays itself out vividly in nineteenth century characterizations of “induction.” In the process of induction, mathematics contributes the appropriate form for scientific arguments, while observation and experimentation provide the necessary content to verify the form. The constant check and balance between experience and reason is a key influence on the inductive process, dictating not only the steps by which mathematical argument might gain credibility, but also what arguers and audiences perceive to be the strengths and weaknesses of mathematical arguments.
For Herschel and Whewell, induction involved two distinct activities: the determination of causes, and the description of effects. Though both were interrelated, the development and use of mathematical arguments was directly implicated in the latter activity while only tangentially important to the former. As a consequence, their discussions of mathematical argument focus primarily on efforts to describe effects and their relationships to one another.
In combination, Whewell and Herschel identify four steps in the quantitative inductive process: quantification, formulization, verification, and extrapolation.6 By examining these steps in detail, it is possible to understand how Herschel, Whewell, and presumably other natural researchers, perceived the possible strengths and potential weaknesses of mathematical argument, and how they could be raised from hypothetical to authoritative statements about nature.
Step One: Quantification
Both Herschel and Whewell are adamant that, without quantification, knowledge could not be considered “scientific.” Whewell writes, for example, “We cannot obtain any sciential truths respecting the comparison of sensible qualities, till we have discovered measures and scales of the qualities which we have to consider” (Philosophy 1: 321). Herschel argues that, without quantification, argument could not be scientific because it would not have the necessary level of precision. Because human senses are not always sufficient to make the distinctions necessary to discover or describe changes in small or large-scale phenomena, the natural philosopher had to depend on precise quantification to establish reliable knowledge about nature. Herschel elaborates this point when he writes:
In all cases that admit of numeration or measurement, it is of the utmost consequence to obtain precise numerical statements, whether in the measure of time, space, or quantity of any kind. To omit this, is, in the first place, to expose ourselves to illusions of sense which may lead to the grossest errors. (122)
Without precise quantitative data, Herschel explains, a scientific argument can never be considered reliable: “But it is not merely in preserving us from exaggerated impressions that numerical precision is desirable. It is the very soul of science; and its attainment affords the only criterion, or at least the best, of the truth of theories, and the correctness of experiments” (122).
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
