We have thus pointed out to us, as the great, and indeed only ultimate source of our knowledge of nature and its laws EXPERIENCE, by which we mean not the experience of one man only, or of one generation, but the accumulated experience of all mankind in all ages, registered in books or recorded by tradition. (Discourse 76)
Whereas the collective, communal experience of nature was the wellspring of understanding for Herschel, Whewell located the universal principles of nature in the human mind. For Whewell the physical world as we perceive it presents us with data but not with the principles to comprehend the underlying relationships between phenomena. These principles could only be supplied by the mind. As a consequence, the mind and its faculties became the ultimate source of natural knowledge. The goal of science, therefore, was to uncover and clarify the vast and hidden laws of nature in the mind by observing and comparing data:
In order to obtain our inference, we travel beyond the cases which we have before us; we consider them as mere exemplifications of some ideal case in which the relations are complete and intelligible. We take a standard and measure the facts by it; and this standard is constructed by us, not offered by Nature. (Philosophy 1: 49)
Though Herschel and Whewell supported two different positions on the ultimate source of knowledge about nature, they agreed that both experience and cognition were necessary complements in the construction of scientific knowledge. Proper science was the balance between the two. On the one hand, experience of natural phenomena was required because, without it, the products of reason, no matter how rationally rigorous, were simply elaborate fictions without purchase in nature. On the other hand, without the higher power of human reason, the hidden relationships between natural phenomena would be eternally locked away from view.
Questions about the appropriateness of mathematical argument and its benefit to the development of natural knowledge are caught up in this debate. Mathematics resided naturally on the mind/reason side of the Cartesian mind/body, reason/experience duality. This point is conceded by both Herschel and Whewell, and epitomized by Herschel in A Preliminary Discourse on the Study of Natural Philosophy when he writes:
Abstract [mathematical] science is independent of a system of nature—of a creation—of everything, in short, except memory, thought, and reason. Its objects are, first, those primary existences and relations which we cannot even conceive not to be, such as space, time, number, order, &c. (18) 5
Despite their position outside of nature, however, mathematical principles, operations, and symbols still had value in its characterization because it was with these conceptual tools that the invisible relationships between physical phenomena could be discovered. Herschel makes this point in the previous passage when he explains that relations of phenomena that have purchase in nature space, time, number, etc. can be conceived of in the abstract science of mathematics. Whewell makes the same point when he writes:
All objects in the world which can be made the subjects of our contemplation are subordinate to the conditions of Space, Time, and Number; and on this account, the doctrines of pure mathematics have most numerous and extensive applications in every department of our investigations of nature. (Philosophy 1: 153)
Just as Herschel and Whewell agree that mathematical reasoning has a place in the interpretation of natural phenomena, both also agree that it only has validity if it is based on evidence from experience of nature. Herschel recognizes the necessity of experience to mathematical reasoning when he writes,
A clever man, shut up alone and allowed unlimited time, might reason out for himself all the truths of mathematics. . . . But he could never tell, by any effort of reasoning, what would become of a lump of sugar if immersed in water, or what impression would be produced on his eye by mixing the colors yellow and blue. (76)
Despite his opinion that the mind was the ultimate source of natural knowledge, Whewell also acknowledges the limitations of mathematics without experience. In an eloquent passage in volume one of Philosophy of the Inductive Sciences, he makes the point that without experience, mathematical knowledge of nature is impossible, and without mathematics, understanding the changes in natural phenomena is inconceivable.
If there were not such external things as the sun and the moon I could not have any knowledge of the progress of time as marked by them. And however regular were the motions of the sun and moon, if I could not count their appearances and combine their changes into a cycle, or if I could not understand this when done by other men, I could not know anything about a year or month. (Philosophy 1: 18)
Though Herschel and Whewell emphasize different sides of the Cartesian split, both agree that experience and reason are necessary components of scientific knowledge. Reason—mental operations which reveal the hidden relationships between phenomena—opened 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