students to focus their attention on the study of nature by relieving them of the extraordinary burden of having to have expert knowledge of mathematics to obtain honors in their studies (Herivel xiii).3
In addition to being in the avant-garde of institutional reform, Whewell was also a leader in nineteenth century discussions about the history, philosophy, and methodology of science and its relationship to mathematics. In 1837 he published History of the Inductive Sciences, which was “aimed at being, not merely a narration of the facts in the history of science, but a basis for the philosophy of science” (viii). In the text, Whewell traces the history of natural philosophy from the Greeks through the Middle-Ages and into the nineteenth century, critiquing the shortcomings and praising the advancements in scientific thought as it developed.
Whereas History of the Inductive Sciences explored and assessed the empirical development of a proper system for obtaining scientific knowledge, Whewell elaborates the epistemological characteristics of that system in Philosophy of the Inductive Sciences (1840). In part one, “Of Ideas,” he offers an expansive discussion of scientific epistemology that addresses the relationship of thought and experience to the production of credible scientific knowledge. In part two, “Of Knowledge,” Whewell describes the process by which he believed scientific knowledge was constructed and the specific methods by which knowledge of nature is obtained.
Although the philosophical positions in Whewell’s History and Philosophy proved to be more controversial than Herschel’s Discourse, they were nonetheless seriously regarded as important scholarship on scientific history, philosophy, and method by nineteenth century scientists. Both History and Philosophy ran three editions, and contributed to a lively debate about the foundations of scientific knowledge, which engaged important nineteenth century figures such as John Stuart Mill, Charles Darwin, and John Herschel.4
Because of their substantial influence on Victorian science and their attention to the role of mathematics in making scientific argument, the philosophies of science by Whewell and Herschel can be considered credible guides for understanding the challenges and benefits of making scientific arguments with mathematics. In addition, they represent opinions on two sides of an important philosophical division in the nineteenth century: between nativism, whose adherents believed that the source of knowledge about nature resides in the mind of the scientist; and empiricism, whose supporters suppose that the truth of nature inhered in nature itself, and could only be uncovered through experience and experimentation (Richards 2–3). Because they represented opposing sides of this debate, Whewell, nativism, and Herschel, empiricism, a combined analysis of their work affords a comprehensive view of the spectrum of opinion on correct procedure in Victorian science as well as common ground on the role of mathematics in making scientific arguments.
Rationality and Reality: The Stakes for Mathematical Argument
Whewell and Herschel took two divergent positions on the question of how reliable knowledge about nature could be attained. For Herschel the truths of nature existed in nature itself. Though the human mind was necessary for decoding the communications of nature, he did not consider the mind the source of true knowledge about nature. Instead, the ultimate source of natural knowledge was experience:
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
