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2 A Proper Science: Mathematics, Experience, and Argument in Nineteenth-Century Science

Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.

—Galileo 1

In the last twenty years, two major trends have emerged in the analysis of the rhetorical features of science, one taxonomic and another constructivist. In taxonomic approaches to analysis, scientific argument is identified and its effects explained using traditional concepts and terms from canonical treatises of rhetoric. This particular approach is employed in well-known rhetoric of science investigations, such as Lawrence Prelli’s Rhetoric of Science: Inventing Scientific Discourse, and Jeanne Fahnestock’s Rhetorical Figures in Science. While taxonomic approaches rely on established catalogues of topoi, tropes, and figures for analysis, constructivist investigations of argument, such as those undertaken by genre and action network theorists, focus on the features of communication and argument as they emerge and change in response to shifting needs within communities. In their analyses of scientific communication and argument, scholars like Carol Berkenkotter, Thomas Huckin, and John Swales, examine the conventions for scientific communication as well as how social, cultural, and institutional circumstances actively shape them (Genre Knowledge; Genre Analysis).

The analytical approach used in this book combines both of these theoretical perspectives and contributes a new set of resources for argument analysis. Throughout the text, taxonomic methods are employed to describe and explain the strategies for arguments used by researchers attempting to advance mathematical approaches to the study of variation, evolution, and heredity. For example, Chapter 3 examines Darwin’s use of the commonplace “the more and the less” to make his case for dynamic variation in species.

Whereas the taxonomic aspect of this analysis provides language and a conceptual framework for describing argument, its constructivist dimension seeks real-world evidence of the conventions for arguing with mathematics in science. To understand this facet of argument, I turn to philosophies/methodologies of science—which have not, to my knowledge, been exploited as analytical resources—to understand conventions for arguing mathematically in science. To provide a context for the discussion in later chapters, this chapter examines in detail the two, nineteenth-century works on the philosophy and methodology of science: John Herschel’s Preliminary Discourse on the Study of Natural Philosophy (1831) and William Whewell’s Philosophy of the Inductive Sciences (1840). The conventions of mathematical argument described in these works provide a context for assessing not only the legitimacy of strategies used by arguers to advance mathematical approaches to variation, evolution, and heredity, but also the reasons for the success and failure of those strategies with nineteenth century, scientific audiences.

John Herschel and William Whewell

Scientific philosophies provide a valuable resource for understanding the choices scientists make when they argue. The influence of such philosophical texts was particularly strong in the nineteenth century, a period when the philosophy of science was not divorced from its practice. Philosophies of natural science were written by influential educators and practitioners who included in their works the latest information about the methods and state of knowledge in a broad range of scientific fields. As a result, they were not only read as theoretical documents, but also as handbooks describing the state of the discipline and the practice of science.

John Herschel (1792–1871) and William Whewell (1794–1866) were two of the nineteenth century’s most eminently qualified writers on natural philosophy. They were both actively engaged in scientific research and publication, and both vigorously participated in developing important institutions of British science.2 John Herschel is, and was regarded, as one of the great figures of Victorian science not only because of his tireless efforts in discovering and cataloguing astronomical phenomena, but also because of his ability to write lucidly about the finest points of scientific philosophy (Partridge xii-xiv). His skills of adaptation are exemplified in Preliminary Discourse on the Study of Natural Philosophy (1830), in which he presents readers with a thorough introduction to scientific philosophy and a clear explanation of scientific method.

The significant influence that the book had on Victorian science is evidenced not only by the fact that the text went through twelve editions, but also by the quality of the Victorian thinkers who vouched for its importance in the development of their own scientific thought. John Stuart Mill, for example, used it as the basis of his own work on scientific theory in his System of Logic (1843) (Partridge xiv; Canon, “John Herschel” 220–221). It also influenced James Clerk Maxwell’s work in the Discourse on Molecules, and William Whewell’s Philosophy (Canon, “John Herschel” 220; Kemsley).

Although not as eminent a producer of scientific knowledge as Herschel, William Whewell had a significant impact on mathematical and scientific education and natural philosophy in Britain. As an influential member of the faculty and administration of Cambridge from 1828–1866, Whewell pushed for the introduction of analytical mathematics at Cambridge, a move which brought the archaic mathematics curriculum at Cambridge up to date with Continental mathematical practices. He also supported the creation of a new Tripos for the natural sciences, which allowed 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 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 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 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 of trials (n) that are conducted (Chatterjee 168).

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 mathematics helped it to persist in biological investigations of variation, evolution, and heredity despite general disagreements over the applicability of mathematical laws to biological phenomena. Conclusions supported by quantified data or mathematical operations could be considered more precise and rigorous than those that did not.

Despite obvious strengths, mathematical reasoning could be challenged on the grounds that it did not accurately reflect experience. As a consequence, mathematical formulae and reasoning had to be tested against evidence from repeated observations and experiments under a variety of conditions. As both modern and historical cases reveal, it is only in the presence of data that mathematical applications and arguments thrive in science. Chapter 4, for example, examines how Mendel’s work fell into obscurity because it lacked a broader data set to support its conclusions, and Chapter 5 explores how Galton’s work succeeded in part because of his herculean efforts to collect data in support of his theory of inheritance.

In addition to describing the qualities that make mathematical argument robust, this chapter has also illustrated the stages by which mathematical knowledge achieves legitimacy. Understanding where argument is perceived to be in this process provides insight into why a particular argument may or may not be considered rhetorical, and for what reasons. When I use the term “rhetorical” here, I am talking about argument which is probable rather than certain; argument which produces agreement from a variety of sources, which includes, but is not limited to, emotions, beliefs, and values; and finally, argument that relies on a number of general strategies/tools for argument, including figures, tropes, and topoi as means to secure agreement and establish understanding. Scientific arguments at the beginning stages of mathematization take on a rhetorical dimension because they rely heavily on the prestige accorded to mathematical deductive rigor and precision to make their scientific case, which initially has only a limited amount of inductive, empirical evidence to support it. In the middle stages of the process, the rhetorical dimension of mathematical arguments shifts from a reliance on the ethos of rigor and precision of mathematics to a dependence on analogy to establish understanding and secure agreement. Finally, in the last stages of quantitative induction, fused analogies are no longer rhetorical because they can be used as a common ground for further argument. However, because the possibility of a challenge always exists, they have the potential to lose their status as reliable warrants for scientific argument and fall once again into the realm of the probable, the rhetorical.

In the chapters that follow, the conventions for mathematical argument set out in Herschel’s and Whewell’s philosophies provide an epistemological framework for assessing the strategies of arguers as they attempt to advance mathematical programs for the study of variation, evolution, and heredity, and their successes or failures in making their cases. These investigations illustrate the utility of scientific philosophies and methodologies in understanding the epistemological context of mathematical argument in science, and it’s possible rhetorical dimensions.