The rhetoric in mathematics: Newton, Leibniz, the Calculus, and the rhetorical force of the infinitesimal

Quarterly Journal of Speech

Volumn 90, Issue 2, 2004, Pages 163-188

  Reyes, G Mitchell ^a  

^a CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

Leibniz; Mathematics; Newton; Rhetoric; Science

Indexed keywords

EID: 3042799706     PISSN: 00335630     EISSN: None     Source Type: Journal    
DOI: 10.1080/0033563042000227427     Document Type: Article

References (119)

1. W. B. Weimer's "Science as a Rhetorical Transaction: Toward a Nonjustificational Conception of Rhetoric," Philosophy and Rhetoric 10 (1977): 1-29, points to the way Aristotle positioned rhetoric in relation to doxa and logic and science in relation to episteme such that rhetoric does not operate within the pure realm of science. Weimer then shows that that binary has recently begun to break down and attempts to reconfigure science in a way that opens it up to rhetorical critique.
2. I am attempting to characterize the general perspective called formalism or mathematical foundationalism. Within the field of mathematics this perspective still holds force, but other philosophical stances challenge it. For an informative book on the different philosophical positions held by mathematicians, see P. Davis and R. Hersh, The Mathematical Experience (Boston, MA: Birkhauser, 1981).
3. Throughout the essay I will capitalize Newton's and Leibniz's methods to distinguish them from calculus in the generic sense. I will also refer to "Newton's and Leibniz's Calculus," but this is meant to simplify presentation, not to indicate that Newton and Leibniz worked collaboratively on developing the Calculus. To the contrary, they were bitter rivals. See A.R. Hall, Philosophers at War: The Quarrel Between Newton and Leibniz (Cambridge: Cambridge University Press, 1980). This is not important here because this essay focuses on the constitutive rhetoric that animates and gives "substance" to the concept of the infinitesimal and how that "substance" calls forth the situational rhetoric in which Newton and Leibniz engaged. In any case, Newton and Leibniz developed their own distinctive terminologies out of significantly different contexts (Newton in Britain and Leibniz on the continent), but these differences in terminology cannot be linked directly to a contest over origination. See I. Newton, "The Method of Fluxions and Infinite Series," Artis Analyticae Spedmina, ed. John Colson (London: H. Woodfall Publishers, 1736); G. W. Leibniz, "A New Method for Maxima and Minima as Well as Tangents Which is Neither Impeded by Fractional nor Irrational Quantities, and a Remarkable Type of Calculus for Them," in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Cambridge, MA: Harvard University Press, 1969), 271-280.
4. Throughout the essay I will capitalize Newton's and Leibniz's methods to distinguish them from calculus in the generic sense. I will also refer to "Newton's and Leibniz's Calculus," but this is meant to simplify presentation, not to indicate that Newton and Leibniz worked collaboratively on developing the Calculus. To the contrary, they were bitter rivals. See A.R. Hall, Philosophers at War: The Quarrel Between Newton and Leibniz (Cambridge: Cambridge University Press, 1980). This is not important here because this essay focuses on the constitutive rhetoric that animates and gives "substance" to the concept of the infinitesimal and how that "substance" calls forth the situational rhetoric in which Newton and Leibniz engaged. In any case, Newton and Leibniz developed their own distinctive terminologies out of significantly different contexts (Newton in Britain and Leibniz on the continent), but these differences in terminology cannot be linked directly to a contest over origination. See I. Newton, "The Method of Fluxions and Infinite Series," Artis Analyticae Spedmina, ed. John Colson (London: H. Woodfall Publishers, 1736); G. W. Leibniz, "A New Method for Maxima and Minima as Well as Tangents Which is Neither Impeded by Fractional nor Irrational Quantities, and a Remarkable Type of Calculus for Them," in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Cambridge, MA: Harvard University Press, 1969), 271-280.
5. Throughout the essay I will capitalize Newton's and Leibniz's methods to distinguish them from calculus in the generic sense. I will also refer to "Newton's and Leibniz's Calculus," but this is meant to simplify presentation, not to indicate that Newton and Leibniz worked collaboratively on developing the Calculus. To the contrary, they were bitter rivals. See A.R. Hall, Philosophers at War: The Quarrel Between Newton and Leibniz (Cambridge: Cambridge University Press, 1980). This is not important here because this essay focuses on the constitutive rhetoric that animates and gives "substance" to the concept of the infinitesimal and how that "substance" calls forth the situational rhetoric in which Newton and Leibniz engaged. In any case, Newton and Leibniz developed their own distinctive terminologies out of significantly different contexts (Newton in Britain and Leibniz on the continent), but these differences in terminology cannot be linked directly to a contest over origination. See I. Newton, "The Method of Fluxions and Infinite Series," Artis Analyticae Spedmina, ed. John Colson (London: H. Woodfall Publishers, 1736); G. W. Leibniz, "A New Method for Maxima and Minima as Well as Tangents Which is Neither Impeded by Fractional nor Irrational Quantities, and a Remarkable Type of Calculus for Them," in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Cambridge, MA: Harvard University Press, 1969), 271-280.
6. 2 (the square of an infinitesimal error). Therefore dy = 2xe.
7. Prior to the Calculus, Euclidean geometry provided the method of exhaustion, but this method was cumbersome and limited to the simplest of geometric objects. The Calculus, on the other hand, could approximate any variety of complex curvatures, which made it ideal for the study of rates of change.
8. H. Eves, Great Moments in Mathematics (after 1650) (Washington, D.C.: The Mathematical Association of America, 1981), 23.
9. Eves, Great Moments, 23.
10. I am offering a description of rhetorical practice and not rhetoric as such.
11. M. Heidegger, "Modern Science, Metaphysics, and Mathematics," in Basic Writings, ed. David Farrell Drell, trans. W.B. Barton, Jr. and Vera Deutsch (San Francisco: Harper Collins Publishers, 1977), 262.
12. G. Desilet, "Physics and Language-Science and Rhetoric: Reviewing the Parallel Evolution of Theory on Motion and Meaning in the Aftermath of the Sokal Hoax," Quarterly Journal of Speech 85 (1999): 343.
13. I do not intend to reinvigorate a substance/appearance dichotomy. I mean to explode that binary by exploring a mathematical concept that is excessive to such thinking. Dilip Gaonkar's "The Idea of Rhetoric in the Rhetoric of Science," in Rhetorical Hermeneutics: Invention and Interpretation in the Age of Science, ed. A. G. Gross and W. M. Keith (Albany, NY: State University of New York Press, 1997), 25-88, claims that rhetoric often is positioned parasitically by scholarship in the rhetoric of science. Here I show that rhetoric is the "material" out of which a new system of mathematics emerges.
14. I do not claim that Newton and Leibniz invented mathematical rhetoric, but that the use of infinitesimals called forth several forms of rhetoric.
15. Philip Wander's pioneering work, for example, dealt with the influence science has on the public sphere: "Science or the use of science in public deliberation," Wander wrote, "begs rhetorical investigation." See P. Wander, "The Rhetoric of Science," Western Journal of Speech Communication 40 (1976): 229. Wander argued that science had been used to intimidate the public and to elevate scientific knowledge to the point that it stifled the layman's voice.
16. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
17. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
18. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
19. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
20. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
21. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
22. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
23. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
24. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
25. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
26. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
27. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
28. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
29. The breakthrough essay on rhetoric and the practices of science is W.B. Weimer's "Science as a Rhetorical Transaction." See also W. B. Weimer, "Why All Knowledge is Rhetorical," Journal of the American Forensic Association 20 (1983): 63-71. In critical practice these two concerns are often interwoven. See A. G. Gross, The Rhetoric of Science (Cambridge, MA: Harvard University Press, 1990); A. G. Gross, "Rhetoric of Science Without Constraints," Rhetorica 9 (1991): 283-299; J. A. Campbell, "Reply to Gaonkar and Fuller," Southern Communication Journal 58 (1993): 312-318; J. A. Campbell and K. R. Benson, "The Rhetorical Turn in Science Studies," Quarterly Journal of Speech 82 (1996): 74-109; L. Ceccarelli, "A Masterpiece in a New Genre: The Rhetorical Negotiation of Two Audiences in Schroedinger's What is Life?" Technical Communication Quarterly 3 (1994): 7-17; L. Ceccarelli, "Polysemy: Multiple Meanings in Rhetorical Criticism," Quarterly Journal of Speech 84 (1998): 395-415; L. Ceccarelli, "A Rhetoric of Interdisciplinary Scientific Discourse: Textual Criticism of Dobzhansky's Genetics and the Origin of Species," Social Epistemology 9 (1995): 91-111. Two excellent starting points for understanding the issues raised by the rhetoric of science are M. Pera, The Discourses of Science (Chicago, IL: University of Chicago Press, 1994), and A. Gross and W. Keith, eds., Rhetorical Hermeneutics (this compilation contains expanded versions of several essays mentioned above). In addition, see T. Melia, "And Lo the Footprint: Selected Literature in Rhetoric and Science," Quarterly Journal of Speech 70 (1984): 303-313; R. M. Bokeno, "The Rhetorical Understanding of Science: An Explication and Critical Commentary," Southern Speech Communication Journal 52 (1987): 285-311; L. J. Prelli, A Rhetoric of Science: Inventing Scientific Discourse (Columbia, SC: University of South Carolina Press, 1989); L. J. Prelli, "Rhetorical Logic and the Integration of Rhetoric and Science," Communication Monographs 57 (1990): 315-322.
30. P. Davis and R. Hersh, "Rhetoric and Mathematics," in The Rhetoric of the Human Sciences: Language and Argument in Scholarship and Public Affairs, ed. J. Nelson, A. Megill, and D. McCloskey (Madison, WI: University of Wisconsin Press, 1987), 54.
31. Davis and Hersh, "Rhetoric and Mathematics," 54.
32. See M. Allen, "Words and Numbers: Mathematical Dimensions of Rhetoric," Southern Communication Journal 55 (1990): 337-354, and P. Ernest, "Forms of Knowledge in Mathematics and Mathematics Education: Philosophical and Rhetorical Perspectives," Educational Studies in Mathematics 38 (1990): 67-83, respectively.
33. See M. Allen, "Words and Numbers: Mathematical Dimensions of Rhetoric," Southern Communication Journal 55 (1990): 337-354, and P. Ernest, "Forms of Knowledge in Mathematics and Mathematics Education: Philosophical and Rhetorical Perspectives," Educational Studies in Mathematics 38 (1990): 67-83, respectively.
34. Brian Rotman's work on semiotics and mathematics is an exception to this rule. Rotman attempts to disclose the semiotic ungroundedness of certain foundational mathematical concepts. See B. Rotman, Ad Infinitum: The Ghost in Turing's Machine (Stanford, CA: Stanford University Press, 1993); B. Rotman, Signifying Nothing: The Semiotics of Zero (London: Macmillan, 1987).
35. Brian Rotman's work on semiotics and mathematics is an exception to this rule. Rotman attempts to disclose the semiotic ungroundedness of certain foundational mathematical concepts. See B. Rotman, Ad Infinitum: The Ghost in Turing's Machine (Stanford, CA: Stanford University Press, 1993); B. Rotman, Signifying Nothing: The Semiotics of Zero (London: Macmillan, 1987).
36. The unique rhetorical status of the infinitesimal begs the question of the rhetorical status of mathematics in general. Although that topic is beyond the scope of this paper, I would agree (at least initially) with Brian Rotman (Ad Infinitum; Signifying Nothing) that mathematics operates on an ideal plane and is informed by a Platonic philosophy that has its own rhetorical features. That said, most mathematical concepts emerge from within the boundaries of the established mathematical logic of the time (that is, from the rules that govern the action allowable in the mathematical world). Once in a while, however, a mathematical concept cornes along that is unique and, thus, transgresses the boundaries of mathematical practice, often requiring novel rhetorical arguments as its substance and in its advocacy.
37. Euclid, The Thirteen Books of Euclid's Elements, vol. 2, trans. and ed. T. L. Heath (New York: Dover, 1956), 114.
38. For an extended treatment of these issues, see Douglas Jesseph, "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-244.
39. K. Burke, A Rhetoric of Motives (Berkeley: University of California Press, 1969). This is intended to describe the ways that the concept of the infinitesimal (its rhetorical makeup) called forth the situational rhetoric explored in the following pages. In a certain sense both layers.of rhetoric are constitutive of the infinitesimal, but I am using constitutive here to designate the rhetoric that governs the mathematical meaning of the infinitesimal, calling forth the debates over the Calculus during the seventeenth century.
40. The Analyst, The Works of George Berkeley, vol. 3, ed. A.C. Fraser (Oxford: Clarendon Press, 1901); B. Nieuwentijdt, cited in C. Boyer, The History of the Calculus and its Conceptual Development (New York: Dover Publications, 1949), 213.
41. The Analyst, The Works of George Berkeley, vol. 3, ed. A.C. Fraser (Oxford: Clarendon Press, 1901); B. Nieuwentijdt, cited in C. Boyer, The History of the Calculus and its Conceptual Development (New York: Dover Publications, 1949), 213.
42. George Berkeley leveled a scathing critique precisely along these lines, arguing that if one asks what "things" infinitesimals express, one "shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not," Berkeley goes on, infinitesimals create "direct Impossibilities and Contradictions." In Berkeley, The Analyst, 4.
43. Rotman, Ad Infinitum, 41-42.
44. M. Bishop, "Semantic Flexibility in Scientific Practice: A Study of Newton's Optics," Philosophy and Rhetoric 32 (1999): 225.
45. On the clarity of the Calculus, Bernard de Fontenelle, a highly respected French mathematician and philosopher, wrote, "it [the Calculus] does not cease still to cast us into the abyss of a profound darkness, or at the very least into realms where the daylight is extremely weak." B. Fontenelle, Éléments de la Géométrie de l'Infini (Paris: De l'Imprimerie Royale, 1727), iv.
46. Eric Havelock speaks of a similar concept when he discusses the "stretching" of language toward the abstract by the Presocratics. See E. A. Havelock, "The Linguistic Task of the Presocratics," in Language and Thought in Early Greek Philosophy, ed. K. Robb (Lasalle, IL: Manist Library, 1983), 7-41.
47. Rotman, Ad Infinitum, 145.
48. Rotman, Ad Infinitum, 44.
49. See B. Nieuwentijdt, who began the criticisms of the Calculus, in Boyer, The History of the Calculus, 213; Fontenelle, Éléments; J. Raphson, The History of Fluxions, Showing in a Compendious Manner the First Rise of, and Various Improvements Made in That Incomparable Method (London: T. Braddyll, 1715).
50. See B. Nieuwentijdt, who began the criticisms of the Calculus, in Boyer, The History of the Calculus, 213; Fontenelle, Éléments; J. Raphson, The History of Fluxions, Showing in a Compendious Manner the First Rise of, and Various Improvements Made in That Incomparable Method (London: T. Braddyll, 1715).
51. See B. Nieuwentijdt, who began the criticisms of the Calculus, in Boyer, The History of the Calculus, 213; Fontenelle, Éléments; J. Raphson, The History of Fluxions, Showing in a Compendious Manner the First Rise of, and Various Improvements Made in That Incomparable Method (London: T. Braddyll, 1715).
52. Along these lines, Berkeley wrote: "For to consider the proportion or Ratio of things implies that such things have Magnitude;" but to do this with infinitesimals, Berkeley argued, "is to talk unintelligibly." Berkeley, The Analyst, 15. The Analyst is considered by many the most powerful critique of the Calculus; see also A.A. Luce, Berkeley and Malebranche: A Study in the Origins of Berkeley's Thought (London: Oxford University Press, 1934).
53. Along these lines, Berkeley wrote: "For to consider the proportion or Ratio of things implies that such things have Magnitude;" but to do this with infinitesimals, Berkeley argued, "is to talk unintelligibly." Berkeley, The Analyst, 15. The Analyst is considered by many the most powerful critique of the Calculus; see also A.A. Luce, Berkeley and Malebranche: A Study in the Origins of Berkeley's Thought (London: Oxford University Press, 1934).
54. Boyer, The History of the Calculus, 209.
55. Newton's treatment of infinitesimals in the Principia changed throughout his career, but Newton did not enjoy spirited philosophical exchange or criticisms of his work, especially those he felt were based on peripheral issues.
56. Boyer, The History of the Calculus, 218.
57. Newton has been quoted by many as claiming later in his career that "the very smallest errors in mathematical matters are not to be neglected." In F. Cajori, "Appendix," in Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, ed. F. Cajori, trans. A. Motte (Berkeley, CA: University of California Press, Berkeley, 1962), 654.
58. I. Newton, cited in E. Burtt, The Metaphysical Foundations of Modern Physical Science: A Historical and Critical Essay (London: Compton Printing, 1967), 223.
59. There are two forms of foundationalism, one scientific and concerned primarily with strict empiricism and the other mathematical and concerned primarily with strict adherence to Euclidean geometry. Cristoph Clavius, an important astronomer and mathematician of the early seventeenth century, expresses well the perspective of mathematical foundationalism: "The theorems of Euclid and the rest of the mathematicians, still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstration ... and thus so much do the mathematical disciplines desire, esteem, and foster truth, that they reject not only whatever is false, but even anything mere probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations." C. Clavius, in J. A. Lattis, Between Copernicus and Galileo: Christopher Clavius and the Collapse of Ptolemaic Astronomy (Chicago, IL: University of Chicago Press, 1994). This was not a univocal position of all seventeenth-century mathematicians, 35. Cavalieri's Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota (Bononiae: Ex Typographia de Duciis, 1653) figures as an early, although tentative, departure from mathematical foundationalism; it was, however, the generally accepted view of rigor in mathematics that caused distrust of the Calculus among scientists and mathematicians. As Douglas Jesseph recently remarked, "they [infinitesimals] ... violated widely accepted philosophical canons which declared the infinite to be incomprehensible and infinitesimal methods inadmissible." In "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215. In addition, Morris Kline recently argued, "many of the English mathematicians, perhaps because they were in the main still tied to the rigor of Greek geometry, distrusted all the work on the calculus." In Mathematical Thought from Ancient to Modem Time (New York: Oxford University Press, 1972), 389.
60. There are two forms of foundationalism, one scientific and concerned primarily with strict empiricism and the other mathematical and concerned primarily with strict adherence to Euclidean geometry. Cristoph Clavius, an important astronomer and mathematician of the early seventeenth century, expresses well the perspective of mathematical foundationalism: "The theorems of Euclid and the rest of the mathematicians, still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstration ... and thus so much do the mathematical disciplines desire, esteem, and foster truth, that they reject not only whatever is false, but even anything mere probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations." C. Clavius, in J. A. Lattis, Between Copernicus and Galileo: Christopher Clavius and the Collapse of Ptolemaic Astronomy (Chicago, IL: University of Chicago Press, 1994). This was not a univocal position of all seventeenth-century mathematicians, 35. Cavalieri's Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota (Bononiae: Ex Typographia de Duciis, 1653) figures as an early, although tentative, departure from mathematical foundationalism; it was, however, the generally accepted view of rigor in mathematics that caused distrust of the Calculus among scientists and mathematicians. As Douglas Jesseph recently remarked, "they [infinitesimals] ... violated widely accepted philosophical canons which declared the infinite to be incomprehensible and infinitesimal methods inadmissible." In "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215. In addition, Morris Kline recently argued, "many of the English mathematicians, perhaps because they were in the main still tied to the rigor of Greek geometry, distrusted all the work on the calculus." In Mathematical Thought from Ancient to Modem Time (New York: Oxford University Press, 1972), 389.
61. There are two forms of foundationalism, one scientific and concerned primarily with strict empiricism and the other mathematical and concerned primarily with strict adherence to Euclidean geometry. Cristoph Clavius, an important astronomer and mathematician of the early seventeenth century, expresses well the perspective of mathematical foundationalism: "The theorems of Euclid and the rest of the mathematicians, still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstration ... and thus so much do the mathematical disciplines desire, esteem, and foster truth, that they reject not only whatever is false, but even anything mere probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations." C. Clavius, in J. A. Lattis, Between Copernicus and Galileo: Christopher Clavius and the Collapse of Ptolemaic Astronomy (Chicago, IL: University of Chicago Press, 1994). This was not a univocal position of all seventeenth-century mathematicians, 35. Cavalieri's Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota (Bononiae: Ex Typographia de Duciis, 1653) figures as an early, although tentative, departure from mathematical foundationalism; it was, however, the generally accepted view of rigor in mathematics that caused distrust of the Calculus among scientists and mathematicians. As Douglas Jesseph recently remarked, "they [infinitesimals] ... violated widely accepted philosophical canons which declared the infinite to be incomprehensible and infinitesimal methods inadmissible." In "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215. In addition, Morris Kline recently argued, "many of the English mathematicians, perhaps because they were in the main still tied to the rigor of Greek geometry, distrusted all the work on the calculus." In Mathematical Thought from Ancient to Modem Time (New York: Oxford University Press, 1972), 389.
62. There are two forms of foundationalism, one scientific and concerned primarily with strict empiricism and the other mathematical and concerned primarily with strict adherence to Euclidean geometry. Cristoph Clavius, an important astronomer and mathematician of the early seventeenth century, expresses well the perspective of mathematical foundationalism: "The theorems of Euclid and the rest of the mathematicians, still today as for many years past, retain in the schools their true purity, their real certitude, and their strong and firm demonstration ... and thus so much do the mathematical disciplines desire, esteem, and foster truth, that they reject not only whatever is false, but even anything mere probable, and they admit nothing that does not lend support and corroboration to the most certain demonstrations." C. Clavius, in J. A. Lattis, Between Copernicus and Galileo: Christopher Clavius and the Collapse of Ptolemaic Astronomy (Chicago, IL: University of Chicago Press, 1994). This was not a univocal position of all seventeenth-century mathematicians, 35. Cavalieri's Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota (Bononiae: Ex Typographia de Duciis, 1653) figures as an early, although tentative, departure from mathematical foundationalism; it was, however, the generally accepted view of rigor in mathematics that caused distrust of the Calculus among scientists and mathematicians. As Douglas Jesseph recently remarked, "they [infinitesimals] ... violated widely accepted philosophical canons which declared the infinite to be incomprehensible and infinitesimal methods inadmissible." In "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215. In addition, Morris Kline recently argued, "many of the English mathematicians, perhaps because they were in the main still tied to the rigor of Greek geometry, distrusted all the work on the calculus." In Mathematical Thought from Ancient to Modem Time (New York: Oxford University Press, 1972), 389.
63. Cajori, "Appendix", 654.
64. A. De Morgan, "On the Early History of Infinitesimals in England," Philosophical Magazine 4 (1852): 321-330, cited in Cajori, "Appendix," 653.
65. A. De Morgan, "On the Early History of Infinitesimals in England," Philosophical Magazine 4 (1852): 321-330, cited in Cajori, "Appendix," 653.
66. Cajori, "Appendix," 654.
67. Cajori, "Appendix," 654.
68. Cajori, "Appendix," 654.
69. Cajori, "Appendix," 654.
70. Boyer, The History of the Calculus, 213.
71. I. Newton, Isaaci Newtoni Opera quae Exstant Omnia, ed. S. Horsley, vol. 1 (Londini: Excudeba J. Nichols, 1779-1785), 250-251.
72. Newton, Isaaci Newtoni Opera quae Exstant Omnia, ed. S. Horsley vol. 2 (Londini: Excudeba J. Nichols, 1779-1785), 40-41.
73. Newton, Sir Isaac Newton's Mathematical Principles, 38.
74. Newton, Sir Isaac Newton's Mathematical Principles, 38.
75. Kline suggests that "neither Newton or Leibniz clearly understood nor rigorously defined his fundamental concepts." Mathematical Thought, 384. Ian Steward argues that Newton's "logic seems faulty. Over an interval of zero seconds, the position changes by zero, so the calculation becomes 0/0, and every mathematician knows that 0/0 can be anything you like." In "Sweet Nothings," New Scientist 173 (26 Jan 2002): 27. A somewhat commonplace conclusion among contemporary scholars is that Newton and Leibniz found the concept of the infinitesimal extremely useful but did not clearly understand it.
76. Kline suggests that "neither Newton or Leibniz clearly understood nor rigorously defined his fundamental concepts." Mathematical Thought, 384. Ian Steward argues that Newton's "logic seems faulty. Over an interval of zero seconds, the position changes by zero, so the calculation becomes 0/0, and every mathematician knows that 0/0 can be anything you like." In "Sweet Nothings," New Scientist 173 (26 Jan 2002): 27. A somewhat commonplace conclusion among contemporary scholars is that Newton and Leibniz found the concept of the infinitesimal extremely useful but did not clearly understand it.
77. Criticism ranged from Fontenelle's rather mild comments to Berkeley's indictment of infinitesimals as absurdities. Berkeley asked, "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?" in The Analyst, 18. See also Michel Rolle's criticisms from a dispute in the French Academy of Sciences in 1701 over the Calculus's
78. Criticism ranged from Fontenelle's rather mild comments to Berkeley's indictment of infinitesimals as absurdities. Berkeley asked, "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?" in The Analyst, 18. See also Michel Rolle's criticisms from a dispute in the French Academy of Sciences in 1701 over the Calculus's foundations as referenced in M. P. Crosland, Science Under Control: The French Academy of Sciences (New York: Cambridge University Press, 1992).
79. Newton, Sir Isaac Newton's Mathematical Principles, 38.
80. Berkeley supports the reading that Newton's textual changes were a response to the problems caused by the infinitesimal. He writes, "It is curious to observe, what subtlety and skill this great Genius [Newton] employs to struggle with an insuperable Difficulty; and through what Labyrinths he endeavors to escape the Doctrine of Infinitesimals." The Analyst, 8.
81. See Boyer, The History of the Calculus, 210-215.
82. I. Newton, "Newton's Letter to Oldenburg, June 2, 1672," in Isaaci Newtoni Opera quae Exstant Omnia, ed. S. Horsley vol. 4 (Londini: Excudeba J. Nichols, 1779-1785), 314-315.
83. One can demonstrate geometrically the decrease in error as the number of rectangles estimating the area under a curve increases (as with figure 1), but what one cannot do geometrically is take this process to its logical and intuitive conclusion, that is, sum up infinitesimal rectangles. As a result, one cannot prove the Calculus with mathematical rigor as long as one's notion of rigor is based on Euclidean geometry.
84. Newton, Sir Isaac Newton's Mathematical Principles, 397.
85. Boyer, The History of the Calculus, 219.
86. Boyer, The History of the Calculus, 209.
87. G. W. Leibniz, Philosophische Schriften, ed. C. I. Gerhardt, vol. 4 (Berlin: Weidmann, 1875-1890), 218.
88. G. W. Leibniz, "On the Principles of Continuity [1702]," in Leibniz Selections, ed. P.P. Weiner (New York: Charles Scribner's Sons, 1951), 185.
89. G. W. Leibniz, The Early Mathematical Manuscripts of Leibniz, ed. and trans. J. M. Child (Chicago, IL: The Open Court, 1920), 147.
90. Boyer, The History of the Calculus, 213.
91. Leibniz, Philosophische Schriften, vol. 6, 90.
92. Berkeley's The Analyst offers the most substantial critique of this; see also D. Jesseph, "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-244.
93. J. M. Child, "Critical and Historical Notes," in The Early Mathematical Manuscripts of Leibniz, ed. J. M. Child (Chicago, IL: The Open Court, 1920), 149-150.
94. Leibniz, Early Mathematical Manuscripts, 155.
95. Berkeley, The Analyst, 16.
96. Child, "Critical and Historical Notes," 147.
97. Leibniz, "On the Principles of Continuity," 188.
98. G. W. Leibniz, "On Some Philosophical Axioms and Mathematical Fictions [1692]," in Leibniz Selections, 71.
99. J. Benardete, Infinity: An Essay in Metaphysics (Oxford: Clarendon Press, 1964), 18.
100. G. W. Leibniz, Opera Omnia, ed. Louis Dutens, vol. 3 (Genevae: apud Fratres de Tournes, 1768), 500.
101. Rhetoric and fiction are not synonymous; rather, rhetoric as a critical vocabulary can describe the process by which certain fictions (for example, narratives, beliefs, ideas, the concept of the infinitesimal) come to dominate the meaning of truth and the means of human understanding.
102. Leibniz, "On Some Philosophical Axioms and Mathematical Fictions," 72.
103. N. Ya Vilenkin, In Search of Infinity, trans. A. Shenitzer (Boston, MA: Birkhauser, 1995), 12-13.
104. Scientists and mathematicians of the seventeenth century generally accepted Euclidean forms of mathematical proof as sufficient verification because each symbol in Euclidean geometry has a corresponding geometric object to which it refers; that abstract reference to a spatial relation can then easily be mapped onto physical processes and verified empirically. Infinitesimals presented a special problem because they could not be represented spatially.
105. Newton's work testifies to this, as he first completed the Principia using his method of fluxions and then translated his work back into acceptable forms of Euclidean geometry. Cavalieri makes a similar effort (although to a lesser degree) in the second half of his Geometria.
106. Boyer, The History of the Calculus, 227.
107. Stewart, "Sweet Nothings," New Scientist 173 (26 January 2002): 28.
108. See F. Bacon, The Philosophical Works of Francis Bacon, ed. J. M. Robertson (Freeport, NY: Book for Libraries Press, 1970).
109. In The Mathematical Experience Davis and Hersh quote Soloman Feferman, who attests to the decline of foundationalism in mathematics, admitting that "the mathematician at work relies on surprisingly vague intuitions and proceeds by fumbling fits and starts with all too frequent reversals" (357).
110. Rotman, Ad Infinitum, 157.
111. I. B. Cohen, "Commentary," Isaac Newton's Papers and Letters on Natural Philosophy and Related Documents, ed. I. B. Cohen (Cambridge, MA: Harvard University Press, 1978), 274.
112. Vilenkin, In Search of Infinity, 13.
113. Cohen, "Commentary," 274.
114. See M. Foucault's The Order of Things, trans. (New York: Routledge Classics, 2002).
115. Vilenkin, In Search of Infinity, 13.
116. Heidegger, "Modern Science, Metaphysics, and Mathematics," 269.
117. The Bolzmann equation, which describes how a cloud of tiny particles changes in density as the particles interact and helps to predict the motion of stars in the galaxy, was considered questionable by scientists because no one knew if it would suddenly go haywire. No one could prove it was a stable equation. In 1984 Lars Arkeryd used non-stardard analysis to prove the stability of the Bolzmann equation. In addition, non-standard analysis, which is based on infinitesimals, has helped scientists and mathematicians understand Brownian motion, the Jordan theorem, and improve computer graphics.
118. Foundationalism is not dead among scientists and mathematicians, but it is no longer the dominant view.
119. R. Rorty, Contingency, Irony, Solidarity (Cambridge: Cambridge University Press, 1989), 6.