Leibniz on the indefinite as infinite

Review of Metaphysics

Volumn 51, Issue 4, 1998, Pages 848-873

  Bassler, O Bradley ^a  

^a UNIVERSITY OF GEORGIA   (United States)

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References (124)

1. The term "infinitesimal calculus" is that most commonly used by Leibniz to refer to his invention, but it is unfortunately misleading, since the fundamental object of the Leibnizian calculus is not the infinitesimal but rather the differential. See the excellent presentation in H. J. M. Bos, "Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus," Archive for History of Exact Sciences 14 (1974): 1-90, especially 16.
2. Here Leibniz later added a note: "Rather (N.B.) one proves by this only that a series is endless."
3. G. W. Leibniz, De Summa Rerum: Metaphysical Papers, 1675-1676, trans. G. H. R. Parkinson (New Haven: Yale University Press, 1992), 31-3.
4. Leibniz, De Summa Rerum, 33. The end of this reading ("possible") is conjectural, but in any case the point is clear from the first clause of the sentence.
5. Leibniz, De Summa Rerum, 31.
6. Ibid.
7. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
8. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
9. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
10. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
11. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
12. Leibniz's philosophical notes from 1676 have been universally recognized as notoriously difficult and perplexing. The general tendency in recent secondary literature has been to attempt to systematize the views Leibniz presents here. This has been attempted by G. H. R. Parkinson in his paper "Leibniz's De Summa Rerum: A Systematic Approach," Studia Leibnitiana 16 (1986): 132-51; and also by Christia Mercer in her joint article with R. C. Sleigh, Jr., "Metaphysics: The early period to the Discourse on Metaphysics," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press), 67-123. See especially 84-107. Mercer, for example, organizes her investigation of Leibniz's early philosophy in terms of certain "basic metaphysical assumptions" about which he is "rarely explicit," but which are "discernible as the implicit premises and unstated assumptions of his arguments . . ."; Mercer, "Metaphysics," 72. Yet, in my opinion, this desire to systematize is most ill-chosen for understanding a period in which a thinker's views are in such radical flux; what is needed instead is an investigation taking as its point of departure motivating concerns. Indeed, Mercer's drive to systematize leads her so far as to declare at one point that "since Leibniz maintains his characteristic silence about his deep motivations, the case for this proposal must be circumstantial, based on clues that Leibniz leaves along the way"; Mercer, "Metaphysics," 93. But at least one main motivation behind Leibniz's metaphysical reflections is perfectly clear: the questions with which he has been presented by the success of the infinitesimal calculus. Another major motivating concern is Leibniz's desire to come to terms with Cartesianism. Mercer has little to say about either of these in her discussion of the Paris period writings. On Leibniz's 1675 De Cartessi erroribus, see the excellent article by Yvon Belaval, "Premières Animadversions sur les 'Principes' de Descartes," reprinted in his Études leibniziennes (Paris: Éditions Gallimard, 1976), 57-85, especially p. 62 and the following pages for a discussion of the indefinite in Descartes's thought.
13. Galileo Galilei, Dialogue Concerning Two New Sciences, trans. Stillman Drake (Madison: University of Wisconsin Press, 1974), 43. make use of this translation despite the greater availability and felicity of the translation of Henry Crew and Alfonso de Salvio (New York: Dover, 1954) due to the greater sensitivity of the former translation to nuances of Galileo's linguistic usage. For ease of reference, however, I will refer to the Galileo Opera numbers which are given in both translations; for example, the above reference will now be given as Galileo, Opera, 81.
14. Galileo Galilei, Dialogue Concerning Two New Sciences, trans. Stillman Drake (Madison: University of Wisconsin Press, 1974), 43. make use of this translation despite the greater availability and felicity of the translation of Henry Crew and Alfonso de Salvio (New York: Dover, 1954) due to the greater sensitivity of the former translation to nuances of Galileo's linguistic usage. For ease of reference, however, I will refer to the Galileo Opera numbers which are given in both translations; for example, the above reference will now be given as Galileo, Opera, 81.
15. Ibid.
16. In his article, "Galileo's Theory of Indivisibles: Revolution or Compromise," Journal of the History of Ideas 37 (1976), 571-88, A. Mark Smith asserts that Galileo's position regarding the continuum represents a modified Aristotelianism. In a critical, but unfortunately obscure, passage Smith seems to interpret Galileo's declaration that the number of parts in the continuum is indefinite as indicating a potentially indefinite division of the continuum, concluding that "an indefinite 'intermediate term' (every assigned number), neither finite nor infinite, must serve to describe the Aristotelian continuum of spatial and temporal processes" (577). Here Smith omits, however, Salviati's explicit declaration that the parts may be taken to be either actual or potential; so far as I can see this passage greatly vitiates his interpretation.
17. René Descartes, Correspondence, in The Philosophical Writings of Descartes, 3 vols., trans. John Cottingham, Robert Stoothoff, Dugald Murdoch, and Anthony Kenny (Cambridge: Cambridge University Press, 1991), 3:319-20. The original letter may be found in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, new presentation in 12 vols. (Paris: J. Vrin, 1964-76), 5:51-2. This passage is briefly discussed with reference to the doctrine of Nicholas of Cusa in Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins University Press, 1957), 6. Descartes's conception of the indefinite is discussed at length in the fifth chapter of Koyré's book, "Indefinite Extension or Infinite Space," 110-24.
18. René Descartes, Correspondence, in The Philosophical Writings of Descartes, 3 vols., trans. John Cottingham, Robert Stoothoff, Dugald Murdoch, and Anthony Kenny (Cambridge: Cambridge University Press, 1991), 3:319-20. The original letter may be found in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, new presentation in 12 vols. (Paris: J. Vrin, 1964-76), 5:51-2. This passage is briefly discussed with reference to the doctrine of Nicholas of Cusa in Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins University Press, 1957), 6. Descartes's conception of the indefinite is discussed at length in the fifth chapter of Koyré's book, "Indefinite Extension or Infinite Space," 110-24.
19. René Descartes, Correspondence, in The Philosophical Writings of Descartes, 3 vols., trans. John Cottingham, Robert Stoothoff, Dugald Murdoch, and Anthony Kenny (Cambridge: Cambridge University Press, 1991), 3:319-20. The original letter may be found in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, new presentation in 12 vols. (Paris: J. Vrin, 1964-76), 5:51-2. This passage is briefly discussed with reference to the doctrine of Nicholas of Cusa in Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins University Press, 1957), 6. Descartes's conception of the indefinite is discussed at length in the fifth chapter of Koyré's book, "Indefinite Extension or Infinite Space," 110-24.
20. René Descartes, Correspondence, in The Philosophical Writings of Descartes, 3 vols., trans. John Cottingham, Robert Stoothoff, Dugald Murdoch, and Anthony Kenny (Cambridge: Cambridge University Press, 1991), 3:319-20. The original letter may be found in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, new presentation in 12 vols. (Paris: J. Vrin, 1964-76), 5:51-2. This passage is briefly discussed with reference to the doctrine of Nicholas of Cusa in Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins University Press, 1957), 6. Descartes's conception of the indefinite is discussed at length in the fifth chapter of Koyré's book, "Indefinite Extension or Infinite Space," 110-24.
21. There is, however, a subtle, but critical, point to be made here. In "On the Secrets of the Sublime," Leibniz declares that the number of finite numbers cannot be infinite; what he does not consider is the possibility of nonetheless admitting that there is an (indefinite) infinity of finite numbers. That is, although Leibniz recognizes the finite whole numbers as progressing indefinitely he does not consider understanding this indefinite progression as infinite. From the perspective of the Leibniz of the late 1690's maintaining that there is an infinity of finite numbers is weaker than maintaining that there is an infinite number; as we will see, while in the late 1690's Leibniz will accept that there is an infinity of finite numbers he will remain agnostic on whether there are infinite numbers. These issues will be considered in more detail below in the context of Leibniz's correspondence with Bernoulli.
22. G. W. Leibniz, New Essays on Human Understanding, trans. and ed. Peter Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1981), 157.
23. Ibid.
24. The distinction is standard in the medieval scholastic tradition. Ockham, for example, says: "Categorematic terms have a definite and fixed signification, as for instance the word 'man' (since it signifies all men) and the word 'animal' (since it signifies all animals), and the word 'whiteness' (since it signifies all occurrences of whiteness). Syncategorematic terms, on the other hand, as 'every', 'none', 'some', 'whole', 'besides', 'only', 'in so far as', and the like, do not have a fixed and definite meaning, nor do they signify things distinct from the things signified by categorematic terms. Rather, just as, in the system of numbers, zero standing alone does not signify anything, but when added to another number gives it a new signification; so likewise a syncategorematic term does not signify anything, properly speaking, but when added to another term, it makes it signify something or makes it stand for some thing or things in a definite manner, or has some other function with regard to a categorematic term"; William of Ockham, Philosophical Writings: A Selection, trans. Philotheus Boehner, O.F.M., rev. Stephen Brown (Indianapolis: Hackett, 1990), 51. Thus Leibniz's indefinite infinite is syncategorematic in the sense that the term "infinite" only signifies when applied to finite numbers, that is, "more than any given finite number." Leibniz also aligns the distinction between syncategorematic and categorematic with the distinction between distributive and collective uses of the term "infinite": Leibniz's position is that there is no collective quantitative infinite. See, for example, G. W. Leibniz, Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, (Berlin, 1875-90; reprint, Hildesheim: Olms, 1961) 2:314. The attempt to use the distinction between the syncategorematic and the categorematic to resolve sophismata pertaining to the infinite is traditional as well. An excellent discussion of one such account, that of Albert of Saxony, is given in Joël Biard's article, "Albert de Saxe et les sophismes de l'infini," in Sophisms in Medieval Logic and Grammar, ed. Stephen Read (Dordrecht: Kluwer, 1993), 288-303 (hereafter cited as "Albert de Saxe"). Biard focuses on the sophisma, "Infinita sunt finita" ("Infinites are finite"), treated by Albert of Saxony but going back at least to the De solutionibus sophismatum, circa 1200 (see Biard, "Albert de Saxe," 288). Henri de Gand proves this sophism as follows: "Infinita sunt finita. Probatio: duo sunt finita, tria sunt finita, et sic in infinitum; ergo infinita sunt finita," ("Infinites are finites. Proof: two are finite, three are finite, and thus to infinity; therefore infinites are finite"): for references, see Biard, "Albert de Saxe," 291 n. 18. A close analogue of this proof recurs in Leibniz's correspondence with Bernoulli, and will be discussed below.
25. The distinction is standard in the medieval scholastic tradition. Ockham, for example, says: "Categorematic terms have a definite and fixed signification, as for instance the word 'man' (since it signifies all men) and the word 'animal' (since it signifies all animals), and the word 'whiteness' (since it signifies all occurrences of whiteness). Syncategorematic terms, on the other hand, as 'every', 'none', 'some', 'whole', 'besides', 'only', 'in so far as', and the like, do not have a fixed and definite meaning, nor do they signify things distinct from the things signified by categorematic terms. Rather, just as, in the system of numbers, zero standing alone does not signify anything, but when added to another number gives it a new signification; so likewise a syncategorematic term does not signify anything, properly speaking, but when added to another term, it makes it signify something or makes it stand for some thing or things in a definite manner, or has some other function with regard to a categorematic term"; William of Ockham, Philosophical Writings: A Selection, trans. Philotheus Boehner, O.F.M., rev. Stephen Brown (Indianapolis: Hackett, 1990), 51. Thus Leibniz's indefinite infinite is syncategorematic in the sense that the term "infinite" only signifies when applied to finite numbers, that is, "more than any given finite number." Leibniz also aligns the distinction between syncategorematic and categorematic with the distinction between distributive and collective uses of the term "infinite": Leibniz's position is that there is no collective quantitative infinite. See, for example, G. W. Leibniz, Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, (Berlin, 1875-90; reprint, Hildesheim: Olms, 1961) 2:314. The attempt to use the distinction between the syncategorematic and the categorematic to resolve sophismata pertaining to the infinite is traditional as well. An excellent discussion of one such account, that of Albert of Saxony, is given in Joël Biard's article, "Albert de Saxe et les sophismes de l'infini," in Sophisms in Medieval Logic and Grammar, ed. Stephen Read (Dordrecht: Kluwer, 1993), 288-303 (hereafter cited as "Albert de Saxe"). Biard focuses on the sophisma, "Infinita sunt finita" ("Infinites are finite"), treated by Albert of Saxony but going back at least to the De solutionibus sophismatum, circa 1200 (see Biard, "Albert de Saxe," 288). Henri de Gand proves this sophism as follows: "Infinita sunt finita. Probatio: duo sunt finita, tria sunt finita, et sic in infinitum; ergo infinita sunt finita," ("Infinites are finites. Proof: two are finite, three are finite, and thus to infinity; therefore infinites are finite"): for references, see Biard, "Albert de Saxe," 291 n. 18. A close analogue of this proof recurs in Leibniz's correspondence with Bernoulli, and will be discussed below.
26. The distinction is standard in the medieval scholastic tradition. Ockham, for example, says: "Categorematic terms have a definite and fixed signification, as for instance the word 'man' (since it signifies all men) and the word 'animal' (since it signifies all animals), and the word 'whiteness' (since it signifies all occurrences of whiteness). Syncategorematic terms, on the other hand, as 'every', 'none', 'some', 'whole', 'besides', 'only', 'in so far as', and the like, do not have a fixed and definite meaning, nor do they signify things distinct from the things signified by categorematic terms. Rather, just as, in the system of numbers, zero standing alone does not signify anything, but when added to another number gives it a new signification; so likewise a syncategorematic term does not signify anything, properly speaking, but when added to another term, it makes it signify something or makes it stand for some thing or things in a definite manner, or has some other function with regard to a categorematic term"; William of Ockham, Philosophical Writings: A Selection, trans. Philotheus Boehner, O.F.M., rev. Stephen Brown (Indianapolis: Hackett, 1990), 51. Thus Leibniz's indefinite infinite is syncategorematic in the sense that the term "infinite" only signifies when applied to finite numbers, that is, "more than any given finite number." Leibniz also aligns the distinction between syncategorematic and categorematic with the distinction between distributive and collective uses of the term "infinite": Leibniz's position is that there is no collective quantitative infinite. See, for example, G. W. Leibniz, Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, (Berlin, 1875-90; reprint, Hildesheim: Olms, 1961) 2:314. The attempt to use the distinction between the syncategorematic and the categorematic to resolve sophismata pertaining to the infinite is traditional as well. An excellent discussion of one such account, that of Albert of Saxony, is given in Joël Biard's article, "Albert de Saxe et les sophismes de l'infini," in Sophisms in Medieval Logic and Grammar, ed. Stephen Read (Dordrecht: Kluwer, 1993), 288-303 (hereafter cited as "Albert de Saxe"). Biard focuses on the sophisma, "Infinita sunt finita" ("Infinites are finite"), treated by Albert of Saxony but going back at least to the De solutionibus sophismatum, circa 1200 (see Biard, "Albert de Saxe," 288). Henri de Gand proves this sophism as follows: "Infinita sunt finita. Probatio: duo sunt finita, tria sunt finita, et sic in infinitum; ergo infinita sunt finita," ("Infinites are finites. Proof: two are finite, three are finite, and thus to infinity; therefore infinites are finite"): for references, see Biard, "Albert de Saxe," 291 n. 18. A close analogue of this proof recurs in Leibniz's correspondence with Bernoulli, and will be discussed below.
27. The distinction is standard in the medieval scholastic tradition. Ockham, for example, says: "Categorematic terms have a definite and fixed signification, as for instance the word 'man' (since it signifies all men) and the word 'animal' (since it signifies all animals), and the word 'whiteness' (since it signifies all occurrences of whiteness). Syncategorematic terms, on the other hand, as 'every', 'none', 'some', 'whole', 'besides', 'only', 'in so far as', and the like, do not have a fixed and definite meaning, nor do they signify things distinct from the things signified by categorematic terms. Rather, just as, in the system of numbers, zero standing alone does not signify anything, but when added to another number gives it a new signification; so likewise a syncategorematic term does not signify anything, properly speaking, but when added to another term, it makes it signify something or makes it stand for some thing or things in a definite manner, or has some other function with regard to a categorematic term"; William of Ockham, Philosophical Writings: A Selection, trans. Philotheus Boehner, O.F.M., rev. Stephen Brown (Indianapolis: Hackett, 1990), 51. Thus Leibniz's indefinite infinite is syncategorematic in the sense that the term "infinite" only signifies when applied to finite numbers, that is, "more than any given finite number." Leibniz also aligns the distinction between syncategorematic and categorematic with the distinction between distributive and collective uses of the term "infinite": Leibniz's position is that there is no collective quantitative infinite. See, for example, G. W. Leibniz, Die Philosophische Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt, (Berlin, 1875-90; reprint, Hildesheim: Olms, 1961) 2:314. The attempt to use the distinction between the syncategorematic and the categorematic to resolve sophismata pertaining to the infinite is traditional as well. An excellent discussion of one such account, that of Albert of Saxony, is given in Joël Biard's article, "Albert de Saxe et les sophismes de l'infini," in Sophisms in Medieval Logic and Grammar, ed. Stephen Read (Dordrecht: Kluwer, 1993), 288-303 (hereafter cited as "Albert de Saxe"). Biard focuses on the sophisma, "Infinita sunt finita" ("Infinites are finite"), treated by Albert of Saxony but going back at least to the De solutionibus sophismatum, circa 1200 (see Biard, "Albert de Saxe," 288). Henri de Gand proves this sophism as follows: "Infinita sunt finita. Probatio: duo sunt finita, tria sunt finita, et sic in infinitum; ergo infinita sunt finita," ("Infinites are finites. Proof: two are finite, three are finite, and thus to infinity; therefore infinites are finite"): for references, see Biard, "Albert de Saxe," 291 n. 18. A close analogue of this proof recurs in Leibniz's correspondence with Bernoulli, and will be discussed below.
28. Leibniz, New Essays, 157.
29. Translated in G. W. Leibniz, Philosophical Papers and Letters, ed. and trans. Leroy Loemker (Boston: Reidel, 1969), 31. The original passage may be found in Leibniz, Die Philosophische Schriften, 2:314. See also the discussion in Loemker's introduction to Philosophical Papers, 31, and also 514 n. 2 and 541 n. 21.
30. Translated in G. W. Leibniz, Philosophical Papers and Letters, ed. and trans. Leroy Loemker (Boston: Reidel, 1969), 31. The original passage may be found in Leibniz, Die Philosophische Schriften, 2:314. See also the discussion in Loemker's introduction to Philosophical Papers, 31, and also 514 n. 2 and 541 n. 21.
31. Translated in G. W. Leibniz, Philosophical Papers and Letters, ed. and trans. Leroy Loemker (Boston: Reidel, 1969), 31. The original passage may be found in Leibniz, Die Philosophische Schriften, 2:314. See also the discussion in Loemker's introduction to Philosophical Papers, 31, and also 514 n. 2 and 541 n. 21.
32. Leibniz, New Essays, 158.
33. Ibid.
34. Ibid.
35. Ibid.
36. For a fuller discussion of Leibniz's chapter on infinity in the New Essays, and especially on Leibniz's opposition to Locke's account of the infinite, see Antonio Lamarra, "Leibniz on Locke on infinity," in L'Infinito in Leibniz: Problemi e Terminologia, ed. Antonio Lamarra (Rome: Edizioni dell'Ateneo, 1990), 173-91. For another discussion of Leibniz on the indefinite, see Hans Poser, "Die Idee des Unendlichen und die Dinge. Infinitum und immensum bei Leibniz," in L'Infinito in Leibniz, 225-33. Poser's excellent discussion is delimited, however, by his exclusion of the debates regarding infinitesimals from his purview.
37. For a fuller discussion of Leibniz's chapter on infinity in the New Essays, and especially on Leibniz's opposition to Locke's account of the infinite, see Antonio Lamarra, "Leibniz on Locke on infinity," in L'Infinito in Leibniz: Problemi e Terminologia, ed. Antonio Lamarra (Rome: Edizioni dell'Ateneo, 1990), 173-91. For another discussion of Leibniz on the indefinite, see Hans Poser, "Die Idee des Unendlichen und die Dinge. Infinitum und immensum bei Leibniz," in L'Infinito in Leibniz, 225-33. Poser's excellent discussion is delimited, however, by his exclusion of the debates regarding infinitesimals from his purview.
38. For a fuller discussion of Leibniz's chapter on infinity in the New Essays, and especially on Leibniz's opposition to Locke's account of the infinite, see Antonio Lamarra, "Leibniz on Locke on infinity," in L'Infinito in Leibniz: Problemi e Terminologia, ed. Antonio Lamarra (Rome: Edizioni dell'Ateneo, 1990), 173-91. For another discussion of Leibniz on the indefinite, see Hans Poser, "Die Idee des Unendlichen und die Dinge. Infinitum und immensum bei Leibniz," in L'Infinito in Leibniz, 225-33. Poser's excellent discussion is delimited, however, by his exclusion of the debates regarding infinitesimals from his purview.
39. Galileo, Opera, 78-85. It is important to note that Galileo has Salviati repeatedly deliver a series of disclaimers along with his remarks about the infinite. Salviati, for example, says he is "going to produce a fantastic idea of mine which, if it concludes nothing necessarily, will at least by its novelty occasion some wonder"; Galileo, Opera, 73. In another passage Salviati speaks of "marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the natures of these have no necessary relation between them"; Galileo, Opera, 83. See also passages at Galileo, Opera, 96 and 105.
40. Galileo, Opera, 78-85. It is important to note that Galileo has Salviati repeatedly deliver a series of disclaimers along with his remarks about the infinite. Salviati, for example, says he is "going to produce a fantastic idea of mine which, if it concludes nothing necessarily, will at least by its novelty occasion some wonder"; Galileo, Opera, 73. In another passage Salviati speaks of "marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the natures of these have no necessary relation between them"; Galileo, Opera, 83. See also passages at Galileo, Opera, 96 and 105.
41. Galileo, Opera, 78-85. It is important to note that Galileo has Salviati repeatedly deliver a series of disclaimers along with his remarks about the infinite. Salviati, for example, says he is "going to produce a fantastic idea of mine which, if it concludes nothing necessarily, will at least by its novelty occasion some wonder"; Galileo, Opera, 73. In another passage Salviati speaks of "marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the natures of these have no necessary relation between them"; Galileo, Opera, 83. See also passages at Galileo, Opera, 96 and 105.
42. Galileo, Opera, 78-85. It is important to note that Galileo has Salviati repeatedly deliver a series of disclaimers along with his remarks about the infinite. Salviati, for example, says he is "going to produce a fantastic idea of mine which, if it concludes nothing necessarily, will at least by its novelty occasion some wonder"; Galileo, Opera, 73. In another passage Salviati speaks of "marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites; for the natures of these have no necessary relation between them"; Galileo, Opera, 83. See also passages at Galileo, Opera, 96 and 105.
43. G. W. Leibniz, Sämtliche Schriften und Briefe, 2d ser. (Berlin: Akademie Verlag, ongoing), 1:226.
44. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
45. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
46. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
47. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
48. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
49. Ibid. As Margaret Baron has emphasized, Leibniz's interpretation of zero is critical to the mathematical results he presents in the Accessio. See her The Origins of the Infinitesimal Calculus (New York: Dover, 1987), 270-2, for a readable presentation of the mathematical results of the Accessio. There is also an elementary presentation of this material without explicit reference to the Accessio in C. H. Edwards, The Historical Development of the Calculus (New York: Springer Verlag, 1979), 234-9. For a more detailed account the reader may consult Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity (Cambridge: Cambridge University Press, 1974), 12-22. Finally, Leibniz himself gives an account of these results in the "Historia et Origo calculi differentialis" of 1714 (G. W. Leibniz, Mathematische Schriften, ed. C. I. Gerhardt [reprint, Hildesheim: Olms, 1971], 3:321-2), which is available in English translation in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: The Open Court Publishing Company, 1920), 22-58.
50. Leibniz, Sämtliche Schriften, 6th ser., 2:482-3. For a discussion of this proof in the Demonstratio propositionum primarum see Ezequiel de Olaso, "Scepticism and the infinite," in L'Infinito in Leibniz, 95-118, especially 107 and the following pages. See also the footnote below regarding an analogous presentation of this argument in the correspondence with Bernoulii.
51. Leibniz, Sämtliche Schriften, 6th ser., 2:482-3. For a discussion of this proof in the Demonstratio propositionum primarum see Ezequiel de Olaso, "Scepticism and the infinite," in L'Infinito in Leibniz, 95-118, especially 107 and the following pages. See also the footnote below regarding an analogous presentation of this argument in the correspondence with Bernoulii.
52. Galileo, Opera, 78.
53. In those contexts in which Leibniz wishes to suppress the more subtle, and technical, issues surrounding the consideration of the infinite, he often substitutes for the proof that there is no greatest number a proof that a fastest motion is absurd. For example, in the Meditations on Knowledge, Truth, and Ideas, Leibniz proves the absurdity of a fastest motion as follows: "For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail's motion is not the fastest, contrary to the hypothesis"; G. W. Leibniz, Philosophical Essays, trans. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), 25. The original text may be found in Leibniz, Die Philosophische Schriften, 4:424. This proof is closely related to debates about what would happen were space to be limited. For a discussion in a variety of early modern contexts, see Koyré, The Infinite Universe.
54. In those contexts in which Leibniz wishes to suppress the more subtle, and technical, issues surrounding the consideration of the infinite, he often substitutes for the proof that there is no greatest number a proof that a fastest motion is absurd. For example, in the Meditations on Knowledge, Truth, and Ideas, Leibniz proves the absurdity of a fastest motion as follows: "For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail's motion is not the fastest, contrary to the hypothesis"; G. W. Leibniz, Philosophical Essays, trans. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), 25. The original text may be found in Leibniz, Die Philosophische Schriften, 4:424. This proof is closely related to debates about what would happen were space to be limited. For a discussion in a variety of early modern contexts, see Koyré, The Infinite Universe.
55. In those contexts in which Leibniz wishes to suppress the more subtle, and technical, issues surrounding the consideration of the infinite, he often substitutes for the proof that there is no greatest number a proof that a fastest motion is absurd. For example, in the Meditations on Knowledge, Truth, and Ideas, Leibniz proves the absurdity of a fastest motion as follows: "For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail's motion is not the fastest, contrary to the hypothesis"; G. W. Leibniz, Philosophical Essays, trans. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), 25. The original text may be found in Leibniz, Die Philosophische Schriften, 4:424. This proof is closely related to debates about what would happen were space to be limited. For a discussion in a variety of early modern contexts, see Koyré, The Infinite Universe.
56. For an alternative account of Leibniz's debate with Bernoulli, centered on the question of the existence of infinitesimals, see George MacDonald Ross, "Are There Real Infinitesimals in Leibniz's Metaphysics?" in L'Infinito in Leibniz, 125-41.
57. Leibniz presents Bernoulli with a proof of this axiom in a part of the correspondence earlier than that under consideration here. See Leibniz's letter of 23 August 1696 in Leibniz, Mathematische Schriften, 3:321-2. This presentation largely follows the argument given in 1671 cited above. For a logical analysis of the argument as presented in this letter to Bernoulli, see H. G. Knapp, "Some Logical Remarks on a Proof by Leibniz," Ratio 12 (1970): 125-37.
58. Leibniz presents Bernoulli with a proof of this axiom in a part of the correspondence earlier than that under consideration here. See Leibniz's letter of 23 August 1696 in Leibniz, Mathematische Schriften, 3:321-2. This presentation largely follows the argument given in 1671 cited above. For a logical analysis of the argument as presented in this letter to Bernoulli, see H. G. Knapp, "Some Logical Remarks on a Proof by Leibniz," Ratio 12 (1970): 125-37.
59. Leibniz, Mathematische Schriften, 3:535. Unless otherwise noted, translations from this source are my own.
60. Leibniz, Mathematische Schriften, 3:536.
61. Ibid.
62. The situation regarding the existence of infinitesimals bears considerable structural analogy to the situation concerning the ontological proof of the existence of God. Leibniz criticized the Cartesian ontological argument on the grounds that it merely proves that if God's existence is possible, then it is actual, but it does not give a proof of the possibility of God's existence. Leibniz was much preoccupied with such a proof of the possibility of God's existence during 1976, and returns to it later in writings of 1678, the mid-1680's, and 1714. The issues involved in the provision of such a proof are extremely intricate and consequently well beyond the bounds of this paper. The reader is referred to the detailed discussion of these issues in Robert Merrihew Adams's Leibniz: Determinist, Theist, Idealist (Oxford: Oxford University Press, 1994). See especially 141 and the following pages.
63. Leibniz, Mathematische Schriften, 3:529.
64. Leibniz, Mathematische Schriften, 3:529.
65. This argument is in fact precisely the one that Galileo uses to motivate his position that the number of parts in the continuum is neither finite nor infinite: ". . . the quantified parts in the continuum, whether potentially or actually there, do not make its quantity greater or less. But it is clear that quantified parts actually contained in their whole, if they are infinitely many, make it of infinite magnitude; whence infinitely many quantified parts cannot be contained even potentially except in an infinite magnitude. Thus in the finite, infinitely many quantified parts cannot be contained either actually or potentially"; Galileo, Opera, 80-1. Galileo commits to the existence of infinite magnitudes in a way that Leibniz will not, but he cannot see any way to account for the number of parts in the continuum in terms of such an infinite magnitude. Consequently he assigns to them a magnitude intermediate between the finite and the infinite, analogous to what Leibniz refers to as the indefinite, but which I suggest might more appropriately be referred to in Galileo's case as the parafinite. Leibniz, on the one hand, cannot accept the Galilean infinite, since it fails the axiom of identity, but on the other hand understands the indefinite as infinite. Why though, we may ask, would Galileo believe that infinitely many quantified parts cannot be contained (either potentially or actually) except in an infinite magnitude? Consider, for example, the case of a line segment one unit long divided into successive parts, disjoint except for their endpoints, of successive lengths 1/2, 1/4, 1/8,. . . . Is this not a perfectly good example of a finite magnitude containing an infinite number of quantified parts? The answer, presumably, is "no," because the number of parts given is, in fact, neither finite nor infinite, but indefinite.
66. This is of course a sufficient, but not a necessary condition, as Leibniz was well aware. Already in 1673 Leibniz took the series 1/1 + 1/2 + 1/3 + 1/4 + . . . to diverge. For a discussion see Hofmann, Leibniz in Paris, 21.
67. Leibniz, Mathematische Schriften, 3:536.
68. Leibniz, Mathematische Schriften, 3:555.
69. The other choice is to assume that Bernoulli changes his position in this particular regard. However, Bernoulli does not give any explicit indication that he concedes Leibniz's point that an infinitude of finite parts need not necessarily sum to an infinite magnitude. As I point out in a later note, there are other regards in which Bernoulli's position does seem to shift - specifically, regarding the probability that infinitesimals exist. But it would do him little good to shift in this regard, for if he were to admit that an infinite number of finite magnitudes may sum to a finite magnitude, then there would be little reason to insist that an infinity of terms being given requires that infinitesimals exist. This, of course, is precisely what Leibniz is trying to get Bernoulli to agree to. Since he does not do it, this should serve as indirect confirmation that Bernoulli would take the number of finite terms to be finite. One might also cite Bernoulli's assertion in his letter of 1698 that "if there are no infinitesimals in nature, then certainly the number of terms will be finitely many [tantum finitus]. . ."; Bernoulli, in Leibniz, Mathematische Schriften, 3:555. Although this could be taken as supporting evidence, it does not tell us directly what the situation would be with the finite terms were there indeed to exist infinitesimals in nature. As such, it is not thoroughly conclusive. A more serious objection, I believe, is that it may not have occurred to Bernoulli to ask how many finite terms there are: the emphasis, after all, is on the infinitesimals filling out the actual infinity of terms. I believe this latter assertion is true; yet if Bernoulli did not consciously consider the question concerning the number of finite terms, it is nonetheless quite close to the surface of the issues being debated. I am not suggesting that it makes no difference whether Bernoulli recognized this question explicitly - in fact, I think it makes a great deal of difference, and it is probably quite significant that he does not broach this issue explicitly. But on the other hand, posing the question explicitly may help us to focus more clearly on Bernoulli's conception of the infinite, and at worst we are filling out Bernoulli's position in a way that would not have occurred to him.
70. This latter would be close in certain regards to the Galilean notion of the indefinite, which in a note above I have suggested could appropriately be characterized as parafinite. Yet as I point out below, Bernoulli is unwilling to admit any tertium quid between the finite and the infinite. On this basis I find it unlikely that Bernoulli would be willing to countenance such an indeterminate finite number.
71. The article by Ross, "Are there real infinitesimals?" considers these dynamical issues in detail.
72. Bernoulli, in Leibniz, Mathematische Schriften, 3:545.
73. Leibniz, Mathematische Schriften, 3:551.
74. As Hofmann points out, Leibniz's correspondents failed to see the effectiveness of this axiom, and "not one of the correspondents to whom Leibniz sent his demonstration [of this axiom] approved of it; in particular Johann Bernoulli made it clear in his reply of September 22, 1696 that he considered the conclusion to be circular" (Leibniz in Paris, 14 n. 13). Since it depends on the use of this axiom, it is small wonder, then, that Leibniz's assertion that a largest magnitude is impossible is so infrequently accompanied by an explicit proof.
75. Leibniz, Mathematische Schriften, 3:563.
76. Leibniz, Mathematische Schriften, 3:566.
77. Leibniz, Mathematische Schriften, 3:571.
78. The shortness of Bernoulli's response is indeed a likely indication of frustration. In his previous letter, Leibniz defers his response regarding the argument of Bernoulli cited in the text above until a postscript, in which Leibniz begins by declaring that he "almost forgot" to respond to Bernoulli! After Bernoulli's brief response (the passage just cited in the text), Leibniz again responds to Bernoulli, but Bernoulli does not continue the discussion. Leibniz attempts a second time to rejuvenate the discussion, but without any success.
79. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
80. In fact, Leibniz first shifted to a commitment to the number of parts in
81. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
82. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
83. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
84. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
85. In fact, Leibniz first shifted to a commitment to the number of parts in the continuum being indefinite during an intensive reading of Galileo's Two New Sciences in 1672 or 1673. That he made this shift due to his reading of Galileo is clear from the notes Leibniz made regarding the "First Day" of Galileo's Two New Sciences. Here Leibniz notes that to the question whether the parts of the continuum are finite or infinite, Galileo responds that they are neither, but are rather more than any given number. To this resumé of Galileo's position that the parts of the continuum are greater than any given magnitude Leibniz appends in parentheses: "or indefinite"; Leibniz, Sämtliche Schriften, 6th ser., 3:168. Hence while Leibniz rejects as impossible what Galileo takes to be infinite, he will nonetheless appropriate Galileo's "indefinite" and eventually understand this as infinite. Previously, in the Theoria Motus Abstracti of 1671 Leibniz had rejected the indefinite division of the continuum, saying that "the indefinite of Descartes is not in the thing but in the thinker"; Leibniz, Philosophical Papers, 139; original text in Leibniz, Sämtliche Schriften, 6th ser., 2:264. discuss this transition in Leibniz's position and the subsequent development of his metaphysics during the early Paris period in a paper currently in progress. A brief discussion is also given in Herbert Breger's essay, "Das Kontinuum bei Leibniz," in L'Infinito in Leibniz, 53-67. See especially 60-1. Breger's penetrating analysis of Leibniz's views on the nature of the continuum in this and related articles has considerable bearing on the issues I discuss in this paper. In addition to the article cited above, see also "Leibniz, Weyl und das Kontinuum," in Studia Leibnitiana Supplementa 26 (1986): 316-30 and "Leibniz" Einführung des Transzendenten," in Studia Leibnitiana Sonderheft 14 (1986): 119-32.
86. Leibniz, New Essays, 158.
87. It might be objected that what Leibniz refers to here is not infinitely small magnitudes, but minimal magnitudes in opposition to maximal wholes. This interpretation is, I believe, extremely implausible, however, because Leibniz goes on to speak of just these infinite wholes and their infinitesimal counterparts as what do have a place in geometrical calculation sub specie imaginationis. I believe the conclusion is inescapable that Leibniz is here declaring the impossibility of the existence of infinitesimals.
88. Leibniz, Philosophical Papers, 542-6; original text in Leibniz, Mathematische Schriften, 4:91-5 and 4:104-6.
89. Leibniz, Philosophical Papers, 542-6; original text in Leibniz, Mathematische Schriften, 4:91-5 and 4:104-6.
90. Leibniz, Philosophical Papers, 542-6; original text in Leibniz, Mathematische Schriften, 4:91-5 and 4:104-6.
91. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
92. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
93. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
94. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
95. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
96. Leibniz, Philosophical Papers, 542-3; original text in Leibniz, Mathematische Schriften, 4:91. identify what we can conceive with what can be conceived simpliciter. But this is clearly repugnant from the Cartesian perspective. Nonetheless, it is also important to point out that the debate concerning infinitesimals in the Paris Academy involved factionalization in ways which cannot be understood solely on the basis of the intellectual affiliations of the disputants. Douglas Jesseph makes much of the points at issue here in his discussion of Leibniz in his "Philosophical Theory and Mathematical Practice in the Seventeenth Century," Studies in History and Philosophy of Science 20 (1989): 215-44. See especially 238-43. Although Jesseph is right to stress Leibniz's desire to achieve rigor in his presentation of the infinitesimal calculus, his suggestion that we can use Leibniz's attitude concerning the segregation of metaphysical from mathematical problems as a resolution of the "ambiguity" in Leibniz's use of infinitesimals (he refers here specifically to the article of Earman, for references see footnote below) seems to me hand-waving at best and circular at worst. See, for example, "In the Leibnizian scheme, true mathematical principles will be found acceptable on any resolution of the metaphysical problems of the infinite. Thus, Leibniz's concern with matters of rigor leads him to propound a very strong thesis indeed, namely no matter how the symbols "dx" and "dy" are interpreted, the basic procedures of the calculus can be vindicated"; Jesseph, "Philosophical Theory," 243. The problem, of course, is that such a "radical thesis" itself requires a metaphysical justification. In the last clause Jesseph presumably means "no matter how the symbols 'dx' and 'dy' are interpreted metaphysically." For a discussion emphasizing the problems already involved at the mathematical level, see Bos, "Differentials," 53-66, and also Eberhard Knobloch, "L'infinie dans les mathématiques de Leibniz," in L'Infinito in Leibniz, 33-51.
97. Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.
98. Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.
99. On the basis of such passages John Earman proposes that Leibniz is in fact speaking of two different sorts of infinitesimals, and that his denial of one sort is in fact a "cover" for his commitment to the other. See his "Infinities, Infinitesimals, and Indivisibles: The Leibnizian Labyrinth," in Studia Leibnitiana 7 (1975): 236-51. Although I disagree with many of the details of Earman's analysis, and although I ultimately reject the distinction he attempts to make, it nonetheless seems to me that the point of his distinction is closely related to many of the issues which I am attempting to discuss in this paper.
100. Failing, that is, a proof that infinite magnitudes exist.
101. Pierre Costabel has tried to explain why this project was never drafted in his "De Scientia Infiniti," in Leibniz 1646-1716, aspects de l'homme et de l'oeuvre (Paris: Aubier-Montagne, 1966), 105-17. Although I would tend to agree with Costabel that the greatest barriers to this project were of a practical (and perhaps, as Costabel also suggests, psychological) nature, I would suggest that there may also have been metaphysical reasons why this project was never pursued past the preliminary stages, much less completed.
102. For example, in a letter from 1716 to Samuel Masson: "The infinitesimal calculus is useful with respect to the application of mathematics to physics; however, that is not how I claim to account for the nature of things. For I consider infinitesimal quantities to be useful fictions"; Leibniz, Philosophical Essays, 230; original text in Leibniz, Die Philosophische Schriften, 6:629.
103. For example, in a letter from 1716 to Samuel Masson: "The infinitesimal calculus is useful with respect to the application of mathematics to physics; however, that is not how I claim to account for the nature of things. For I consider infinitesimal quantities to be useful fictions"; Leibniz, Philosophical Essays, 230; original text in Leibniz, Die Philosophische Schriften, 6:629.
104. In his Eloge de M. Leibnitz (1716, edited 1718), Fontenelle writes: "He [Leibniz] understood this infinity of orders of the infinitely small always infinitely smaller the one than the other, and that in geometrical rigor, and the greatest geometers have adopted this idea in all its rigor. It seems however that he then scared himself, and that he believed that these different orders of the infinitely small were only incomparable magnitudes due to their extreme inequality, as are a grain of sand and the globe of the earth, the earth and the sphere including the planets, and so forth. But this would only be a great inequality, but not infinite, such as one establishes in this system . . ." (translation mine). I have used the citation of this passage given in Michel Blay's article, "Du fondement du calcul différentiel au fondement de la science du mouvement dans les «Elémens de la géométrie de l'infini» de Fontenelle," in Studia Leibnitiana Sonderheft 17 (1989): 99-122. The quotation from Fontenelle translated above is given at 100 n. 6.
105. In his Eloge de M. Leibnitz (1716, edited 1718), Fontenelle writes: "He [Leibniz] understood this infinity of orders of the infinitely small always infinitely smaller the one than the other, and that in geometrical rigor, and the greatest geometers have adopted this idea in all its rigor. It seems however that he then scared himself, and that he believed that these different orders of the infinitely small were only incomparable magnitudes due to their extreme inequality, as are a grain of sand and the globe of the earth, the earth and the sphere including the planets, and so forth. But this would only be a great inequality, but not infinite, such as one establishes in this system . . ." (translation mine). I have used the citation of this passage given in Michel Blay's article, "Du fondement du calcul différentiel au fondement de la science du mouvement dans les «Elémens de la géométrie de l'infini» de Fontenelle," in Studia Leibnitiana Sonderheft 17 (1989): 99-122. The quotation from Fontenelle translated above is given at 100 n. 6.
106. In the letter to Varignon, Leibniz uses this to defend his calculus as a calculus of the infinite: "Yet we must not imagine that this explanation debases the science of the infinite and reduces it to fictions, for there always remains a 'syncategorematic' infinite, as the Scholastics say"; Leibniz, Philosophical Papers, 542; original in Leibniz, Mathematische Schriften, 4:93.
107. In the letter to Varignon, Leibniz uses this to defend his calculus as a calculus of the infinite: "Yet we must not imagine that this explanation debases the science of the infinite and reduces it to fictions, for there always remains a 'syncategorematic' infinite, as the Scholastics say"; Leibniz, Philosophical Papers, 542; original in Leibniz, Mathematische Schriften, 4:93.
108. Leibniz, New Essays, 158.
109. Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.
110. Leibniz, Philosophical Papers, 543; original text in Leibniz, Mathematische Schriften, 4:91.
111. Ibid.
112. See, for example, Leibniz, Philosophical Papers, 511; original text in Leibniz, Mathematische Schriften, 3:551. Bernoulli, on the other hand, is inclined to think that infinitesimals do exist, although he agrees with Leibniz that they have not been demonstrated to exist. See his letter of August/September 1698 in Leibniz, Mathematische Schriften, 3:539. Bernoulli's declaration in this letter that it is more probable that infinitesimals exist seems to mark a shift from the position he took in his previous letter of 16/26 August 1698. There he reports astonishment at the fact that he takes Leibniz to suggest that "it [is] possible for such as act among us to be infinite and infinitely small . . ."; Bernoulli, in Leibniz, Mathematische Schriften, 3:529. Here too in this earlier letter Bernoulli grants neither a proof for or against such infinite and infinitely small things, but here his attitude toward them seems, on the whole, skeptical.
113. See, for example, Leibniz, Philosophical Papers, 511; original text in Leibniz, Mathematische Schriften, 3:551. Bernoulli, on the other hand, is inclined to think that infinitesimals do exist, although he agrees with Leibniz that they have not been demonstrated to exist. See his letter of August/September 1698 in Leibniz, Mathematische Schriften, 3:539. Bernoulli's declaration in this letter that it is more probable that infinitesimals exist seems to mark a shift from the position he took in his previous letter of 16/26 August 1698. There he reports astonishment at the fact that he takes Leibniz to suggest that "it [is] possible for such as act among us to be infinite and infinitely small . . ."; Bernoulli, in Leibniz, Mathematische Schriften, 3:529. Here too in this earlier letter Bernoulli grants neither a proof for or against such infinite and infinitely small things, but here his attitude toward them seems, on the whole, skeptical.
114. See, for example, Leibniz, Philosophical Papers, 511; original text in Leibniz, Mathematische Schriften, 3:551. Bernoulli, on the other hand, is inclined to think that infinitesimals do exist, although he agrees with Leibniz that they have not been demonstrated to exist. See his letter of August/September 1698 in Leibniz, Mathematische Schriften, 3:539. Bernoulli's declaration in this letter that it is more probable that infinitesimals exist seems to mark a shift from the position he took in his previous letter of 16/26 August 1698. There he reports astonishment at the fact that he takes Leibniz to suggest that "it [is] possible for such as act among us to be infinite and infinitely small . . ."; Bernoulli, in Leibniz, Mathematische Schriften, 3:529. Here too in this earlier letter Bernoulli grants neither a proof for or against such infinite and infinitely small things, but here his attitude toward them seems, on the whole, skeptical.
115. See, for example, Leibniz, Philosophical Papers, 511; original text in Leibniz, Mathematische Schriften, 3:551. Bernoulli, on the other hand, is inclined to think that infinitesimals do exist, although he agrees with Leibniz that they have not been demonstrated to exist. See his letter of August/September 1698 in Leibniz, Mathematische Schriften, 3:539. Bernoulli's declaration in this letter that it is more probable that infinitesimals exist seems to mark a shift from the position he took in his previous letter of 16/26 August 1698. There he reports astonishment at the fact that he takes Leibniz to suggest that "it [is] possible for such as act among us to be infinite and infinitely small . . ."; Bernoulli, in Leibniz, Mathematische Schriften, 3:529. Here too in this earlier letter Bernoulli grants neither a proof for or against such infinite and infinitely small things, but here his attitude toward them seems, on the whole, skeptical.
116. Leibniz, Philosophical Papers, 544; original in Leibniz, Mathematische Schriften, 4:93-4.
117. Leibniz, Philosophical Papers, 544; original in Leibniz, Mathematische Schriften, 4:93-4.
118. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
119. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
120. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
121. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
122. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
123. Leibniz, Mathematische Schriften, 4:110 (translation mine). This passage is discussed by Ross, "Are There Real Infinitesimals?" 127, although the translation he gives of the above passage slightly obscures the strength of what Leibniz says, which is that the proof he thinks he can give is strong enough to show that such beings are impossible. No proof is indicated, howeven my conjecture is that this is because Leibniz is in fact reassessing the strength of his old proof that maxima and minima are impossible. Admittedly my conjecture is based only on circumstantial evidence. (Curiously, although Ross discusses both the Bernoulli correspondence and the correspondence with Varignon, he does not explicitly discuss the shift in Leibniz's position.) In his book Architectonique Disjonctive Automates Systémiques et Idéalité Transcendantale dans l'Oeuvre de G. W. Leibniz (Paris: J. Vrin, 1986), 292, André Robinet provides the text of the sketch of a proof in which Leibniz attempted to show that infinitely small magnitudes are impossible. The proof is, however, crossed out and not resumed elsewhere. Robinet says that this proof is the proof referred to in the letter to Varignon, but he does not give any indication why he thinks so. The proof involves attempting to take successive mean proportionals between an infinitely small and a finite quantity; the details are not entirely clear to me. In any case, the proof appears to have been abandoned, whereas Leibniz's commitment to the impossibility of infinitely small quantities appears to have grown. This indicates to me that the ultimate explanation of Leibniz's position must be sought elsewhere, as I have attempted to do in considering Leibniz's views on the indefinite as infinite. My view is that the fundamental question is what Leibniz takes to be the relation between the indefinite and the infinite, on the one hand, and the actual and the potential on the other. The author who comes closest to recognizing the relevance of these issues (although proceeding along a different route) is Paolo Mancosu, in his book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), 144-5. Mancosu asks whether the limited infinite, such as an infinitely long line terminated on both ends, "must necessarily be read as an actual infinite." Mancosu then responds: "This is a difficult question and, unfortunately, Leibniz does not address the issue directly. But I believe he has the resources to handle these results in terms of potential infinity"; Mancosu, Philosophy of Mathematics, 144-5. In the mathematical results Mancosu considers I would agree, but I would add that it is precisely this conflict which leads Leibniz to reject the existence (and even the possibility) of the limited infinite. In his article, "Das Kontinuum bei Leibniz," Breger also tends to think Leibniz rejects a limited infinite, but declines to pose the question what relation this bears to the distinction between the actually versus the potentially infinite: "Whether the opposition between infinitum terminatum and infinitum interminatum has something to do with the opposition between potential and actual infinity need not be decided here"; "Das Kontinuum bei Leibniz," 65, my translation. I would argue, however, that the Leibnizian treatment of the indefinite as infinite stands or falls with this question.
124. Leibniz, Die Philosophische Schriften, 6:542.