Publicado en Leibniz und Europa, VI. Internationaler Leibniz-Kongress, 361-371.
Gottfried-Wilhelm-Leibniz-Gesellschaft e. V. Hannover., 1994.

The Continuity of Continuity:
A Theme in Leibniz, Peirce and Quine

Begoña Ilarregui y Jaime Nubiola
Universidad de Navarra

The notion of continuity is of vital importance for an understanding of Peirce's metaphysical ideas and his Leibnizian heritage, and affords an insight into the shortcomings of the specializing scientism of our century. In this paper we aim to trace some of the landmarks in the history of the principle of continuity, not only in its original mathematical formulation, but in its broad metaphysical and epistemological scope as a central component of Leibniz's thought. This perspective contrasts with Quine's contemporary scientist naturalism which, though maintaining that a continuity exists between science and philosophy, ends up by reducing the latter to the former. According to this framework, our discussion is divided into four sections: 1) The history of continuity; 2) Continuity in Leibniz; 3) Continuity in Peirce; and 4) Continuity in Quine*.

1. The history of continuity

The origin of the subject of continuity can be traced back to the problem of the continuum in Greek philosophy. The subject of the continuum is in turn related to that of movement. As Raven wrote "theories of motion depend inevitably on theories of the nature of space and time; and two opposed views of space and time were held in antiquity. Either space and time are infinitely divisible, in which case motion is continuous and smooth-flowing; or else they are made up of indivisible minima, in which case motion is what Lee aptly calls 'cinematographic'"1. From this we can see that the subject of the continuous arose out of the subject of infinity and divisibility, just like that of the one and the many, which "becomes evident at the very heart of continuity"2. Zeno of Elea, Leucippus, Democritus, Anaxagoras and Aristotle devote special attention to analyzing this problem. Zeno of Elea, influenced by Parmenides, considered that unity and divisibility, and therefore continuity, must go together. Zeno's paradoxes try to show that this is so. Leucippus and Democritus hold that reality is made up of infinitely small particles: atoms, which are indivisible. As Aristotle says (De Caelo, 4, 303a 5), they "state that the first magnitudes are infinite in number and indivisible in magnitude, and that plurality does not arise out of unity, nor does unity out of plurality, but that all things are generated by the joining or dispersal of these first magnitudes". The atoms, then, are infinitesimal magnitudes considered from a quantitative point of view.

Anaxagoras conceived of matter as being composed of particles, each of which is irreducible (quality) but not indivisible (quantity). Anaxagoras drew on Zeno's paradoxes, especially the so-called dichotomy, and on the reflections of Leucippus and Democritus concerning matter as being made up of atoms. Anaxagoras found that the infinite is in both the large and the small, and reached the conclusion that everything is in everything "in such a way that even though nothing is the same as anything else, there are infinite degrees of variation between one thing and another"3. Frank compares Anaxagoras' view with that of Leibniz, who also "associated infinitesimal calculus with monadology and, while being fully aware of the scarcely discernible differences between individual beings, asserted the principle of universal continuity with the greatest rigour"4.

The complexity and interest of continuity is obvious. The Gordian knot of the problem lies in whether the existence of continuity should be admitted, or reality should be thought of as an aggregate of points and atoms which are related to each other in some accidental way. According to Aristotle, "the extremes of things can be together without even being one, and can be one without necessarily being together" (Phys V, 3, 227a 22-24). That is, continuity does not simply mean that things are contiguous in space, or even successive in time, but that there may be continuity in all that is real. Our present purpose is to trace the notion of continuity in all its philosophical extension. Continuity is to contiguity as the genus is to the species. From this perspective, we approach the continuity of the sciences, which is a consequence of the continuity of reality understood with all the breadth of Aristotle's definition. Interest in this philosophical issue is heightened by the debate raging between continuism and discontinuism in contemporary scientific theories.

2. Continuity in Leibniz

The principle of continuity, or "lex continuitatis", is one of Leibniz's most interesting contributions to western science and philosophy. The principle of continuity is not an Aristotelian first principle; strictly speaking, what constitutes a principle is that something follows it, but nothing precedes it. The principle of continuity is therefore open to proof, as are Leibniz's other principles (principle of identity of the indiscernibles, principle of sufficient reason). This characteristic, as well as giving Leibniz's system a unique dynamism which involves "la réfutation des conceptions mécaniques de Descartes et de ses adeptes"5, removes it from complications such as that of Euclid, in which the principles are immovable, unquestionable pillars of the system. The axioms are the first truths, basic principles which cannot be proven without falling into a process which runs back ad infinitum. In Leibniz's thinking, "the principle is radically different from that of contradiction, it is an architectonic principle, the opposite of which does not imply contradiction but only imperfection or indeterminateness, absence of order, which in Leibniz's thought amounts to the same"6.

Above we mention the Leibnizian principle of the identity of indiscernibles. Both principles are apparently incompatible. The principle of identity thus lies "in the thesis that there are no two substances which resemble one another entirely, differing only numerically, because their 'complete concepts' would otherwise coincide"7. This means that every monad, every substance, contains in it all that is real. The monads therefore become qualitative infinites. It is here, however, that the problems arise. How is it possible to reconcile the theory of monads with the thesis of universal continuity? Does the principle of identity not lead to a kind of atomism, to what might be called a qualitative atomism? On a perfunctory reading we might jump to the conclusion that continuity and identity are irreconcilable.

This problem is a modern version of Anaxagoras' reflections. It is a question of reconciling qualitative identity with divisibility, the basis for infinitesimal calculus. It is precisely here that the mathematical angle of this problem emerges: how are the rules of algebra to be brought into harmony with "geometrical" phenomena? Leibniz extricates himself from this with an appeal to the Divinity: "the opposition of principle of identity and the continuity principle appears, within the philosophy of Leibniz as in Plato's as contrast between God's mind and His activity"8. Mathematics, metaphysics and theodicy flow together.

Leibniz seeks to advance in knowledge. A condition of this progress is flexibility in the principles and propositions of science. Human knowledge is not not made by perfectly logical buildings in which there is no scope for variation. "The basis for all system-construction, the law of continuity, is the principle which guarantees the ability to invent and advance constantly, the progress of the system"9. The image which best conveys Leibniz's idea of knowledge is that of the ocean, which also figures in Peirce's writing: "The entire corpus of the sciences can be regarded as an ocean, which is continuous in all its parts, with no interruption or partition, even though men conceive of parts within it and give them names at their own convenience"10. As Lovejoy wrote "Leibniz had a horror vacui which he was certain that Nature shared. In its internal structure the universe is a plenum, and the law of continuity, the assumption that 'nature makes no leaps', can with absolute confidence be applied in all the sciences, from geometry to biology and psychology"11. The ocean of knowledge has as its correlate the ocean of reality. Far from being fragmented, reality is endowed with a remarkable unity. It would therefore be wrong to approach reality with an attitude tending to favour the insularity of the sciences. The principle of continuity thus becomes a universal law governing knowledge. But the principle of continuity is not the principle of identity, which means that the unity and continuity of the sciences does not break down this distinction. The sciences can be distinguished from one another, even though there are common features. The relationship of one science with another is not a relationship of subsidiarity, nor is it one of implication. "The systematic order of the multiple does not boil down to a deductive or linear order starting from certain obvious principles". The image of the net -which will appear in Quine as a web- is a "better symbol of Leibniz's systematic model than the chain"12. In Leibniz's view, human knowledge resembles a woven fabric rather than a thread. The Leibnizian image explains why the internal coherence of a theory does not suffice as a proof of its truth, even if it is expressed in mathematical language. "Mathematical methods do not enable us to act without observing the world of phenomena. The various mathematically coherent systems should be considered hypotheses just like any others, which are acceptable or not according to whether they stand up to empirical contradiction"13.

In the last analysis, the principle of continuity led Leibniz to put forward an optimistic vision of human knowledge. Reality is given to the human being as an entirety, in which there are no leaps, and is impregnated with an order which is the work of God. "Der höhere begriffliche Gesichtspunkt ist allerdings nicht unmittelbar gegeben, sondern muss erst im Fortschritt der Wissenschaft gewonnen werden. Aber eben darin stellt das Kontinuitätsprinzip dem Denken eine wichtige Aufgabe und erweist sich als positiv fruchtbar für die Entwicklung der Begriffe"14.

3. Continuity in Peirce

The founder of pragmatism and contemporary semiotics, Charles S. Peirce (1839-1914), has quite often been described as "the American Leibniz"15. This attribution is founded not solely on a similar breadth of interest, but also on a special affinity which comes to light very clearly in the notion of continuity. In 1972, Max Fisch, the greatest expert on Peirce of the day, concluded his study of the relations between the two philosophers by indicating that "Peirce identified himself more closely with Leibniz than with any other thinker", as Leibniz, of all the great philosophers, was not only a mathematician, logician and scientist, but also a metaphysician16. Peirce came to believe that his own contribution to logic was on a par with that made by Leibniz17. Peirce has been said to have known Leibniz better than any other American of the time18, but in addition to this, Peirce's ambitions in logic and metaphysics can be understood in a certain sense as being the development of Leibniz's aspirations towards total science, or even as an updated version of Leibniz's Scientia Generalis, integrating the advances made by nineteenth-century science.

Just as the principle of continuity can be viewed as the fulcrum of all Leibnizian thought, for Charles S. Peirce, above all in his thought of the later period, the idea of continuity is "the keystone of the arch", as he wrote to William James on 25 November 1902 (CP 8.257)19. Potter and Shields demonstrated the persistent oscillation of Peirce's reflections between 1860 and 1911 regarding the mathematical definition of continuity. Peirce's attempts show a clear development marked by an increasing sophistication20. Michael Otte recently tried to shed light on Peirce's development of the Leibnizian principle of continuity, taking Peirce's own texts with reference to three examples: individual existence, the forms of philosophy, and geometrical demonstration21. Ketner and Putnam also recently showed how Peirce's idea of continuity, though beginning with the metaphysico-mathematical analysis of the continuity of the line, culminates in a metaphysical generalization which will strike the reader of our day as completely unexpected. Not only does Peirce state that there are many continuous phenomena in nature, and many continuous functions in physics, but he also establishes "a metaphysics which identifies ideal continuity with the notion of inexhaustible and creative possibility"22.

"Of all conceptions Continuity is by far the most difficult for Philosophy to handle." These were the opening words in Peirce's eighth Cambridge Lecture of 1898, which was entitled "The Logic of Continuity"23. The notion of continuity, along with the notions of chance, as really operative in the universe, and love, as "the great evolutionary agency of the universe", are the three key doctrines of Peirce's comprehensive evolutionary cosmology, which he developed in the last two decades of his life24. In the advertisement of his projected Principles of Philosophy: or Logic, Physics and Psychics considered as a unity in the light of the Nineteenth Century, Peirce announced that in the first volume "a definite affinity is traced between all these ideas [of the century in very different areas], and is shown to lie in the principle of continuity. The idea of continuity traced through the history of the Human Mind, and shown to be the great idea which has been working itself out"25. In accordance with this, Peirce liked to call all his philosophy synechism, "because it rests on the study of continuity"26.

Peirce wishes to show the leading methodological role of the law of continuity in philosophical reflection and in the metaphysical evolution of the universe. Peircean reflection on continuity stems from mathematics and geometry, but the metaphysical generalization of the principle of continuity to the human mind and the universe has its origin in the inadequacy of mechanicist scientific explanations: "the universe is not a mere mechanical result of the operation of blind law. The most obvious of all its characters cannot be so explained. It is the multitudinous facts of all experience that show us this; but that which has opened our eyes to these facts is the principle of fallibilism" (CP 1.162). The idea of continuity, which is the "leading idea" of differential calculus and of all the useful branches of mathematics, and which has a decisive role in scientific thought, is the master key which unlocks the arcana of philosophy (CP 1.163). Peirce asserts that continuity involves infinity in the strictest sense, and infinity goes beyond the possibility of direct experience. Peirce's weightiest reason for believing the reality of continuity is interaction between minds. How can one mind act upon another mind? This is an unintelligible fact within a mechanicist scientific framework.

For Peirce there is a natural affinity between the principle of continuity and the doctrine of fallibilism: "The principle of continuity is the idea of fallibilism objectified. For fallibilism is the doctrine that our knowledge is never absolute but always swims, as it were, in a continuum of uncertainty and of indeterminacy. Now the doctrine of continuity is that all things so swim in continua" (CP 1.171). The advance of knowledge, the development of science and philosophy, are evolution, growth, a gain in relational generality27. Logical analysis applied to mental phenomena shows that there is but one law of mind, that ideas tend to spread continuously and to affect other ideas, and in this spreading they lose intensity as they gain generality (CP 6.104). "Wherever ideas come together they tend to weld into general ideas; and wherever they are generally connected, general ideas govern the connection; and these general ideas are living feelings spread out" (CP 6.143). To say that mental phenomena are governed by law is to say that there is a living idea, a conscious continuum of feeling, which pervades mental phenomena, and to which they are docile (CP 6.152). When the principle of continuity is embraced, no kind of explanation of things will satisfy you except that they grew. While the infallibilist naturally thinks that everything always was substantially as it is now and that the laws of nature are absolutely blind and inexplicable, the fallibilist thinks that there is no reason to think that they are absolute and he "asks may these forces of nature not be somehow amenable to reason? May they not have naturally grown up? (...) If all things are continuous, the universe must be undergoing a continuous growth from non-existence to existence." (CP 1.175).

As Hausman has stressed, in Peirce's thought the idea of spontaneity and radical creativity is interwoven with his view on continuity: "Both continuity and spontaneity are constitutive of the universe through the function of infinitesimals." The extensively quoted texts from Peirce suggest his Leibnizian heritage and the strong consistency of his philosophy understood as evolutionary realism28.

4. Continuity in Quine

"Science is self-conscious common sense. And philosophy in turn, as an effort to get clearer on things, is not to be distinguished in essential points of purpose and method from good and bad science"29. This statement suggests that Quine in some way shares Leibniz's vision of the continuity of the sciences, but in fact this is illusory. Quine's view of science is imbued with epistemological naturalism which makes it radically different from that of Leibniz. Quinean naturalism, as Susan Haack notes, was born of "his repudiation of foundationalism, which motivates his naturalism"30. Quine turns against the foundationalist intentions of Carnap, whose attempt to establish a basis for the sciences led him to assert a privileged status for mathematics and logic. Mathematics and logic are foundational because they are a priori. Quine refuses to accept that mathematics and logic should have any such privileged status because "it is nonsense, and the root of much nonsense, to speak of a linguistic component and a factual component in the truth of any individual statement"31. In fact, Quine defends gradualism in the sciences: the entire body of the sciences forms a whole which appears before the tribunal of experience in complete unity. This is what is meant by Quinean holism. Thus "the correction of each statement, -except for observational statements- turns out to be an argument against the existence of a priori knowledge, a form of knowledge which tenets like that of the distinction between analytic and synthetic, or of true statements by convention, purport to explain"32. The rejection of foundationalism, and therefore of the concept of the system of the sciences as Aufbau, led Quine to opt for a continuist naturalism between philosophy, science and all our beliefs about the world.

It would seem, then, that Quine inherits and makes his own the Leibnizian project of the continuity of knowledge. Both the Quinean perspective, and the epistemological presuppositions on which his naturalism rests, cut short this attempt. The Quinean metaphor which most resembles the image of the ocean of the sciences is that of the force-field: "Total science is like a field of force whose boundary conditions are experience"33. The statements which make up the field of science mingle together, forming webs, which means that one assertion cannot be taken isolated and subjected to experience; it is the whole of science which is thus subjected. "This vision of science as a framework, a network to which everything that is stated belongs, is the basic aspect of Quine's generalized holism"34. Despite the apparent analogy with Leibniz, Quine's point of view has a naturalist slant.

According to Haack35, if Quine is analyzed, two versions come to light within his naturalism, a modest naturalism, the "consequence of Quine's repudiation of a priori", which is manifested in "his gradualism and his affinity with fallibilism", and a "scientific naturalism that does not follow from his critique of apriorism". These two ways of using the term "naturalism" lead to two different goals. From the point of view of modest naturalism, the idea of continuity between science, philosophy, common sense and all human activity is admissible without any kind of reductionism. Yet from the standpoint of scientific naturalism the only way out is via reductionist scientism, which leaves no room for any system of the sciences. Both usages of the word "naturalism" are supported by Quine's ambiguous use of the word "science". In the narrower usage, "science" refers exclusively to the natural sciences, whereas speaking more broadly it is taken to mean our empirical beliefs in general, including common sense: "Sciences, after all, differ from common sense only in degree of methodological sophistication"36. In the last analysis, the problem lies in whether or not to admit the autonomy of particular disciplines, which is the sine qua non of continuity. There is no case for establishing continuity without distinction. At bottom, what here is on the stage is the possibility of a first philosophy, not necessarily in the foundationalist sense, -of a solid foundation on which the system of the sciences can be built up-, but rather as an ontology which goes beyond epistemology. At times, Quine seems to reject this possibility: "Knowledge, mind and meaning are part of the same world that they have to do with, and they are to be studied in the same empirical spirit that animates natural science. There is no place for a prior philosophy"37."I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat -a boat which, to revert to Neurath's figure as I so often do, we can rebuild at sea while staying afloat in it. There is no external vantage point, no first philosophy"38.

There is no privileged position for philosophy. "What I have been discussing under the head of philosophy is what I call scientific philosophy (...). By this vague heading I do not exclude philosophical studies of moral and aesthetic values"39. Quinean scientism reflects the contemporary standard scientistic view: there are no grounds for conceiving of a philosophy without science. But as Haack remarked recently "the continuity theme is not only different from, but actually incompatible with, the scientistic theme. Philosophy cannot be both continuous with, and at the same time assimilated to, or displaced by science"40. The reason why the Leibnizian project is curtailed is, in the final analysis, the conception of philosophy as subsidiary to science. Perhaps it is time to come back to Peirce's principle of continuity to recover the dream of a Leibnizian Scientia Generalis.


* This joint paper on the occasion of the VI Leibniz Congress is the first fruits of the doctoral dissertation project of B. Ilarregui on the continuity between science and philosophy in Quine and of J. Nubiola's study on Peircean communication theory.

1. G. S. Kirk and J. E. Raven: The Presocratic Philosophers, Cambridge University Press, London, 1964, p. 292.

2. J. J. Rodríguez Rosado: El problema del continuo y la gnoseología, Real Monasterio, El Escorial, 1965, p. 1.

3. R. Mondolfo: El infinito en el pensamiento de la antigüedad clásica, Imán, Buenos Aires, 1952, pp. 246-247.

4. E. Frank: Plato und die sogenannten Pythagoreer, Niemeyer, Halle, 1923; R. Mondolfo, loc. cit.

5. E. Stipanic: "L' idee de continuité chez G. W. Leibniz (1646-1716) et chez R. J. Boskovic (1711-1787)", in Leibniz Tradition und Aktualität, V. Internationaler Leibniz-Kongress, Hannover, 1988, p. 949.

6. A. Currás Rábade: "La teoría leibniziana del método", Anales del Seminario de Metafísica, VII (1972), p. 146.

7. M. Otte: "Two Principles of Leibniz's Philosophy in relation to the History of Mathematics", Theoria, VIII (1993), p. 113.

8. Ibid., p.115.

9. A. Currás Rábade: "La teoría leibniziana del método", p. 143.

10. C , 530-531.

11. A. O. Lovejoy: The Great Chain of Being, Harvard University Press, Cambridge, Mass., 1936, p. 58.

12. M. Serres: Le systèm de Leibniz et ses modèles mathématiques, Presses Universitaires de France, Paris, 1968, pp. 13-19.

13. I. Murillo: El sentido de la ciencia en Leibniz, Universidad Complutense, Madrid, 1983, p. 36.

14. E. Cassirer: Leibniz System in seinen Wissenschaftlichen Grunlagen, Elwert, Marburg, 1902, p. 176.

15. P. Weiss: "Charles Sanders Peirce", in Dictionary of American Biography, edited by D.Malone, Scribner, New York, vol. 14, 403.

16. M. Fisch: "Peirce and Leibniz", in Peirce, Semeiotic, and Pragmatism, edited by K. L. Ketner and Ch. J. W. Kloesel, Indiana University Press, Bloomington, p. 258.

17. Peirce's ms L 387 y L 482. Schröder himself, writing in 1896 to Paul Carus, editor of The Monist, about the review of the third volume of his Algebra der Logik, declared that Peirce's reputation would one day shine alongside that of Leibniz or Aristotle. Quoted by Fisch: "Peirce and Leibniz", p. 259.

18. G. W. Leibniz: Philosophical Papers and Letters, edited by L. E. Loemker, Reidel, Dordrecht, 1969, p. 57. Quoted by Fisch: "Peirce and Leibniz", p. 259.

19. Peirce writings are quoted as usual as CP referring to the Collected Papers of Charles Sanders Peirce, vols. 1-8, edited by Ch. Hartshorne and P. Weiss, Harvard University Press, Cambridge, Mass., 1931-1958.

20. V. Potter and P. B. Shields: "Peirce's definitions of continuity", Transactions of the Charles S. Peirce Society, XIII (1977), pp. 20-34.

21. M. Otte: "Das Prinzip der Kontinuität", Mathematische Semesterberichte, XXXIX (1992), pp. 105-125.

22. C. S. Peirce: Reasoning and the Logic of Things. The Cambridge Conferences Lectures of 1898, edited by K. L. Ketner, Harvard University Press, Cambridge, Mass., 1992, p. 37.

23. C. S. Peirce: Reasoning and the Logic of Things, p. 242.

24. N. Houser and Ch. J. W. Kloesel (eds.): The Essential Peirce, vol. I, Bloomington, Indiana University Press, 1992, p. xxii.

25. C. S. Peirce: Reasoning and the Logic of Things, p. 13.

26. C. S. Peirce: Reasoning and the Logic of Things, p. 261.

27. C. S. Peirce: Reasoning and the Logic of Things, p. 258.

28. C. R. Hausman: Charles S. Peirce's Evolutionary Philosophy, Cambridge University Press, New York, 1993, p. 190.

29. W. V. Quine: Word and Object, MIT Press, Cambridge, Mass., 1960, pp. 3-4.

30. S. Haack: “The Two Faces of Quine’s Naturalism”, Synthese, 94 (1993), p. 339.

31. W. V. Quine: From a Logical Pont of View, Harvard University Press, Cambridge, Mass., 1961, 2nd ed., p. 42.

32. A. Pérez Fustegueras: La epistemología de Quine, Fundación Juan March, Madrid, 1988, p. 17.

33. W. V. Quine: From a Logical Point of View, p. 42.

34. T. Calvo: "Experiencia y holismo: El Deúteros Plous de Quine", in Symposium Quine, edited by J. J. Acero and T. Calvo, Universidad de Granada, 1987, p. 55.

35. S. Haack: “The Two Faces of Quine’s Naturalism”, pp. 335-356, and Evidence and Inquiry, Blackwell, Oxford, 1993, pp. 118-138.

36. W. V. Quine: Ontological Relativity and Other Essays, Columbia University Press, New York, 1969, p. 129.

37. W. V. Quine: Ontological Relativity, p. 26.

38. W. V. Quine: Ontological Relativity, pp. 126-127.

39. W. V. Quine: Theories and Things, Harvard University Press, Cambridge, Mass., 1981, p. 193.

40. Personal communication. Cfr. S. Haack: Evidence and Inquiry, chap. 6.

Última actualización: 27 de agosto 2009

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