Abstract
The rapid development of mathematical analysis in the eighteenth century had not concealed the fact that its underlying concepts not only lacked rigorous definition but were even (e.g. in the case of differentials and infinitesimals) of doubtful logical character. The lack of precision in the notion of continuous function—still vaguely understood as one which could be represented by a formula and whose associated curve could be smoothly drawn—had led to doubts concerning the validity of a number of procedures in which that concept figured. For example when Lagrange had formulated his method for “algebraizing” the calculus he had implicitly assumed that every continuous function could be expressed as an infinite series by means of Taylor’s theorem. Early in the nineteenth century this and other assumptions began to be questioned, thereby initiating an inquiry into what was meant by a function in general and by a continuous function in particular.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Bolzano (1950), p. 123.
- 2.
Ibid., p. 30.
- 3.
Ibid. pp. 129–131.
- 4.
Ibid., p. 109.
- 5.
Ibid., p. 124. Bolzano’s calculation here is formally equivalent to the procedure adopted in smooth infinitesimal analysis in which squares and higher powers of infinitesimals (but not the infinitesimals themselves) are set to zero: see Chapter 10 below.
- 6.
Ibid., p. 125.
- 7.
Ibid., p. 127.
- 8.
Ibid., pp. 139–40.
- 9.
Ibid., pp. 172–73.
- 10.
Quoted in Kline (1972), p. 951.
- 11.
This had been previously given by Bolzano .
- 12.
Kline (1972), p. 963.
- 13.
Quoted ibid., pp. 951–2.
- 14.
Boyer (1959), p. 283.
- 15.
Boyer (1959), p. 284. Fisher (1978) argues that here and there in his work Cauchy did “argue directly with infinitely small quantities treated as actual infinitesimals.” More recently Bair et al. (to appear) have shown that in a number of places Cauchy “used [infinitesimals] neither as variable quantities nor as sequences but rather as numbers”.
- 16.
The term “differential geometry ” was introduced in 1894 by the Italian mathematician Luigi Bianchi (1856–1928).
- 17.
Cauchy too made important contributions to differential geometry , but he was much more circumspect than his fellows in the use of infinitesimals.
- 18.
In the words of Hermann Weyl :
The principle of gaining knowledge of the external world from the behaviour of its infinitesimal parts is the mainspring of the theory of knowledge in infinitesimal physics as in Riemann’s geometry ,and, indeed, the mainspring of all the eminent work of Riemann (1950, p. 92).
- 19.
On the Hypotheses which Lie at the Foundations of Geometry, published 1868, translated in A Source Book in Mathematics, Smith (1959), pp. 411–25.
- 20.
Ibid., pp. 412–13.
- 21.
Ibid., pp. 424–25.
- 22.
According to Hobson (1907, p. 22), “the term ‘arithmetization’ is used to denote the movement which has resulted in placing analysis on a basis free from the idea of measurable quantity , the fractional, negative, and irrational numbers being so defined that they depend ultimately upon the conception of integral number.”
- 23.
It would perhaps be too much of a conceptual stretch to regard arithmetization as a kind of neo-Pythagoreanism.
- 24.
Boyer (1959), p. 286. This property—now known as density in itself—later came to be regarded as too weak to characterize a continuum; see the remarks on Cantor below.
- 25.
The concept of function had by this time been greatly broadened: in 1837 Dirichlet suggested that a variable y should be regarded as a function of the independent variable x if a rule exists according to which, whenever a numerical value of x is given, a unique value of y is determined. (This idea was later to evolve into the set-theoretic definition of function as a set of ordered pairs.) Dirichlet’s definition of function as a correspondence from which all traces of continuity had been purged, made necessary Weirstrass’s independent definition of continuous function .
- 26.
The notion of uniform continuity for functions was later introduced (in 1870) by Heine: a real valued function f is uniformly continuous if for any ε > 0 there is δ > 0 such that |f(x) – f(y)| < ε for all x and y in the domain of f with |x – y| < δ. In 1872 Heine proved the important theorem that any continuous real-valued function defined on a closed bounded interval of real numbers is uniformly continuous.
- 27.
Ewald (1999), p. 770.
- 28.
Ibid., p. 771.
- 29.
Ibid., p. 771.
- 30.
Ibid., p. 772.
- 31.
Ibid.
- 32.
Ibid.
- 33.
Ibid.
- 34.
Ibid., p. 773.
- 35.
Ibid., p. 776.
- 36.
Ibid., p. 778.
- 37.
The following observations in 1900 of A. Schoenflies, one of set theory’s earliest contributors, are pertinent in this connection. He saw the emergence of set theory, and the eventual discretization of mathematics, as issuing primarily from the effort to clarify the function concept, and only secondarily as the result of the struggle to tame the continuum:
The development of set theory had its source in the effort to produce clear analyses of two fundamental mathematical concepts, namely the concepts of argument and of function. Both concepts have undergone quite essential changes through the course of years. The concept of argument, specifically that of independent variable, originally coincided with [the] no further defined, naive concept of the geometric continuum; today it is common everywhere, to allow as domain of arguments any chosen value-set or point-set, which one can make up out of the continuum by rules defined in any way at all. Even more decisive is the change which has befallen the notion of function. This change may be tied internally to Fourier’s theorem, that a so-called arbitrary function can be represented by a trigonometric series; externally it finds expression in the definition which goes back to Dirichlet, which treats the general concept of function, to put it briefly, as equivalent to an arbitrary table … It was left to Cantor to find the concepts which proved proper for a methodical investigation, and which made it possible to force infinite sets under the dominion of mathematical formulas and laws … (Quoted in McLarty (1988), p. 83.)
- 38.
Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Mathematische Annalen 5, 123–132.
- 39.
Dauben (1979), p. 38.
- 40.
Quoted ibid., p. 39.
- 41.
Quoted ibid., p. 40.
- 42.
Quoted ibid., p. 40.
- 43.
Ewald (1999), p. 844.
- 44.
Kline (1972), p. 981.
- 45.
In fact nondenumerably many, although Cantor did not make this fact explicit until later.
- 46.
Quoted in Dauben (1979), p. 53.
- 47.
Ewald (1999), p. 850.
- 48.
“I see it, but I don’t believe it.” Ibid., p. 860.
- 49.
Ibid.
- 50.
Ibid., p. 863.
- 51.
Ibid.
- 52.
Ibid., pp. 863–4.
- 53.
Ibid., p. 863.
- 54.
Ibid., p. 864.
- 55.
Ibid..
- 56.
Ibid.
- 57.
Ibid.
- 58.
At the same time Cantor ’s recognition that the problem of defining dimension depends on the possibility of restricting the correspondences between the structures concerned is indicative of his great prescience as a mathematician. For the idea of taking as primary data correspondences between mathematical structures, as opposed to the structures themselves, was, through category theory, to prove seminal in the mathematics of the twentieth century.
- 59.
Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84, 242–58.
- 60.
Mächtigkeit. Cantor mentions that the origin of this concept lies in the work of the Swiss geometer Jakob Steiner , who used it in his study of invariants of conic sections.
- 61.
Quoted in Dauben (1979), p. 59.
- 62.
Ibid., p. 60.
- 63.
Ibid., p. 60.
- 64.
Dauben (1979), pp. 70–72
- 65.
Ibid., pp. 72–76. The matter was only placed beyond doubt in 1911 when Brouwer showed definitively that the dimension of a Euclidean space is a topological invariant.
- 66.
In modern terminology, spaces like A are arcwise connected.
- 67.
Quoted in Dauben (1979), p. 86.
- 68.
Ibid.
- 69.
Ewald (1999), p. 895. Cantor refers specifically here to the concept of the natural numbers.
- 70.
Loc. cit.
- 71.
Loc. cit.
- 72.
Or, perhaps, vice-versa.
- 73.
Ibid., p. 918.
- 74.
Ibid., p. 896.
- 75.
Cantor goes on to assert, in a famous pronouncement, “the essence of mathematics lies precisely in its freedom.”.
- 76.
Here Cantor inserts a characteristic footnote, in which he puts forward a theory of the formation of concepts reminiscent of Plato’s doctrine of anamnesis:
The procedure in the correct formulation of concepts is in my opinion everywhere the same. One posits a thing with properties that at the outset is nothing other than a name or a sign A, and then in an orderly fashion gives it different, or even infinitely many, intelligible predicates whose meaning is known on the basis of ideas already at hand, and which may not contradict one another. In this way one determines the connection of A to the concepts that are already at hand, in particular to related concepts. If one has reached the end of this process, then one has met all the preconditions for awakening the concept A which slumbered inside us, and it comes into being accompanied by the intrasubjective reality which is all that can be demanded of a concept; to determine its transient meaning is then a matter for metaphysics. (Ewald 1999, p. 918}
- 77.
Ewald (1999), p. 896.
- 78.
“from parts divisible without end”. It may strike one as odd to find the atomists Leucippus and Democritus here bracketed with Aristotle as upholders of divisionism ; presumably Cantor is implicitly distinguishing the formers’ material atomism from their probable, or at least possible, assent to theoretical divisionism. Cf. Heath (1981), vol. I, p. 181, where the idea that Democritus may have upheld geometric indivisibles is dismissed on the grounds that “Democritus was too good a mathematician to have anything to do with such a theory.”
- 79.
Ewald (1999), p. 903.
- 80.
Ibid., p. 904.
- 81.
Ibid.
- 82.
This, Cantor’s continuum hypothesis , is actually stated in terms of the transfinite ordinal numbers introduced in previous sections of the Grundlagen.
- 83.
In the terminology of general topology, a set is perfect if it is closed and has no isolated points.
- 84.
This set later became known as the Cantor ternary set or the Cantor discontinuum.
- 85.
Ewald (1999), p. 906.
- 86.
Ibid, p. 919.
- 87.
Cantor later turned to the problem of characterizing the linear continuum as an ordered set. His solution was published in 1895 in the Mathematische Annalen. For a modern presentation, see §3 of Ch. 6 of Kuratowski-Mostowski (1968).
- 88.
- 89.
For an account, see Hallett (1984).
- 90.
- 91.
Fisher (1981), p. 118.
- 92.
Dauben (1979), p. 233.
- 93.
Fisher (1981), p. 118.
- 94.
In this connection Fraenkel observes (1976 , p. 123) that Cantor’s cardinals and ordinals themselves constitute nonarchimedean domains for the simple reason that, when α is finite and β is transfinite, then nα < β for any n. While admitting the legitimacy of the sort of nonarchimedean domains put forward by mathematicians such as Veronese and du Bois-Reymond , Fraenkel distinguishes between Cantor’s from the other nonarchimedean domains through the fact that in the former any cardinal or ordinal can be reached by the repeated addition of unity (if necessary, transfinitely), while in the latter such a procedure is not even definable. Fraenkel concludes: “This contrast explains in what sense other non-Archimedean domains contain ‘relatively infinite’ magnitudes while the transfiniteness of cardinals and ordinals is an ‘absolute’ one.”
It is worth quoting Fraenkel’s full endorsement of Cantor’s rejection of infinitesimal numbers :
Cantor , when undertaking a “continuation of the series of real integers beyond the infinite” and showing the usefulness of this generalization of the process of counting, refused to consider infinitely small magnitude beyond the “potential infinite” of analysis based on the concept of limit. The ‘infinitesimals’ of analysis, as is well known, refer to an infinite process and not to a constant positive value which, if greater than zero, could nit be infinitely small…
Opposing this attitude, some schools of philosophers… and later sporadic mathematicians proposed resuming the vague attempts of most 17th and 18th century mathematicians to base calculus on infinitely small magnitudes, the so-called infinitesimals or differentials. After the introduction of transfinite numbers by Cantor such attitudes pretended to be justified by set theory because there ought to be reciprocals (inverse ratios) to the transfinite numbers, namely the ostensible infinitesimals of various degrees representing the ratios of finite to transfinite numbers.
These views have been thoroughly rejected by Cantor and by the mathematical world in general. The reason for this uniformity was not dogmatism, which is a rare feature in mathematics and then almost invariably fought off; nobody has pleaded more ardently than Cantor himself that liberty of thought was the essence of mathematics and that prejudices had a very short life. The argument was not even that the admission of infinitesimals was self-contradictory, but just that it was sterile and useless…This uselessness contrasts strikingly with the success of the transfinite numbers regarding both their applications and their task of generalizing finite counting and ordering. (Fraenkel 1976, pp. 121–2.)
But it has to be said (as remarked by Fisher 1981) that concerning infinitesimals Cantor did display a dogmatic attitude and did argue, in effect, that the admission of infinitesimals was self-contradictory. While Cantor ’s intolerant attitude towards the infinitesimal is not, strictly speaking, inconsistent with the “freedom” of mathematics in his sense, it does seem to reflect his deep-seated conviction (reported in Dauben 1979, pp. 288–91) that his transfinite set theory was the product of “divine inspiration”, so that anything in conflict with it must be anathematized.
- 95.
Fisher (1981), p. 118. Even so, Cantor’s argument (in amended form) seems to have convinced a number of influential mathematicians, including Peano and Russell , of the untenability of infinitesimals.
- 96.
Dauben (1979), p. 235.
- 97.
Ibid., p. 236.
- 98.
A related point is made by Fraenkel (1976), p. 123.
- 99.
Russell (1964), p. 259. Earlier (in Ch. XXIII) Russell presents an argument to show that, while continuity, infinity and the infinitesimal have been traditionally associated with the category of Quantity, in fact they are more properly regarded as being ordinal and arithmetical in nature.
- 100.
Ibid., p. 260.
- 101.
Ibid., p. 270.
- 102.
Ibid.
- 103.
Ibid., p. 271.
- 104.
Indeed, in Hobson (1957), we find Russell’s definition of real number (introduced, with due acknowledgment, as Russell’s “form” of the definition) included in the section of the book entitled “The Dedekind Theory of Irrational Numbers” (loc. cit., pp. 23–27). Hobson refers to Russell’ s “segments” as “lower segments”; and in fact Russell ’s real numbers are the same as those lower segments of Dedekind cuts which possess no largest member.
- 105.
Hobson (1957), p. 24.
- 106.
Russell (1964), p. 287.
- 107.
Russell calls this property “compactness”; it is now usually referred to as “density ”.
- 108.
Russell (1964), p. 288.
- 109.
Written while Russell was in prison for his pacifist activities during the First World War.
- 110.
Russell (1995) p. 105.
- 111.
With the later subsumption of both order and metric under general topology, Cantor ’s two definitions of continuity, when referred to ordered and metric topological spaces , respectively, become equivalent.
- 112.
This too receives a sparkling introduction which is worth quoting:
The mathematical theory of infinity may almost be said to begin with Cantor. The Infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world. Cantor has abandoned this cowardly policy and has brought the skeleton out of its cupboard. He has been emboldened in this course by denying that it is a skeleton. Indeed, like many other skeletons, it was wholly dependent on its cupboard, and vanished in the light of day. (Ibid., p. 304).
- 113.
While Russell ’s disdain for the infinitesimal may have tempered somewhat his estimation of Leibniz , he never wavered in his regard for the latter as “one of the supreme intellects of all time” (1945, p. 581).
- 114.
Russell (1964)., p. 325.
- 115.
Ibid., p. 325–6.
- 116.
Ibid., p. 326.
- 117.
Ibid., p. 331.
- 118.
Ibid.
- 119.
Ibid., p. 332.
- 120.
Ibid., p. 151.
- 121.
Ibid., pp. 332–3. This example is essentially the same as that mentioned above by Fraenkel.
- 122.
Loc. cit, p. 335.
- 123.
Loc. cit, p. 337.
- 124.
Loc. cit, p. 338.
- 125.
For an illuminating discussion of this work and its historical context see Moynahan (2003).
- 126.
Op. cit., p. 44.
- 127.
Ibid., p. 45.
- 128.
Ibid.
- 129.
In this connection it is worth mentioning that no less a figure than Leo Tolstoy ascribed similar importance to the continuous and the infinitesimal. In Book 3, Part 3 of War and Peace we read:
To elicit the laws of history we must leave aside kings, ministers, and generals, and select for study the homogeneous, infinitesimal elements which influence the masses. No one can say how far it is possible for a man to advance in this way to an understanding of the laws of history; but it is obvious that this is the only path to that end…
It is impossible for the human intellect to grasp the idea of absolute continuity of motion. Laws of motion only become comprehensible to man when he can examine arbitrarily selected units of that motion. But at the same time it is this arbitrary division of continuous motion into discontinuous units which gives rise to a large proportion of human error.
A new branch of mathematics [i.e., the calculus], having attained the art of reckoning with infinitesimals, can now yield solutions in … complex problems of motion which before seemed insoluble. … This new branch of mathematics … by admitting the conception, when dealing with problems of motion, of the infinitely small and thus conforming to the chief condition of motion (absolute continuity), corrects the inevitable error which the human intellect cannot but make if it considers separate units of motion instead of continuous motion.
In the investigation of the laws of historical movement precisely the same principle operates. The maarch of humanity, springing as it does from an infinite multitude of individual wills, is continuous. The discovery of the laws of this continuous movement is the aim of history.
Only by assuming an infinitesimally small unit of observation—a differential of history (that is, the common tendencies of men)—and arriving at the art of integration (finding the sum of the infinitesimals) can we hope to discover the laws of history.
- 130.
An English translation of Frege’s review may be found in Frege (1984), pp. 108–12.
- 131.
Loc. cit., pp. 109–10. In 1906 Cohen’s student Ernst Cassirer summed up Cohen’s outlook more sympathetically:
The basic idea of Cohen’s work can be stated quite briefly: if we want to achieve a true scientific grounding of logic, we should not begin from any sort of completed existence. What naive intuition takes as its obvious and secure possession, this is for logic a real problem; what it assumes as directly ‘given’, this is what must be critically analyzed and taken apart in its crucial conditions for thought. We should not begin with any objective Being, no matter of what sort and no matter [in] what relation we place ourselves to it: for every “being” is in the first place a product and a result which the operation of thought and its systematic unity has as a presupposition. A foundational conceptual setting of this sort, an intellectual tradition in which we can first speak of “reality” in a scientific sense, is found by Cohen in the idea of the infinitesimal as it is detailed and fixed in modern mathematics. (Quoted in Moynahan 2003, p. 42.)
- 132.
Russell (1964), p. 339.
- 133.
Ibid., p. 344.
- 134.
Ibid.
- 135.
Ibid., p.345.
- 136.
Ibid. Russell would have been greatly surprised to learn that Cohen’s conception of infinitesimals as intensive magnitudes can in fact be given a precise mathematical sense. See Chap. 10 below.
- 137.
Ibid., p. 347.
- 138.
Ibid., pp. 347–8.
- 139.
Ibid., p. 348.
- 140.
Ibid.
- 141.
Ibid., p. 350.
- 142.
Ibid., p. 350.
- 143.
Ibid., p. 351.
- 144.
Ibid., p. 351–2.
- 145.
Ibid., p. 353.
- 146.
Ibid., pp. 353–4.
- 147.
See Chap. 10 below.
- 148.
See Chap. 10 below.
- 149.
Hobson (1957), pp. 54–5.
- 150.
Hobson (1957), pp. 53–4.
- 151.
See Chap. 5 below.
- 152.
Hobson (1957), pp. 57–8.
- 153.
Ibid., p. 58.
- 154.
McLarty (1988), p. 87.
- 155.
Ibid.
Bibliography
Bolzano, B. 1950. Paradoxes of the Infinite. Trans. Prihovsky. Routledge & Kegan Paul.
Boyer, C. 1959. The History of the Calculus and Its Conceptual Development. New York: Dover.
Dauben, J. 1979. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge: Harvard University Press.
Ewald, W. 1999. From Kant to Hilbert, A Source Book in the Foundations of Mathematics. Vol. I and II. Oxford: Oxford University Press.
Fisher, G. 1978. Cauchy and the infinitely small. Historia Mathematica 5: 313–331.
———. 1981. The infinite and infinitesimal quantities of du Bois-Reymond and their reception. Archive for History of Exact Sciences 24 (2): 101–163.
Fraenkel, A. 1976. Abstract Set Theory. Fourth edition of first (1953) edition, revised by A. Levy. North-Holland.
Frege, G. 1984. In Collected Papers on Mathematics, Logic and Philosophy, ed. B. McGuinness. Oxford: Blackwell.
Hallett, M. 1984. Cantorian Set Theory and Limitation of Size. Oxford: Clarendon Press.
Heath, T. 1981. A History of Greek Mathematics. Vol. 2. New York: Dover.
Hobson, E.W. 1957. The Theory of Functions of a Real Variable. New York: Dover.
Hocking, J.G., and G.S. Young. 1961. Topology. Reading: Addison-Wesley.
Kline, M. 1972. Mathematical Thought from Ancient to Modern Times. Vol. 3 volumes. Oxford: Oxford University Press.
Kuratowski, K., and A. Mostowski. 1968. Set Theory. Amsterdam: North-Holland.
McLarty, C. 1988. Defining sets as sets of points of spaces. Journal of Philosophical Logic 17: 75–90.
Moynahan, G. 2003. Hermann Cohen’s Das Prinzip der Infinitesimal-Methode und Seine Geschichte, Ernst Cassirer, and the Politics of Science in Wilhelmine Germany. Perspectives in Science 11 (1): 35–75.
Russell, B. 1945. A History of Western Philosophy. New York: Simon and Schuster.
———. 1964. The Principles of Mathematics. London: Allen and Unwin.
———. 1995. Introduction to Mathematical Philosophy. London: Routledge.
Smith, D.E. 1959. A Source Book in Mathematics. New York: Dover.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bell, J.L. (2019). The Reduction of the Continuous to the Discrete in the Nineteenth and Early Twentieth Centuries. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-18707-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-18706-4
Online ISBN: 978-3-030-18707-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)