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Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

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Abstract

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.

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Notes

  1. It is often claimed that the hyperreals require the resources of model theory. See Appendix for a more nuanced view.

  2. See main text at footnote 7.

  3. Edwards also surmises that Democritus saw that a triangular pyramid could be completed to form a triangular prism with the same base and height by adding two more prisms, each with the same base and height (ibid.).

  4. Kepler also used both infinitesimals of the same dimension, treating a circle, e.g., as consisting of infinitesimal triangles, and indivisibles, treating an ellipse, e.g., as consisting of its radii (Boyer 1959, pp. 108–109).

  5. Difficulties of this sort are what lead thinkers to conceive of continuous magnitudes kinematically.

  6. G. Schubring attributes the first systematic use of infinitesimals as a foundational concept, to Johann Bernoulli (see Schubring 2005, pp. 170, 173, 187). To note the fact of such systematic use by Bernoulli is not to say that Bernoulli’s foundation is adequate, or it could distinguish between manipulations with infinitesimals that produce only true results and those manipulations that can yield false results. One such infinitesimal distinction between two types of convergence was provided by Cauchy in 1853 (see Cauchy 1900), thereby resolving an ambiguity inherent in his 1821 “sum theorem” (see Bråting 2007; Katz and Katz 2011; Borovik and Katz 2011; Błaszczyk et al. 2012).

  7. Jesseph’s evocation of Hilbert connects well with a viewpoint expressed by Robinson (see main text at footnote 2).

  8. See further in footnotes 38 and 66.

  9. Boyer claims that Leibniz used this formulation of LC in “a letter to [Pierre] Bayle in 1687” (Boyer 1959, p. 217). Boyer’s claim contains two errors. First, the work in question is not a letter to Bayle but rather the Letter of Mr. Leibniz on a general principle useful in explaining the laws of nature, etc. (Leibniz 1687). Second, while this letter does deal with Leibniz’ continuity principle, it does not contain the formulation In any supposed continuous transition, ending in any terminus, etc.; instead, it postulates that an infinitesimal change of input should result in an infinitesimal change in the output (this principle was popularized by Cauchy in 1821 as the definition of continuity in Cauchy 1821, p. 34). Boyer’s erroneous claims have been reproduced by numerous authors, including Kline (1972, p. 385).

  10. This is consistent with Child’s translation: “In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included” (Child 1920, p. 147). We have reinstated the adjective continuous modifying transition (deleted by Child possibly in an attempt to downplay a perceived logical circularity of defining LC in terms of continuity itself). Jorgensen (2009, p. 228) cites Child’s translation and claims in footnote 21 that “this passage says nothing about continuity”.

  11. One scholar who was so misled was Boyer (see further in footnote 23).

  12. Bos goes on specifically to criticize the Bourbaki’s limite wording “(Leibniz) se tient très près du calcul des différences, dont son calcul différentiel se déduit par un passage à la limite” (Bourbaki 1960, p. 208).

  13. Specifically, Leibniz treats in detail an inassignable quantity he refers to as status transitus (see Sect. 4.4).

  14. Equality up to an infinitesimal is a state of transition from inequality to equality (this anticipates the law of homogeneity dealt with in Sect. 5.3).

  15. Robinson’s attribution in Robinson (1966, p. 262) contains a misprint: “(Leibniz 1701)” should be read as “(Leibniz 1702)”.

  16. Laugwitz pointed out that this law “contains an a priori assumption: our mathematical universe of discourse contains both finite objects and infinite ones” (Laugwitz 1992a, p. 145). As we have already discussed, identifying distinct A- and B-methodologies in Leibniz does not require realist commitments.

  17. A syncategorematic expression has no referential function. Thus, the phrase ‘the present king of France is bald’ is a syncategorematic expression, in that it doesn’t refer to any concrete individual. A syncategorematic expression serves to reveal logical relations among those parts of the sentence which are referential. In Arthur and Levey’s interpretation, the infinitesimal “serves to reveal logical relations” by tacitly encoding a quantifier applied to ordinary real values. But Leibniz clearly does not have real values in mind when he exploits the term status transitus. His status transitus is something between real values of the variable, on the one hand, and its limiting real value, on the other. Leibniz’s observation that the metaphysical (i.e., ontological) status of infinitesimals is “open to question” should apparently have put to rest any suspicions as to their alleged syncategorematic nature. After all, if an infinitesimal is merely meant as shorthand for talking about relations among sets of real values, what is the point of the lingering doubts expressed by Leibniz as to the ontological legitimacy of infinitesimals? Certainly the absence of a concrete individual counterpart of the bald king is a closed and shut question, rather than being “open to question”. See further in footnotes 38 and 66.

  18. The notation “st” parallels that for the standard part function in the context of the hyperreals (see Appendix).

  19. When the general reasoning being transferred to the infinite realm is generalized to encompass arbitrary elementary properties (i.e. first order properties), one obtains the Łoś-Robinson transfer principle (see Appendix).

  20. Child incorrectly transcribes formula (4.7) from Gerhardt, replacing the equality sign in Gerhardt by a plus sign. Note that Leibniz himself used the sign \(\ulcorner\!\urcorner\) (see McClenon 1923, p. 371).

  21. Child’s transcription of formula (4.8) contains numerous errors: the numerator of the fraction \(\frac{x}{a}\) is missing; the expression \(\frac{ddv}{ddx}\) appears with a y in place of v in the numerator; the expression \(\frac{2dx ddv}{a ddx}\) appears with a ddx in place of ddv in the numerator.

  22. See further in footnotes 38 and 66.

  23. L’Huillier (1750–1840) understood Leibniz’s law of continuity similarly: “if a variable quantity at all stages enjoys a certain property, its limit will enjoy the same property” (L’Huilier 1786, p. 167). L’Huilier, writing a century before Weierstrass, is using the term limit in its generic sense close to terminus/status transitus. Blinded by the modern limit doctrine, Boyer comments as follows: “The falsity of this doctrine is immediately apparent from the fact that irrational numbers may easily be defined as the limit of sequences of rational numbers, or from the observation that the properties of a polygon inscribed in a circle are not those of the limiting figure—the circle” (Boyer 1959, p. 256). But Boyer’s “limiting figure” is an anachronistic imposition, of a post-triumvirate variety, upon both L’Huilier and Leibniz. What Leibniz had in mind was a status terminus whose shadow is the circle (see also footnote 11).

  24. Our analysis of Berkeley’s criticism of the proof of the product rule for differentiation appears in Sect. 7.

  25. Kline opines that “In response to criticism of his ideas, Leibniz made various, unsatisfactory replies” (Kline 1972, p. 384), and proceeds to quote a passage from a letter to Wallis from 30 March 1699 (Kline reports an incorrect year 1690): “It is useful to consider quantities infinitely small such that when their ratio is sought, they may not be considered zero but which are rejected as often as they occur with quantities incomparably greater […] Thus if we have x + dx,  dx is rejected […] Similarly we cannot have xdx and dx dx standing together. Hence if we are to differentiate xy we write (x + dx)(y + dy) − xy = xdy − xy = xdy + ydx + dx dy. But here dx dy is to be rejected as incomparably less than than xdy + ydx” (Leibniz 1699, p. 63). This summary of the law of homogeneity is dismissed as “unsatisfactory” by Kline. In fairness it must be added that Kline wrote two years before the appearance of the seminal study by Bos (1974).

  26. Levey elaborates his position as follows: “The syncategorematic analysis of the infinitely small is […] fashioned around the order of quantifiers so that only finite quantities figure as values for the variables. Thus,

    (3) the difference |a − b| is infinitesimal

    does not assert that there is an infinitely small positive value which measures the difference between a and b. Instead it reports,

    (3*) For every finite positive value \(\varepsilon, \) the difference |a − b| is less than \(\varepsilon\).

    Elaborating this sort of analysis carefully allows one to express the now-usual epsilon-delta style definitions, etc.” (Levey 2008, pp. 109–110). To summarize: no B-methodology, syncategorematically speaking. What support does Levey provide for his nominalistic interpretation? Leibniz’s comments in April, 1676.

  27. See footnote 26 for Weierstrassian epsilontic details in Levey.

  28. Thus, Boyer writes: “The traditional view […] ascribes the invention of the calculus to […] Newton and […] Leibniz” (Boyer 1959, p. 187).

  29. Berkeley’s spelling.

  30. Sherry’s dichotomy was picked up by (Jesseph 2005, p. 124).

  31. Three additional significant aspects of Berkeley’s criticism could be mentioned: (3) a belief in naive indivisibles (this a century after Cavalieri), i.e., a rejection of infinite divisibility that was already commonly accepted by mathematicians as early as Wallis and others; (4) empiricism, i.e., a belief that a theoretical entity is only meaningful insofar as it has an empirical counterpart, or referent; this belief ties in with Berkeley’s theory of perception which he identifies with a theory of knowledge; (5) Berkeley’s belief that Newton’s attempt to escape a reliance on infinitesimals is futile (see the epigraph to this Sect. 6). The latter belief is contrary to a consensus of modern scholars. Thus, Pourciau (2001) argues that Newton possessed a clear kinetic conception of limit similar to Cauchy’s, and cites Newton’s lucid statement to the effect that “Those ultimate ratios … are not actually ratios of ultimate quantities, but limits … which they can approach so closely that their difference is less than any given quantity…” See Newton (1946, p. 39; 1999, p. 442). The same point, and the same passage from Newton, appeared a century earlier in Russell (1903, item 316, pp. 338–339).

  32. See Sects. 8 and 9 for more detailed comments on Berkeley’s philosophy in relation to infinitesimals.

  33. Note here the similarity between the empiricist dogma and the syncategorematic interpretation of infinitesimals. Both assert that meaningfulness consists ultimately in referring to some favored type of entity.

  34. Berkeley uses extension in the sense of what we would call today a continuum.

  35. The reference is to Richard Whately (1787–1863).

  36. Accuracy.

  37. Leibniz had two laws of homogeneity, one for dimension and the other for the order of infinitesimalness. Bos states that they ‘disappeared from later developments’ (Bos 1974, p. 35), referring to Euler and Lagrange.

  38. Leibniz freely inverts his infinitesimals, making it difficult to interpret his infinitesimals in terms of modern nilsquare ones, as Arthur attempts to do in Arthur (see also footnote 66).

  39. This is not to say that Leibniz’s system for differential calculus satisfied modern standards of rigor. Rather, we are rejecting the claim by Berkeley and triumvirate historians to the effect that Leibniz’s system contained logical fallacies.

  40. See footnote 34.

  41. Weierstrass’s nominalistic reconstruction (as C. S. Peirce called it, and as Burgess (1983) might have) was analyzed in Katz and Katz (2012).

  42. C. Boyer refers to Cantor, Dedekind, and Weierstrass as “the great triumvirate” (see Boyer 1949, p. 298).

  43. To note Klein’s appreciation of the tradition of the B-continuum is not to imply that he was referring to modern theories thereof; see footnote 57 for a fuller discussion of Klein’s views.

  44. Dismissing Bell’s martial flourishes as merely verbal excesses would be missing the point. Bell has certainly been criticized for other fictional excesses of his purportedly historical writing (thus, Rothman writes: “[E. T.] Bell’s account [of Galois’s life], by far the most famous, is also the most fictitious” (Rothman 1982, p. 103)); however, his confident choice of martial imagery here cannot but reflect Bell’s perception of a majority view among professional mathematicians. Bell is convinced that Berkeley refuted infinitesimals only because triumvirate historians and mathematicians told him so.

  45. See footnote 6.

  46. Simon Stevin’s decimals are at the foundation of the common number system; see footnote 62 below for additional details.

  47. In this context, it may be interesting to note that a close relationship exists between Cantor’s construction of the usual A-continuum in terms of equivalence classes of Cauchy sequences, on the one hand, and one of the more straightforward constructions of the B-continuum, on the other; see Appendix.

  48. Schubring (2005, p. 454) notes that both Cauchy and Carnot approached infinitesimals dynamically, in terms of sequences (sometimes referred to as “variables”, understood as a succession of values) which tend to zero. A. Youschkevitch quotes Carnot to the effect that an infinitesimal is a variable quantity all of whose values are determinate and finite (see Gillispie 1971, p. 242). To note the fact of the identical definition of infinitesimals found in Carnot in Cauchy is not to imply total agreement; thus, Cauchy rejected Carnot’s definition of the differential.

  49. Not every hyperreal field can be obtained this way, though a more general construction called limit ultrapower can be used to construct a maximal class hyperreal field (see Ehrlich 2012; Borovik et al. 2012).

  50. From a strict set-theoretic viewpoint, each of the successive number systems \({{\mathbb N}\subset{\mathbb Z}\subset{\mathbb Q}\subset{\mathbb R}}\) can be reduced to the previous one by the familiar set-theoretic constructions. Yet it is generally recognized that each successive enlargement, when additional entities come to be viewed as individuals (or atomic entities), constitutes a conceptual advantage over the previous one, with a gain in problem solving power.

  51. The notion of infinitesimal in Russell’s time was a heuristic concept that has not been defined yet. The entire enterprise by Cantor and Russell (to prove the non-well-foundedness of a heuristic concept that has not been defined yet) retroactively strikes one as ill-conceived.

  52. Note that the second sense ties in well with Louis Narens’ approach to measurement where he transforms it into a relative notion: certain entities are measurable compared to others. This allows him to do measurement theory in non-Archimedean contexts (see Narens 1976).

  53. To elaborate, Earman is simply mistaken to think that higher-order infinitesimals require higher order hyperreals. Thus, given a ‘first-order’ infinitesimal \({dx\in{\mathbb R}^{*},}\) a second-order infinitesimal in Leibniz’s sense would correspond to the square \({dx^2\in{\mathbb R}^{*},}\) so that we don’t need to consider \({{\mathbb R}^{**}}\) at all.

  54. See Appendix. Robinson did not work with the “st” notation, explained in Sects. 8, 9, and Appendix.

  55. We reproduce Bos’ passage in first person.

  56. Leibniz did consider expressions like d 1/2 x and even gave an explanation of this expression; see his letter to L’Hospital from 30 september 1695 (Leibniz 1695).

  57. It is interesting to note a criterion of success of a theory of infinitesimals as proposed by Adolf Abraham Fraenkel and, before him, by Felix Klein. In 1908, Klein formulated a criterion of what it would take for a theory of infinitesimals to be successful. Namely, one must be able to prove a mean value theorem for arbitrary intervals, including infinitesimal ones (Klein 1932, p. 219). In 1928, Fraenkel (1946, pp. 116–117) formulated a similar requirement in terms of the mean value theorem. Such a Klein-Fraenkel criterion is satisfied by the Hewitt-Łoś-Robinson theory by the transfer principle (see Appendix).

  58. Such was indeed the tenor of a recent referee report, see http://u.cs.biu.ac.il/∼katzmik/straw2.html.

  59. Fowler (1992, p. 733) notes that “Stevin described an algorithm for finding the decimal expansion of the root of any polynomial, the same algorithm we find later in Cauchy’s proof of the intermediate value theorem”. The matter is discussed in detail in Błaszczyk et al. (2012). See also footnote 62.

  60. Cauchy’s notation for the two sequences is \(x_0, x_1, x_2, \ldots\) and \(X, X^{\prime}, X^{\prime\prime}, \ldots\) (Cauchy 1821, p. 462).

  61. Comte’s notes of Cauchy’s lectures have been preserved (see Schubring 2005, p. 437).

  62. Fearnley-Sander writes that “the modern concept of real number […] was essentially achieved by Simon Stevin, around 1600, and was thoroughly assimilated into mathematics in the following two centuries” (Fearnley-Sander 1979, p. 809). Fearnley-Sander’s sentiment is echoed by van der Waerden (1985, p. 69). Stevin had anticipated Cauchy’s proof of the intermediate value theorem, and produced a fine-tuned version of the iteration, where each step of the iteration produces an additional digit of the decimal expansion of the solution. The algorithm is discussed in more detail in Stevin (1958, §10, pp. 475–476). Stevin subdivides the interval into ten equal parts, resulting in a gain of a new decimal digit of the solution at every iteration of the algorithm. Who needs the “existence” of the real numbers when Stevin constructs an explicit decimal representation of the solution? See also footnote 59.

  63. The pioneers of infinitesimal calculus were aware of the non-uniqueness of decimal representation (at least) as early as 1770, see Euler (1840, p. 170).

  64. Once the real numbers have been defined, considerable technical difficulties remain in the definition of the multiplication and other algebraic operations. They were overcome by Dedekind (see Fowler 1992).

  65. On occassion, Cauchy uses inequalities, rather than equations involving infinitesimals, as, for instance, in Theorems I and II in Sect. 3 of chapter 2 of the Cours d’analyse (see Grabiner 1981). However, the thrust of his foundational approach, pace Grabiner, is to use infinitesimals (generated by null sequences) as inputs to functions, i.e., as individuals/atomic entities; to define continuity in terms of infinitesimals; and to apply infinitesimals to a range of problems, including an infinitesimal definition of the “Dirac” delta function (see Freudenthal 1971, p. 136; Laugwitz 1992b).

  66. It is therefore puzzling to find Arthur insisting, in J. Bell’s name, on similarities between Leibniz’s approach and that of Smooth Infinitesimal Analysis (SIA). Arthur analyzes Bell’s notion of an (intuitionistic) infinitesimal x as satisfying the relation \(\neg\neg x=0\). Bell describes such an x as “indistinguishable from 0”. In more detail, Bell’s infinitesimal x satisfies NOT(NOT(x = 0)). In classical logic this would imply x = 0, but not in intuitionistic logic. This can certainly sound like a “fictional” entity to a classically-trained audience, but no intuitionist has been known to have embraced fictionalism about anything in mathematics (certainly not E. Bishop—see Katz and Katz 2011c); indeed, many intuitionists insist that all mathematical expressions refer to constructible objects), and at any rate such “fictionality” certainly has nothing to do with Leibniz’s. Certainly Arthur’s claim that SIA infinitesimals are variable quantities unlike Robinson’s is incorrect. Arthur proceeds to describe Bell’s infinitesimals as “fictional”, and bases his analogy with Leibniz on the latter term, which procedure strikes us as an unconvincing pun (see also footnote 38).

  67. Leibniz is apparently referring to the fourth definition.

  68. See footnote 6.

  69. Note that both the term “hyper-real”, and an ultrapower construction of a hyperreal field, are due to Hewitt in 1948 (p. 74). Luxemburg (1964) also clarified its relation to the competing construction of Schmieden and Laugwitz (1958), also based on sequences, which used a different kind of filter.

  70. Namely, the traditional construction of the real field, usually attributed to Cantor, views a real number as an equivalence class of Cauchy sequences of rational numbers. Null sequences comprise the equivalence class corresponding to the real number \({0\in{\mathbb R}. }\)

  71. Such elements could be called “infinitesimal” to the extent that they violate the Archimedean property suitably interpreted, but the ring they are elements of has unsatisfactory properties.

  72. Analyzing \(\mathcal{M}, \) one discovers that its structure is controlled by a free ultrafilter on \({{\mathbb N}. }\) The order relation on \(\hbox{I\!I\!R}\) is defined relative to the ultrafilter. Additional details may be found in Błasczcyk et al. (2012, Appendix A).

  73. See footnote 72.

  74. Some examples were provided in Sect. 4.5 following formula (4.4).

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Acknowledgments

We are grateful to H. Jerome Keisler for helpful remarks that helped improve an earlier version of the manuscript. The influence of Hilton Kramer (1928–2012) is obvious.

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  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Mikhail G. Katz

  2. Department of Philosophy, Northern Arizona University, Flagstaff, AZ, 86011, USA

    David Sherry

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Appendix: Rival Continua

Appendix: Rival Continua

The historical roots of infinitesimals go back to Cauchy, Leibniz, and ultimately to Archimedes. Cauchy’s approach to infinitesimals is not a variant of the hyperreals. Rather, Cauchy’s work on the rates of growth of functions anticipates the work of late nineteenth century investigators such as Stolz, du Bois-Reymond, Veronese, Levi-Civita, Dehn, and others, who developed non-Archimedean number systems against virulent opposition from Cantor, Russell, and others (see Ehrlich 2006; Katz and Katz 2012 for details). The work on non-Archimedean systems motivated the work of T. Skolem on non-standard models of arithmetic (Skolem 1934), which subsequently stimulated work culminating in the hyperreals of Hewitt, Łos, and Robinson.

The relation between the rival theories of the continuum distinguished by Felix Klein (see Sect. 8.1) can be summarized as follows. A Leibnizian definition of the differential quotient

$$ \frac{\Updelta y}{\Updelta x}, $$

whose logical weakness was criticized by Berkeley, was modified by A. Robinson by exploiting a map called the standard part, denoted “st”, from the finite part of a “thick” B-continuum (i.e., a Bernoullian continuum),Footnote 68 to a “thin” A-continuum (i.e., an Archimedean continuum), as illustrated in Figs. 2 and 4. The derivative is defined as \(\hbox{st} \left( \frac{\Updelta y}{\Updelta x} \right), \) rather than the differential quotient \(\frac{\Updelta y}{\Updelta x}\) itself. Robinson wrote that “this is a small price to pay for the removal of an inconsistency” (Robinson 1966, p. 266). However, the process of discarding the higher-order infinitesimals has solid roots in Leibniz’s law of homogeneity (see Sect. 5.3).

Fig. 4
figure 4

Zooming in on infinitesimal \(\epsilon\) (here st\((\pm \epsilon)=0\))

Full size image

1.1 Hyperreals via Maximal Ideals

We summarize a twentieth century implementation of an alternative to an Archimedean continuum, namely an infinitesimal-enriched continuum. Such a continuum is not to be confused with incipient notions of such a continuum found in earlier centuries. We refer to such a continuum as a B-continuum (see footnote 6). We begin with a heuristic representation of a B- or “thick” continuum, denoted \(\hbox{I\!I\!R}, \) in terms of an infinite resolution microscope (see Fig. 4). One presentation of such a structure is in Robinson (1961). Such an infinitesimal-enriched continuum is suitable for use in calculus, analysis, and elsewhere. Robinson built upon earlier work by Hewitt (1948), Łoś (1955) and others. In 1962, Luxemburg (1964) popularized a presentation of Robinson’s theory in terms of the ultrapower construction,Footnote 69 in the mainstream foundational framework of the Zermelo–Fraenkel set theory with the axiom of choice (ZFC).

The construction can be viewed as a relaxing, or refining, of Cantor’s construction of the reals. This can be motivated by a discussion of rates of convergence as follows. In Cantor’s construction, a real number u is represented by a Cauchy sequence \({\langle u_n : n\in{\mathbb N}\rangle}\) of rationals. But the passage from \(\langle u_n\rangle\) to u in Cantor’s construction sacrifices too much information. We would like to retain a bit of the information about the sequence, such as its “speed of convergence”. This is what one means by “relaxing” or “refining” Cantor’s construction of the reals (cf. Giordano and Katz 2011). When such an additional piece of information is retained, two different sequences, say \(\langle u_n\rangle\) and \(\langle u'_n\rangle, \) may both converge to u, but at different speeds. The corresponding “numbers” will differ from u by distinct infinitesimals. If \(\langle u_n\rangle\) converges to u faster than \(\langle u'_n\rangle, \) then the corresponding infinitesimal will be smaller. The retaining of such additional information allows one to distinguish between the equivalence class of \(\langle u_n\rangle\) and that of \(\langle u'_n\rangle\) and therefore obtain distinct hyperreals infinitely close to u.

At the formal level, we proceed as follows. We construct a hyperreal field as a quotient of the collection of arbitrary sequences, where a sequence

$$ \langle u_1, u_2, u_3, \ldots \rangle $$
(13.1)

converging to zero generates an infinitesimal (the kernel of the quotient homomorphism is the maximal ideal \(\mathcal{M}\) described below). Arithmetic operations are defined at the level of representing sequences; e.g., addition and multiplication are defined term-by-term. Thus, we start with the ring \({{\mathbb Q}^{\mathbb N}}\) of sequences of rational numbers. Let

$$ {\mathcal{C}}_{\mathbb Q} \subset {\mathbb Q}^{\mathbb N} $$
(13.2)

denote the subring consisting of Cauchy sequences. The reals are by definition the quotient field

$$ {\mathbb R}:= {\mathcal{C}}_{\mathbb Q} / {\mathcal{F}}_{\!n\!u\!l\!l}, $$
(13.3)

where \({{\mathcal{F}_{\!n\!u\!l\!l}}}\) is the ideal containing all null sequences (i.e., sequences tending to zero).Footnote 70 Note that \({{\mathbb Q}}\) is imbedded in \({{\mathbb Q}^{\mathbb N}}\) by constant sequences. An infinitesimal-enriched extension of \({{\mathbb Q}}\) may be obtained by modifying (13.3). Now consider the subring

$$ {\mathcal{F}}_{ez}\subset{{\mathcal{F}}_{\!n\!u\!l\!l}} $$

of sequences that are “eventually zero”, i.e., vanish at all but finitely many places. Then the quotient \({\mathcal{C}_{\mathbb Q} / \mathcal{F}_{ez}}\) naturally surjects onto \({{\mathbb R}= \mathcal{C}_{\mathbb Q} / {\mathcal{F}_{\!n\!u\!l\!l}}}\). The elements in the kernel of the surjection

$$ {\mathcal{C}}_{\mathbb Q}/{\mathcal{F}}_{ez} \to {\mathbb R} $$

are prototypes of infinitesimals.Footnote 71 Note that the quotient \({\mathcal{C}_{\mathbb Q}/\mathcal{F}_{ez}}\) is not a field, as \(\mathcal{F}_{ez}\) is not a maximal ideal. To obtain a field, we must replace \(\mathcal{F}_{ez}\) by a maximal ideal.

It is more convenient to describe the modified construction using the ring \({{\mathbb R}^{\mathbb N}}\) rather than \({\mathcal{C}_{\mathbb Q}}\) of (13.2).

We therefore redefine \(\mathcal{F}_{ez}\) to be the ring of real sequences in \({{\mathbb R}^{\mathbb N}}\) that eventually vanish, and choose a maximal proper ideal \(\mathcal{M}\) so that we have

$$ {\mathcal{F}}_{ez}\subset{\mathcal{M}}\subset{\mathbb R}^{\mathbb N}. $$
(13.4)

Then the quotient

$$ \hbox{I\!I\!R}:={\mathbb R}^{\mathbb N}/{\mathcal{M}} $$
(13.5)

is a hyperreal field. The foundational material needed to ensure the existence of a maximal ideal \(\mathcal{M}\) satisfying (13.4) is weaker than the axiom of choice. This concludes the construction of a hyperreal field \(\hbox{I\!I\!R}\) in the traditional foundational framework, ZFC.

The construction is equivalent to the usual ultrapower construction as popularized by Luxemburg.Footnote 72 Thus it is not entirely accurate to suppose, as Jesseph does, that a consistent theory of infinitesimals requires the resources of model theory. The resources of a rigorous undergraduate course in abstract algebra suffice.

1.2 Example

To give an example, the sequence

$$ \left\langle \tfrac{1}{n} : n\in {\mathbb N}\right\rangle $$
(13.6)

represents a nonzero infinitesimal, in the sense that its class \(\left[\tfrac{1}{n} \right]\) in (13.5) is nonzero and satisfies \(\left[\tfrac{1}{n} \right]<r\) for every positive real number r.Footnote 73

1.3 Construction of Standard Part

In the field \(\hbox{I\!I\!R}\) of (13.5), consider the subring \(I\subset\hbox{I\!I\!R}\) consisting of infinitesimal elements (i.e., elements e such that \(|e|<\frac{1}{n}\) for all \({n\in{\mathbb N}}\)). Denote by I −1 the set of inverses of nonzero elements of I. The complement \(\hbox{I\!I\!R} \setminus I^{-1}\) consists of all the finite (sometimes called limited) hyperreals. Constant sequences provide an inclusion \({{\mathbb R}\subset\hbox{I\!I\!R}. }\) Every element \(x\in \hbox{I\!I\!R}\setminus I^{-1}\) is infinitely close to some real number \({x_0\in{\mathbb R}}\). The standard part function, denoted “st”, associates to every finite hyperreal, the unique real infinitely close to it:

$$ \hbox{st}:\hbox{I\!I\!R}\setminus I^{-1} \to {\mathbb R}, \hbox {with}\,x\,\mapsto\,x_0. $$

The real x 0 is sometimes called the shadow of x. If x happens to be the equivalence class of a Cauchy sequence \({\langle x_n:n\in{\mathbb N} \rangle,}\) then the shadow of x is the limit of \(\langle x_n\rangle: \)

$$ \hbox{st}(x)=\lim_{n\to\infty} x_n. $$

As explained in Sect. 5.3, the standard part function can be seen as an implementation of Leibniz’s transcendental law of homogeneity.

1.4 The Transfer Principle

The transfer principle is a mathematical implementation of Leibniz’s heuristic law of continuity: “what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa” (see Robinson 1966, p. 266). The transfer principle, allowing an extention of every first-order real statementFootnote 74 to the hyperreals, is a consequence of the theorem of Łoś in 1955, and can therefore be referred to as a Leibniz-Łoś transfer principle. A Hewitt-Łoś framework allows one to work in a B-continuum satisfying the transfer principle.

A helpful “semicolon” notation for presenting an extended decimal expansion of a hyperreal was described by Lightstone (1972). See also Roquette (2010) for infinitesimal reminiscences. A discussion of infinitesimal optics is in Stroyan (1972), Keisler (1986), Tall (1980), Magnani and Dossena (2005), Dossena and Magnani (2007) and Bair and Henry (2010). Applications of the B-continuum range from aid in teaching calculus (Ely 2010; Katz and Katz 2010a, b; Katz and Tall 2012; Tall 1991, 2009) (see illustration in Fig. 5) to the Bolzmann equation (see Arkeryd 1981, 2005); modeling of timed systems in computer science (see Rust 2005); Brownian motion and economics (see Anderson 1976); mathematical physics (see Albeverio et al. 1986); etc. The hyperreals can be constructed out of integers (see Borovik et al. 2012). The traditional quotient construction using Cauchy sequences, usually attributed to Cantor, can be factored through the hyperreals (Giordano and Katz 2011).

Fig. 5
figure 5

Differentiating y = f(x) = x 2 at x = 1 yields \(\tfrac{\Updelta y}{\Updelta x} = \tfrac{f(.9..) - f(1)}{.9..-1} = \tfrac{(.9..)^2 - 1}{.9..-1} = \tfrac{(.9.. - 1)(.9.. + 1)}{.9..-1} = .9.. + 1 \approx 2\). Here ≈ is the relation of being infinitely close. Hyperreals of the form. 9.. are discussed in Katz and Katz (2010b)

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Katz, M.G., Sherry, D. Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond. Erkenn 78, 571–625 (2013). https://doi.org/10.1007/s10670-012-9370-y

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