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Proofs and Retributions, Or: Why Sarah Can’t Take Limits


The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history.

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Fig. 1
Fig. 2


  1. See Heyting (1956) and Pourciau (1999) for the literary career of this character.

  2. This character made her debut in Robert Ely’s study (Ely 2010). Ely was a TA in a calculus course. One of his students, Sarah, spontaneously re-invented Leibniz-style infinitesimals so as to make sense of what was going on in the calculus course (needless to say, this was taught using epsilon, delta).

  3. See note 58 for details on the foundational status of Noncommutative Geometry.

  4. The term was introduced by Hewitt (1948, p. 74). A hyperreal number \(H\) equal to its own integer part, or \(H=\lfloor H \rfloor \), is called a hyperinteger. The hyperintegers form a set \({\mathbb Z}^*\). The elements of the complement \({\mathbb Z}^* \setminus {\mathbb Z}\) are called infinite hyperintegers. The hyperreals can be constructed directly out of integers; see Borovik et al. (2012).

  5. In the “semicolon” notation of Lightstone (1972), such a number would appear as the extended terminating infinite decimal \(.999\ldots ;\ldots 9\), with the last nonzero digit appearing at infinite decimal rank \(H\). See note 56 on interpretations of the symbol “\(.999\ldots \)”.

  6. See note 130 for a more nuanced view. In 1908, Felix Klein described a rivalry of two types of continua in the following terms. Having outlined the developments in real analysis associated with Weierstrass and his followers, Klein pointed out that “The scientific mathematics of today is built upon the series of developments which we have been outlining. But an essentially different conception of infinitesimal calculus has been running parallel with this [conception] through the centuries (Klein 1908, p. 214). Such a different conception, according to Klein, “harks back to old metaphysical speculations concerning the structure of the continuum according to which this was made up of \(\ldots \) infinitely small parts” (ibid.). Thus according to Klein there is not one but two separate tracks for the development of analysis: [A] the Weierstrassian approach (in the context of an Archimedean continuum); and [B] the approach with indivisibles and/or infinitesimals (in the context of what could be called a Bernoullian continuum).

  7. Recent work on Leibniz and infinitesimals includes (Katz and Sherry 2012, 2013; Sherry and Katz 2012; Tho 2012). A seminal study of Leibnizian methodology by Bos notes that Robinson’s hyperreals (see Robinson 1966) provide a “preliminary explanation of why the calculus could develop on the insecure foundation of the acceptance of infinitely small and infinitely large quantities” (Bos 1974, p. 13). In addition to this positive assessment, the article by Bos also contains a brief Appendix 2 where Bos criticizes Robinson’s reading of Leibniz. The appendix, written as it was by a fresh Ph.D. in history with apparently limited training in mathematics (not to speak of mathematical logic), contains numerous misunderstandings of the hyperreal framework. Bos’s technical errors were detailed by Katz and Sherry (2013, Section 11.3), and include a misreading of Robinson’s transfer (i.e., the transfer principle). See further in note 8.

  8. Urquhart commented that “some of Bos’ criticisms of Robinson involve absurd and impossible demands—for example, his first criticism (Bos 1974, p. 83) is that Robinson proves the existence of his infinitesimals, whereas Leibniz does not!” (Urquhart 2006). See further in note 9.

  9. Bos writes: “the most essential part of non-standard analysis, namely the proof of the existence of the entities it deals with, was entirely absent in the Leibnizian infinitesimal analysis, and this constitutes, in my view, so fundamental a difference between the theories that the Leibnizian analysis cannot be called an early form, or a precursor, of non-standard analysis” (Bos 1974, p. 83). Bos’s comment fails to appreciate the crucial dichotomy of mathematical practice (or procedures) versus mathematical ontology (or foundations). Leibnizian procedures exploiting infinitesimals find suitable proxies in the procedures in the hyperreal framework (see Reeder 2013 for a related discussion in the context of Euler). The relevance of such hyperreal proxies is in no way diminished by the fact that set-theoretic foundations of the latter (“proof of the existence of the entities”, as Bos put it) were obviously as unavailable in the seventeenth century as set-theoretic foundations of the real numbers. See further in note 10.

  10. In the context of Bos’s discussion of “present-day standards of mathematical rigor”, Bos writes: “\(\ldots \) it is understandable that for mathematicians who believe that these present-day standards are final, nonstandard analysis answers positively the question whether, after all, Leibniz was right” (Bos 1974, p. 82, item 7.3). The context of the discussion makes it clear that Bos’s criticism targets Robinson. If so, Bos’s criticism suffers from a strawman fallacy, for Robinson specifically wrote that he does not consider set theory to be the foundation of mathematics, and being a formalist does not subscribe to the view attributed to him by Bos that “present-day standards are final”. See further in note 11.

  11. Robinson expressed his position on the status of set theory as follows: “an infinitary framework such as set theory \(\ldots \) cannot be regarded as the ultimate foundation for mathematics” (Robinson 1969, p. 45; see also Robinson 1966, p. 281). Furthermore, contrary to Bos’s claim, Robinson’s achievement was not to show that “Leibniz was right” (see note 10), but rather to provide hyperreal proxies for the inferential procedures commonly found in Leibniz as well as Euler and Cauchy. Leibniz’s procedures, involving as they do infinitesimals and infinite numbers, seem far less puzzling when compared to their hyperreal proxies than from the viewpoint of the traditional A-track frameworks (see note 6 on tracks A and B). See further in note 12.

  12. Some decades later, a mellower Bos distanced himself from his flawed Appendix 2 (see notes 7, 8, 9, 10, and 11) in the following terms (in response to a question from one of the authors of the present text): “An interesting question, what made me reject a claim some 35 years ago? I reread the appendix and was surprised about the self assurance of my younger self. I’m less definite in my opinions today—or so I think. You’re right that the appendix was not sympathetic to Robinson’s view. Am I now more sympathetic? If you talk about “historical continuity” I have little problem to agree with you, given the fact that one can interpret continuity in historical developments in many ways; even revolutions can come to be seen as continuous developments” (Bos 2010).

  13. See Fig. 1 and note 22 for a discussion of a generalized notion of equality in Leibniz.

  14. See note 99 on wheels, infinitesimals, and limits.

  15. Sarah will pursue the matter further; see main text at note 60.

  16. A view of the \((\epsilon ,\delta )\) approach as a nominalistic paraphrase, or reconstruction, of analysis was elaborated by Katz and Katz (2012a). They note in Katz and Katz (2012b): “If our students are being dressed to perform multiple-quantifier epsilontic logical stunts on the pretense of being taught infinitesimal calculus, it is because infinitesimals are assumed to be either metaphysically dubious or logically unsound”.

  17. See Nabokov (1962) and main text at note 78.

  18. See The AnalystBerkeley (1734) for the famous criticism of infinitesimal calculus in terms of “the ghosts of departed quantities”. Berkeley claimed calculus was based on an inconsistency of the type \((dx\not =0)\wedge (dx=0)\). See further in note 22. See also note 36 for Robinson’s comment on Berkeley. Sherry (1987) dissected Berkeley’s criticism into its metaphysical and logical components.

  19. See Stewart (2009).

  20. See note 130.

  21. See Keisler (1986, p. 43). To define the real derivative of a real function \(f\) in this approach, one can bypass an infinite limiting process as in Weierstrass’s approach. Instead, one sets \(f'(x) = \mathrm{st} \left( \frac{f(x+\epsilon )-f(x)}{\epsilon } \right) \), where \(\epsilon \) is infinitesimal, yielding the standard real number in the cluster of the hyperreal argument of “st” (the derivative exists if and only if the value above is independent of the choice of the infinitesimal).

  22. See Robinson (1966, p. 266). Katz and Sherry (2012, 2013) argue that the inconsistency alleged by Berkeley in Leibniz, namely \((dx\not =0)\wedge (dx=0)\), was not there in the first place, as Leibniz repeatedly indicated that he is working with a generalized notion of equality “up to” a negligible term. Such a principle was dubbed the transcendental law of homogeneity by Leibniz (1710).

  23. See Richman (1996, p. 249); see note 31 below for the source of the adjective.

  24. See Bishop (1968, p. 54).

  25. See note 30 for details on Bishop’s talk at Stanford University.

  26. See Weber (1893, p. 15).

  27. Hellman (1993) introduces a dichotomy within constructivism, between liberal constructivism and radical constructivism. The former views constructivism as a companion to classical mathematics. The latter views constructivism as an alternative to classical mathematics. The “fundamentalist” comment leans toward the latter variety.

  28. Based on personal conversations with Bishop, Hill (2013) indicated that there was a definite connection in Bishop’s mind between his rejection of what he felt was his fundamentalist upbringing, “ in a strict fundamentalist Protestant situation”, and his eventual rejection of classical mathematics. In a poem he wrote around 1973, Bishop compared classical mathematics to “sawdust”, and depicted Formalism as decidedly diabolical; see main text at note 106.

  29. Bishop’s rejection of classical mathematics followed his work on the existence of holomorphic disks; see Bishop (1965a). The work exploited nonconstructive fixedpoint theorems. As Hill (2013) relates, Bishop attempted to exhibit such a disk explicitly, and was unable to do so. He then attempted to exhibit a single point on such a disk, and was still unable to do so. Even an attempt to exhibit a single coordinate of such a point failed. Bishop was apparently struck by an allegedly fundamentalist nature of a mathematician’s belief in entities he is unable to exhibit. This eventually led to his abandoning complex variable research, and switching to constructive mathematics.

  30. The table (except for the colors) is a replica of the first blackboard from the Stanford University colloquium lecture (Bishop 1965b), according to an eyewitness account (Hill 2009). See further in note 31.

  31. Bishop indicated in the Stanford colloquium that the target of his rebellion was a perceived fundamentalism (see note 28). The term Idealism is a euphemism employed in his later writings; see notes 81, 87, and 90. The original term remains in use in constructivist circles, as in the expression fundamentalist excluded thirdist; see note 23.

  32. Berkeley (1734) wrote as follows of the general public: “With this bias on their Minds, they submit to your Decisions where you have no right to decide. And that this is one short way of making Infidels I am credibly informed.”

  33. See Keisler (1986, p. 28) and Herzberg (2013).

  34. See notes 65, 66 below.

  35. See Richman (1996, p. 257). The obliterating comment leans toward radical constructivism; see note 27.

  36. Robinson’s comment on Berkeley appeared in (Robinson 1966, p. 280).

  37. See Pourciau (1999).

  38. See Netz et al. (2001) and note 99, as well as Roquette (2010). The latter text quotes a charming definition of continuity from a 1912 calculus textbook by Kiepert [67]: “If some function is given by \(y=f(x)\) then, in general, infinitely small changes of \(x\) will give rise to infinitely small changes of \(y\).” See further in note 40.

  39. But see notes 123 and 135.

  40. Some time after World War 1, Kiepert’s textbook (see note 38) seems to have been edged out of the market by Courant’s. Courant (1937, p. 101) describes infinitesimals as (1) incompatible with the clarity of ideas; (2) entirely meaningless; (3) vague mystical ideas; (4) fog which hung round the foundations; (5) hazy idea. Courant was unable to peer through the hazy mystical fog the way Robinson would. It should be kept firmly in mind that Courant’s criticism predated Robinson’s framework, unlike certain criticisms of more recent vintage.

  41. See note 99.

  42. See Bishop (1975, pp. 513–514) and note 44.

  43. See Bishop (1977, p. 208).

  44. See Bishop and Keisler (1977). Bishop’s claims need to be understood in the context of his anti-fundamentalist ideology; see notes 28 and 64. Keisler (1977) asked why E. Bishop was chosen as the reviewer in the first place. In his measured reply to Bishop’s vitriolic review Bishop and Keisler (1977) of the first edition of Elementary calculus, An infinitesimal approachKeisler (1986, 1977) asked: “\(\ldots \) why did P. Halmos, the Bulletin book review editor, choose a constructivist as the reviewer?”See further in note 45.

  45. Halmos’ answer to Keisler’s question came in the form of an editorial pointer on p. 271 of the same issue, referring the reader to Halmos’ outline of his editorial philosophy on p. 283: “As for judgments, the reviewer may \(\ldots \) say (or imply) what he thinks.” In other words, a reviewer may use the review as a springboard for developing his own ideological agenda. See further in note 46.

  46. According to a close associate of Halmos’ Ewing (2009), Halmos’ strategy was to confront opposing philosophies in the goal of livening up the debate. One of his goals was to boost lagging sales that were plaguing the publisher at the time (see Halmos 1985). The bottom-line issue, combined with Halmos’ own unflattering opinion of Robinson’s framework as “a special tool, too special” (Halmos 1985, p. 204), apparently made the choice of Halmos’ student (Bishop) as the reviewer, appealing to the editor. The result was a review Bishop and Keisler (1977) that was short on pedagogy and long on vitriol. See Katz and Katz (2011) for further details.

  47. See main text at note 128 for such an invocation.

  48. See Bishop (1977, p. 208).

  49. See notes 44, 64, 65, no match for Kinbote’s diligence (note 17).

  50. Katz et al. (2013) reexamine Fermat’s contribution to the problems of maxima and minima, tangents, and variational techniques.

  51. Sherry and Katz (2012) argue that Leibniz treated infinitesimals as akin to imaginary numbers: Both are fictions, but well-founded fictions because they contribute to discovery and systematization.

  52. See Bair et al. (2013), Bascelli et al. (2014) and Reeder (2013). Euler’s inferential moves exploiting infinitesimals and infinite numbers find natural proxies in the hyperreal framework. Thus Euler’s procedures are only “puzzling” from the viewpoint of an A-track framework (see note 6).

  53. See Błaszczyk et al. (2013) and Borovik and Katz (2012).

  54. Mormann and Katz (2013, p. 224, Section 2.3) argue that at a time when the followers of Cantor, Dedekind and Weierstrass and their philosophical henchmen like Russell and Carnap sought to ban infinitesimals as pseudo-concepts, Hermann Cohen and the Marburg school of neo-Kantian philosophy sought to develop the foundations of a working logic of the infinitesimal. Cohen’s thought is known to have influenced A. Fraenkel, whose student A. Robinson ultimately brought the idea to full fruition. Fraenkel explicitly linked Cohen and Robinson in his memoirs: “my former student Abraham Robinson had succeeded in saving the honour of infinitesimals—although in quite a different way than Cohen and his school had imagined” (Fraenkel 1967, p. 107). Of course, to Cohen, Logik was a philosophical discipline akin to philosophy of science. According to Fraenkel, Robinson’s work on infinitesimals was only an indirect offspring of the concept that Cohen and his school had in mind.

  55. See Bishop (1967, p. viii). Bishop’s expression “peculiar pragmatic content”, connoting an alleged lack of empirical validity of classical mathematics, was analyzed by Billinge (2003, p. 179).

  56. The standard real decimal \(.999\ldots =1\) is defined as the limit of the sequence \((.9, .99, .999 \ldots )\). The class of the same sequence in the ultrapower \({\mathbb R}^{{\mathbb N}}/\mathcal {U}\) gives a hyperreal that falls infinitesimally short of \(1\), providing an alternative interpretation closer to student intuitions; see (Katz and Katz 2010b, a). The hyperreal \(h= [(.9, .99, .999, \ldots )]\), represented by the sequence \((.9, .99, .999, \ldots )\), is an infinite terminating string of \(9\)s, with the last nonzero digit occurring at a suitable infinite hypernatural rank \(N\). The latter is represented by the sequence listing all the natural numbers \((1,2,3,\ldots )\), and \(h=1-\frac{1}{10^{N}}\).

  57. See notes 69 and 72.

  58. See Connes (1995, p. 6207). Note that Connes’ Noncommutative Geometry relies on nonconstructive foundational material such as free ultrafilters, Dixmier trace, and the Continuum Hypothesis. For an analysis of Connes’ critique, see Kanovei et al. (2013) as well as Katz and Leichtnam (2013). See also

  59. See Bishop (1973/1985, p. 1).

  60. This is a follow-up of the discussion in main text at note 15.

  61. G. Cantor had convinced himself of the impossibility of a rigorous justification of infinitesimals, and referred to them as the “cholera bacillus” of mathematics; see Meschkowski (1965, p. 505), Dauben (1995, p. 353), Dauben (1996, p. 124), Ehrlich (2006).

  62. See Connes (2001, p. 16).

  63. Robinson (1968, p. 921) characterized Bishop’s “attempt to describe the philosophical and historical background of [the] remarkable endeavor” of the constructive approach to mathematics, as “more vigorous than accurate”.

  64. Bishop failed to acknowledge in his essay Bishop and Keisler (1977) that his criticism of Keisler’s textbook based on infinitesimals was motivated by Bishop’s foundational preoccupation with the extirpation of the law of excluded middle (see notes 44 and 66).

  65. Bishop’s criticisms apply in equal measure to all of classical mathematics, relying as it does on classical logic (see note 66). Feferman (2000) made a related point in the following terms: “[Bishop] called non-constructive mathematics ‘a scandal’, particularly because of its ‘deficiency in numerical meaning’.”

  66. Classical logic incorporates the law of excluded middle, unlike intuitionistic logic, favored by constructivists.

  67. See Dauben (1996, p. 132). A. Robinson had been scheduled as keynote speaker at the Workshop on the evolution of modern mathematics in 1974 (see Birkhoff 1975) but did not live to deliver his lecture. In a last-minute change, the organizers replaced his lecture in the section on foundations, by Bishop’s.

  68. See Bishop (1968, p. 53). This text was reviewed for MathSciNet by R. L. Goodstein, who commented on the text’s “curiously old-fashioned air”, and “avoidance of the concept of an algorithm and apparent ignorance of almost everything that has been done in constructive mathematics in the past thirty years” (Goodstein 1970).

  69. See Bishop (1975, p. 3).

  70. See Bishop (1967, p. viii).

  71. On procedures vs ontology see note 9.

  72. See Bishop (1967, p. ix).

  73. See Robinson (1968, p. 920).

  74. An imagined fear or threat, or a fear presumed larger than it really is.

  75. See Bishop (1967, p. 6).

  76. See Bishop (1967, p. 6).

  77. See Bishop (1967, p. 10).

  78. See Timon of Athens (Shakespeare 1623), act IV, scene iii, and note 17.

  79. See Robinson (1968, p. 921).

  80. See Heijting (1973, p. 136). Maddy (Maddy 1989, pp. 1121–1122) quotes Heyting somewhat out of context, implying that Heyting is a radical constructivist; see note 27. However, a closer examination reveals conclusively that Heyting is a liberal constructivist, who declared that “intuitionistic mathematics is no longer isolated from classical mathematics\(\ldots \) The two subjects become more and more intertwined” (Heijting 1973, p. 135).

  81. See Bishop (1967, p. ix). See note 31 above for the connotation of the term Idealism in Bishop’s ideology.

  82. See Kolmogorov (2006).

  83. See Bishop (1967, p. 6). See Goodstein’s comment in note 68.

  84. See Maddy (1989) for an analysis of mathematical Platonism in relation to other doctrines.

  85. See (Billinge 2003, p. 183).

  86. See Billinge (2003, p. 183).

  87. See Bishop (1967, pp. 3–4).

  88. See Weyl (1921), a seminal Intuitionist text.

  89. Half a century later, E. Bishop penned his own “crisis” essay Bishop (1975). See also Novikov (2002b, a).

  90. See Bishop (1967, p. 10) where one reads: “Weyl \(\ldots \) suppressed his constructivist convictions [and] expressed the opinion that idealistic mathematics finds its justification in its applications to physics”.

  91. See Bishop (1967, pp. 1–2).

  92. The race to conquer space between the United States and the Soviet Union in the 60s, captured the popular imagination and formed the historical backdrop for Bishop’s another universe comment.

  93. The field is known as Plateau’s problem.

  94. See Beeson (1985, p. 22) and Katz et al. (2014).

  95. See Hellman (1998, pp. 426–427 and p. 432).

  96. See note 65 on Bishop’s quarrel with classical mathematics.

  97. See Bishop (1968, p. 54).

  98. See Bishop (1973/1985, p. 14).

  99. Reinventing the wheel (see main text at note 14) is a metaphor that fits the history of the calculus with uncanny accuracy. The recent work of Netz et al. (see Netz et al. 2001, 2002, pp. 118–119) on the Archimedes Codex reveals that not only have elements of integral calculus been invented by Archimedes, but that Archimedes based his arguments on infinitary concepts involving infinite sums in his Method. As vintage wine that only improves with age, the infinitary idea of the calculus lay dormant for over a millennium, making a comeback in the seventeenth century. Two centuries later, the reinvention of the calculus wheel on the basis of \(\epsilon , \delta \) and limits was completed by Weierstrass. The Weierstrassian track did not displace the infinitary approach, in spite of the ever-rising stridency of the anti-infinitesimal rhetoric, ranging from the cholera bacillus of Cantor (see note 61 for references) to the entirely meaningless hazy fog of Courant (see Courant 1937, p. 101 and note 38).

    As a vintage wine, the infinitary idea endured several decades of post-Weierstrassian scorn, before being clarified by A. Robinson (see note 130 for Robinson’s comments on infinitary processes). The infinitary idea persevered through teetotaller (expression used by Keisler 1977; see note 44) vitriol (term used by Dauben 1996, p. 139 to describe Bishop’s condemnation of Robinson’s framework) of a reluctant guru (expression used by Halmos 1985, p. 162), enjoyed a refreshing endorsement by the intuitionist Heyting (who particularly appreciated Robinson’s insight into the Dirac delta function; see Katz and Katz 2011; Katz and Tall 2013 for a discussion), and has passed the reality check of thousands of publications in economics, engineering, mathematics, and physics.

    Getting back to Archimedes, an argument in the proof of Proposition 14 from Method, “derives a proportion of solids and areas, from a proportion of areas and lines, based on on a rule of summation of proportion[, namely,] Lemma 11” (Netz et al. 2002, p. 119). Archimedes compares a pair of infinite sums that are “equal in multitude” (ísos plethei), i.e. equal in the number of summands. The argument derives the equality of the infinite sums, from the equality of the corresponding summands. Netz et al conclude that Archimedes was explicitly calculating with infinitely great numbers.

  100. A gun should have been hung on the wall in the first act and fired in the third, according to Chekhov’s rule, which we have bent.

  101. The splinter will complete its task in the main text at note 119. Russian-speaking readers may be reminded of Pushkin’s visionary quip to a fellow revolutionary, “K Chaadayevu”: “our names will be inscribed on the splinters of autocracy.” Cf.

  102. See note 61 for references on the Cantor bacillus.

  103. See note 23 for an explanation of the term.

  104. See Richman (1987).

  105. In intuitionistic logic, \(\lnot \lnot P\) does not imply \(P\), or in symbols \(\lnot \lnot P \not \rightarrow P\); see Fig. 2.

  106. See Bishop (1973/1985, p. 14).

  107. See Connes (2007).

  108. See epigraph.

  109. Bishop’s essay Bishop (1975) is cast in the form of an imaginary conversation between Hilbert and Brouwer. G. Birkhoff described non-standard analysis and constructivism as two opposing schools; see Dauben (1996, p. 132).

  110. See Katz and Katz (Katz and Katz (2011), Section 3).

  111. See Dauben (1995, p. 461) quoting A. Robinson’s acceptance speech for the Brouwer medal.

  112. See note 130.

  113. To be more precise, one could ask whether a nonstandard object can be uniquely defined (as e.g., \(\sqrt{2}\) or \(\pi \) are uniquely defined by their usual definitions). In other words, is there a formula \(\phi (x)\) such that 1) \(\exists !x\;\phi (x)\), and 2) such an \(x\) is nonstandard. The answer tends to be in the negative, at least in IST (Internal Set Theory; see note 123) and related theories (e.g., Hrbáček 1978). With IST as the background nonstandard setup, there are two “degrees” of the negative answer.

    First, if \(\phi \) is assumed to be an internal formula (namely, no occurrence of the st-ness predicate) then the fact that its unique solution is standard is an elementary consequence of the Transfer Principle of IST.

    Furthermore, if \(\phi \) is not necessarily internal, then the Transfer argument does not work, but a much less trivial argument (see 3.4.16 in Kanovei and Reeken 2004) yields the standardness of the unique solution of \(\phi \) anyway.

  114. Since there are only countably many definable irrationals, a generic irrational is undefinable. One could ask (following early intuitionists like Poincaré) why one needs non-definable mathematical objects at all? This question makes sense not only in the context of validation of nonstandard methods, of course.

    The answer is that, first of all, by Tarski’s undefinability result, the informal property of definability cannot be soundly described by a mathematical formula. One can observe that a real number \(x\), say \(x=\pi \), is definable in virtue of the mere fact that speaking of it we have in mind its canonical definition, but it turns out that we cannot form “the set of all definable reals” on the basis of Zermelo–Fraenkel axioms (with or without choice) alone. See further in note 115.

  115. On the other hand, various particular types of definability do form legitimate sets of accordingly definable reals, among them:

    • the set COMP of computable reals,

    • the (bigger) set HYP of hyperarithmetic reals,

    and many others, whose common property is that restricting the real line to one of such sets leads to a failure of basic mathematical results.

    In particular, pretending that there is no real outside of COMP, one obtains the failure of the intermediate value theorem (asserting that if \(a < b\), \(f\) is continuous, \(f(a) < 0\), and \(f(b) > 0\) then \(f(x) = 0\) for a suitable \(x\) with \(a < x < b\)).

    Pretending that there is no real outside of HYP, one obtains the failure of the Cantor principle of comparability of any pair of countable wellordered sets; see Simpson (2009). See further in main text at note 135.

  116. See Richard (1964).

  117. See Tarski (1936).

  118. See Cantor (1892).

  119. See note 61 for sources for each of these epithets.

  120. See Keisler (1977, p. 269).

  121. See Keisler (1994).

  122. See Kanovei and Shelah (2004) as well as Keisler (2007, pp. 23–31).

  123. Nelson (1977) introduced a syntactic enrichment into set theory by means of a unary predicate “standard” (st(\(x\)), meaning “\(x\) is standard”), which allows one to detect both non-standard (i.e., infinitely large) integers within the ordinary ZFC (see note 124) integers, and infinitesimals within the ordinary ZFC reals. In particular, Nelson’s internal set theory IST contains an axiom schema called Transfer, which guarantees that all standard sets obey the same mathematical rules as do all sets, standard and nonstandard combined together.

  124. Zermelo-Fraenkel set theory with the Axiom of Choice.

  125. See Dauben (2003, p. 243).

  126. Robinson has been quoted as saying that he would like to get into Leibniz’s head.

  127. In a 2 feb. 1702 letter to Varignon, Leibniz formulated the law of continuity, described as a “souverain principe”, as follows: “il se trouve que les règles du fini réussissent dans l’infini\(\ldots \) et que vice versa les règles de l’infini réussissent dans le fini” (Leibniz, p. 350). This formulation was cited in (Robinson 1966, p. 262), and connected with the Transfer Priciple. To summarize: the rules of the finite succeed in the infinite, and conversely.

  128. See Robinson (1966, p. 2).

  129. See Dauben (1995, p. 461).

  130. Robinson wrote: “[T]he infinitely small and infinitely large numbers of a non-standard model of Analysis are neither more nor less real than, for example, the standard irrational numbers \(\ldots \) both standard irrational numbers and non-standard numbers are introduced by certain infinitary processes” (Robinson 1966, p. 282). See note 99 for infinitary considerations in Archimedes.

  131. See Euler (1748), Kanovei (1988), McKinzie and Tuckey (1997) and Bair et al. (2013).

  132. See Ribenboim (1999) as well as McLarty (2011).

  133. See Hrbacek (2005), Kanovei and Reeken (2004) and Kanovei and Lyubetskii (2007).

  134. For a recent analysis motivated by the foundational status of Fermat’s Last Theorem, see McLarty (2010, 2011).

  135. See notes 113, 114, 115.

  136. See

  137. See Sanders (2014) and


  • Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D., & Shnider, S. (2013). Is mathematical history written by the victors? Notices of the American Mathematical Society, 60(7), 886–904.,

  • Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., Nowik, T., Sherry, D., & Shnider, S. (2014). Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow. Notices of the American Mathematical Society, 61(8), 848–864.

  • Beeson, M. (1985). Foundations of constructive mathematics. Metamathematical studies. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Vol. 6). Berlin: Springer.

    Google Scholar 

  • Berkeley, G. (1734). The analyst: A discourse addressed to an infidel mathematician.

  • Billinge, H. (2003). Did Bishop have a philosophy of mathematics? Philosophia Mathematica (3), 11(2), 176–194.

    Article  Google Scholar 

  • Birkhoff, G. (Ed.) (1975). Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, MA, 1974). Historia Mathematica, 2(4).

  • Bishop, E. (1965a). Differentiable manifolds in complex Euclidean space. Duke Mathematical Journal, 32, 1–21.

    Article  Google Scholar 

  • Bishop, E. (1965b). Colloquium lecture at Stanford University, Circa.

  • Bishop, E. (1967). Foundations of constructive analysis. New York-Toronto, Ontario-London: McGraw-Hill.

    Google Scholar 

  • Bishop, E. (1968). Mathematics as a numerical language. 1970 Intuitionism and Proof Theory (Proc. Conf., Buffalo, New York) pp. 53–71. Amsterdam: North-Holland.

  • Bishop, E. (1973). Schizophrenia in contemporary mathematics. [Published posthumously; originally distributed in 1973]. In Errett Bishop: reflections on him and his research (San Diego, CA, 1983), 1–32, Contemp. Math., 39. Providence, RI: American Mathematical Society (1985).

  • Bishop, E. (1975). The crisis in contemporary mathematics. Historia Mathematica, 2(4), 507–517.

    Article  Google Scholar 

  • Bishop, E. (1977). Review: H. J. Keisler, elementary calculus. Bulletin of the American Mathematical Society, 83, 205–208.

  • Błaszczyk, P., Katz, M., & Sherry, D. (2013). Ten misconceptions from the history of analysis and their debunking. Foundations of Science, 18(1), 43–74. doi:10.1007/s10699-012-9285-8.

  • Borovik, A., Jin, R., & Katz, M. (2012). An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals. Notre Dame Journal of Formal Logic, 53(4), 557–570. doi:10.1215/00294527-1722755.

  • Borovik, A., & Katz, M., (2012). Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Foundations of Science 17(3), 245–276. doi:10.1007/s10699-011-9235-x.

  • Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.

    Article  Google Scholar 

  • Bos, H. (2010). Private communication, 2 Nov (2010).

  • Cantor, G. (1892). Ueber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutsche Mathematiker-Vereinigung, 1, 75–78.

    Google Scholar 

  • Connes, A. (1995). Noncommutative geometry and reality. Journal of Mathematical Physics, 36(11), 6194–6231.

    Article  Google Scholar 

  • Connes, A. Non-standard stuff. Blog (2007).

  • Connes, A., Lichnerowicz, A., & Schützenberger, M. (2001). Triangle of thoughts. Translated from the 2000 French original by Jennifer Gage. Providence, RI: American Mathematical Society.

  • Courant, R. (1937). Differential and integral calculus (Vol. I). Translated from the German by E. J. McShane. Reprint of the second edition. Wiley Classics Library. A Wiley-Interscience Publication. New York: Wiley (1988).

  • Dauben, J. (1995). Abraham Robinson. The creation of nonstandard analysis. A personal and mathematical odyssey. With a foreword by Benoit B. Mandelbrot. Princeton, NJ: Princeton University Press.

  • Dauben, J. (1996). Arguments, logic and proof: mathematics, logic and the infinite. History of mathematics and education: Ideas and experiences (Essen, 1992), pp. 113–148, Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik, 11, Vandenhoeck and Ruprecht, Göttingen.

  • Dauben, J. (2003) Abraham Robinson. Biographical memoirs V. 82. National Academy of Sciences.

  • Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences, 60(1), 1–121.

    Article  Google Scholar 

  • Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41(2), 117–146.

    Google Scholar 

  • Euler, L. (1748). Introductio in Analysin Infinitorum, Tomus primus. SPb and Lausana.

  • Euler, L. (1988). Introduction to analysis of the infinite. Book I. Translated from the Latin and with an introduction by John D. Blanton. New York: Springer, 1988 [translation of (Euler 1748)].

  • Ewing, J. (2009). Private communication, 27 January.

  • Feferman, S. (2000). Relationships between constructive, predicative and classical systems of analysis. Proof theory (Roskilde. (1997). 221–236, Synthese Library, 292. Dordrecht: Kluwer

  • Fraenkel, A. (1967). Lebenskreise. Aus den Erinnerungen eines jüdischen Mathematikers. Stuttgart: Deutsche Verlags-Anstalt.

    Google Scholar 

  • Gerhardt, C. I. (Ed.) (1850–1863). Leibnizens mathematische Schriften. Berlin and Halle: Eidmann)

  • Goodstein, R. L. (1970) Review of (Bishop 1968). See

  • Halmos, P. (1985). I want to be a mathematician. An automathography. New York: Springer.

    Book  Google Scholar 

  • Heijting, A. (1973). Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof. A. Robinson on the 26th April 1973. Nieuw Archief voor Wiskunde (3), 21, 134–137.

    Google Scholar 

  • Hellman, G. (1993). Constructive mathematics and quantum mechanics: Unbounded operators and the spectral theorem. Journal of Philosophical Logic, 12, 221–248.

    Article  Google Scholar 

  • Hellman, G. (1998). Mathematical constructivism in spacetime. The British Journal for the Philosophy of Science, 49(3), 425–450.

    Article  Google Scholar 

  • Herzberg, F. (2013). Stochastic calculus with infinitesimals. Lecture notes in mathematics (Vol. 2067). Heidelberg: Springer.

  • Hewitt, E. (1948). Rings of real-valued continuous functions. I. Transactions of the American Mathematical Society, 64, 45–99.

    Article  Google Scholar 

  • Heyting, A. (1956). Intuitionism. An introduction. Amsterdam: North-Holland.

    Google Scholar 

  • Hill, D. (2009). Personal communication, 21 January 2009.

  • Hill, D. (2013). Personal communication, 12 March 2013.

  • Hrbáček, K. (1978). Axiomatic foundations for nonstandard analysis. Fundamenta Mathematicae, 98(1), 1–19.

    Google Scholar 

  • Hrbacek, K. (2005). Remarks on nonstandard class theory. (Russian) Fundam. Prikl. Mat. 11(5), 233–255; translation in Journal of Mathematical Science (N.Y.) 146(1), 5608–5621 (2007).

  • Kanovei, V. (1988). The correctness of Euler’s method for the factorization of the sine function into an infinite product. Russian Mathematical Surveys, 43, 65–94.

    Article  Google Scholar 

  • Kanovei, V., Katz, M., Mormann, T. (2013). Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics. Foundations of Science, 18(2), 259–296. See doi:10.1007/s10699-012-9316-5 and

  • Kanovei, V., & Lyubetskii, V. (2007). Problems of set-theoretic nonstandard analysis. Uspekhi Mat. Nauk, 62(1) (373), 51–122 (Russian); translation in Russian Math. Surveys, 62(1), 45–111.

  • Kanovei, V., & Reeken, M. (2004). Nonstandard analysis, axiomatically. Springer monographs in mathematics Berlin: Springer.

  • Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. Journal of Symbolic Logic, 69(1), 159–164.

    Article  Google Scholar 

  • Katz, K., & Katz, M. (2010a). When is.999... less than 1? The Montana Mathematics Enthusiast 7(1), 3–30.

  • Katz, K., & Katz, M. (2010b). Zooming in on infinitesimal \(1-.9..\) in a post-triumvirate era. Educational Studies in Mathematics, 74(3), 259–273.

  • Katz, K., & Katz, M. (2011). Meaning in classical mathematics: Is it at odds with Intuitionism? Intellectica, 56(2), 223–302.

  • Katz, K., & Katz, M. (2012a). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51–89. doi:10.1007/s10699-011-9223-1.

  • Katz, K., & Katz, M. (2012b). Stevin numbers and reality. Foundations of Science, 17(2), 109–123. doi:10.1007/s10699-011-9228-9.

  • Katz, K., Katz, M., & Kudryk, T. (2014). Toward a clarity of the extreme value theorem. Logica Universalis, 8(2), 193–214. doi:10.1007/s11787-014-0102-8.

  • Katz, M., & Leichtnam, E. (2013). Commuting and noncommuting infinitesimals. American Mathematical Monthly, 120(7), 631–641. doi:10.4169/amer.math.monthly.120.07.631.

  • Katz, M., Schaps, D., & Shnider, S. (2013). Almost equal: The method of adequality from Diophantus to Fermat and beyond. Perspectives on Science, 21(3), 283–324.

  • Katz, M., & Sherry, D. (2012). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550–1558.

  • Katz, M., & Sherry, D. (2013). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78(3), 571–625. doi:10.1007/s10670-012-9370-y.

  • Katz, M., & Tall, D. (2013). A Cauchy–Dirac delta function. Foundations of Science, 18(1), 107–123. doi:10.1007/s10699-012-9289-4.,

  • Keisler, H. J. (1977). Letter to the editor. Notices of the American Mathematical Society, 24, 269.

    Google Scholar 

  • Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach (2nd ed.). Boston: Prindle, Weber and Schimidt.

    Google Scholar 

  • Keisler, H. J. (1994). The hyperreal line. In Real numbers, generalizations of the reals, and theories of continua (pp. 207–237). Synthese Lib., 242, Dordrecht: Kluwer.

  • Keisler, H. J. (2007). Foundations of infinitesimal calculus. On-line Edition.

  • Kiepert, L. (1912). Grundriss der Differential- und Integralrechnung. I. Teil: Differentialrechnung. Helwingsche Verlagsbuchhandlung. Hannover, 12th edn. XX, 863 S.

  • Klein, F. (1908). Elementary mathematics from an advanced standpoint. Vol. I. Arithmetic, algebra, analysis. Translation by E. R. Hedrick and C. A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908).

  • Kolmogorov, A. (2006). Modern debates on the nature of mathematics. With a commentary by V. A. Uspenskii. Reprinted from Nauchnoe Slovo 1929, no. 6, 41–54. Problemy Peredachi Informatsii, 42(4), 129–141; translation in Problems of Information Transmission, 42(4), 379–389 (2006).

  • Lakatos, I. (1976). Proofs and refutations. The logic of mathematical discovery. Edited by John Worrall and Elie Zahar. Cambridge University Press, Cambridge-New York-Melbourne.

  • Leibniz, G. Letter to Varignon, 2 Feb 1702. In Gerhardt (see item (1850–1863)) IV, pp. 91–95.

  • Leibniz, G. (1710). Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali. In Gerhardt [(1850–1863), vol. V, pp. 377–382]

  • Lightstone, A. H. (1972). Infinitesimals. American Mathematical Monthly, 79, 242–251.

    Article  Google Scholar 

  • Luxemburg, W. (1964). Nonstandard analysis. Lectures on A. Robinson’s Theory of infinitesimals and infinitely large numbers. Pasadena: Mathematics Department, California Institute of Technology’ second corrected ed.

  • Maddy, P. (1989). The roots of contemporary Platonism. Journal of Symbolic Logic, 54(4), 1121–1144.

    Article  Google Scholar 

  • Mancosu, P. (Ed). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.

  • McKinzie, M., & Tuckey, C. (1997). Hidden lemmas in Euler’s summation of the reciprocals of the squares. Archive for History of Exact Sciences, 51, 29–57.

    Article  Google Scholar 

  • McLarty, C. (2010). What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory. Bulletin of Symbolic Logic, 16(3), 359–377.

    Article  Google Scholar 

  • McLarty, C. (2011). A finite order arithmetic foundation for cohomology, preprint.

  • Meschkowski, H. (1965). Aus den Briefbuchern Georg Cantors. Archive for History of Exact Sciences, 2, 503–519.

    Article  Google Scholar 

  • Mormann, T., & Katz, M. (2013). Infinitesimals as an issue of neo-Kantian philosophy of science. HOPOS: The Journal of the International Society for the History of Philosophy of Science, 3(2), 236–280.

  • Nabokov, V. (1962). Pale fire. USA: G. P. Putnam’s Sons.

  • Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.

    Article  Google Scholar 

  • Netz, R., Saito, K., & Tchernetska, N. (2001). A new reading of method proposition 14: Preliminary evidence from the Archimedes Palimpsest (Part 1). In SCIAMVS (Vol. 2, pp. 9–29).

  • Netz, R., Saito, K., & Tchernetska, N. (2002). A new reading of method proposition 14: Preliminary evidence from the Archimedes Palimpsest (Part 2). In SCIAMVS 3 (pp. 109–125).

  • Novikov, S. P. (2002a). The second half of the 20th century and its conclusion: crisis in the physics and mathematics community in Russia and in the West. Translated from Istor.-Mat. Issled. (2), 7(42), 326–356, 369; by A. Sossinsky. American Mathematical Society Transl. Ser. 2, 212, Geometry, topology, and mathematical physics, 1–24, Providence, RI: American Mathematical Society (2004).

  • Novikov, S. P. (2002b). The second half of the 20th century and its results: The crisis of the society of physicists and mathematicians in Russia and in the West. Istor.-Mat. Issled. (2) 7(42), 326–356, 369 (Russian).

  • Pourciau, B. (1999). The education of a pure mathematician. American Mathematical Monthly, 106(8), 720–732.

    Article  Google Scholar 

  • Reeder, P. (2013). Internal set theory and Euler’s Introductio in Analysin Infinitorum. M.Sc. Thesis, Ohio State University.

  • Ribenboim, P. (1999). Fermat’s last theorem for amateurs. New York: Springer.

    Google Scholar 

  • Richard, J. (1964). Les principes des mathématiques et le problème des ensembles. Revue gńérale des sciences pures et appliquées, 16 (1905), 541–543. Translated in Heijenoort, J. van, (Eds.) Source book in mathematical logic 1879–1931 (pp. 142–144). Cambridge, MA: Harvard University Press.

  • Richman, F. (1987). The last croak. The Mathematical Intelligencer, 9(3), 25–26.

    Article  Google Scholar 

  • Richman, F. (1996). Interview with a constructive mathematician. Modern Logic, 6(3), 247–271.

    Google Scholar 

  • Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland.

    Google Scholar 

  • Robinson, A. (1968). Reviews: Foundations of constructive analysis. American Mathematical Monthly, 75(8), 920–921.

    Article  Google Scholar 

  • Robinson, A. (1969). From a formalist’s points of view. Dialectica, 23, 45–49.

    Article  Google Scholar 

  • Roquette, P. (2010). Numbers and models, standard and nonstandard. Mathematische Semesterberichte, 57, 185–199.

    Article  Google Scholar 

  • Sanders, S. (2014). Algorithm and proof as \(\Omega \)-invariance and transfer: A new model of computation in nonstandard analysis.

  • Shakespeare, W. (1623). Timon of Athens.

  • Sherry, D. (1987). The wake of Berkeley’s Analyst: Rigor mathematicae? Studies in History and Philosophy of Science, 18(4), 455–480.

    Article  Google Scholar 

  • Sherry, D., & Katz, M. (2012). Infinitesimals, imaginaries, ideals, and fictions. Studia Leibnitiana, 44(2), 166–192.

  • Simpson, S. (2009). Subsystems of second order arithmetic, 2nd edn. Perspectives in logic. Cambridge: Cambridge University Press; Poughkeepsie, NY: Association for Symbolic Logic.

  • Stewart, I. (1996). From here to infinity. A retitled and revised edition of The problems of mathematics [New York: Oxford University Press (1992). With a foreword by James Joseph Sylvester. New York: The Clarendon Press, Oxford University Press.

  • Stewart, I. (2009). Professor Stewart’s Hoard of mathematical treasures. Profile Books.

  • Tarski, A. (1936). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405.

    Google Scholar 

  • Tho, T. (2012). Equivocation in the foundations of Leibniz’s infinitesimal fictions. Society and Politics, 6(2), 70–98.

    Google Scholar 

  • Urquhart, A. (2006). Mathematics and physics: Strategies of assimilation. In Mancosu (see item (2008)) (pp. 417–440).

  • Weber, H. (1893). Leopold Kronecker. Mathematische Annalen, 43(1), 1–25.

    Article  Google Scholar 

  • Weyl, H. (1921). Über die neue Grundlagenkrise der Mathematik. (Vorträge, gehalten im mathematischen Kolloquium Zürich.). [J] Mathematische Zeitschrift, 10, 39–79.

    Article  Google Scholar 

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The work of V. Kanovei was partially supported by RFBR grant 13-01-00006. M. Katz was partially funded by the Israel Science Foundation grant no. 1517/12. The authors are grateful to Antonio Montalban for expert advice in matters of note 115. The influence of Hilton Kramer (1928–2012) is obvious.

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Kanovei, V., Katz, K.U., Katz, M.G. et al. Proofs and Retributions, Or: Why Sarah Can’t Take Limits. Found Sci 20, 1–25 (2015).

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  • Errett Bishop
  • Abraham Robinson
  • Infinitesimals
  • Hyperreals
  • Intuitionists
  • Petard