Denying Infinity: Pragmatism in Abraham Robinson’s Philosophy of Mathematics

Abstract

Abraham Robinson is well-known as the inventor of nonstandard analysis, which uses nonstandard models to give the notions of infinitesimal and infinitely large magnitudes a precise interpretation. Less discussed, although subtle and original–if ultimately flawed–is Robinson's work in the philosophy of mathematics. The foundational position he inherited from David Hilbert undermines not only the use of nonstandard analysis, but also Robinson's considerable corpus of pre-logic contributions to the field in such diverse areas as differential equations and aeronautics. This tension emerges from Robinson's disbelief in the existence of infinite totalities (any mention of them is ‘literally meaningless’) and the fact that much of his work involves them. I argue that he treats infinitary avenues of mathematics as useful tools to avoid this difficulty, but that this is not successful to the extent that these tools must be justified by a conservative extension from finitary mathematics. While Robinson provides a compelling and unorthodox pragmatic justification for the role of formal systems in mathematical practice despite their apparent infinitary presuppositions, he deflates mainstream mathematics to a collection of games that occasionally produces meaningful results. This amounts to giving up on a commitment to reconciling his finitism with his mathematical practice.

KEYWORDS:

• Abraham Robinson
• finitism
• nonstandard analysis
• potential infinity

Acknowledgments

I wish to thank Haim Gaifman for encouraging me to read Formalism 64 and for numerous extensive conversations on the subject. Thanks also to Achille Varzi, Roman Kossak, Artha Lufie, Alan Weir, and an anonymous referee for helpful comments, and to Justin Clarke-Doane for feedback on an early draft that resulted in my framing the paper as a historical project.

Notes

1 For Robinson, it is an ‘empirical fact’ that mathematicians believe certain problems have objective answers because they believe in the ‘objective existence of mathematical objects’ (Robinson 1964/1979b, p. 230). Haim Gaifman writes, ‘[It is no] wonder that the vast majority of mathematicians hate to entertain the possibility that the problem they work on has no solution because it is independent of ZFC. I once heard Paul Erdös say in a lecture on number theory, “hopefully this kind of problem does not fall prey to the monster of independence”’ (Gaifman 2004, p. 502).

2 A well-known theory in Quine 1948 holds that we are ontologically committed to the objects over which we quantify. Not only does this remain a dominant view, but as will be seen below, this is the sort of theory that Robinson implicitly treats as the background in his own philosophy of mathematics, and is hence the perspective we will adopt, while granting that many constructivists and other mathematicians would dispute Quine and further argue that much mathematics can be done without a commitment to infinite totalities in this sense.

3 It must be emphasized at the outset that though Robinson is remembered chiefly as the mathematician who invented nonstandard analysis, he was far from philosophically naive. This article is in part motivated by the belief that his philosophical work has been unduly neglected, perhaps as a product of bias against non-professional philosophers in the community. Georg Kreisel, in a note critical of Robinson’s position, begins thus: ‘[…] one can’t expect too much when [mathematical logicians] publish on philosophical, foundational matters’ (Kreisel 1971, p. 189). This prejudice is not uniquely directed at Robinson—Gödel is another logician whose writings have been treated more as a curiosity than legitimate philosophy—though some eminent figures have sympathized with the former’s views. Witness Paul Cohen: ‘In my own case, I can say that it was the address by Abraham Robinson at Jerusalem in 1964 which caused me to make my choice [of the Formalist position] quite explicit’ (Cohen 1971, p. 9). What follows is intended to briefly defend the above contention about Robinson’s philosophical acumen. Many of the biographical details come from Dauben 1995. Already by the time Robinson was a teenager, his friend Chimen Abramsky said that he possessed, ‘an enormous, wide erudition in every branch of German philosophy. I sat almost spellbound at his immense knowledge of Kant and Hegel, and we discussed various problems until far in the night’ (Dauben 1995, p. 40). During his first term at the Hebrew University of Jerusalem Robinson took a seminar on the foundations of mathematics with his future mentor, Abraham Fraenkel, and in addition to studying Greek, he took courses broadly across philosophy, from epistemology to logic and a seminar on Leibniz. Not only did Robinson study philosophy, but he taught it. During much of his tenure at UCLA he held a joint position in the mathematics and philosophy departments, where he lectured on logic and the philosophy of mathematics, including one course co-taught with David Kaplan. In the year before his death, he co-taught a seminar titled Philosophical Foundations of Mathematics with Stephan Körner at Yale. Robinson also had a background in the history of mathematics, with particular interests in Euclid, Leibniz, Hilbert, and the history of analysis. He possessed an uncanny ability to link technical developments with their philosophical implications. For some of Robinson’s writings on history, see Robinson 1967/1979b, 1968b/1979b, 1996.

4 Robinson 1964/1979b, pp. 231–232. Robinson ends a letter to Gödel concerning these very topics thus: ‘[…] as far as the main thrust of my paper is concerned, with which you are bound to disagree, I can only quote Martin Luther: “Hier stehe ich, ich kann nicht anders”’ (Robinson 1973b). This is Robinson’s acknowledgment that his disagreement with Gödel is so basic that even their expertise cannot adjudicate their differences.

5 Consider Cantor’s continuum problem: Is there an intermediate cardinality between the naturals and the reals? If you are a realist about infinite totalities, then the continuum problem is an obvious question to ask. However, ZFC + CH and ZFC + CH are both consistent, which Robinson takes as implying that there must be no universe of sets. A platonist will respond that we have simply not yet found the axioms stating the basic properties of the universe of sets sufficient to decide CH one way or the other. This is a valid objection, but it does not sway Robinson.

6 This was not always Robinson’s view. At the outset of a talk given at the 1950 International Congress of Mathematicians that covered some of the initial work in model theory, he preceded his technical discussion with the following proviso: ‘The argument will be developed from the point of view of a fairly robust philosophical realism in mathematics, and it is left to those to whom this point of view is unacceptable to interpret our undoubtedly positive results according to their individual outlook. Thus we shall attribute full “reality” to any given mathematical structure, and we shall use our formal language merely to describe the structure, but not to justify its “reality” or “existence” which is taken for granted’ (Robinson 1950/1979a, p. 3). J. W. Dauben, Robinson’s biographer, speculates that this stance may have been a product of his thesis advisor Paul Dienes’ mathematical realism (Dauben 1995, p. 176), or the fact that his mentor when at the Hebrew University of Jerusalem was Abraham Fraenkel. There is a large gap before Robinson next explicitly mentions his philosophical views, at which point they have shifted to the formalism discussed below. However, as early as 1954 it appears that Robinson was already contemplating the philosophical implications of arithmetical independence. In a draft manuscript prepared for the International Congress for Philosophy of Science, Robinson wondered, ‘Is there a specific statement about positive integers, and formulated, preferably, in the first order predicate calculus, such that the truth of the former can be neither confirmed nor denied?’ (Dauben 1995, p. 212). An affirmative answer, he speculated ten years later in Formalism 64, would support formalism over realism (see footnote 39 for his 1973 discussion of the same topic). And toward the end of Robinson’s life he remarked, ‘[…] I cannot imagine that I shall ever return to the creed of the true platonist, who sees the world of the actual infinite spread out before him and believes that he can comprehend the incomprehensible’ (Robinson 1969a, p. 49). A year before his untimely death he wrote, ‘Subsequent exchanges, both oral and in published writings, have not induced me to change my views’ (Robinson 1973a; p. 557).

7 Bernays (1983, p. 258). This statement is not so justified today as it was in 1934. Bernays’ ‘complete security’ has been challenged, not only by philosophers, but mathematicians like Edward Nelson (2005, 2007, 2011, 2015), who believed Peano arithmetic was inconsistent, and the Fields Medal winner Vladimir Voevodsky (2014), who proposed to employ a variety of type theory in his univalent foundations, an alternative to ZFC. In this latter case, the foundational questions at issue do not hinge on infinity, but rather the expressive powers of predicate logic, set theory, and category theory. While at first blush neither instance appears relevant to Robinson, this is not the case, and is discussed in the final section. Nonetheless, most working mathematicians believe the foundations of mathematics are secure.

8 Robinson remarks that Aristotle’s critique of infinite totalities remained relevant to his own philosophy (Robinson 1964/1979b, p. 230).

9 Though some things Aristotle accepts appear to suggest the existence of an infinite totality, such as the boundlessness of time or the infinite divisibility of a line segment, he reduces these examples to instances of potential rather than actual infinity, and definitively rejects the latter (Aristotle 1993, pp. 203b15–203b31).

10 Hilbert’s mathematical acumen notwithstanding, he was not always precise in his philosophical argumentation concerning potential and actual infinity. Hilbert’s imprecision in this respect is visible in his ode to Weierstrass for the latter’s success with the epsilon-delta definition of limit: ‘Just as operations with the infinitely small were replaced by operations with the finite which yielded exactly the same results and led to exactly the same elegant formal relationships, so in general must deductive methods based on the infinite be replaced by finite procedures which yield exactly the same results; i.e. which make possible the same chains of proofs and the same methods of getting formulas and theorems’ (Hilbert 1926, p. 184). This is not technically correct, for the epsilon-delta definition of limit still contains quantification over infinite totalities, an important fact that Hilbert glosses over. Nonetheless, barring this reference to actual infinity, the epsilon-delta definition can be viewed as an infinitary proof each stage of which is finite—resembling an infinite process, as in Aristotle’s conception of potential infinity—making it far preferable on Hilbert’s view to the exotic infinitesimals introduced by Leibniz, which themselves constituted infinitary objects. It remains a topic of historical debate to determine just what his positions were. Robinson (1969a) points out that in the same address Hilbert (1926) states both that, ‘Nobody shall drive us out of Cantor’s Paradise’ (p. 191) and ‘the Infinite is not realized anywhere; it does not exist in the physical world, nor is it admissible as a foundation for our intelligent thinking […]’ (p. 201).

11 There is a crucial philosophical distinction between the novel way Hilbert meant to prove this consistency and the established mode of consistency proof, semantic modeling, which facilitates the introduction of new objects to a structure while guaranteeing the consistency of the resulting system. When a logician expands a structure by introducing new objects, she justifies this by first showing that the new entities can be modeled in an accepted structure. For instance, one can extend (ℕ, ×, +) to the integers (ℤ, ×, +) by modeling the integers as pairs of natural numbers. The key point is that semantic modeling in this fashion already commits one to an infinite totality in the form of the original structure in which the new entities are being modeled. This contrasts with what Hilbert referred to as the ‘axiomatic’ method (Hilbert 1904, p. 131), which was supposed to provide consistency proofs without an appeal to semantics—thus avoiding the commitments of semantic modeling. Instead, his method would be committed only to potential infinity in the form of the concrete signs used in the formal system, and particularly the syntactic proof that one could not derive a contradiction in classical mathematics. Note that there is dispute among some philosophers as to whether Hilbert should be construed as a concretist in this sense, but his attempt to use potential infinity in this way is well-known.

12 This is an oversimplification. The entirety of mathematics may not have been formalized and proven consistent with finitary methods, but the work that went into the attempt—and that which continued afterward—had lasting impacts. Richard Zach (2019) points out that proof theory continued with the development of new methods (such as Gentzen’s ordinal analysis) and results that were not absolute in the pre-incompleteness sense, but relative to various axiomatic systems. Furthermore, there has been work to preserve elements of Hilbert’s Program, as in Feferman 1988, who pursues a relativized Hilbert Program in which stronger theories are reduced to weaker but not necessarily finite systems, and reverse mathematics (Friedman 2006; Simpson 1988), which, as opposed to attempting a justification of all classical mathematics on the basis of finitary assumptions, seeks to determine just which theories of classical mathematics can be so justified. There remain some (see Detlefsen 1986) who, undeterred, remain hopeful that Hilbert’s Program will still be completed.

13 For example, the inductive definition for well-formed formulae in a first-order language whose only connectives are ¬ and ∧:

1. All atoms are well-formed formulae

2. If α is a well-formed formula, then ¬α is a well-formed formula

3. If α and β are well-formed formulae, then α ∧ β is a well-formed formula

4. If α is a well-formed formula and x is a variable, then ∀xα is a well-formed formula

 A syntactic string is a well-formed formula if these rules are used a finite number of times, ‘a finite number of times’ being taken as implicitly understood. The natural number structure is produced in corresponding fashion with 0 taken as primitive and the successor operation applied ‘a finite number of times’. The set of natural numbers is not philosophically well-founded, for ‘the set of all finite items reached by applying the successor operation a finite number of times’ is defined in the very same terms as its members. This is also true for the set of all well-formed formulae in a language like the above.

14 Hilbert writes, ‘[…] we find writers insisting, as though it were a restrictive condition, that in rigorous mathematics only a finite number of deductions are admissible in a proof—as if someone had succeeded in making an infinite number of them’ (Hilbert 1926, p. 184). It should be noted that this does not satisfactorily answer the charge, as Robinson’s admission confirms.

15 See Weir 2021 for a survey.

16 A natural second example comes from Robinson himself. While Leibniz had developed a calculus of infinitesimals, it lacked a foundation to guarantee validity. Criticisms from Berkeley and others are well-documented, and infinitesimals were ridiculed even by modern mathematicians like Cantor or Peano. Robinson wrote of Leibniz that ‘what was lacking at the time was a formal language which would have made it possible to give a precise expression of, and delimitation to, the laws which were supposed to apply equally to the finite numbers and to the extended system including infinitely small and infinitely large numbers as well’ (Robinson 1996; p. 266).

17 Robinson 1973c/1979a, p. 235. Robinson also mentions other criteria beyond interpretability that might justify our accepting infinitary portions of mathematics; applicability to natural science, consistency, or beauty. This utilitarian outlook distinguishes his attitude from that of other philosophically-motivated groups—most notably the constructivists—for whom these alternative rationales are not compelling. It is for now sufficient to note that Robinson is willing to disregard his concerns about areas of mathematics that reference infinity as long as they can be shown to have value based on criteria like the above, though this strategy will in the end prove unsatisfying.

18 For more discussion, see Zach 2019; section 2.

19 Robinson 1973a, p. 561, italics in the original. Not only was Hilbert’s proof theory developed to parallel our understanding, but Frege and then Russell and Whitehead—the former who created the formal system and the latter pair who took it to the extreme with the extent of their proofs and program—had similar beliefs about formal languages. In a letter Robinson wrote in 1946 he indicated a close reading of the Principia, as well as his interest in expressing ‘ideas in a more “formal” way’ because of the ‘clarity’ it offered (Robinson 1946). Dauben agrees that ‘for Robinson, the precise if austere focus that axiomatizations furnished was both an aesthetic and a practical matter’ (Dauben 1995; p. 135).

20 See Bourbaki 2004, pp. 7–8, for a description of this process.

21 Another drawback to Robinson’s account is that it apparently leads to an infinite regress. We formalize the language in the metalanguage to clarify our understanding of our informal work. But this construction appeals to actual infinity, and because it cannot be interpreted, to clarify our understanding of this further system we must formalize again in the metametalanguage. This process could continue forever. As Robinson sees it, since we are treating formal systems as tools, the problem can be avoided, for we are expressly not viewing them as a subject of study requiring an ultimate foundation. Haim Gaifman notices that Robinson ‘[distinguishes] between formalism as a method, or tool, and formalism as a subject of study’ (Gaifman 2004, p. 20, italics in the original). When viewed as a subject, formal systems commit us to the totalities Robinson repudiates, but once we begin to see them as a tool, ‘we can then adopt […] the “useful fiction” attitude towards our subject of study’. But this move is not successful insomuch as we might reasonably respond that despite Robinson’s intentions, to use a metatheory with an infinite language or that requires an infinite domain tacitly affirms what he seeks to reject.

22 Körner 1979, p. xliv. At this point it may naturally be wondered whether Robinson should be classified as a fictionalist. Certainly he did not suggest such a connection himself, for this would have been anachronistic given Field’s introduction of fictionalism in 1980. Moreover, Balaguer (2018) notes that fictionalism is a species of mathematical nominalism, which Robinson emphatically repudiates (Robinson 1964/1979b, pp. 230–231). In opposition to nominalists (and a fortiori, fictionalists), Robinson is a devout realist with regard to each individual natural number. This is evidenced by his commitment to potential infinity, and the observation that there are sentences concerning them with objectively determined truth values that we may never know, due to lack of time or sufficiently powerful computers. Though this is part of a larger discussion, Robinson would be better classified as an instrumentalist or pragmatist with respect to actual infinity, for while actual infinity may be regarded as ‘fictional’ in some loose sense, referring to Robinson as a fictionalist would associate him with a more general thesis that he rejects.

23 Though one need not work in a formal system to do mathematics, a formal system is necessary to pursue certain mathematical avenues, such as independence results, which are by definition oriented around axiomatic frameworks and rules of proof. Consider Gödel’s incompleteness theorems and their implications about provability in sufficiently strong systems of arithmetic, which would be meaningless without a well-defined formal system. It was earlier thought by Hilbert and others that a finitary consistency proof of mathematics would be possible, but Gödel’s results shattered that belief. See footnote 12 for caveats. This sort of discovery exemplifies the objective significance of formalization, as it dramatically expands our mathematical knowledge in ways that could not be done otherwise.

24 See footnote 14, and the possibility that Robinson was alluding to an old joke of Hilbert’s.

25 See Section 4 for an account of why this is unjustified for a finitist of Robinson’s sort. However, his observation concerning the clarificatory role of formalization remains relevant regardless of the association to finitism (discussed further in Section 5).

26 See Tait 1981 for the classic article on what constitutes finitist reasoning. Tait defends an earlier thesis that ‘finitist reasoning is essentially primitive recursive reasoning in the sense of Skolem’ (Tait 1981, p. 524). However, he does not mention Robinson, nor does Robinson himself mention PRA. The purpose of this passage is to explicitly link Robinson—as well as his use of ‘meaningful’ and ‘concrete’—to Skolem’s calculus.

27 The paper in which Skolem introduced PRA was not intended to be part of a finitist agenda (or a formalist one directed by Hilbert, for that matter), but rather the author had been influenced by the Principia, and wanted to demonstrate that arithmetic ‘can be founded in a rigorous way without use of Russell and Whitehead’s notions “always” and “sometimes”. This can also be expressed as follows: a logical foundation can be provided for arithmetic without the use of apparent logical variables’ (Skolem 1923, p. 304). It must also be noted that in Skolem’s original article PRA is deliberately not presented as a formal system. Presenting the theory in this way would naturally pose problems for Robinson, for as with the other formal languages mentioned its use would entail postulating infinite totalities. Yet how he would justify this has already been discussed, and a discussion of whether that justification should be judged satisfactory is forthcoming.

28 It should be noted that there are philosophers and mathematicians who would disagree, suggesting perhaps that infinite domains could be done away with or that a different reading of the quantifiers may enable the logician to avoid infinitary commitments.

29 See van Heijenoort 1967, pp. 302–303, for further discussion.

30 More specifically, because ∃yP(x) $\to$ [P(y) ∧ (x < y) ∧ (y ≤ [x! + 1])] only contains bounded search, P, and the less-than function, which are all primitive recursive, it can be rewritten to eliminate the existential quantifier in favor of the statement that f(x)> 0 for a certain primitive recursive function f. For example, f(x) returns 1 if x is not prime; if x is prime, f(x) either returns the largest y such that y ≤ [x! + 1] and P(y) and x < y or, if there is no such y, returns 0. We can then prove that f(x)> 0 for all x, because for any quantifier-free formula φ(x), the induction schema for PRA is (0) ∧ ∀x(φ(x) $\to$ φ(S(x))) $\to$ ∀xφ(x). In the case of f(x) > 0, we can show that f(0) > 0 and that f(x) > 0 $\to$ f(x + 1) > 0 by decomposing the latter into more elementary facts about prime numbers, such that for all x—and also x! + 1—x is either prime or has a prime factor smaller than x, and that x! + 1 has remainder 1 when divided by any number less than or equal to x, which means that any prime factor of x! + 1 must be greater than x. Then an application of the induction schema proves that ∀x∃yP(x) $\to$ [P(y) ∧ (x < y) ∧ (y ≤ [x! + 1])] holds in PRA. Thanks to Juanhe (TJ) Tan for helping to make this connection explicit.

31 See theorem 10 of Skolem 1923, p. 311, for the proof.

32 See footnote 10, where Hilbert praises Weierstrass’s definition of limit for being committed only to potential infinity.

33 Though others, such as du Bois-Reymond (1877) and Chwistek (1935), had worked on rigorous applications of infinitesimals, and Laugwitz and Schmieden (1961) and Laugwitz (2001) constructed them a foundation in an extension of the rational field, it was Robinson who gave the lasting, canonical justification of Leibniz’s intuitive system. In his textbook on nonstandard analysis Robinson writes, ‘Finally, we should record that […] Skolem’s works on non-standard models of Arithmetic was the greatest single factor in the creation of Non-standard Analysis’ (Robinson 1996, p. 278). Adding to the irony—and irresolvable tension in Robinson’s work and philosophy—is that, due to its predication on the transfinite, Robinson himself says that ‘the entire notion of standardness must be meaningless […]’ (Robinson 1964/1979b, p. 242). In addition to the aforementioned mathematicians, Robinson also credits Artin-Schreier, Hewitt, Erdös, Gillman, and Henriksen for contributions toward the development of a calculus of infinitesimals (Robinson 1961/1979b, p. 4).

34 The axiom is stated as an assumption in Archimedes’ On the Sphere and Cylinder: ‘Further, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with [it and with] one another’ (Archimedes 2002, p. 4). A variation was first presented in Euclid’s Elements Book V: ‘Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another’ (Euclid 1956, p. 114). However, in this case it is not an axiom, but a definition of what constitutes a magnitude.

35 The following outlines Robinson’s reconstruction of the Leibnizian model: let T be the theory of analysis and $\mathfrak{L}$ its language. First, one adds a constant c to $\mathfrak{L}$, and second, an infinite set of sentences to T that tell us c is greater than each finite number n. Call this new theory T’. Every finite subset of T’ will be consistent—ℝ itself models this theory—and so by the compactness theorem there exists an infinite model of T’ which contains an element larger than every standard real. c designates this infinite number, and the reciprocal of c becomes Leibniz’s infinitesimal delta. Not only does $\mathfrak{L}$ itself represent an infinite totality, as described in section 3, but any model of such a theory must also contain transfinite elements.

36 For instance, W. A. J. Luxemburg writes of one such happy circumstance, in which an empirical problem couched in the language of potential infinity (witness ‘without bound’) is reframed in transfinitary terms: ‘[…] Robinson came into contact with the mathematical economist D. J. Brown of Yale University. From their discussions on the problems concerning exchange economies in which the set of traders is allowed to grow without bound, Robinson immediately realized that a more natural approach would be to consider exchange economies in which the set of traders is a nonstandard integer. This nonstandard approach led to the resolution of Edgeworth’s conjecture that, if the set of traders increases, the core approaches the set of competitive equilibrium […]’ (Luxemburg 1979, pp. xxxv–xxxvi). Dauben points out that despite shared interests in metamathematics, Robinson always differed from his other contemporaries—e.g. Tarski or Henkin—in that his purpose was to use logic as a tool for solving problems of pure or applied mathematics rather than to study logic itself. In this case, his insight was that in an economy with a boundless set of members any single participant would have an infinitesimal impact on the economy as a whole, and thus that his nonstandard approach was an intuitively appealing way to treat such scenarios (Dauben 1995, pp. 156–161, 444).

37 Several months after Robinson and Allen Bernstein proved the first remarkable result of nonstandard analysis—the invariant subspace theorem for Hilbert space, published in Robinson & Bernstein & Robinson 1966/1979b—Paul Halmos found a proof by standard means. He wrote the following twenty years later: ‘However admirable some of the accomplishments of the logicians may be, the mathematician doesn’t need them, cannot use them, in his daily work. The logic proofs of mathematics theorems, all of them so far as I know, are dispensable: they can be (and they have been) replaced by proofs using the language and techniques of ordinary mathematics. The Bernstein-Robinson proof uses non-standard models of higher order predicate languages, and when Abby sent me his preprint I really had to sweat to pinpoint and translate its mathematical insight. Yes, I sweated, but, yes, it was a mathematical insight and it could be translated’ (Halmos 1985, p. 320).

38 See Mario Bunge’s comments during a London colloquium on the philosophy of science in 1965 where Robinson gave a keynote address, in which he points out that ‘the physicist may feel relieved [… for he] can now refer to non-standard analysis for the rigorous justification of his intuitive infinitesimals […]’ (Robinson 1967/1979b, p. 553–554). As it so happened, Robinson would later collaborate on work in nonstandard quantum theory. See Keleman & Robinson 1972/1979b.

39 Jin 2000, p. 332. There are many uses for the nonstandard methods that stem from Robinson’s development of nonstandard analysis. For two more examples, see Leth 1988a, 1988b for a treatment in combinatorial number theory, and Keisler & Leth 1991 for applications in descriptive set theory.

40 Contrary to this pretense, Robinson actually appears to believe in arithmetical independence. In his twelfth metamathematical problem, posed to parallel Hilbert’s famous address, Robinson asks, ‘Is there a well-formed first order assertion about the natural numbers which can be neither proved nor refuted by a formal or by a generally acceptable informal argument?’ (Robinson 1973c/1979a, p. 57). He affirms that he believes the answer is yes when he later adds, ‘While others are still trying to buttress the shaky edifice of set theory, the cracks that have opened up in it have strengthened my disbelief in the reality, categoricity or objectivity, not only of set theory but also of all other infinite mathematical structures, including arithmetic’ (Robinson 1973c/1979a). Robinson believes that the existence—when proven—of arithmetically independent statements will support his position on the unreality of the totality of natural numbers.

41 There is an analogy to the infinite regress of formalization mentioned in footnote 21; though computers may verify our proofs, who is to verify theirs? This is not merely a hypothetical problem. In 2011, after 25 years of work, Edward Nelson (2015) published his proof of the inconsistency of Peano arithmetic—which, if correct, would have been the most explosive result in mathematics of all time, dwarfing even the discovery of the irrational numbers—only to recall it after Terence Tao and Daniel Tausk discovered an error a few days later. This error was missed by Nelson’s proof checker, and he went back to working on a revised version the next day.

42 In this sense mathematics parallels the empirical sciences. Just as experimentation is vital to chemistry or physics as a methodology for testing hypotheses and answering questions, the method of agreement in mathematics is proof from axioms. If there is no way of answering questions, whether in physics or mathematics, there is no longer a discipline.

43 See Robinson 1973a/1979b, pp. 561, 563–564, for evidence of his own conviction that logic, the brain, and mathematics are importantly linked and require deeper analysis.