Formalism 25

Abstract

Abraham Robinson’s philosophical stance has been the subject of several recent studies. Erhardt following Gaifman claims that Robinson was a finitist, and that there is a tension between his philosophical position and his actual mathematical output. We present evidence in Robinson’s writing that he is more accurately described as adhering to the philosophical approach of Formalism. Furthermore, we show that Robinson explicitly argued against certain finitist positions in his philosophical writings. There is no tension between Robinson’s mathematical work and his philosophy because mathematics and metamathematics are distinct fields: Robinson advocates finitism for metamathematics but no such restriction for mathematics. We show that Erhardt’s analysis is marred by historical errors, by routine conflation of the generic and the technical meaning of several key terms, and by a philosophical parti pris. Robinson’s Formalism remains a viable alternative to mathematical Platonism.

1 Formalism and Finitism

Abraham Robinson’s intellectual profile continues to inspire passions fifty years after his passing. A number of publications over the past few decades have analyzed his philosophical position, including Benis-Sinaceur (1988), Gaifman (2012), Sherry and Katz (2012), Sanders (2020), Weir (2024), and most recently Erhardt (2025). A majority of commentators take it for granted that Robinson’s mature position is best described as mathematical Formalism. However, Erhardt reads Robinson as a finitist (see Sect. 1.2 for a discussion of distinct meanings of the term) who rejects infinite totalities.

1.1 Meaning and Reference

Significantly, there is a key observation in Robinson’s 1975 article that is missing from Erhardt’s text (though he does cite the article). The observation helps understand Robinson’s take on meaning. In 1975, Robinson clarified his position by stating that

mathematical theories that, allegedly, deal with infinite totalities have no detailed meaning, i.e. reference. (Robinson 1975, emphasis added; reprinted in Robinson 1979, 557)

Namely, the term infinite totality has no reference (or referent) in either the physical world or any Platonic realm of mathematical abstracta. Robinson’s main goal here was to distance himself from mathematical Platonism. He did not believe that expressions such as infinite totality lacked meaning in the sense of being ‘pointless’ or ‘devoid of significance’, as we will see below; he only intended that such expressions lacked a reference, as we explained. In his earlier philosophical text Formalism 64 he used the term direct interpretation in place of reference:

I [...] regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot possess the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations.¹

Having outlined Robinson’s position, let us examine Erhardt’s presentation thereof. Erhardt misrepresents Robinson’s position by quoting Robinson out of context. Thus, Erhardt claims the following:

In Formalism 64, Robinson’s most fully developed philosophical paper, he writes (Robinson 1964/1979b, pp. 230-231, italics in the original), “[...] the notion of a particular class of five elements, e.g. of five particular chairs, presents itself to my mind as clearly as the notion of a single individual [...]. By contrast, I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...]”²

We stress the profusion of ellipses ‘[...]’ in Erhardt’s quotation from Robinson. Superficially, the abridged passage may suggest a finitist position. It is instructive to explore the issue whether Erhardt’s deletions alter the meaning of the passage as intended by Robinson. On the face of it, the passage as quoted by Erhardt sounds rather odd. A modern mathematician who claims to be “unable to grasp” the concept of an actual infinite totality certainly sounds peculiar. We will analyze Robinson’s passage in its context in Sect. 1.4.

1.2 Gaifman’s Reading of Robinson

Erhardt thanks Gaifman for encouraging him to read Robinson’s text Formalism 64 Erhardt (2025, 446). Gaifman may have been the original source of a characterisation of Robinson as a finitist. Indeed, Gaifman claims in 2012:

Abraham Robinson, who was a finitist, or something very near to it, realized the seriousness of the limitations that his position implied with regard to syntactic concepts that required quantification over infinite domains. (Gaifman 2012, 488, emphasis added)

Gaifman’s own position is analyzed in Sect. 3.6. Was Robinson a finitist as claimed by Gaifman, and does Robinson’s position entail serious ‘limitations’ as Gaifman claims? To address the issue, it is crucial to distinguish between finitism in a narrow sense and finitism in a broad sense. In its broad sense, finitism denotes opposition to the use of infinitary concepts at the metamathematical level (as in Hilbert’s program). In its narrow sense, finitism is characterized by an opposition to the use of infinite totalities at both the metamathematical and the mathematical level. Thus, many intuitionists and constructivists were opposed to the use of certain infinite totalities in mathematical practice.

Erhardt’s claim of tension in Robinson’s work stems from a conflation of these two meanings of finitism. The fact that Robinson’s position is more accurately described as Formalism than finitism is evident from his recommendation concerning the business of mathematics:

[W]e should continue the business of Mathematics “as usual,” i.e., we should act as if infinite totalities really existed.

(Robinson 1965, 230, emphasis added)

Finitists (in the narrow sense) never “act as if infinite totalities really existed.” The idea that Robinson was not a finitist in the narrow sense is not merely a matter of our opinion against Erhardt’s. Indeed, Robinson explicitly argued against some finitists’ rejection of the use of infinitary terms in mathematics:

Those who adopt this attitude [including the Intuitionists] think that a concept, or a sentence, or an entire theory, is acceptable only if it can be understood properly and that a concept, or sentence, or a theory, is understood properly only if all terms which occur in it can be interpreted directly, as explained. By contrast, the formalist holds that direct interpretability is not a necessary condition for the acceptability of a mathematical theory. (Robinson 1965, 234)

One such intuitionist is Dummett, who claimed the following:

Constructivist philosophies of mathematics insist that the meanings of all terms, including logical constants, appearing in mathematical statements must be given in relation to constructions which we are capable of effecting, and of our capacity to recognise such constructions as providing proofs of those statements; [...] (Dummett 1975, 301, emphasis added)

Robinson clearly disagreed with Dummett’s claim that all terms must be assigned such a direct meaning. Robinson concluded:

To sum up, the direct interpretability of the terms of a mathematical theory is not a necessary condition for its acceptability; a theory which includes infinitary terms is not thereby less acceptable or less rational than a theory which avoids them. (Robinson 1965, 235, emphasis added)

Here Robinson endorsed the Formalist position and moreover contrasted it with finitism in the narrow sense.

1.3 Standard Model

The source of Platonists’ discomfort with Formalism (and their proclivity to paint Formalists as finitists) is identified in section 10 of Robinson’s text Formalism 64:

In particular, I will mention here the assumption that there exists a standard or intended model of Arithmetic or (alternatively, but relatedly) of Set Theory. Clearly, to the formalist, the entire notion of standardness must be meaningless, in accordance with our first basic principle.³

As discussed in Sect. 1.1, Robinson denied the existence of a referent for such a standard model (a.k.a. intended interpretation) of $\mathbb N$ or $\mathbb R$, in either the physical or any Platonic realm. To Platonists, this may appear as a narrow finitist stance, but they may well ponder why Robinson did not name his article “Finitism 64”.

Erhardt goes on to comment as follows:

There is, however, another alternative: more strongly committing to finitism. This is the path taken by various constructivists, who rather than play the uninterpretable game of symbols choose, in a philosophically principled manner, to adopt a weaker form of mathematics. (Erhardt 2025, 444, emphasis added)

Here, Erhardt may be right about constructivists (rather than Robinson) describing classical mathematics as an ‘uninterpretable game of symbols’,⁴ though Erhardt does not admit in his article that Robinson himself does not describe it in these terms. Erhardt’s first sentence is based on the unfounded assumption that Robinson was a finitist, as we have already discussed; Erhardt compounds his error – of attributing to Robinson the position of narrow finitism⁵ – by taking it upon himself to offer advice as to how to practice the latter.

1.4 Robinson’s Passage in Context

In Sect. 1.1, we saw that Erhardt claimed that Robinson was a finitist based on a passage from Robinson’s text Formalism 64. Examining Robinson’s passage in context reveals a rather different picture. Robinson begins by mentioning the traditional positions of mathematical philosophy (Formalism, Intuitionism, Logicism) on page 228 of his text Formalism 64. On page 230, he mentions a philosophical school of nominalism.⁶ Robinson does not treat nominalism in much detail, on the grounds that to him it represents “little depth from the mathematical point of view” (Robinson 1965, 231). Robinson claims that to a nominalist, there is not much difference between a set of five objects and an infinite set. Both are ‘illusory’ to a nominalist, according to Robinson:

To a nominalist, the existence of a set of five elements is no less illusory than the existence of the totality of all natural numbers. At the other end of the scale are the so-called platonic realists or platonists who believe in the ideal existence of mathematical entities in general, including the existence of transfinite sets of arbitrarily large cardinal numbers to the extent to which they can be introduced at all by means of suitable axioms. (Robinson 1965, 230, emphasis added)

By the time Robinson gets to the “five particular chairs” (as quoted by Erhardt) at the bottom of page 230, it is clear that his goal is to stress the difference between his position and that of nominalists. After noting the difference between “five particular chairs” and an infinite set, Robinson emphasizes on page 231 that describing infinite totalities as ‘meaningless’ does not mean that “such a theory is therefore pointless or devoid of significance” (Robinson 1965, 231).

As noted in Sect. 1.1, Robinson clarified his adjective ‘meaningless’ in his 1975 text where he stated that what he has in mind is an absence of a reference; namely, the term infinite totality does not refer to anything in either the real or a Platonic realm. Already in Formalism 64 he emphasized that he rejected the idea that infinite totalities exist “either really or ideally” (Robinson 1965, 230). Here ‘really’ refers to the physical realm, and ‘ideally’ to a Platonic realm, as he states explicitly in the passage quoted above. It emerges that Erhardt’s presentation of Robinson’s position, as when Erhardt claims that it “places Robinson at odds with typical mathematical practice” (Erhardt 2025, 432), conflates the generic meaning of terms such as to grasp and meaningless, with the precise technical meaning attributed to such terms by Robinson.

1.5 Fermat’s Last Theorem: Is the Proof Legitimate?

The conflation of the generic and the technical meaning of the term meaningless has a further effect of leading Erhardt to a preposterous misrepresentation of Robinson’s position, as when Erhardt claims:

Though Fermat’s Last Theorem is a general result proven by illegitimate methods and that purports to say something about all natural numbers – constituting an illicit reference to actual infinity – one can derive from it a potentially infinite number of legitimate, material claims. Yet the fact that the proof is illegitimate on Robinson’s view is crucial to our appraisal of his view. (Erhardt 2025, 441, emphasis added)

Erhardt’s illegitimacy claim stems from his mistaken identification of Robinson as a narrow finitist. But Wiles’ proof of Fermat’s Last Theorem would not be ‘illegitimate on Robinson’s view’ as Erhardt claims. While Wiles’ proof may seem illegitimate to those finitists who view proofs involving infinite totalities as meaningless in the generic sense of the term, it is certainly legitimate on Robinson’s view, collapsing what Erhardt describes as “our appraisal of his view” due to Erhardt’s conflation of distinct meanings of the term meaningless.

While discussing Robinson’s commitment to potential infinity, Erhardt claims that Robinson is a ‘devout realist’ about each natural number:

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, [...].⁷

Robinson himself would have likely rejected such a description of his position. Being able to grasp a concept (such as a potential infinity of metalanguage integers) does not commit one to a reality of a Platonic object. It emerges that Erhardt is trying to foist an untenable position on Robinson.

With regard to the issue of nominalism and fictionalism mentioned by Erhardt above, we note a pecularity of Erhardt’s approach. Erhardt is keen to cast Robinson as an instrumentalist, rather than one who regards infinite classes as useful fictions. Erhardt dismisses the suggestion that Robinson was a fictionalist, in spite of acknowledging that Leibniz, an avowed fictionalist, “had a profound effect” on Robinson (Erhardt 2024, 8 and note 22). The rationale for such a dismissal is most peculiar: Erhardt considers that it would be anachronistic for Robinson, in 1964, to see himself as a fictionalist since Field’s ‘introduction of fictionalism’ occurred only in 1980 (see Field 1980). Erhardt is right to note that Field’s fictionalism, a form of nominalism, conflicts with Robinson’s repeated rejection of nominalism. But such a conflict arises only if fictionalism is assumed to be necessarily a nominalist proposal. Leibniz’s fictionalism offers an alternative to nominalism. Leibniz treated infinitesimals, along with negative and imaginary numbers, as well-founded fictions; see Sherry and Katz (2012). Although he sometimes suggested that infinitesimals are eliminable in the manner of Archimedean exhaustion arguments, he made no such suggestion for negatives and imaginaries. In all three cases, it is the contribution to systematicity that establishes the mantle ‘well-founded fiction’, rather than any sort of nominalist reduction.

1.6 Symptomatic Keyword List and Imaginary Tensions

Already Erhardt’s keyword list is symptomatic of a problem with his text: the keyword finitism appears, but formalism does not. Erhardt misinterprets Robinson’s comment about being ‘unable to grasp’ actual infinite collections, by viewing it as a finitist stance. As analyzed in Sect. 1.4, Robinson merely sought to distance himself from nominalism and Platonism. Erhardt’s Abstract claims to detect a ‘tension’ between Robinson’s philosophy and his mathematical practice:

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 contributions⁸ 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 [...] (Erhardt 2025, Abstract, emphasis added)

Erhardt is referring to the book Non-standard Analysis (Robinson 1966). However, Erhardt’s argument for his claim is too strong because it would apply to all Formalists. Erhardt’s Abstract accuses Robinson of “giving up on a commitment to reconciling” his philosophy and his mathematical practice. Possibly a Platonist would tend to view all Formalists in such a fashion. This may betray Erhardt’s own philosophical stance.

In sum, Erhardt’s depiction of Robinson’s philosophical position falls prey to a straw man fallacy. Erhardt’s claim in his abstract that Robinson’s philosophical position is somehow at odds with his work in applied mathematics, is unfounded (see further in Sect. 3.5).

2 Twin Prime Conjecture and Types of Infinite Totalities

In his introduction, Erhardt claims that

even basic conjectures of number theory, such as the twin prime conjecture, presuppose the existence of infinite totalities. (Erhardt 2025, 431)

However, such a claim involves a conflation of distinct meanings associated with the term infinite totality; namely, of different levels of language or theory. The twin prime conjecture (TPC) can be formulated in Peano Arithmetic (PA). This can be done, for example, by the formula on page 442, line 4 in Erhardt (2025), which we will express as follows:

$$\begin{aligned} \forall x \, \exists y \, \phi (x,y), \end{aligned}$$

(2.1)

where the formula $\phi$ says that $y>x$ and both y and $y+2$ are prime. Note that PA is a theory that does not know any infinite sets and that is even biinterpretable with the theory obtained from ZF by replacing the axiom of infinity by its negation, and adding $\in$-induction (see Ackermann 1937; Kaye and Wong 2007). The claim that the TPC presupposes the existence of infinite totalities is incorrect with respect to the object theory PA in which the TPC is formulated (because PA does not know any infinite totalities).

On the other hand, such a claim is true with respect to a metatheory in which PA is interpreted in an appropriate structure and thus given a semantics. On this reading, the term infinite totality refers to quantification over an infinite domain. Since formulation (2.1) involves such quantification, it can be said to involve infinite totalities in such a different sense. The special status of $\Sigma ^0_1$ and $\Pi ^0_1$-sentences is discussed in Sect. 3.6.

With regard to the first sense of the term infinite totality, note that it is not the twin prime conjecture that presupposes the existence of infinite totalities, but rather the assumption that there must be a determinate answer to whether it is true or false, based on the belief that the entire universe of all natural numbers exists as a completed infinity somewhere. Formalists, including Robinson, refrain from making such assumptions.

3 Fallacies and Misrepresentations

We document several fallacies and misrepresentations in Erhardt’s text.

3.1 A Logical Fallacy

In Formalism 64, Robinson presents his disagreement with his Platonist opponents in the form of an Alice/Bob-type exchange (Robinson 1965, 231–232). Erhardt misrepresents Robinson’s Alice/Bob-type presentation by attributing Alice’s position to Robinson himself, and failing to mention the more substantive of the two Alice/Bob issues. To elaborate, on page 231, Robinson mentions two objections that Alice (the opponent) might raise:

• (i) the ‘superior brain’ argument (which Robinson quickly dismisses), and

• (ii) the ‘postulation of physical/Platonist infinity’ argument.

Alice’s ‘superior brain’ argument (i) involves the idea that, while Robinson’s brain may have limitations ruling out a “clear conception of all sorts of infinite totalities,” his opponent may possess a superior conception of the said totalities.

Objection (ii) involves a postulation of infinite totalities in either the physical or a platonic realm.⁹

Robinson quickly dismisses the first objection, pointing out that the postulation of such superiority “does not permit any further direct debate of the issue,” (Robinson 1965, 231–232) and proceeds to comment on the second objection. Neither the noun psychology nor the adjective psychological occurs in Robinson’s discussion of either objection, or for that matter elsewhere in his text Formalism 64.

Surprisingly, Erhardt attributes Alice’s argument (i) to Robinson himself, and misleadingly presents the “no further direct debate" comment as Robinson’s final word in the issue. Erhardt writes:

While Robinson has no difficulty grasping the possibility of arbitrarily large finite sets, which motivates his acceptance of the natural numbers taken individually, he reports that it is simply a psychological fact that he cannot grasp infinitary objects, obviating the possibility of finding common ground with disputants.¹⁰

But Robinson ‘reports’ no such thing. As we already mentioned, the ‘psychological’ thing is a fabrication.

3.2 Robinson’s Letter to Gödel

Erhardt’s discussion of Robinson’s letter to Gödel contains further misrepresentations:

In addressing a claim that someone with Gödel’s outlook might make–that he can fathom the infinite even if Robinson himself cannot–Robinson states that this objection ‘does not permit any further direct debate of the issue’. (Robinson 1965, 231–232)

Contrary to what Erhardt’s quotation (taken out of context) suggests, Robinson is perfectly willing to engage in a debate – of the pertinent objection (ii) (see Sect. 3.1).

In his footnote 4, Erhardt similarly misrepresents Robinson’s comment “Hier stehe ich, ich kann nicht anders” in his letter to Gödel,¹¹ by quoting it out of context. Robinson’s point is not to concede purported brain limitations but rather to reject Gödel’s realist assumptions. Moreover, Robinson gives his reasons for a reluctance to accept Gödel’s realist views concerning the hyperreals:

[T]he present evidence for the uniqueness of a non standard $\omega$-ultrapower of the reals is not strong.¹²

Erhardt’s attribution of Alice’s position to Robinson constitutes a logical fallacy¹³ and does not inspire confidence in his analysis.

3.3 Verbal Excesses from Kreisel to Erhardt

Similar remarks apply to certain verbal excesses, such as Erhardt’s claim that

Robinson has already admitted that such totalities are meaningless, and without an argument defending the use of these languages, formalism becomes an inconsistent project (Erhardt 2024, 6, emphasis added).

Erhardt does not explain how exactly Formalism would become an ‘inconsistent’ project in the absence of an argument defending the use of infinite totalities. Where is the inconsistency? His claim appears to be untenable.

Kreisel published his “Observations on popular discussions of foundations" in 1971 (Kreisel 1971). Reviewer James D. Halpern for Mathematical Reviews described Kreisel’s “Observations” as “an unrestrained attack on P. J. Cohen [...] and on A. Robinson” and noted that

Most readers interested in foundations will probably find the previously mentioned papers of Cohen and Robinson profitable reading, the disparagement of these in the paper under review notwithstanding. (Halpern 1971)

As noted by Robinson,

Bernays, in an article published in Dialectica, criticized my attitude in his usual gentle manner, while Kreisel had stated his disagreement with me previously, also in his usual manner. (Robinson 1973b, 515)

The reference is respectively to Bernays (1971) and Kreisel (1971). In a critical response to Robinson’s text (Robinson 1969), Bernays claimed that “there is no fundamental obstacle to attributing objectivity sui generis to mathematical objects.” (Bernays 1971, 178, our translation). Robinson appears to have responded in 1975 by including such ‘objectivity’ of infinitary entities alongside ‘reference’ as claims that a Formalist would reject (see, e.g., Robinson 1975, 49).

In his note 3, Erhardt seeks to distance himself from Kreisel’s verbal excesses, and describes Kreisel’s attack as ‘bias’ and ‘prejudice’. It is therefore disappointing to find Erhardt himself engaging in regrettable verbal excesses.

3.4 Whose Game Was It Anyway?

Erhardt lodges the following claim concerning Robinson’s view of mathematics:

“He explains non-finitary mathematics as a collection of ‘uninterpretable games with symbols’ (Robinson 1969a, p. 47).” (Erhardt 2025, 439)

Looking up the original, one finds that Erhardt has applied a technique that is already familiar from Sect. 3.1: he attributes Robinson’s opponent’s position (in this case, ‘the intuitionist’) to Robinson himself. For the sake of completeness, we reproduce the full passage from Robinson’s text From a formalist’s point of view:

[T]he intuitionist may believe that the classical mathematican, whatever his underlying philosophy, is wasting his time in developing uninterpretable games with symbols, the sin of the formalist being the greater because he does so deliberately and consciously. (Robinson 1969, 47)

It emerges that, according to Robinson, it is the intuitionist (not Robinson himself) who is wont to make pejorative remarks about classical mathematicians and formalists allegedly developing ‘uninterpretable games with symbols.’¹⁴ Erhardt has again misrepresented Robinson’s position.

3.5 Meanings of Standardness

As already mentioned at the end of Sect. 1.4, Erhardt appears to have difficulty keeping apart the generic and the technical meaning of terms like meaningless and to grasp. Following a discussion of Skolem’s nonstandard models and Robinson’s nonstandard analysis, involving the technical distinction between standard and nonstandard numbers, Erhardt writes:

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).” (Erhardt 2025, 442, note 33, emphasis added)

The irony, according to Erhardt, is that Robinson himself talks ‘standard and nonstandard’, and then comes out swinging against ‘the entire notion of standardness’. However, the context – not clarified by Erhardt – of Robinson’s comment on ‘the entire notion of standardness’ is a discussion of the Platonist idea of a standard or intended model of arithmetic or set theory, as analyzed in Sect. 1.3. To reject such an idea, Robinson used the term ‘standard’ in its generic sense, whereas Erhardt misleadingly presents it as if what is involved is the technical sense used in nonstandard analysis.

In more detail, theories of nonstandard analysis exploit a mathematical concept called the standardness predicate, which distinguishes between standard and nonstandard numbers (and more general sets). Since this distinction can be thought of as a formalisation of Leibniz’s distinction between assignable and inassignable numbers,¹⁵ the predicate could be referred to also as the assignability predicate. Meanwhile, Platonists believe in (and Robinson rejects) the existence of standard models a.k.a. intended interpretations of ${\mathbb N}$, ${\mathbb R}$, and other mathematical entities. When the issue is translated into the terminology of the assignability predicate and the intended interpretation, it is obvious that there is no connection between them, contrary to the impression Erhardt seeks to create.

3.6 Gaifman’s Realism

Gaifman (2012) claims to use the term realism in the sense of realism in truth-value. In relation to the natural numbers, this means assuming that every PA sentence is a factual statement, i.e., has an objective truth value (even if it is undecidable in PA itself). As Robinson pointed out almost half a century earlier,

as a matter of empirical fact the platonists believe in the objective truth of mathematical theorems because they believe in the objective existence of mathematical entities. (Robinson 1965, 230)

There is internal evidence in Gaifman’s article that such is indeed his position, as illustrated by his snide comment about Hamkins and Robinson:

Another possibility has emerged from the multiverse conception proposed by Hamkins (2011). On this view, the model of natural numbers depends on the set-theoretic universe containing the model. What it comes to is that we have a clear enough conception of models of PA, or of some extensions of PA, but we have no clear distinction between standard and non-standard models. Perhaps Abraham Robinson, the founder of non-standard mathematics, who was not a set theoretician, would agree to that. (Gaifman 2012, 489)

Gaifman’s professed truth-value realism appears to be consistent with the belief that the ‘standard or intended model of arithmetic’ is a meaningful notion. Recall that Robinson rejects such a notion (see Sect. 1.3).

Note that “$\phi$ is provable” and “T is consistent” are respectively $\Sigma ^ 0_1$ and $\Pi ^0_1$ sentences (when encoded in PA). Gaifman apparently holds that one commits oneself to realism in truth value (in relation to the natural numbers) when one uses the terms provable and consistent in metamathematics. He seems to find it philosophically unacceptable to assume that metamathematical statements about the provability of a sentence or the consistency of a theory could be undecidable. Meanwhile, for a formalist practicing finitistic metamathematics in Primitive Recursive Arithmetic (PRA) such a posture toward undecidability is a natural consequence. The “finitistic semantics” of PRA is naturally incomplete (see Tait 1981). A $\Pi ^0_1$ sentence is ‘true’ if it can be proved by induction; it is ‘false’ if one can provide a counterexample. A $\Sigma ^0_1$ sentence is ‘true’ if one can provide an example; it is ‘false’ if one can provide an inductive proof of the corresponding $\Pi ^0_1$ sentence expressing its negation (see further in Simpson 2009).

Similarly to intuitionism, truth in this semantics is defined by provability, and falsity by refutability. Consequently, in metamathematics, on the basis of potential infinity, there is no tertium non datur. Such a position is unacceptable for Gaifman.

Such a metamathematics can nevertheless prove meaningful statements. For instance, as explained in Hrbacek and Katz (2021) the conservativity of the theory SPOT over ZF can be proved in $\hbox {WKL}_0$ (which is not finitistic). But this conservativity result is a $\Pi ^0_2$ sentence, and according to the results of reverse mathematics (Simpson 2009), there is therefore a proof in PRA, where the result is formulated as $\phi (m, f(m))$ with a primitive recursive function. Such a formulation is finitistic.

The fact that realist positions (like Gaifman’s) make it harder to appreciate the hidden resources of the theory of the real number line was noted by Massas in the following terms:

[T]he conservativity results in [Hrbacek and Katz]¹⁶ would likely fall short of convincing anyone who thinks that any existence claim regarding Robinsonian infinitesimals is simply false. (Massas 2024, 263)

Paul Cohen seemed to regard it as a prerequisite for his progress in foundational research to abandon the idea that there is a unique (intended) interpretaion of $\mathcal P({\mathbb N})$ that conforms to our intuition:

I can assure that, in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one could use natural mathematical intuition. (Cohen 2002, 1072)

3.7 Knowledge-Transcendent Truth?

With regard to the relation between independence results and mathematical realism, Gaifman mentions ‘knowledge-transcendent truth’:

If an independence result indicates that some mathematical truths outstrip our capacities of knowing them, then it points to knowledge-transcendent truth, hence to realism. (Gaifman 2012, 498)

Objectively speaking, the implication would seem to go the other way around: it is only if one wishes to defend realism that one may be led to postulate knowledge-transcendent truth. Such ‘knowledge-transcendent truth’ is mentioned twice in the article (the second occurrence is in the concluding section). It emerges that when Gaifman claims that something points toward realism, what is he has in mind is that it points toward transcendent truth. However, Gaifman’s argument appears to be circular. Namely, the starting assumption that “independence results indicate that some truths outstrip our capacity” itself depends on a realist standpoint. An anti-realist has little reason to assume that independence results would indicate such a thing. So the full Gaifmanian circularity would run as follows:

$$\begin{aligned} \begin{aligned} { \text {realism}}&\Rightarrow { \text {independence results indicate that truths outpace our capacities}} \\&\Rightarrow { \text {there exist transcendent truths}} \\&\Rightarrow { \text {realism}}. \end{aligned} \end{aligned}$$

Gaifman claims further:

I am mostly concerned with independence results for first-order arithmetical statements. My general view is that these results do not affect our conception of the standard natural numbers. Hence, they reinforce any realism that is based on this conception. (Gaifman 2012, 509)

Gödel’s second incompleteness theorem as applied to PA is a classical arithmetic independence result. Gaifman appears to claim that such a result would “reinforce realism.” Accordingly, the independence of CON(PA) of PA would reinforce realism. One wonders what Gaifman would make of Artemov’s recent proof of $\hbox {CON}^S$(PA) within PA.¹⁷ If even independence reinforces realism, lack of independence (in the sense of Artemov’s result) would seem to also reinforce realism. This would suggest that Gaifman’s variety of realism is unfalsifiable.

All known undecidable first-order arithmetic sentences have a preferred truth value. For example, CON(PA) is generally considered to be true (and is of course provable in a stronger system such as ZFC), unlike higher-order statements such as CH. Some mathematicians view this as an argument in favor of realism about PA. However, such an argument does not appear in Gaifman (2012).

4 Conclusion

While Erhardt goes on eventually to discuss Robinson’s mathematical Formalism, his opening presentation of Robinson’s position as finitism is off the mark, and his attempt to present Robinson’s opponent’s position as Robinson’s own does not inspire sufficient confidence to give credence to his speculations concerning Robinson’s analysis of potential infinity.

Erhardt’s endorsement of a naive realism about a standard model of ZFC, including his faith in a determinate truth value of CH (yet to be ‘discovered’), fails to take into account the recent literature in the philosophy of the foundations of mathematics, such as the multiverse perspective of Hamkins (2012); see also (Hamkins 2021, Sect. 8.14). Due to his misreading of Robinson’s terms meaningless, to grasp, and standard, Erhardt sees an irony and an unresolvable tension between Robinson’s use of nonstandard models of the real numbers and his statement that “the entire notion of standardness must be meaningless.” However, there is neither irony nor tension here, since Robinson is referring to his previous statement that infinite sets, including both standard and non-standard models, lack reference. He did not mean that they are pointless or devoid of significance.

In his 1969 text, Robinson envisions

[...] a solution which involves the equal acceptance of several kinds of set theory. At this point, it is natural to recall the historical development of Euclidean geometry whose analogy with the recent development of axiomatic set theory is perhaps closer than is generally appreciated. (Robinson 1969, 46–47, emphasis added)

The comparison between the existence of distinct set theories and the existence of distinct geometries was pursued in more detail by Cohen and Hersh (1967). Different set theories of this type were indeed developed shortly after Robinson’s passing. Hrbacek (1978) and Nelson (1977) independently developed axiomatic approaches to nonstandard analysis as an alternative to model-theoretic approaches. In fact, Robinson’s visionary insight is arguably more likely to emerge from a Formalist than a Platonist philosophical stance.¹⁸

While traditional set theory is formulated in the $\in$-language, the axiomatic approaches enrich the language of set theory by means of the introduction of a one-place predicate ‘standard’ or ‘st’, and are thus formulated in the st-$\in$-language. Such theories are conservative over ZFC.  Suitable axioms govern the interaction of the new predicate with the traditional ZFC axioms. In the axiomatic approaches, infinitesimals are found within ${\mathbb R}$ itself (rather than in an extension, as in the model-theoretic approaches to nonstandard analysis), and nonstandard integers in ${\mathbb N}$ itself. This challenges Platonist notions about ${\mathbb N}$ and ${\mathbb R}$ as determinate (or even mind-independent) entities. Robinson’s Formalism remains a viable alternative to mathematical Platonism.