Erhardt's misreading of Abraham Robinson

R. Erhardt attempts to examine Abraham Robinson's philosophical posture. In 1975 (MR0392492, page 42), Robinson clarified his philosophical position as follows:

"Mathematical theories that, allegedly, deal with infinite totalities have no detailed meaning, i.e. reference."

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 he clarified in his 1965 article Formalism 64 (MR0214431, page 231). He only intended that such expressions lacked a reference, as he explained in 1975. Whenever Robinson used the term meaningless to refer to infinite totalities, he intended it in this precise technical sense of an absence of reference. In Formalism 64 (page 231), Robinson 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."

Robinson's position is recognizably that of mathematical Formalism. The reader may therefore be surprised to find Erhardt describing Robinson as a finitist. Erhardt was encouraged by Gaifman to read Robinson's 1965 article Formalism 64. The following question naturally arises. Where did Erhardt get the surprising idea of describing Robinson as a finitist? Guessing the answer is not hard. Indeed, Gaifman wrote in his 2012 text On ontology and realism in mathematics (MR2970698, page 488): "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." How well-supported is the surprising Gaifman-Erhardt claim that Robinson was a finitist? Here it is helpful to distinguish between finitism in a narrow sense and finitism in a broad sense. Finitism in the broad sense involves 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. Finitism in the narrow sense is often associated with positions of those intuitionists and constructivists who reject the use of certain infinite totalities in mathematical practice. In Formalism 64 (page 230), Robinson wrote:

"We should continue the business of Mathematics 'as usual', i.e., we should act as if infinite totalities really existed."

Since finitists in the narrow sense do not ``act as if infinite totalities really existed'' it is evident that Robinson was not a finitist in the narrow sense. Indeed, in Formalism 64 (page 234), 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."

One such intuitionist is Michael Dummett, who argued in 1975 (Dummett, Michael. "Wang's paradox." Synthese 30 (1975), 301--324) on page 301 that "Constructivist philosophies of mathematics insist that the meanings of all terms appearing in mathematical statements must be given in relation to constructions which we are capable of effecting." In Formalism 64 (page 235), Robinson explicitly rejects such a requirement concerning `all terms':

``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.''

What evidence does Erhardt provide for his surprising claim? On page 441, Erhardt claims the following concerning Wiles' proof of Fermat's Last theorem: "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." But would Wiles' proof of Fermat's Last Theorem 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 would be legitimate on Robinson's view. This makes it clear that Erhardt mistakenly considers Robinson a finitist in the narrow sense. Erhardt's mistaken conclusion is based on a conflation of the generic meaning of the term meaningless and the technical meaning assigned to it by Robinson. As is well known, Robinson was influenced by Leibniz. Leibniz's fictionalism offers an alternative to both Platonism and nominalism. Leibniz treated infinitesimals, negatives and imaginaries as well-founded fictions (see e.g., Sherry, David; Katz, Mikhail. "Infinitesimals, imaginaries, ideals, and fictions." Studia Leibnitiana 44 (2012), no. 2, 166--192 or MR4786387 or MR4722258 or Katz, M.; Kuhlemann, K. "Leibniz's contested infinitesimals: Further depictions." Gaṇita Bhāratī 45 (2023), no. 1, 77--112. {https://doi.org/10.32381/GB.2023.45.1.4}, {https://arxiv.org/abs/2501.01193}). 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. In the context of a discussion of the Continuum Hypothesis (note 5 on page 432), Erhardt observes the following: "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." He goes on to describe such an observation as a `valid objection'. This is a Platonist position. Evaluating Robinson's stance from an avowedly Platonist viewpoint is not likely to produce meaningful insight. Erhardt's interpretation has been influenced by Gaifman, who wrote the following in his text On ontology and realism in mathematics (page 498) in connection with arithmetic statements: "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." But consider Gaifman's assumption that an independence result (such as G\"odel's Incompleteness Theorem as applied to Peano Arithmetic) "indicates that truths outstrip our capacities, etc." Such an assumption itself depends on a realist posture. Therefore Gaifman's argument is circular. Erhardt's text contains numerous additional errors, which are analyzed in the article Katz, M.; Kuhlemann, K.; Sanders, S.; Sherry, D. "Formalism 25." Journal for General Philosophy of Science 57 (2026), 169-184. {https://doi.org/10.1007/s10838-025-09726-8}, {https://arxiv.org/pdf/2502.14811} Robinson's philosophy of mathematical Formalism remains a viable alternative to mathematical Platonism.