3.2.1 Salience

Whitehead and Russell (Reference Whitehead and Russell1910–13), I, 11, themselves acknowledge that: “in spite of the fact that definitions are theoretically superfluous [ … ] a definition usually implies that the definiens is worthy of careful consideration. Hence the collection of definitions embodies our choice of subjects and our judgement as to what is most important.”

Different combinations of expressions can be used as definienta for abbreviation, and we are at liberty to pick. Often, in the course of a proof, abbreviations are given for convenience. But other abbreviations can encode salient or crucial notions. For instance, the fact that in a given set-theory we choose to use ‘ $\{\varnothing ,\{\varnothing \},\{\varnothing ,\left\{\varnothing \right\}\left\}\right\}$’ as a definiens for ‘3’ points to the fact that that particular expression, possibly opposed to other available choices, plays a meaningful role in our attempt to recover arithmetic set-theoretically. We can abbreviate anything, but some abbreviations are more important than others.

3.2.2 Conceptual Analysis

Salience pertains to definienda too. In mathematics, definienda will often be inherited from a varied, partly symbolic and partly informal tradition, and a possibly unsystematic practice. Explicit definitions may then be the final step of a process of conceptual analysis. Whitehead and Russel (Reference Whitehead and Russell1910–13), Introduction, 12, mention Cantor’s analysis of the notion of continuum as “the statement that what he is defining is the object which has the properties commonly associated with the word ‘continuum,’ though what precisely constitutes these properties had not before been known.” A definition aims at “making definite” (coherently with the greek term ‘’, meaning ‘boundary’): “it gives definiteness to an idea which had previously been more or less vague.” Countless examples could be mentioned, starting from definitions of simple geometrical figures in Euclidean geometry. In general, rigorous (possibly formalized) definitions will provide informal notions with a sharp meaning by determining the extent and limits of their application. Here, definiendum and analysandum coincide: the explanatory direction of the definition within the theory, from definiens to definiendum, reverses the prior process of analysis from an insufficiently clear notion back to its basic constituents. Conceptual analysis has a long history both within and outside mathematics (Otte and Panza Reference Otte and Panza1997; Beaney Reference Beaney and Zalta2021), is a central ingredient to foundational projects, and can be given different readings, some of which will be discussed in what follows.

3.2.3 Ontological Reduction: Identification

Explicit definitions may be a tool for ontological reduction. Suppose we have a translation procedure from an $F$-theory to a $G$-theory, allowing – via some suitable mapping – to translate talk of $F$’s (e.g. natural numbers) into talk of $G$’s (e.g. sets, or categories). By itself, this only tells us that $G$-talk can be used – possibly even only for practical purposes – to express things about $F$’s. Many cases of mathematical modeling in the sciences behave this way. But one can also take the translation to provide the real content of $F$-statements, hence entailing that to be an $F$ is to be a $G$, that we need not consider $F$’s as additional entities with respect to $G$’s, for they simply are some of the $G$’s under different names.

Part of the process of arithmetization of analysis which, during the nineteenth century, led to ultimately base analysis on arithmetic, can be interpreted along these lines. The stepwise introduction of higher number systems from lower ones mirrors a process of ontological identification of higher numbers with set-theoretical constructions out of lower ones, thus combining conceptual analysis and ontological reduction. Following Hilbert, this is called the genetic method – although Hilbert thought only axiomatic presentations could offer proper foundations, and attributed to this method primarily an heuristic function (Hilbert Reference Hilbert and Ewald1900; Landry Reference Landry2013). A sketch of this process is given here.

3.2.4 Ontological Reduction: Elimination

Explicit definitions can be used to show that reference to a kind of objects can be dispensed with. This can be motivated by a principle of ontological parsimony, or by epistemological concerns with the disputed objects.

One leading conception of the subject matter of mathematics is mathematical platonism. According to platonists, mathematical theories (their axioms and theorems) describe facts holding of domains of sui generis mathematical objects. These are conceived as abstract, that is, noncausal, nonspatiotemporally located, and possibly necessarily existent objects, whose existence is independent of the mental and linguistic activities of human subjects (Hale Reference Hale1988; Dummett Reference Dummett1991, chapter 18; Falguera, Martínez-Vidal, and Rosen Reference Falguera, Martínez-Vidal, Rosen and Zalta2022). The opposing view, rejecting the existence of (some or all) abstract mathematical objects, is called (respectively local or global) nominalism (Burgess and Rosen Reference Burgess and Rosen1997). Platonism immediately raises the worry of how we can have knowledge of such objects, especially if one demands such knowledge to fall within the limits of the broadly empirical ways in which we access objects in the natural world. This epistemological problem of access traces back to Plato’s times, but was then emphasized by Benacerraf (Reference Benacerraf1973) (Linnebo Reference Linnebo2006; Liggins Reference Liggins2010; Panza and Sereni Reference Panza and Sereni2013; Linnebo Reference Linnebo, Zalta and Nodelman2023; Nutting Reference Nutting2024). It heavily hinges on the theory of knowledge being assumed in the background. Benacerraf initially endorsed a causal theory of knowledge (Goldman Reference Goldman1967) requiring a causal connection between a (true) belief that $p$ and the fact that $p$, something which is obviously precluded for facts involving abstract objects. In an attempt to avoid reliance on any epistemological assumption, Field (Reference Field1989) reformulated the challenge as the request for an explanation of mathematicians’ reliability in their belief-forming processes, apparently making the challenge even stronger (but see Sereni Reference Sereni and Pataut2016). In what follows, we’ll encounter several ways in which definitions can contribute an answer to Benacerraf’s challenge, by providing mathematical knowledge or facilitating assess to mathematical objects.

One way (among others) to argue for platonism goes through semantic analysis of mathematical language. To take Bencerraf’s example, consider the sentences (A) ‘There are at least three large cities older than New York’, and (B) ‘There are at least three perfect numbers greater than 17’. Provided we take the surface-grammar of sentences at face value and as conducive of their logical form, (A) and (B) share the same logical form, that is, that there are at least mutually distinct objects $x$, $y$, and $z$ which stand in a relation $R$ with an object $a$ which is such that $F\left(a\right)$ and $G\left(a\right)$. Under a standard (referential, Tarskian) semantics, singular terms and constants denote first-order objects. For (A) and (B) to be true, the objects being the referent of ‘ $a$’ and the values of $x$, $y$, $z$ must exist.

The referential import of numerical expressions may not be evident at first. In the applied arithmetical statement:

$\text{There are two moons of Mars.}\left(M^a\right)$

‘two’ occurs in adjectival position and, as any predicate, it should denote a property (being two). However, one can follow Frege in believing that the grammatical form of $M^a$ is misleading, its proper logical form being that of an identity statement where numerical expressions occur in substantival position:

$\text{The number of the concept}┌\text{Moons of Mars}┐=2.\left(M^s\right)$

Since $M^s$, if true, entails (in a non-free logic and in negative free logic) by existential generalization that the referents of the terms flanking the identity sign exist, then – assuming numerical objects are abstract – the truth of $M^s$ entails a form of platonism.

To avoid this conclusion, a nominalist may want to explain the meaning of $M^s$ while doing away with reference to objects. This can be done by reverting to $M^a$ and defining it explicitly in non-arithmetical language:

$\exists x\exists y\left[M\right(x)\wedge M(y)\wedge x̸=y\wedge \forall z(M\left(z\right)\to z=x\vee z=y\left)\right].\left(M^{nq}\right)$

$M^{nq}$ tells us that $x$ and $y$ are distinct $M$’s and anything which is an $M$ is either $x$ or $y$ – expressing in non-arithmetical terms what we say arithmetically with $M^s$ (substantivally) and $M^a$ (adjectivally). The occurrence of ‘ $n$’ in any sentence of the form “There are $nF$’s” should be seen as a meaningless syntactic string in a complex unstructured predicate, while the whole sentence gets replaced by a corresponding application of a numerical quantifier explicitly defined in non-arithmetical language (Frege Reference Frege1884, §§55–56; Field Reference Field1980/2016, 21–23), for example:

$\begin{array}{cc}& {\exists }_{0}xF\left(x\right){=}_{df}\neg \exists xF\left(x\right),\left({\exists }_0^{nq}\right)\\ & {\exists }_{1}xF\left(x\right){=}_{df}\exists xF\left(x\right)\wedge \forall y\left(F\right(y)\to y=x).\left({\exists }_1^{nq}\right)\end{array}$

These numerical quantifiers are explicitly (and recursively) definable. Eliminative definitions of this kind restrict or simplify the language of a theory through the elimination of symbols, and logical analysis is used to dispense with the apparent commitments to disputed objects.

3.2.5 Reduction, Elimination, and Multiple Realizability

A challenge to using explicit definitions for ontological reduction is raised by Benacerraf (Reference Benacerraf1965). Suppose we want to reduce arithmetic to set-theory, explicitly defining numerical terms via set-theoretical ones. To claim that natural numbers are sets, we must be able to tell which sets they are: given a numerical term ‘ $n$’ and terms referring to distinct sets $s_1$ and $s_2$, then either $n=s_1$ or $n=s_2$, but not both. To reconstruct arithmetic, a subject will need to identify a subset of the set-theoretical universe of the same cardinality as $N$ (${\aleph }_0$), a set as initial element (i.e. to play the role of 0), and an injective function on sets which behaves as the successor function on naturals. This delivers a set of sets isomorphic to the natural numbers, whose sequence is called progression, or $\omega$-sequence (two theories $S$ and $S^{*}$ are isomorphic if there is a bijection mapping their domains and vocabularies which is structure-preserving, that is, such that all theorems of $S$ about $S$-objects are mapped, or translated, into theorems of $S^{*}$ about $S^{*}$-objects, and vice versa). The subject will then need to explain how numbers can be used not only to count intransitively, that is, to list numerals in the correct order, but also transitively, that is, to state how many objects are in a given collection or set. Two procedures are available. One subject (Ernie, in Beancerraf’s story) follows von Neumann, starting with $\varnothing$ and applying a function which provides sets which are the union set of their predecessors: $\varnothing$, $\left\{\varnothing \right\}$, $\{\varnothing ,\{\varnothing \left\}\right\}$, $\{\varnothing ,\{\varnothing \},\{\varnothing ,\left\{\varnothing \right\}\left\}\right\}$… Another subject (Johnny) follows Zermelo, using the singleton function instead, delivering sets which are singletons of their predecessor: $\varnothing$, $\left\{\varnothing \right\}$, $\left\{\right\{\varnothing \left\}\right\}$, $\left\{\right\{\left\{\varnothing \right\}\left\}\right\}$, …  These sequences are provably different, since for any two nonconsecutive sets $a$ and $b$, ‘ $a\in b$’ will be true for Ernie (von Neumann’s sets contain all their predecessors), and false for Johnny (the unique member of singleton sets is their immediate predecessor). However, since both set-theoretical reconstructions are isomorphic to the natural numbers, arithmetic can be translated into either. But which is which? Is $2=\{\varnothing ,\{\varnothing \left\}\right\}$ or is 2 = $\left\{\right\{\varnothing \left\}\right\}$? Since arithmetic itself isn’t able to settle the question, we cannot say which of the two sets 2 is, hence we cannot say that 2 is a set. Numbers could not be sets. The argument works for any system of objects isomorphic to the natural numbers, hence the conclusion generalizes: numbers could not be objects. Benacerraf ’s conclusion (Reference Benacerraf1965, 70) that arithmetic is not a study of objects, but rather the study of “the abstract structure that all progressions have in common merely in virtue of being progressions,” echoes Dedekind’s work (Section 5.1.1) and lies at the heart of mathematical structuralism.

One can demote this challenge (as Wright Reference Wright1983, §xv, did) by seeing it as an exemplification in mathematics of the argument for the indeterminacy of translation raised by Quine (Reference Quine1960), chapter 2, according to which common patterns of use of a term always license different and possibly incompatible semantic interpretations: Benacerraf’s challenge would not raise any specific problem with mathematics then, but would simply remind us that mathematics is, on this score, on a par with the rest of language. This notwithstanding, reduction through explicit definitions seems to require univocal identification of individual numbers, but whenever the same structure can be multiply realized, such univocal identification seems precluded, since equally effective alternatives will always be available. Elimination is obtained only in a roundabout way: the availability of incompatible but equally acceptable definitions prevents the disputed terms from standing for objects at all.

Obviously, Benacerraf’s challenge relies on the assumption that the only evidence to decide between alternative constructions can be given by arithmetical considerations. But – as Benacerraf (Reference Benacerraf, Morton and Stich1996) himself acknowledges – broader philosophical or theoretical considerations could be offered. These can then lead to identifying one particular construction as preferred over the others, either in set-theoretical (Section 5.1.2) or category-theoretical (McLarty Reference McLarty1993) terms. An alternative option is to trade the identification of individual numbers for the positing of a unique mathematical structure, this being itself an object that can be differently instantiated. As we will see, this is the strategy adopted in several structuralist views.

3.3.1 Fregean Criteria

As anticipated, broader, not strictly formal, desiderata may be required of definitions. Given their historical and persisting relevance, it is worth considering some of Frege’s suggestions.

Fruitfulness. Definitions must “show their worth” by being fruitful (fruchtbar). In a weak sense, this means that “it must be possible to use them for constructing proofs” (Frege Reference Frege1884, xxi; also, Shieh Reference Shieh2008; Boddy Reference Boddy2021). Fruitfulness is a widely shared desideratum on definitions in mathematical practice (Tappenden Reference Tappenden and Mancosu2008, Reference Tappenden2012). Frege, in particular, connects this to the claim that “Those [definitions] that could just as well be omitted and leave no link missing in the chain of our proofs should be rejected as completely worthless” (Frege Reference Frege1884, §70). Definitions that are not used anywhere in proofs are not properly definitions at all (Frege seems to have used Euclid’s alleged definitions of primitives as an example – as outlined in Section 2.3 – taking them as purely “ornamental definitions”; see Frege and Carnap Reference Frege, Carnap, Reck and Awodey2003, p. 141).

This understanding of Frege’s conception, however, is controversial. After all Frege (Reference Frege and Wright1893–1903 I, §33) endorses both Eliminability and Noncreativity, and these together guarantee that definienda can be eliminated anywhere without loss of content. One way of reconciling this tension is to appeal to passages where Frege suggests that definitions should provide analysis of pretheoretical concepts (e.g. natural number). The fruitfulness of a definition would then be located in the fact that the theory provides the correct definiens for such concepts: its absence would surely leave gaps in proofs, thereby showing the definition to be fruitful. Still, this is in tension with later works of Frege’s, where the emphasis is not on conceptual reconstruction, but rather on theory construction. In Logic in Mathematics (Frege Reference Frege1979b, 209–211), Frege distinguishes between “constructive definitions,” that is, arbitrary explicit stipulations within a theory, and “analytical definitions,” that is, definitions of symbols based on a logical (i.e. conceptual) analysis of previously established uses. However, there is no guarantee that such analysis will preserve meaning. Moreover, the analysis “does not [ … ] form part of the construction of the system, but must take place beforehand.” Analytic definitions are thus no definitions at all, and “the sense in which [a] sign was used before the new system was constructed is no longer of any concern to us.” In such cases, the worth of the analysis does not seem to rely on its capturing exactly the original informal meaning of prior notions, but rather on its being theoretically fruitful. As Carnap reports from Frege’s lectures in Jena (Frege and Carnap Reference Frege, Carnap, Reck and Awodey2003, p. 140), “Such an analysis cannot be proved right; one can only feel that one has hit the nail on the head, and it can prove itself fruitful.” Fruitfulness is then strictly related to the role of Conceptual Analysis in theory construction, as well as to the role of elucidations and explications (see Section 6).

Completeness. Definitions must provide concepts with sharp boundaries of application. As a consequence, they cannot be expanded or modified by adapting them to new entities (see Frege Reference Frege and Wright1893–1903, II, §§56 ff.). We shouldn’t for example define ‘ $+$’ to express sum among naturals and then extend it to sum among integers, rationals, and so on, for we would either be defining a new operation, or claiming that the original was ill-defined. Frege calls such defective definitions “piecemeal”: “This consists in providing a definition for a special case [ … ] and putting it to use and then, after various theorems, following it up with a second explanation for a different case” (Frege Reference Frege and Wright1893–1903, §57). Frege acknowledges that the “development of the science which occurred in the conquest of ever wider domains of numbers, almost inevitably demands such a practice” (§58). But it would then be better to introduce new symbols, since what is acceptable in practice may not be acceptable in logic, that is, in a stable logical reconstruction of a mathematical notion relative to its full domain of application. At each step, piecemeal definitions are only shadows of definitions: “Logic cannot recognise such concept-like constructions as concepts” (§56).

Frege’s insistence on completeness is motivated by his views that concepts (i.e. functions from objects to truth-values) should be defined over a universal domain, and by his adoption of the Principle of Bivalence, which demands that any sentence is either true or false, so that for every object it must be determined whether a given concept does or does not apply. These views led Frege to controversy with fellow mathematicians, especially Peano (see Frege Reference Frege and Wright1893–1903, II, §58, fn. 1, 71), who believed that symbols like ‘ $+$’ or even ‘=’ should be given “conditional definitions”: definitions should only be given over suitably restricted domains, and Frege’s idea that at some stage all such partial definitions can be “combined into the form of a single explanation” would be unreasonable.

Completeness. can be jeopardized for many ordinary concepts by vagueness, which entails an area of borderline cases in which a given concept neither clearly applies nor clearly doesn’t. However, even if one believes that vagueness does not affect mathematical concepts, the criterion can be questioned on more general grounds. Some concepts may be indefinitely extensible (Dummett Reference Dummett1993c, 441), namely such that, “if we can form a definite conception of a totality all of whose members fall under [them], we can, by reference to that totality, characterize a larger [sic] totality all of whose members fall under [them].” A paradigmatic example is the concept $┌$ not being a member of itself $┐$ (Section 5.2.1), but other candidates are those of set, concept, extension, and ordinal and cardinal number. Indefinite extensibility may be at the origins of paradoxes afflicting these notions (Russell Paradox for sets and extensions, Burali-Forti Paradox for ordinals, etc.). To avoid paradoxes, Dummett suggests abandoning classical logic and its requirement for determinate domains of quantification, and adopting intuitionistic logic, thus forfeiting both Completeness and the Principle of Bivalence (Shapiro and Wright Reference Shapiro, Wright, Rayo and Uzquiano2006; Uzquiano Reference Uzquiano2015; Linnebo Reference Linnebo2018a; notice that similar conclusions can be reached in the context of intuitionist mathematics, largely inspired by Brouwer’s works, where definitions must be constructive, i.e. must not merely support a proof that certain objects exist, but must allow for a finite construction procedure for them; Posy Reference Posy2020).

Frege’s rejection of piecemeal definitions may also be resisted by arguing that however strictly we delimit the application of a concept, there may always be new cases for which it is unclear whether the concept applies: concepts can have open texture, or porosity (Waismann Reference Waismann1945, 122–123). Most of our empirical concepts, for instance, “are not delimited in all possible directions” and so their definitions are “always corrigible or emendable.” This phenomenon is different from vagueness, but still leaves a modicum of indeterminacy in our definitions, and it may affect mathematical concepts too, including most basic ones – for example, the concept of number, which groups an always corrigible “family” of more specific concepts (integer, rational, etc.) (Waismann Reference Waismann1951, 235–237). For any porous mathematical concept, Completeness will fail (Shapiro Reference Shapiro2014; Shapiro and Roberts Reference Shapiro and Roberts2021).

Other challenges come from Lakatos’ antifoundationalist views. Lakatos Reference LakatosLakatos (1976) presents a dialogue between characters discussing the (Descartes–)Euler theorem for polyedra, namely that: V(ertices) $-$ E(dges) $+$ F(aces) = 2. In the discussion, several counterexamples are advanced. The effect of these is not just to refute the theorem, but to challenge the very initial definition of ‘polyedron’ on which it rests. Proofs and refutations, in mathematics, not only concern propositional content: they determine the meaning of mathematical terms (a conception evoking some Aristotelian remarks; see Section 2.2). On this picture, definitions, against Completeness, are always modifiable conjectures, part of a fallible mathematical practice (Brown Reference Brown2008; De Toffoli Reference De Toffoli2021).

Notice that these challenges to Completeness have different nature. Indefinite extensibility allows for a univocal identification of the target concepts, but allocates in the very nature of those concepts their impossibility of delimiting a determinate and self-contained domain of objects. Waismann’s open texture and Lakatos’ views put a heavier burden on the analysis of mathematical concepts in general. They suggest that either by the underdetermined nature of concepts, or by the fallibilist nature of conceptual inquiry, a sharp determination of concepts is neither forthcoming nor probably desirable. On this score, they resonate with more nuanced conceptions of analysis, concerned with how informal notions can be differently explicated and how definitions can be modified with time for contextual or pragmatic reasons (see Section 6.2).

Simplicity. A definition must “contain a single sign whose reference it stipulates,” and “one must not [ … ] explain a sign or word by explaining an expression in which it occurs and whose remaining parts are known” (Frege Reference Frege and Wright1893–1903, II, §66). Simplicity entails (a) that it is not possible to jointly define more than one symbol, and (b) that it is not possible to define a symbol $\alpha$ by an expression in which $\alpha$ itself occurs. The latter is distinct from a circularity worry and is meant to rule out methods for defining several undefined expressions at once by means of a matrix of statements in which the expressions occur. Frege (Reference Frege and Wright1893–1903) illustrates this with an algebraic analogy (II, §66):

This requirement had been already discussed in the literature, and relates more generally to implicit definitions, to which we now turn.

4.2.1 Ontological Reduction

Implicit definitions allow translating discourse about $\alpha$ in terms of more elementary expressions ‘ ${\beta }_0,{\beta }_1,\dots {\beta }_n$’ without forcing us to identify $\alpha$ with one of the ${\beta }_i$. As Quine (Reference Quine1969), 72, emphasizes, even outside mathematics this procedure proved helpful, for instance in translating, in empiricist accounts of empirical knowledge talk about physical bodies into talk of sense expressions while avoiding Hume’s controversial identifications of the former with the latter (Quine sees Bentham’s works on fictions in Bentham Reference Bentham and Bowring1962 as a “crucial step” in the development of this procedure).

It is to this definitional procedure that Ayer refers to in the earlier quotation, for the role it plays in Carnap’s Aufbau (Carnap Reference Carnap and by R. A. George1928), which attempts an ambitious foundation of our scientific language by means of a “constructional system of concepts.” Such a system provides a “genealogy of concepts” starting from a limited basis of fundamental concepts (§1), aided by a suitable theory of relations. “Construction takes place through definition” (§38), and provides progressive “ascension” to concepts of objects of higher level from concepts of lower-level objects. In the simplest cases, explicit definitions will do, and “the new object [ … ] remains within one of the already formed object spheres.” In other cases it will only be possible to define an object “relative to the already constructed objects” (§39); Carnap’s term for these is “quasi-objects.” Construction requires that:

Carnap takes the definition of ‘prime number’ as an example: ‘ $x$ is a prime number $\leftrightarrow x$ is a natural number and has only 1 and $x$ as divisors’. Carnap concludes that ‘prime’:

As Carnap’s terminology of “quasi-objects” suggests, ontological reduction is weaker when obtained through implicit definition: we trade identification of objects for translation of talks about them. However, this coheres with the semantic tradition which has characterized a large bulk of analytic philosophy (Coffa Reference Coffa1991; Dummett Reference Dummett1993b): philosophical inquiries, ontological ones included, must proceed through semantic analysis. In this setting, constructions by contextual definitions address every legitimate concern on what speaking of certain objects consists in, while all further metaphysical questions become pseudo-problems.

4.2.2 Rational Reconstruction

Carnap conceived of the Aufbau as a project of rational reconstruction, which he defines as “the searching out of new definitions for old concepts” (Carnap Reference Carnap and by R. A. George1928, Preface, 2nd ed., v). Given a choice of primitives, a rational reconstruction should yield a way for a subject of acquiring a priori the fundamental concepts of a target discourse. This is provided by contextual definitions.

The only evidence for the correctness of a rational reconstruction is its ability to output the targeted discourse. But then nothing prevents in principle different reconstructions, based on different selections of primitives and definitions, from resulting in equally adequate outputs. As Quine (Reference Quine1969), 75, vividly states: “If Carnap had successfully carried such a construction through, how could he have told whether it was the right one? The question would have had no point. [ … ] If there is one way there are many [ … ]”.

Rational reconstructions of mathematical theories by implicit definitions may face the same problem. Once a certain mathematical discourse is recovered, asking which particular reconstruction is the right one may have no point. This exacerbates the multiple realizability problem Benacerraf (Reference Benacerraf1965) raises for explicit definitions.

5.1.1 Outline

In the Euclidean model, one way to picture the epistemic role of primitives and postulates is that they neither admit nor need, respectively, definition and proof, since the basic source of geometrical knowledge is immediately provided by the perceptually accessible physical space we inhabit, whose properties geometry studies (or, in a Kantian perspective, constitutes). This picture was shaken in the nineteenth century by the development of non-Euclidean geometries (mainly through the negation of the fifth postulate), which detached the study of geometrical space and physical space, and removed any kind of extratheoretical ground for postulates and primitives. Geometries started being studied for their formal properties as mathematical theories. The first axiomatic presentation of Euclidean geometry was Hilbert’s, in the Grundlagen der Geometrie (Hilbert Reference Hilbert1899). He considers “three distinct systems of things,” points, straight lines, and planes, and proceeds to discuss geometrical axioms. Although he still presents “the choice of the axioms and the investigation of their relations to one another” as “tantamount to the logical analysis of our intuition of space,” and although he presents his five groups of Axioms (connection, order, parallel, congruence, and continuity) as expressing “certain related fundamental facts of our intuition,” he crucially claims that

In order to define the geometrical primitives, it suffices to determine which relations hold between the elements points, lines, and planes, and everything there is to establish about these relations is provided by the axioms (in either Euclidean or non-Euclidean groupings). The Axioms of Connection “establish a connection between the concepts indicated above; namely, points, straight lines, and planes”; the Axioms of Congruence “define the idea of congruence or displacement”; and so on (emphasis added). Axioms do not simply capture, reconstruct, or recover relations among elements that are given to us prior to the theory, but rather establish them. While subsequent definitions are explicitly given in terms of primitives (e.g. “If $M$ is an arbitrary point in the plane $\alpha$, the totality of all points $A$, for which the segments $MA$ are congruent to one another, is called a circle”; §7), primitives are jointly defined by the axioms.

Lacking extratheoretical evidence, the consistency and independence of axioms must be ascertained through appropriate mathematical procedures. To this aim, Hilbert develops (metamathematical) methods that will then constitute the bulk of model theory. To prove consistency (§9), “it is sufficient to construct a geometry where all of the five groups are fulfilled [i.e. satisfied].” Let points stand for ordered pairs <x, y> from the field of real numbers (or for an appropriately selected enumerable subset $\Omega$ of algebraic numbers), and define lines and planes as constructions on them (that points, lines, surfaces could be represented as constructions out of real numbers was common since Descartes introduced analytic geometry). Then any contradiction in the axioms would surface in the theory of real (or $\Omega$) numbers. Hilbert thus provides a model-theoretic consistency proof of his axioms relative to the consistency of analysis (or the consistency of the arithmetic of $\Omega$), this being assumed as granted. Analogous procedures prove that axioms are mutually independent, and hence that “none of them can be deduced from the remaining ones by any logical process of reasoning” (§10).

In arithmetic, the axiomatic definition of arithmetical primitives was to be developed by Peano and had been anticipated by Dedekind (Reference Dedekind1888). Dedekind sets out to investigate “that part of logic which deals with the theory of numbers” (14). Both he and Frege saw themselves as extending to arithmetic the quest for rigor that dominated the mathematics of their time, they both set out to expel the intuitions of space and time from the foundations of mathematics, and both saw arithmetic as intimately related to logic and as, as Dedekind puts it, “an immediate result of the laws of thought” (Reck Reference Reck2013, Reference Reck, Boccuni and Sereni2021; on different ways of characterizing mathematical rigor, see Burgess Reference Burgess2015; Tanswell Reference Tanswell2024).

Dedekind begins by assuming the notion of an object [thing, Ding] as any object of thought. He then introduces the notion of a system [System] of objects (a proto-set-theoretical notion), having other objects as elements. He then defines subset (part), proper subset (proper part), and intersection [Gemeinheit]. He provides the first systematic treatment of transformations [Abbildungen], namely mappings between a system $S$ and its image $S^{\prime }$, or $\varphi \left(S\right)$, associating an element $s$ of $S$ to an element $s^{\prime }\left(\varphi \right(s\left)\right)$ of $S^{\prime }$. He first defines similar transformations (injective mappings, although he sometimes means bijections), and then automorphism, namely transformations of $S$ into an $S^{\prime }$ which is a (proper or improper) subset of itself. He introduces the crucial notion of chain [Kette] of a system $S\left(S_0\right)$ as the intersection of all those systems which contain $S$ and contain all and only those systems which can be obtained by $S$ through $\varphi$. Apart from ambiguities between a system $S$ and an element $s$, a chain coincides with the smallest inductive set obtained from $s$ of $S$ through a function $\varphi$ (i.e. closed under $\varphi$). Two other major accomplishments are (a) the theorem of complete induction; (b) the first rigorous definition of an infinite set as one that stands in a bijection with a proper subset of itself (subverting a long tradition according to which the whole is greater than the part, as we read in Euclid’s Common Notions). This by itself does not prove that infinite systems (sets) exist. In §66 Dedekind proves that if we take the system $S$ of all things that can be objects of my thought, and the function $\varphi$ mapping each element $s$ of $S$ to the thought that $s$ is an object of my thought (which, being a thought, is itself an object of my thought), then the set $\varphi \left(S\right)$ so obtained is a proper subset of $S$, and hence, by the definition of infinity, $S$ is infinite. To prove that $\varphi \left(S\right)$ is a proper subset of $S$, Dedekind shows that there is at least one thing which is an object of my thought (hence an element of $S$) but is not itself a thought of the form “ $s$ is an object of my thought” (hence not an element of $\varphi \left(S\right)$). Quite unfortunately, he selects “my own Ego” as an example, thus jeopardizing his proof with psychologistic notions. The criticism of psychologism or mentalism is, however, possibly too harsh, and Dedekind’s aim was in fact to provide a logical proof of the existence of infinite sets.

To define natural numbers, Dedekind introduces a specific kind of infinity, simple infinity (§71), and proves that every infinite system $S$ contains as a (proper or improper) part a simply infinite system $N$ (§72). Via the earlier proof that there are infinite systems, this also proves that there are simply infinite systems. These are defined by the following conditions:

The use of ‘ $N$’ and ‘ $1$’ is notationally arbitrary: by themselves, conditions $\alpha -\delta$ only state that a simply infinite system $S$ is ordered [geordnet] by an injective function $\varphi$ from elements of $S$ to elements of $S$, starting from an initial element $s$ which is not in the image $\varphi \left(S\right)$ of $S$. Natural numbers are subsequently characterized via these conditions (§73), through a process of apparently psychological abstraction (but see Reck Reference Reck2013; Yap Reference Yap2014):

§134 contains the proof of the categoricity of (second-order) arithmetic establishing that all systems (models) satisfying $\alpha -\delta$ are isomorphic. Arithmetic can thus disregard the particular nature of the objects composing its models and be concerned only with what follows from those conditions.

In the Arithmetices Principia (Peano Reference Peano1889) and several editions of the Formulaire de Mathématiques (Peano Reference Peano1895), Peano sets up to present arithmetic through a “new method,” which first consists in presenting a background logical language and notation (which will inspire the formal setting of Principia Mathematica). Peano emphasizes that arithmetical “signs” are divided into those that can be defined from other arithmetical signs plus logical ones, and those that cannot. Their properties are characterized axiomatically (Peano Reference Peano1889, pp. 85–86):

Primitive arithmetical notions are number (N), unity (1), the successor of ($\text{a}+1$), and is equal to (=). Peano takes numerical identity as a nonlogical notion to be axiomatically defined, rather than taking identity as a prior logical notion that can then be applied to numbers (as for instance Frege did; Frege Reference Frege1884, §63): as a consequence, four of his axioms are today seen as extra-arithmetical claims concerning identity. The other five, given (as Dedekind’s conditions) within a theory of classes, are the proper axioms of arithmetic (presented here in modified notation):

Axiom 1 states that 1 is a natural number. Axiom 2 states that if $a$ is a natural number, so is its successor. Axiom 3 states that the successor function is injective (different natural numbers have different successors). Axiom 4 states that 1 is not the successor of any natural number. Axiom 5 is the principle of mathematical induction – if $k$ is a class, 1 is in $k$, and if $x$ is a number in $k$ then also $x+1$ is in $k$, then the whole class $N$ is in $k$. As regards Dedekind’s conditions, $\alpha$ corresponds to Axiom 2; $\beta$ entails both Axiom 5 – for the notion of chain is so defined as to encode mathematical induction – and Axiom 1 – since Dedekind’s base-element 1 is contained in $N$ by the definition of chain; $\gamma$ corresponds to Axiom 4; and $\delta$ corresponds to Axiom 3.

As mentioned earlier (Section 3.1), and as is witnessed by the different conceptions of identity just rehearsed, the interplay between logical and nonlogical resources can significantly affect mathematical definitions. Standard modern presentations of the so-called Dedekind–Peano Axioms for arithmetic are given in second-order logic, and take 0 as the initial number:

Second-order arithmetic is categorical. Notice that the Induction Axiom [5] involves quantification over predicates. Since on a standard (Tarskian, referential) semantics second-order quantification over predicates is tantamount to quantification over sets (the sets of individuals to which the corresponding predicates apply), Quine (Reference Quine1970), §5, argued that second-order logic is not logic, but rather set-theory in disguise (“in sheep’s clothing”): it is a mathematical theory itself and shouldn’t be used in the foundations of mathematics. Not everyone shares Quine’s concerns (Shapiro Reference Shapiro1991), but this is still perceived as a serious challenge. Axiom 5 can also be formulated at first-order, avoiding this threat. In this case, however, further axioms need to be added, for addition and multiplication (and hence subtraction and division) can be defined from the PA ${}^2$ Axioms, but must be characterized axiomatically in first-order PA.

Most importantly, first-order arithmetic (due to the compactness of first-order languages) is not categorical, and has nonstandard models (i.e. models not isomorphic to the natural numbers; Boolos et al. Reference Boolos, Burgess and Jeffrey2007, ch. 25), hence the possibility of capturing a unique domain of objects (at least up to isomorphism) is jeopardized. Moreover, Axiom 5 becomes an axiom schema (quantifiers are dropped and predicate letters occur as schematic letters), expressing that any of its instances obtained by replacing schematic letters with suitable arithmetical open formulae is an axiom:

Not only, however, one must clarify what knowing a schema amounts to (McGee Reference McGee1997); also, Axiom 5 now stands for an infinity of sentences: first-order arithmetical axioms are not five, but infinitely many, and it seems to be a reasonable requirement on knowing a theory that it is finitely axiomatizable, so that all of its axioms can be surveyed at once. On the other hand, first-order logic is complete, while second-order logic isn’t, thus making first-order logic a more secure setting to be working with. For these and other reasons, also weaker arithmetical systems are studied (the most relevant being PRA, Primitive Recursive Arithmetic, and Q, Robinson Arithmetic). However, PA and PA ${}^2$ remain standard reference axiomatizations in the foundations of mathematics.

Hilbert, Dedekind, and Peano (and his school; Burali-Forti Reference Burali-Forti1894 ch. IV, §6; Padoa Reference Padoa1900; Cantù and Luciano Reference Cantù and Luciano2021) all contributed to the canonization of the axiomatic method as a standard of mathematical theorizing (soon to be applied, for instance, to presentations of set-theories in the early nineteenth century). They also shared the intuition that axioms jointly define mathematical primitives (and, with Frege and Russell, sought to understand indefinables in logic too; Picardi Reference Picardi and Coliva2022).

5.1.2 Roles and Criteria

Consistency is obviously a crucial requirement of axiomatic definitions (although research programs on inconsistent mathematics are also pursued; Weber Reference Weber2022). When it comes to arithmetic, however, a dedicated discussion is needed. Hilbert’s method to prove the consistency of geometry provided a relative consistency proof: Euclidean geometry is consistent if analysis (i.e. the theory whose objects are used to build the model) also is. The consistency of analysis remained an open question, as did the consistency of arithmetic itself. A relative consistency proof for arithmetic could be given only in logical terms, but the limitations of logicist programs made this hard to obtain. Hilbert hoped to find a direct consistency proof instead. He did this by taking the finitary part of arithmetic to be concerned with signs (syntactic strings) and by claiming that on its basis the whole of arithmetic, once suitably formalized in a system like that of Whitehead and Russell’s Principia Mathematica, could be shown unable to syntactically derive both a formula and its negation (Hilbert Reference Hilbert1926). Gödel’s incompleteness theorems in 1930 and 1931 proved this to be impossible (second theorem), while also showing (first theorem) that any formal system capable of recovering enough arithmetic is incomplete (not every true consequence of the system is syntactically derivable in it). A later proof of the consistency of first-order PA was given by Gentzen, which nonetheless used methods (transfinite induction) that are not formalizable in PA itself. The debate on these issues is vast and cannot be rehearsed here (for surveys, see Raatikainen Reference Raatikainen and Zalta2022; Zach Reference Zach and Zalta2023). Notice, however, that when it comes to the consistency of mathematical axiom systems, often only relative proofs can be obtained, based on other theories whose consistency is not directly proved, but rather taken as plausible enough given the extended time they have been used without engendering contradictions. Also, notice that while we are focusing here on criteria for axioms as definitions, other criteria (such as indeed completeness, soundness, etc.) can be imposed on the metamathematical properties of axiom systems more generally.

As regards other formal criteria, neither Eliminativity nor Non-creativity (Conservativeness) meaningfully apply. For explicit (or implicit) definitions, these criteria guarantee that a newly defined symbol $\alpha$ of the extended language $L^{+}$ can always be replaced by the definiens symbols ${\beta }_0,\dots ,{\beta }_n$ of $L$, or that $L^{+}$-sentences containing $\alpha$ can be replaced by $L$-sentences containing only ${\beta }_0,\dots ,{\beta }_n$. But in axiomatic definitions the definienda are the primitives of $L$: their elimination is tantamount to annihilating the theory altogether.

Axiomatic definitions have been traditionally considered implicit definitions proper (originally by Enriques; Biagioli Reference Biagioli, Cantù and Schiemer2023). An association between the two may have resided in an analogy with the principle of duality which Gergonne himself introduced in projective geometry, since the intersubstitutability of ‘plane’ and ‘line’ makes the individuation of their meaning a relational rather than absolute matter, close in spirit to how meaning is supposedly assigned to primitives by axioms (see also Pasch 1882 and Pollard 2010, Ch. 4). But the association is inadequate (Otero Reference Otero1970; Gabriel Reference Gabriel1978). First, they forfeit $\left(G_3\right)$ (Numerical Balance): generally each undefined symbol occurs in more than one axiom (hence they also fail Fregean Simplicity). Also, they may fail to satisfy $\left(G_4\right)$, Univocality. As foreshadowed by Dedekind and displayed in Hilbert’s independence and consistency proofs, axioms constrain, but do not necessarily fix univocally, the interpretation of the primitives. They only determine a set of relations, leaving room for indeterminacy as to what in fact their relata are. Any way of specifying their meaning will thus prioritize structural features (Giovannini and Schiemer Reference Giovannini and Schiemer2019 aptly call them “structural definitions”), and for this reason many proposals on this score underlie different varieties of mathematical structuralism.

Starting from Dedekind’s work and later suggestions by Benacerraf (see Section 3.2), structuralism has become one of the most developed philosophical accounts of arithmetic and mathematics generally (Hellman and Shapiro Reference Hellman and Shapiro2018). One of its major methodological strengths is that it conjoins philosophical inquiry with a focus on those structural relations which to many seem to matter to professional mathematicians much more than preoccupations with the nature of mathematical objects. Structuralism thus seems especially adequate as a philosophical account of actual mathematics. Also, it is closer in spirit to some algebraic traditions in mathematics (think of algebra, or the views expounded by Bourbaki; Bourbaki Reference Bourbaki1950), as well as to more recent discussions on the foundational role of category theory (Reck and Schiemer Reference Reck, Schiemer and Zalta2023, §3).

For our present purposes (although comparison between structuralism and rival views will resurface below), we are just interested in outlining how axiomatic or structural definitions can provide knowledge of their subject matter, hence how their semantic role can be differently understood, either by reinstating Univocality somehow, or by motivating how to dispense with it. Here are some possible options:

The axiomatic method has now a long tradition, and is pervasive in mathematical practice. For this reason, an exhaustive discussion of its theoretical and philosophical roles would need to consider a vast array of purposes, including its ability to provide rigorous analysis of mathematical notions, of permitting suitable architectonic systematizations of different parts of mathematics, of serving heuristic or even pedagogical functions (Cantù Reference Cantù2023). Here we’ll confine ourselves to some brief remarks on those roles that we have explored in connection with explicit and implicit definitions in general.

Salience of the definienda is straightforward: primitives are necessarily salient within a theory. Salience of the definiens, namely the axiom system, boils down to the motivation for having the theory in the first place.

Ontological Reduction may be accomplished in a peculiar sense. We could first define reals (e.g. following Cantor or Dedekind) as set-theoretical constructions of rationals, and then provide axioms for real number theory. However, these axioms would not introduce reals as primitives, but simply point to objects independently defined: reduction is accomplished beforehand. If, on the contrary, we outright lay down the axioms, no reduction is at stake. An indirect form of reduction is afforded when an axiomatic theory $M$ offers a reduction basis for another theory $M^{*}$, and the primitives of $M^{*}$ are in the end reduced to the objects axiomatically defined in $M$ (as is the case for most mathematical theories with respect to set theory or category theory).

To some extent, axiomatic definitions can provide both Conceptual Analysis and Rational Reconstruction (as well as being the outcome of a process of Explication; see Section 6.2). They can systematize informal notions inherited from practice, or provide a way to reconstruct our knowledge in a certain domain. For a familiar example among the many, Hilbert’s claim that the choice of the axioms of Euclidean geometry “is tantamount to the logical analysis of our intuition of space” could be seen in this light. Frege emphasized that axioms can be ambiguous between analyses of antecedent notions and arbitrary stipulations (Frege Reference Frege1980; Frege Reference Frege and Kluge1971, §3). Both eventually took the connection between informal notions and formal rendition as more inspirational than constitutive: once a formal system has been laid out, the connection between the informal notion and the formalized one can be disregarded (Section 7.3.2).

5.2.1 Outline

Definitions by abstraction are discussed in the foundations of mathematics at least since Frege’s works (and adopted both in Frege’s and earlier times; Mancosu Reference Mancosu2016), and still animate a vast debate (Boccuni and Zanetti Reference Boccuni and Zanettiin press). They are provided by abstraction principles of the form:

$\forall \alpha \forall \beta \left[\Sigma \right(\alpha )=\Sigma (\beta )\leftrightarrow \alpha \sim \beta ]\left(\text{ABS}\right)$

Syntactically, ‘ $\Sigma$’ is a term-forming operator taking expressions like ‘ $\alpha$’ and ‘ $\beta$’ as arguments and delivering singular terms as values, and ‘ $\sim$’ is a relation-term of the underlying language. Semantically, the abstraction function $\Sigma$ takes the denotata of ‘ $\alpha$’ and ‘ $\beta$’ as arguments and delivers the denotata of ‘ $\Sigma \left(\alpha \right)$’ and ‘ $\Sigma \left(\beta \right)$’ as values, and ‘ $\sim$’ stands for an equivalence relation between $\alpha$’s and $\beta$’s. The principle can then be instantiated by different choices of ‘ $\alpha$’, ‘ $\beta$’, and ‘ $\sim$’ in its right-hand side (RHS).

Frege appealed to an instance of ABS in attempting a definition of the concept of (finite) cardinal number. In Frege (Reference Frege1884), after surveying a vast array of rival views, some cornerstone claims are offered:

Concept-Ascriptions: numerical ascriptions – Zahlangaben, namely statements of the form ‘Mars has two moons’ – are statements about concepts. They state that the relevant concept (e.g. $┌$ Moons of Mars $┐$) is such that a particular number is associated to it.

Singular Termhood: the proper logical form of numerical ascriptions is that of identity statements (‘The number of the concept $┌$ Moons of Mars $┐$ = 2’), with numerical expressions occurring as singular terms in substantival position, hence purporting to refer to individual, “self-subsistent” objects.

Cardinality: expressions of the form ‘The number of the concept F’ stand for cardinal numbers (Anzahlen), and answer ‘How many?’-questions about the objects falling under a concept F.

Context Principle: the meaning of words should never be sought for in isolation, but always in the context of a sentence (Frege Reference Frege1884, Introduction and §62) (ruling out, e.g. psychologistic conceptions of linguistic meanings as mental representations).

It follows that the (inherently epistemological) question “How are numbers given to us?” (Frege Reference Frege1884, §62) can only be answered by establishing the meaning of numerical identities. Frege’s insight is that “whenever we speak of objects of any kind, we must have in the background a principle for determining what is to count as the same object of that kind” (Dummett Reference Dummett1991, p. 162). Fixing the meaning of identity statements between numerical terms gives us a criterion of identity for the objects they refer to, insofar as they “express our recognition of a number as the same again” (Frege Reference Frege1884, §62) even when referred to by means of different terms. Such numerical identities should thus be part of the very definition of cardinal number (Anzahl), which Frege (inspired by a passage by Hume) first attempts through what is known today as Hume’s Principle:

$\forall F\forall G(\#F=\#G\leftrightarrow F\approx G).\left(\text{HP}\right)$

Hume’s Principle states that for any two concepts $F$ and $G$, the number of $F$ is identical to the number of $G$ iff $F$ and $G$ are equinumerous, that is, can be put into a one-to-one correspondence relation (bijection).

Hume’s Principle is a second-order abstraction, with terms for second-order entities (concepts) in the RHS, and terms for first-order entities (objects) in the LHS. A first-order abstraction is the Direction Principle (DP) also discussed by Frege:

$\forall a\forall b\left(D\right(a)=D(b)\leftrightarrow a//b)\left(\text{DP}\right)$

stating that, for any line $a$ and $b$, the Direction of $a$ is identical to the Direction of $b$ iff $a$ and $b$ are parallel.

Hume’s Principle and DP contextually define their definienda (‘ $\#F$’, ‘ $D\left(a\right)$’) by means of the sentential conditions on their RHS: Frege suggests (Frege Reference Frege1884, §64) that we “carve up the content” of the RHS “in a way different from the original one, and this yields us a new concept.” How recarving, or reconceptualization, is best understood is controversial (Hale Reference Hale1997; Linnebo Reference Linnebo2018b, ch. 2). According to Frege, we “remov[e] what is specific in the content” of the equivalence in the RHS, and we “divid[e] it” between the two related items. Such logical abstraction turns an equivalence relation into an identity between objects – which are objects of $\alpha$ and $\beta$ (as numbers, for Frege, are numbers of concepts).

Hume’s Principle has significant definitional potential. Its definiens (the RHS) can be formulated logically – in a second-order logic, defining equinumerosity requires quantifying over relations:

$\exists R\left[\forall x\right(Fx\to \exists !y(Gy\wedge R(x,y\left)\right)\wedge \forall x(Gx\to \exists !y(Fy\wedge R(y,x)\left)\right)\left]\right(\approx )$

An initial true instance of HP is obtained by instantiating the RHS with a purely logical concept, $┌$ being non-self-identical $┐$, $┌x̸=x┐$ – where identity, following Leibniz, can be second-order defined by saying that objects are identical iff they share the same properties: $\forall x\forall y(x=y\leftrightarrow \forall F(Fx\leftrightarrow Fy\left)\right)$. This first instantiation then allows the definition of infinitely many cardinals through a procedure known as Fregean Bootstrapping:

Further definitions deliver the notion of finite cardinal, i.e. natural, number. Frege (Reference Frege1879), III, 76, defines the notion of following in a $R$-series, or ancestral of a relation $R$. Take a relation $R$. Define a concept $F$ as hereditary in the $R$-series iff for any two objects $x$ and $y$, if $R(x,y)$, then if $F\left(x\right)$, then $F\left(y\right)$: $F$ is inherited in $R$-series from any $x$ which is $F$ by any $y$ with which $x$ bears $R$. The ancestral of $R$, $R^{+}$, is defined by saying that $y$ follows $x$ in the $R$-series iff $y$ falls under all those $R$-hereditary concepts under which $x$ and any object $z$ such that $R(x,z)$ fall. Compare $┌x$ is the father of $y┐$ ($R$) and $┌x$ is the ancestor of $y┐$ ($R^{+}$). Weak ancestral, $R^{-}$, is then defined as: $y$ follows $x$ in the $R$-series or $y=x$. Compare $┌x$ is the ancestor of $y┐$ ($R^{+}$) with $┌x$ is a member of the lineage of $y┐$ ($R^{-}$), which reflexively applies to the originator of the lineage. Frege then logically defines the relation $S$ of immediate successor between cardinals $m$ and $n$, and can thus define the (ancestral) relation $S^{+}(n,m)┌$ being greater than $┐$ ($m<n$), and (weak ancestral) $S^{-}(n,m)$ as $┌$ being greater than or equal $┐$ ($m\le n$). With $m=0$, this provides the relation of being greater or equal to 0, that is, following in the $S$-series beginning with 0 or being equal to 0. This condition defines the concept of finite cardinal (i.e. natural) number. It does so via logical principles and definitions only. This would qualify arithmetical knowledge, and epistemic access to numbers, as logical and a priori. It also would show (a) that some objects are given to us through purely intellectual processes without appeal to intuition (contra Kant) or mental representations (contra empiricists); (b) that arithmetical statements are analytic in Frege’s sense of being provable by logic and definitions (more on this in Section 5.2.2).

HP provides a simple procedure to establish whether two numerical terms refer to the same cardinal (hence providing conditions of identity). We extract the concepts $F$ and $G$ from ‘ $\#F$’ and ‘ $\#G$’ and check whether they are equinumerous. If they are (the RHS is true, hence the LHS is true, hence), they refer to the same cardinal; if they aren’t (the RHS is false, hence the LHS is false, hence), they refer to different cardinals. Given a well-defined $F$, however, Frege’s Completeness criterion also requires us to be able, for any $x$, to establish whether $x$ is or is not an $F$. But take any sentence of the following form:

$\#F=q,\left(\text{CS}\right)$

where ‘ $q$’ is a singular term not of the form ‘ $\#F$’ (nor previously introduced as abbreviating such a term), such as ‘England’ or ‘Julius Caesar.’ There is no $G$ to be extracted from $q$ and be compared with $F$ as regards equinumerosity. HP does not establish, for any $x$, whether $x$ is a cardinal (hence fails to provide conditions of application). This is known as the Caesar Problem (CP) and is a symptom that HP fails Eliminativity, for it cannot eliminate its definiendum in all sentential contexts, but only in those of the form ‘ $\#F$ = $\#G$’ (see also Section 5.2.2 below).

Frege then discards HP as a definition, and quite surprisingly reverts to an explicit definition. He avails himself of what he initially considers a logical notion, that of extension of a concept, this being an object containing the elements falling under a concept. One way to understand it is as a proto-set-theoretical notion, roughly coinciding with a naive notion of set. Frege (Reference Frege1884), §69, defines ‘ $\#F$’ explicitly as ‘the extension of the concept $┌$ being equinumerous to the concept $F┐$’. If we let ‘ $\epsilon \left(F\right)$’ stand for ‘the extension of the concept $F$’ and ${┌}^{\approx }F┐$ be the second-level concept of concepts $┌$ being equinumerous to the concept $F┐$, Frege’s explicit definition is:

This identifies cardinal numbers with equivalence classes of equinumerous concepts, providing a specific kind of logicist platonism about cardinal numbers (the definition in Frege Reference Frege and Wright1893–1903 differs slightly and defines cardinals as classes of extensions of equinumerous concepts). Being explicit, $N^{=}$ is eliminative: every arithmetical statement becomes a notational variant of a logical statement, and both identity and application conditions are guaranteed, provided we know what extensions are.

Extensions get formal treatment in Frege (Reference Frege and Wright1893–1903). Again, Frege appeals to an abstraction principle. His Basic Law V introduces the crucial notion Frege works with in Grundgesetze, that of value-ranges of functions, stating that two functions have the same value-range iff they have the same values for the same arguments. When applied to those particular functions (from objects to truth-values) that concepts are for Frege, this provides extensions:

$\forall F\forall G\left(ϵ\right(F)=ϵ(G)\leftrightarrow \forall x(Fx\leftrightarrow Gx\left)\right).\left(\text{BLV}\right)$

Any concepts $F$ and $G$ have the same extension iff the same objects fall under both (notice that for reasons internal to the system of Grundgesetze and related to the so-called proof of referentiality, an equivalent CP for extensions may be avoided; (see Frege Reference Frege and Wright1893–1903, §10; Heck Reference Heck1999; Linnebo Reference Linnebo2004; Bentzen Reference Bentzen2019).

Frege shows that all arithmetical truths (including, most notably for us, the PA ${}^2$ Axioms) can be derived as theorems from BLV via $N^{=}$ (and HP, which can itself be derived as a theorem – but see May and Wehmeier Reference May and Wehmeier2019). In his setting we have no proper axioms of arithmetic, that is, unprovable arithmetical first principles: the only basic laws are (supposedly) logical. The Induction Axiom itself is a consequence of the definition of cardinals. If BLV is taken as a definition of extensions, Frege’s logicism can be seen as the view that arithmetical statements are derivable from logical laws as notational variants of logical statements obtained via logical definitions (on varieties of logicism, see Boccuni and Sereni Reference Boccuni and Sereni2021).

Unfortunately, the introduction of BLV in the system of Grundgesetze (Cook Reference Cook, Zalta and Nodelman2023) makes the latter inconsistent, as discovered by Russell (Frege Reference Frege1980, letter 16.06.1902). Extension bears strong resemblance with a naive notion of set (Burge Reference Burge1984), and Russell’s Paradox affects both (hence also Cantor’s, Dedekind’s, and Peano’s settings). Two principles underly the latter:

Principle of Extensionality (PE): $\forall x\forall y\left(\forall z\right(z\in x\leftrightarrow z\in y)\to x=y)$: sets with the same elements are identical.

Unrestricted Comprehension Principle for Sets (UC): $\exists x\forall y(y\in x\leftrightarrow \varphi (y\left)\right)$, where $\varphi$ is any formula not containing $y$ free: for any open formula $\varphi$ not containing $y$ free there exists the set $\{x:\varphi (x\left)\right\}$ of the objects satisfying $\varphi$.