• 1 A position that is better summarized by Frege’s belief in a Third Realm, as in G. Frege, “The Tho (...)

2They couldn’t be psychological entities either, given that our psychological capabilities are just as contingent as physical reality. The solution is to take them as abstract entities independent from this world,1 and therefore, to take mathematical statements as necessary truths about them.

• 2 P. Benacerraf, “Mathematical truth”, The Journal of Philosophy, vol. 70, n^o 19, 1973, p. 661-679.

3After Benacerraf,2 we learned that this is not an easy solution. It is one thing to say that numbers are abstract entities, but to fully grasp truths about them is another. If numbers are abstract entities, how can we have access to them?

• 3 As consequence, many phenomenon related to the practice would be non-essential, collateral, featu (...)

4The thesis that such objects are abstract entities also clashes with mathematical practice. If ultimately they are independent entities, there is not much to the practice other than discover things to be true.3 But given the difficulties of grasping abstract phenomenon, how can this be even possible? Just claiming the opposite seems also not that helpful, as it is doubtful that such objects are humanly created. It would turn (P2) as false, which is a even higher price to pay.

• 4 K. Fine, “Our knowledge of mathematical objects”, in T. Gendler and J. Hawthorne (eds.), Oxford S (...)
• 5 K. Fine, “Mathematics: Discovery or invention?”, Think, vol. 11, n^o 32, 2012, p. 11-27.

5It is on the context of this dilemma that Kit Fine has offered a proposal labeled as “Postulationism” in 20054 and 2012.5 Our goal in this paper is to evaluate Fine’s postulationism. By considering the practice of mathematics as a linguistic practice, we will follow the perspective of Speech Act Theory to evaluate Fine’s main ideas. This is, we believe, a fruitful way to give the practice its value without falling to the ontological and epistemological traps above described. We will conclude that a modification of his perspective, here named Declarationism, is better suited for Fine’s goals.

• 7 Ibid., p. 91.
• 8 Ibid.

9which needs to be read as “introduce an object x conforming to the condition C(x)”.7 The !-sign is an illocutionary force indicating device for a directive speech act. Since directives are usually uttered from a speaker to an hearer that is supposed to perform a task, Fine assumes the presence of an idealized hearer, a “genie”, that ideally is capable of performing any task commanded. Thus, by uttering the Introduction rule, “the genie will introduce an object into the domain that conforms to the condition C(x) if there is not already such an object in the domain and otherwise he will do nothing.”8

• 9 We skip details on this point as it would lead us too far.

12The single simple rule Introduction, together with the four derived rules, form a procedural logic, by adding relevant rules of inference.9 As consequence, procedural logic is an axiom-free system. In fact, in Fine’s postulationism, axioms are consequences of the rules just described:

• 11 For a detailed criticism on these points, see A. Weir, “Honest toil or sheer magic?”, Dialectica, (...)

18However, Fine’s way around this issues is not our main concern here.11 Our main goal is to evaluate the most important ground on postulationism, the Introduction rule. And we will do that from a linguistic perspective, particularly, from the perspective of speech act theory.

• 12 In the 19th century, mathematics shifted to a more objectual perspective, in which constructions (...)
• 13 K. Fine, “Our knowledge of mathematical objects”, art. cit., p. 108.

20Postulationism is also based on actual mathematical practices, as postulates were, at least before the 19th century mathematics,12 commonly used as means for introducing new entities. Thus, Fine sees a good reasons for taking creative postulations as genuine mathematical tools. “The philosopher who rejects postulation must reject standard postulational practice”,13 as he claims.

• 14 Even though the presence of computational devices has been increasing in the practice, such as in (...)
• 15 A similar point is made by Bob Hale. In his assessment of Fine’s postulationism, he raises the fo (...)

21Although Fine’s observation is correct, he misses some key differences. First, the analogy from procedural programming languages does not hold entirely for the practice. Mathematical practice is carried out not by machines, but by human mathematicians, through a mixed language that is partly informal, partly formal.14 Second, postulates were never based on a idealized agent such as genie. Classically, postulates were means for carrying over constructions by human agents.15

• 16 Much of the Speech Act analysis here undertaken is derived from our assumption that mathematical (...)

22For that reason, we find it fruitful to departure from Fine’s analogy, and to evaluate postulationism through the lens of Speech Act Theory. As a matter of fact postulates, as Fine present them, are directives: a specific kind of speech acts. Moreover, Fine builds on an idealized scenario, which hardly matches with practice. For this reason, a linguistic analysis of directive makes justice to Fine’s intuition, while being more faithful to actual mathematical practice.16

• 17 J. Searle, “A taxonomy of illocutionary acts”, in J. Searle, Expression and Meaning: Studies in t (...)

23Speech Act theory finds its standard formulation in the work of John Searle. Searle17 offers a taxonomy of the different speech acts, presenting five classes: assertives, directives, commissives, declaratives and expressives. For present purposes, we are interested in only two: directives and declaratives.

• 18 Things are more complicated than that. Sometimes, mathematical definitions are just means to stip (...)

25The point of a declarative speech act is also to perform a change in the world, but by different means. In a declarative, the speaker manages to perform such change by simply representing the world as being such. In other words, a declarative makes something such by saying it so. Standard examples of declaratives are “You are fired” or “I pronounce you husband and wife”. We can also take mathematical definitions as declaratives. To take a simple example, in defining abelian groups as those groups having commutative operation on its members, I am making it the case that abelian groups are just the groups for which commutativity holds. And this is achieved simply by the act of declaring it so.18 For this reason, declaratives have a double direction of fit between world and word: words modify the world, which is automatically modified to accord to words.

• 19 K. Fine, “Our knowledge of mathematical objects”, art. cit., p. 93.
• 20 If they were directives, mathematical proofs would have the strange feature of being dependent on (...)

31First, in Fine’s words, the rules NUMBER and SET should be read as, respectively, “Let there be Numbers!” and “Let there be Sets!”19 The use of the verb “let” here may be misleading. Fine is certainly reading it from the procedural programming languages, where commands are written in the imperative tense (“do this”, “write that”, and so on). In this sense, and as is the case in most of the English sentences, statements with the verb “let” are directives (“let the man speak”, “let me help you”). But the verb “let” is also highly used within mathematical discourse, and usually as definitions, as in “Let n be a number such that n≠0”, “let f be a function such that f:ℕ→ℕ”. Although these are written in the imperative tense, we believe they are better interpreted as declaratives. As we saw, directives and imperatives are attempts by a speaker to alter the world by requesting something for a hearer. But the directive success in carrying out such changes is dependent ultimately on the hearer side. Either he complies, or for whatever reasons, he doesn’t. In contrast, no such acts are present in mathematical texts. In saying “let x be y”, a mathematician is not asking anything.20 He is actually declaring, with immediate effects, that x is a y. To use the imperative tense in such cases is a mathematical façon de parler. If this proviso is right, the rules NUMBER and SET, if read as “Let there be Numbers!” and “Let there be Sets!”, are declaratives and not directives.

• 21 “It might be thought that the existential requirement is something that should be imposed after t (...)

32Second, Fine’s lengthy discussion on the different ways in which a new object can be introduced in a domain makes clear that we are here dealing with a declarative speech act. Indeed, Fine distinguishes two ways in which this domain extension can go: one consists in renaming a pre-existent object, while another consists in letting an object be in reason of a given way to characterize it. Fine views postulationism as producing the second kind of domain expansion, describing this method as creative and expansive. As a matter of fact, an important aspect of this method is that it provides existence at the same time in which the introduction of the new object (conforming to condition C(x)) is performed.21 With the terminology of speech act theory, this can be rephrased as saying that Fine’s rules have a double direction of fit: not only words fit with the world, but also the world is modified by words at the same time.

• 22 Of course, the latter action is not part of the speech act, but it is important for Fine’s projec (...)

38For a directive speech act to be successful, at least two actions are expected to be performed: the utterance itself, and the hearer’s action that is required by the utterance.22 In this case, each !-expression marks a directive utterance, while each introduction of an entity x marks the hearer’s action (in this case the genie). If we want an infinite number of objects being introduced, we need an infinite number of requests. This is exactly what the iteration rule is supposed to offer. For example, the SUCCESSOR* rule condense an infinite number of requests, that is, an infinite number of directive speech acts being performed. If we can have an infinite number of requests, under the iteration rule, then we can have an infinite number of declarations either. We might ask: why do we have to give instructions, if we can simply declare the objects to exist, under the relevant conditions? This is what the genie is performing in the first place, as we showed in section 4 above.

• 23 This does not mean that the acts of declarations are performed independently of anything. Indeed, (...)
• 24 Notice that this question was already left open by the actions of the genie.

39From this rationale, we could simply change the Introduction from a directive to a declarative, and thus derive a form of Declarationism instead. In fact, this seems even more faithful to Fine’s account. In this way, we don’t have to rely on a ideal genie, as the acts of declaration are performed directly by mathematicians themselves.23 But we do have now to specify in which sense a mathematician can use the Introduction rule to introduce an entity by a simple act of declaration.24 Therefore, this shift from Postulationism to Declarationism is not only a terminological modification, but, instead, it represents a change in perspective that helps to clarify an important aspect of Fine’s position: it’s creative component.

• 25 J. Searle, Speech acts: An essay in the philosophy of language, Cambridge, Cambridge University P (...)
• 26 See A. Giddens, The constitution of society, Berkeley, University of California Press, 1984, and (...)
• 27 See F. Hindriks, “Constitutive rules, language, and ontology”, Erkenntnis, n^o 71, 2009, p. 253-27 (...)

41In order to account for the creative role of language in extending a given mathematical domain, we can resort to “constitutive rules”. These rules are those which constitute a practice, like the game of chess, and are normally defined in opposition to “regulative rules”, that are meant to regulate a practice that precede them, as it is the case for the rules of etiquette. The standard form of the distinction between constitutive and regulative rules can be found in Searle’s work.25 Although very useful in describing the roles of these rules in our practice, this distinction has been also criticised as purely linguistical. While constitutive rules can also regulate,26 regulative rules are capable of constituting new practices.27

• 29 Although Lewis takes the possibility to define a theoretical term of a scientific theory as a vin (...)

43Now, back to mathematical theories, what does our argument that Postulationism is a form of Declarationism in disguise can tell us about an alleged linguistic expansion of a mathematical domains? Probably not much about the existence of mathematical objects,29 but surely something about the objectivity of mathematics. As a matter of fact, to recognise a declarative component in the process of mathematical creation allows us to analyze the latter in terms of the former.

• 30 J. Searle, The construction of social reality, op. cit.

46To see how Declarationism can still warrant mathematical objectivity, we consider Searle’s30 distinction between epistemic and ontological objectivity. The property of epistemic objectivity pertains to assertions, while ontological objectivity pertains to entities and modes of existence. This distinction explains how we can match up different kinds of objectivity or subjectivity in a statement. A few examples:

• 31 Thus, at least partially, mathematical objectivity is institutional, as far as its epistemic dime (...)

48Following this distinction, we can see that objectivity is either a property of mathematical assertions or property of mathematical objects. Mathematics is epistemically objective by virtue of the constitutive rules adopted. That, for example, (ℤ,+) is an instance of an abelian group follows from the definition of abelian groups, viz., it is epistemic objective under the definition provided. Now, ontological objectivity will ultimately depend on the ontology adopted. But one can explain the dual creation-discovery character of mathematics regardless. Through acts of declarations, our practices put forward constitutive rules that are ontologically subjective (creations), as they are dependent on collective intentionality.31 But they still warrant epistemic objective assertions (either in the (OS) or (OO) forms above, depending on the ontology adopted), that is, true discoveries.

• 32 This is formally realized in terms of models. But the fact of considering models as realizations (...)
• 33 With this shift, the question for the consistency of the rules takes a different turn as well. In (...)

49If this reading is correct, Fine’s rules must be updated. NUMBER and SET cannot be stated as rules for simply “adding” new objects, one by one. Instead, they must be taken collectively as declaratives about collections of objects having such and such characteristics.32 The existence of such objects will depend on one’s ontology. However, one still obtain a true domain expansion: not by adding new objects, but by capturing different collections of objects which satisfy the axioms. Therefore, we flip over Fine’s original proposal. Instead of first introducing objects in a domain and then deriving axioms as consequences of the domain, Declarationism first lay down axioms (as an act of declaration) in order to, later, capture possible domains that exemplify the axiomatic system. This may also explain what happens in expanding natural numbers, to integers, rationals, and so on. We are not simply adding more numbers but we are considering new domains of objects entirely. In themselves (whatever what this amounts to) objects can remain the same, but their role change, according to the axioms. Similarly to what happens in re-drawing the borders of a country after a peace treaty.33