Fine’s Postulationism, Objectivity, and Mathematical Creation
Résumés
Nous analysons la proposition faite par Kit Fine d’un postulationnisme procédural à propos des mathématiques. En adoptant une perspective linguistique, nous soutenons que le postulationnisme se laisse mieux comprendre en termes d’actes de langage déclaratifs. Sur la base de cette observation, nous argumentons en faveur d’une forme de déclarationnisme capable de rendre compte à la fois de l’objectivité des mathématiques et de leur dimension créative.
Entrées d’index
Haut de pagePlan
Haut de pageTexte intégral
1. Introduction
1A common belief of many trends of philosophy of mathematics is that, in order to save the objectivity of mathematical statements, mathematical objects must be cast as independent abstract entities. The reasoning is simple:

(P1) if mathematical objects were physical entities, mathematical statements would be only contingently true.

(P2) Mathematical statements are not contingently true.

(C) Therefore, mathematical objects are not physical entities.
 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. 661679.
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 nonessential, 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. 1127.
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.
2. Fine’s Postulationism
6Fine starts with the classical problem of mathematical knowledge: how can we obtain knowledge of such abstract entities? Classical answers either tries to argue that they are things that we somehow discover (e.g., platonists) or things that we create (e.g., constructivists). Fine wants to have it both ways, that is, he wants to argue that they may be both invented and discovered.
7The core idea is not to have axioms as assertions about a given existing domain, but a set of postulates that give instructions for constructing such domains. As he claims, his account
 6 ^{ }K. Fine, “Our knowledge of mathematical objects”, art. cit., p. 89.
[…] shares with traditional forms of postulationism, advocated by Hilbert (1930) and Poincaré (1952), the belief that the existence of mathematical objects and the truth of mathematical propositions are to be seen as the product of postulation. But it takes a very different view of what postulation is. For it takes the postulates from which mathematics is derived to be imperatival, rather than indicative, in form; what are postulated are not propositions true in a given mathematical domain, but procedures for the construction of that domain.6
8For Fine, a mathematical theory can therefore be developed as a set of rules for the construction of the domain of the theory. Fine individuates two kinds of rules: “simple” rules, that are primitive, and “complex”, that are recursively constructed from other rules. The single simple rule is the following:
(i) Introduction: !x.C(x)
 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
10Then, there are four kinds of complex rules, that can be roughly presented as follows:

Composition: if β and γ are rules, then β;γ is a rule, i.e., the execution of β followed by the execution of γ;

Conditionality: if β is a rule and A is an indicative sentence, then A→ β is a rule, i.e. the execution of β under the condition A;

Universality: if β(x) is a rule, then Ɐxβ(x) is also a rule, i.e., the execution of β(x) for every x of the given domain;

Iteration: if β is a rule, so is β*, i.e., the iterated execution of β.
11An important point needs to be stressed. The genie only performs one action: the introduction of a given object. Everything else, as Fine maintain, is just an iteration of this act of introduction. Ultimately, the whole foundation of postulationism depends on this single rule.
 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 axiomfree system. In fact, in Fine’s postulationism, axioms are consequences of the rules just described:
 10 ^{ }Ibid., p. 95.
the postulational rules are not only sufficient to characterize the intuitively given domains of mathematics, but also sufficient to derive the standard axioms for those domains. We obtain in this way a kind of axiomfree foundation for mathematics.10
13And thus, Fine argues that from this set of rules one can introduce a mathematical domain in such a way that axioms can be derived as sentences that describe that domain. Two examples are provided: Arithmetic and SetTheory.
14Both examples depends on the following predicates. For the arithmetical case, “Nx” reads as “x is a number”, “Syx” reads “y is the successor of x”. In the settheoretical case, “Sx” reads as “x is a set”, “∈” is the usual membership relation, and the secondorder variable “F” ranges over pluralities. Then we have the two sets of rules:
Arithmetic:
ZERO: !x.Nx
SUCCESSOR: Ɐx(Nx→ !y.(Ny & Syx))
NUMBER: ZERO; SUCESSOR*.
SetTheory:
POWER: ⱯF!y(Sy & Ɐx(x∈y↔Fx))
SET: POWER*
15Assuming the genie is able to carry over an infinite number of tasks, that is, infinite executions of the Introduction rule, both the domains of numbers and sets are introduced. The ZERO rule, for example, says “Introduce an x such that x is a number”. The SUCCESSOR rule says “For every x in the domain, if x is a number, introduce a y such that y is a number and the successor of x”. NUMBER is the composition of the ZERO rule with the iteration of the SUCCESSOR. With just this complex iteration of the introduction rule, all natural numbers are therefore introduced. The same goes for SetTheory. The POWER rule reads “for all pluralities F of the objects of the domain, introduce an object y such that y is the set containing exactly the objects in the plurality.” SET, as the iteration of the POWER rule, directs the genie to keep adding sets that correspond to the pluralities of the domain. If the domain is empty, the first iteration will introduce the emptyset. The second iteration will introduce its singleton. Eventually, the domain will contain all sets of finite rank.
16What Fine’s is trying to perform is a genuine extension of a domain of objects. He claims that postulationism is ontologically neutral. As the examples show, we start from no objects and derive all numbers in the first case, or all sets in the second. In this sense, he claims that postulationism is the middle ground between discoveries and inventions in mathematical knowledge, as mathematical claims are assertions about objects in a domain, but warranted by the postulations that are responsible for the domain to exist.
17But for that project to be successful, some demands has to be met. For example, the rules must be designed as not resulting an inconsistent domain. And giving the almost divine capabilities of the genie, there should be constrains about the condition C(x) in the introduction rule either.
 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.
3. Speech Act Interlude
19Fine’s postulationism builds from the analogy of procedural programming languages, particularly, dynamic logics. The idea is to provide a set of rules or instructions for the composition and expansion of mathematical domains, just as a computer program is a set of instructions to change states in a machine.
 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.
24The point of a directive speech act is to request a hearer to perform some given action (“Go to your room”, “Pass me the salt”). In such illocutions, the speaker expresses a desire as sincerity condition, while expecting the world to match the propositional content expressed. This relationship between world and words is what Searle calls the “direction of fit”. Directives have the direction worldtoword, in the sense that it is the world that is modified in order to match with words.
 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.
4. Declaratives in Disguise
26Fine’s insight, that postulates are imperatives rather than indicatives, is certainly right if put into an Euclidean perspective. Postulates are most likely directive speech acts, as they lay down instructions for constructing geometrical figures. But are Introduction and all complex rules derived from it really directive speech acts? Do these rules represent instructions that a hearer, like the genie, should follow?
27Let us give a closer look at the Introduction rule. This rule should be read as follows.
(a) Introduce an object conforming to the condition C(x).
28Fine is very explicit about which kind of rule Introduction is: it is a creative rule, in that it extends the domain of objects under consideration. Although interesting, we do not attempt here to analyse Fine’s notion of creativity, since preliminary to that is an assessment of the kind of linguistic actions this rule represents and consequently the kind of actions the genie should perform in executing the rule.
29Let us start by marking an important difference between the instructions contained in Euclid’s postulates and those we find in Fine’s rules. While the Elements provide step by step instructions for the construction of geometrical figures, Fine’s rule are opaque with respect to the way in which the actions they request should be performed. Indeed, the reading of Introduction seems just to be understood as an exhortation to introduce a new object in conformity of the condition C(x), in whichever possible way. This can also be expressed by saying the following.
(b) Let c be an object that conforms to C(x).
30Therefore, we can argue that (a) and (b) have the same illocutionary point, which is not directive, but declarative, as (b) clearly shows. This reading is confirmed by two important pieces of evidence.
 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 preexistent 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.
33With that in mind, how are we to read Fine’s Introduction rule? Indeed, is its imperative form compatible with the declarative task that the rules ask (to the genie) to perform? With D standing for the illocutionary force indicator of a declarative, does the following expression even make sense?
(i.2) Introdution: !Dx.C(x).
34This is perfectly possible. For example, we can order a Father to marry a couple. However, the marriage would still be realized by a declarative speech act. The order is just a means to perform the declaration, but the illocutionary point of the action would still be a declaration, which is the speech act responsible for the introduction, in this case, of a new institutional fact.
35If this reading is plausible, the stack of different acts of speech might introduce unnecessary difficulties for Fine’s postulationism. We might just avoid directives, and adopt a plain declarative rule.
(i.3) Introduction: Dx.C(x).
36We don’t see reasons for not preferring this formulation for the Introduction rule, since the content of Fine’s position is maintained. What this move would effects, however, is the appropriateness of the imperative form in which postulations are presented. Indeed, we take Fine’s postulates as being declaratives in disguise.
5. From Postulationism to Declarationism
37In accepting Fine’s framework we are compelled to assume that the genie is able to perform an infinite number of actions: for every successful utterance of the Introduction rule, the genie will successfully comply in introducing a new object. It is not up to the genie to choose whether to introduce it or not. For the success of the project, he must always comply. But then, why to base Postulationism on imperatives, if we also have to take into account such an idealized setting? A directive speech act that has no chance to fail is hardly a directive.
 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.
40Following Fine, Postulationism is a creative and expansive method, in the sense that, by performing the tasks of its postulates, the genie is really expanding the domain of quantification of our language, introducing new elements. Thus, Declarationism seems to be better suited to account for this creative aspect of language. As a matter of fact, the linguistic actions that bring about the domain expansion (be them in the form of directives or declaratives) can be seen as constitutive of the new special role that mathematical objects acquire, as part of the extended domain. The attempt to account for both invention and discovery suggests to consider the domain expansion proposed by Fine as a linguistic action that is ontologically neutral, while, at the same time, constitutive of some important aspects of the mathematical objects under consideration: namely their belonging to the domain that is being expanded. In other terms, declarations produce new perspectives on mathematical objects, without taking a stance on their (possibly independent) existence. In order to explain this ontological neutrality, we need another Searlian concept.
 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. 25327 (...)
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
42The reason for focusing on this distinction and the debate that it generated is that constitutive rules are often presented in terms of declaratives, while regulative rules in terms of directives. We would like to suggest that the difficulties in assessing the directive or the declarative component of Postulationism is nothing than a reflect of the trouble in clearly separating between regulative and constitutive rules. This ambiguity in using directive or declarative in defining a new practice has been clearly presented by Lewis in the following terms.
 28 ^{ }D. Lewis, “Scorekeeping in a language game”, in D. Lewis, Philosophical papers, Oxford, Oxford Un (...)
The rules of baseball could in principle be formulated as straightforward directives concerning behavior, without the aid of definable terms for score and its components. Or they could be formulated as explicit definitions of the score function, the components of score, and correct play, followed by directives in which the newly defined terms occur. It is easy to see why neither of these methods of formulation has found favor. The first method would pack the entire rulebook into each directive; the second would pack the entire rulebook into a single preliminary explicit definition. Understandably averse to very long sentences, we do better to proceed in our more devious way.28
 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.
44Moreover, our linguistic practices and our methods of definition offer the right tools able to account for the objectivity to mathematics. As a matter of fact, constitutive rules are those rules that allow the creation of new layers of objective reality over and above the physical one. Now, if this is possible with respect to physical reality, why shouldn’t we be able to do the same in a much more rarefied and immaterial context like that of our mathematical theories? We believe that this is not only possible, but that is at the core of Fine’s Postulationism, which, for this reason, should be named Declarationism.
45Constitutive rules are able to account for the domain expansion without producing mathematical objects ex nihilo, since what these rules constitute are not the objects themselves, but their role in our mathematical theorizing. Thus, the focus on declarations allows us to shed light on the creative dimension of Fine’s notion of domain expansion. Without taking a stance on whether the existence of mathematical objects depends on our linguistic practices, this same practice is, nonetheless, able to create new roles for the objects of our mathematical work, in the same way that the rules of chess do not create our bodymoves, but they determine what “moving the queen” consists of. Therefore, declarations create a new status for mathematical objects, without (necessarily) creating them as objects, but only as objectofagiventheory, that is, members of a new extended domain.
 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:
(OO) “Iguaçu falls is bigger than Niagara falls”
(SO) “Iguaçu falls is more beautiful than Niagara falls”
(OS) “John owns George 20 dollars”
47(OO) is both epistemic and ontologically objective, as both the Iguaçu and Niagara falls are independent entities (ontological objectivity). Also, their compared sizes are objectively verifiable and do not dependent on any personal disposition (epistemic objectivity). (SO), albeit ontologically objective, is epistemically subjective, given that it is an assertion about personal preferences. (OS) is an interesting case. First, it is ontologically subjective, given that money is not a selfsubsisting entity. It exists in virtue of some collective intentionality and constitutive rules that creates the institution of money. But that John owns George 20 dollars is an epistemically objective fact that can be verified: either John borrowed 20 dollars from George, or he didn’t.
 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 creationdiscovery 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 redrawing the borders of a country after a peace treaty.33
50In conclusion, Declarationism consists in the position according to which our linguistic practices are able to create new and objective layer of mathematical reality, that, although not sufficient to account for the existence of mathematical objects, are capable of explaining, on the one hand, the objectivity of mathematical knowledge, while on the other hand, why mathematics is partially discovered and partially created.
Notes
1 ^{ }A position that is better summarized by Frege’s belief in a Third Realm, as in G. Frege, “The Thought”, in G. Frege, Collected Papers on Mathematics, Logic and Philosophy, New York, Basil Blackwell, 1984, p. 363.
2 ^{ }P. Benacerraf, “Mathematical truth”, The Journal of Philosophy, vol. 70, n^{o} 19, 1973, p. 661679.
3 ^{ }As consequence, many phenomenon related to the practice would be nonessential, collateral, features.
4 ^{ }K. Fine, “Our knowledge of mathematical objects”, in T. Gendler and J. Hawthorne (eds.), Oxford Studies in Epistemology, Oxford, Clarendon Press, 2005, p. 89109.
5 ^{ }K. Fine, “Mathematics: Discovery or invention?”, Think, vol. 11, n^{o} 32, 2012, p. 1127.
6 ^{ }K. Fine, “Our knowledge of mathematical objects”, art. cit., p. 89.
7 ^{ }Ibid., p. 91.
8 ^{ }Ibid.
9 ^{ }We skip details on this point as it would lead us too far.
10 ^{ }Ibid., p. 95.
11 ^{ }For a detailed criticism on these points, see A. Weir, “Honest toil or sheer magic?”, Dialectica, vol. 61, n^{o} 1, 2007, p. 89115.
12 ^{ }In the 19th century, mathematics shifted to a more objectual perspective, in which constructions plays no role whatsoever.
13 ^{ }K. Fine, “Our knowledge of mathematical objects”, art. cit., p. 108.
14 ^{ }Even though the presence of computational devices has been increasing in the practice, such as in computerassisted proofs, mathematics is still a human activity.
15 ^{ }A similar point is made by Bob Hale. In his assessment of Fine’s postulationism, he raises the following issue: “[…] it is not easy to see how, once we set aside Fine’s helpful (and tireless!) genie, we might be supposed to be able so to use them [the postulates] […] While Fine says ‘let there be numbers!’, it seems a little optimistic to suppose that one of us mere mortals can just say a few words and achieve the desired effect. […] if such talk is not to be taken literally and at face value, how are we to conceive of postulation?” B. Hale, “Definition, abstraction, postulation, and magic”, in J. Leech (ed.), Essence and Existence, Selected Essays, Oxford, Oxford University Press, 2020, p. 177. Our perspective here tries to keep up with Fine’s objectives while, at the same time, not to depart from the practice; in its human limits. This is what justifies the adoption of speech act theory for our analysis.
16 ^{ }Much of the Speech Act analysis here undertaken is derived from our assumption that mathematical practice is inherently axiomatic (and within a modeltheoretic setting) and that axioms are used also as definitions.
17 ^{ }J. Searle, “A taxonomy of illocutionary acts”, in J. Searle, Expression and Meaning: Studies in the Theory of Speech Acts, Cambridge, Cambridge University Press, 1979.
18 ^{ }Things are more complicated than that. Sometimes, mathematical definitions are just means to stipulate the meaning of terms, just as the example above. But sometimes a definition represents an attempt to capture a previous concept in a formal apparatus; like Dedekind’s formal definition of natural numbers. We can call the former stipulative definitions, while the latter contentual definitions, following M. Ruffino, L. San Mauro and G. Venturi, “Speech acts in mathematics”, Synthese, vol. 198, n^{o} 10, 2021, p. 1006310087.
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 the reader’s goodwill to be valid.
21 ^{ }“It might be thought that the existential requirement is something that should be imposed after the rule has been prescribed, but not before. […] The problem with this suggestion is to understand what this intermediate interpretation of the quantifier might be. […] There is therefore no room for an intermediate interpretation of the quantifier; and we see that no sensible requirement of existence can be imposed either as a condition on postulation itself or as a condition on what can be inferred from a postulational rule, once it has been prescribed.” Ibid., p. 104106.
22 ^{ }Of course, the latter action is not part of the speech act, but it is important for Fine’s project nonetheless: the genie is expected to comply.
23 ^{ }This does not mean that the acts of declarations are performed independently of anything. Indeed, as declarations, definitions are dependent on institutional facts in order to be successful. But the institutional requirements for making a declaration in mathematical texts, for example, are very low. The point is that when enough deontic powers are in place, a declaration has immediate effects and does not depend on the performance of a hearer to bring about the desired effects of the utterance.
24 ^{ }Notice that this question was already left open by the actions of the genie.
25 ^{ }J. Searle, Speech acts: An essay in the philosophy of language, Cambridge, Cambridge University Press, 1969 and J. Searle, The construction of social reality, New York, The Free Press, 1995.
26 ^{ }See A. Giddens, The constitution of society, Berkeley, University of California Press, 1984, and G. J. Warnock, The object of morality, London, Methuen & Co, 1971.
27 ^{ }See F. Hindriks, “Constitutive rules, language, and ontology”, Erkenntnis, n^{o} 71, 2009, p. 253275.
28 ^{ }D. Lewis, “Scorekeeping in a language game”, in D. Lewis, Philosophical papers, Oxford, Oxford University Press, 1979, p. 327.
29 ^{ }Although Lewis takes the possibility to define a theoretical term of a scientific theory as a vindication of scientific realism, in D. Lewis, “How to define theoretical terms”, in D. Lewis, Philosophical papers, Oxford, Oxford University Press, 1970.
30 ^{ }J. Searle, The construction of social reality, op. cit.
31 ^{ }Thus, at least partially, mathematical objectivity is institutional, as far as its epistemic dimension is concerned. Indeed, the objectivity of a definition is wholly dependent on the constitutive rules that acts in its performance. Similarly, a bachelor is objectively an unmarried man. However, this epistemic objectivity does not automatically projects on the ontological sphere. Any question about the ontology of mathematics is largely independent (although in principle compatible) with the present perspective. For an account that takes mathematics as ontologically institutional (although from a realist perspective), see J. Cole, “Towards an institutional account of the objectivity, necessity, and atemporality of mathematics”, Philosophia Mathematica, vol. 21, n^{o} 1, 2013, p. 936.
32 ^{ }This is formally realized in terms of models. But the fact of considering models as realizations of axioms is only a byproduct of the logic that we use to express them.
33 ^{ }With this shift, the question for the consistency of the rules takes a different turn as well. In Fine’s case, given that the rules are introducing objects directly, he must explain when and how a new object can be consistently introduced by means of postulation, in order to explain when and how a postulate is effectively used. Differently, by taking declaratives at the center stage of mathematical and axiomatic practice, we take the question about consistency as derivative, meaning that knowing whether our declarations successfully characterize a consistent domain is a matter of further mathematical research, not a previously given restriction on the declaratives themselves. Indeed, the fact that a definition is contradictory does not effect the felicitous character of the definition (but, perhaps, its perlocutionary effect in characterizing an existing mathematical reality). Moreover, we consider this to be a more faithful depiction of the practice, given that mathematicians are free to declare new layers of mathematical reality even when, in retrospect, such layers are proved to be vacuous; Frege’s Basic Law V being the chief example here.
Haut de pagePour citer cet article
Référence papier
Giorgio Venturi et João Vitor Schmidt, « Fine’s Postulationism, Objectivity, and Mathematical Creation », Noesis, 38  2022, 123137.
Référence électronique
Giorgio Venturi et João Vitor Schmidt, « Fine’s Postulationism, Objectivity, and Mathematical Creation », Noesis [En ligne], 38  2022, mis en ligne le 01 juin 2024, consulté le 31 octobre 2024. URL : http://journals.openedition.org/noesis/7139 ; DOI : https://doi.org/10.4000/11xma
Haut de pageDroits d’auteur
Le texte et les autres éléments (illustrations, fichiers annexes importés), sont « Tous droits réservés », sauf mention contraire.
Haut de page