Abstract
Most of the arguments usually appealed to in order to support the view that some abstraction principles are analytic depend on ascribing to them some sort of existential parsimony or ontological neutrality, whereas the opposite arguments, aiming to deny this view, contend this ascription. As a result, other virtues that these principles might have are often overlooked. Among them, there is an epistemic virtue which I take these principles to have, when regarded in the appropriate settings, and which I suggest to call ‘epistemic economy’. My purpose is to isolate and clarify this notion by appealing to some examples concerning the definition of natural and real numbers.
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Notes
- 1.
I thank Andrew Arana, Jeremy Avigad, Francesca Boccuni, Annalisa Coliva, Sorin Costreie, Michael Detlefsen, Sébastien Gandon, Guido Gherardi, Emmylou Haffner, Bob Hale, Brice Halimi, Greeg Landini, Paolo Mancosu, Ken Manders, Daniele Molinini, Julien Ross, Andrea Sereni, Stewart Shapiro, Giorgio Venturi, Sean Walsh, and David Waszek for useful comments and/or suggestions.
- 2.
I am not interested here in discussing the relation between existential parsimony and ontological neutrality. All I shall say of these virtues is independent of this matter.
- 3.
Plausibly, epistemic economy not only applies to definitions or parts of them. However, apart from some short remarks on the epistemic cost of proofs in the next section, I do not investigate this matter further here.
- 4.
By this, I mean that the relevant resources, or a significant part of them, come about in the former case in agreement with an order in which some conceptually depend on others but not vice versa, whereas they come about in the latter case all together at once.
- 5.
Here, I adopt the current neo-logicist interpretation of monadic predicate variables as ranging over first-level concepts. Mutatis mutandis, one could go for another option and interpret such variables as ranging over properties of objects. If first-level concepts are conceived as they are by neo-logicists, then it does not seem to me that this would make any significant difference.
- 6.
Regarding my appraisal of Frege’s conception of logic, I refer the reader to Cadet and Panza (2015).
- 7.
Taking the relevant system of second-order logic to implicitly define first-level concepts (or properties of objects: cf Footnote 5) is not mandatory. One might merely take HP as an implicit definition of a functional constant inputting monadic predicates and outputting terms. However, in this case, adding this principle to the axioms of such a system of logic would merely result in introducing appropriate numerals, rather than in defining cardinal numbers. These numerals being given, defining natural numbers would then require much more than merely singling the latter out among the former, because this selection would at most provide a family of terms.
- 8.
This argument questions the strong neo-logicist thesis because it is only in the presence of second-order logic with comprehension appropriately extended, and of appropriate explicit definitions, that the existence of infinitely many objects follows from HP (through the admission of appropriate explicit definitions).
- 9.
Of course, the existence of natural numbers being analytic is strictly not the same as their being logical objects, as well as a truth being logical is not the same as its being analytic. These important distinctions are not relevant for the issue under discussion so it is not necessary to insist on them here.
- 10.
In short, I say that a stipulation, or a system of stipulations, has an ontological import for objects if the truth of this stipulation, or these stipulations, requires the existence of some objects.
- 11.
As a matter of fact, Hale and Wright only consider the case of definitions given by a single sentence including a single definiendum, but the generalisation to the case of systems (or conjunctions) of sentences including more definienda is as natural as it is necessary to adapt the account to the case of Peano axioms.
- 12.
As a matter of fact, MacFarlane’s paper is also concerned with Hale and Wright’s conception of numerical definite descriptions as singular terms in relation with the sort of logic that FA actually requires (namely whether this logic is classical or free). In Hale and Wright (2009), Hale and Wright also reply to this point, but, though somehow connected with the question I’m discussing, this matter can be left aside here.
- 13.
Neo-logicists have come back to this last point in different ways and in the context of different forms of argumentation. A very compact way to make the same point is found, for example, in Wright (1999), § II.2.
- 14.
Apart for the mention of the extension of comprehension (which neo-logicists seems to consider as existentially anodyne), condition (2) is suggested by Wright’s following remarks (1999, pp. 307 and 310):
Analyticity, whatever exactly it is, is presumably transmissible across logical consequence. So if second-order consequence is indeed a species of logical consequence, the analyticity of Hume’s Principle would ensure the analyticity of arithmetic.
[… ] on the classical account of analyticity the analytical truths are those which follow from logic and definitions. So if the existence of zero, one, etc. follows from logic plus Hume’s Principle, then provided the latter has a status relevantly similar to that of a definition, it will be analytic, on the classical account, that n exists, for each finite cardinal n.
- 15.
Adding to these axioms a first-order axiom-scheme of induction, one gets the system Z1, which provides a convenient version of Peano first-order arithmetic (Simpson 2009, pp. 7–8).
- 16.
Cf. Footnote 18.
- 17.
The most evident case is that of a comprehension axiom-scheme restricted to formulas containing no second-order quantifier, as that involved in \( {\text{ACA}}_{0} \): whatever such formula ‘\( \varphi \left( n \right) \)’ might be, the syntactical complexity of ‘\( {\exists X\forall n\left[ {n \in X \Leftrightarrow \varphi \left( n \right)} \right]} \)’ is greater than that of this formula.
- 18.
It should be noted that what matters here is not merely the way in which \( {\mathfrak{L}}^{{\text{FA}}} \)’s predicates are informally conceived, in particular by neo-logicists, as opposed to the way in which \( {\mathfrak{L}}^{{\text{Z}_{2} }} \)’s predicates are conceived. Indeed, intending the second-order variables of Z2 to range over sets of elements of the range of the first-order ones, and the constant ‘\( \in \)’ as designating the set-theoretic relation of membership is not mandatory. One could rather intend the second-order variables of Z2 to range over the monadic properties of the elements of the range of the first-order ones, and consider that ‘\( n \in X \)’ is nothing but a typographic variant of ‘\( X\left( n \right) \)’ or ‘\( Xn \)’ (that is, merely an alternative way to predicate the property X of the individual n). What matters is rather the way in which predicates work in FA and Z2, respectively. Focusing on the mere definition of natural numbers, the difference is not really significant, because what Z2’s predicates do in relation to this definition can, mutatis mutandis, also be done by FA’s monadic predicates. The difference becomes, instead, quite significant in relation to the definition of real numbers within these theories (which I shall consider in the next section). Indeed, if second-order variables of Z2 are taken to range over monadic properties of the elements of the range of the first-order ones, rather than over sets of these same elements, one can hardly be happy with a definition of real numbers as some particular items within the range of the former of these variables, as suggested by Simpson in relation to ACA0 and RCA0
- 19.
Cf. Footnote 5.
- 20.
Cf. Footnote 18.
- 21.
It should be noted that the notion of a many-one association between appropriate sorts of items is involved in any definition of whatsoever functional constant.
- 22.
Once this definition is immersed within the whole Z2, the items it defines—namely (the items re-casting) real numbers within Z2—provably have many properties that the items it defines when it is immersed within ACA0 or RCA0—namely (the items re-casting) real numbers within ACA0or RCA0—do not provably have. The crux of reverse mathematics (to which Simpson 2009 is entirely devoted) is just which properties of the former items are preserved once real numbers are defined within weaker sub-systems of Z2, like ACA0 or RCA0. However, of course, this is not a matter I can consider here.
- 23.
It is easy to see the essential difference between this fourth step and the three previous ones: whereas in these three steps, the sets \( {\mathbb{N}} \), \( {\mathbb{N}}_{\mathbb{Z}} \), and \( {\mathbb{N}}_{\mathbb{Q}} \) are explicitly defined, the subsets of \( {\mathbb{N}} \) coding real numbers cannot be explicitly defined, in turn, and it is, a fortiori, impossible to define anything working as the set of real numbers. All that one can do is fixing a condition that a subset of \( {\mathbb{N}} \) has to met in order to code a single real number.
- 24.
In fact, integer numbers could be singled out among natural ones by appealing only to addition and multiplication on the latter, by merely stipulating that the former numbers are coded by those of the latter ones which are equal to \( \zeta^{2} + \zeta \) or \( \zeta^{2} \), for some natural number \( \zeta \). In this way, no justification could be offered for this choice. Analogously, rational numbers could be directly singled out among natural numbers, by only appealing, again, to addition and multiplication on the latter, by merely stipulating that the former numbers are coded by those of the latter ones which are equal to \( \left( {\zeta^{2} + \zeta + \vartheta^{2} + \vartheta } \right)^{2} + \zeta^{2} + \zeta \) or \( \left( {\zeta^{2} + \vartheta^{2} + \vartheta } \right)^{2} + \zeta^{2} \), for some pair of coprime natural numbers \( \zeta \) and \( \vartheta \), the second of which is positive. In this case also, no justification could be offered for this choice.
- 25.
Cf. Footnote 23.
- 26.
In other terms, I shall take ‘\( \forall_{\Phi } x\left[ \phi \right] \)’, ‘\( \forall_{\Phi } X\left[ \phi \right] \)’, \( \exists_{\Phi } x\left[ \phi \right] \)’, and ‘\( \exists_{\Phi } X\left[ \phi \right] \)’ to abbreviate ‘\( \forall x\left[ {\Phi \left( x \right) \Rightarrow \phi } \right] \)’, ‘\( \forall X\left[ {\forall x\left[ {X\left( x \right) \Rightarrow\Phi \left( x \right)} \right] \Rightarrow \phi } \right] \)’, ‘\( \exists x\left[ {\Phi \left( x \right) \wedge \phi } \right] \)’, and ‘\( \exists X\left[ {\forall x\left[ {X\left( x \right) \Rightarrow\Phi \left( x \right)} \right] \wedge \phi } \right] \)’ respectively.
- 27.
Hale’s domains of magnitudes, unlike Frege’s, only include, as we see below, positive elements, with the result that only positive real numbers can be defined as ratios on them. Non-positive ones are, then, to be defined by extension.
- 28.
This proof is quite convoluted, but combinatorial in spirit, and epistemically quite economic, because, elementary arithmetic on natural numbers being taken for granted, it does not involve much more than propositional logic applied to predicate (second-order) formulas.
- 29.
Though he generically speaks of numbers (Zahlen), Dedekind’s claims seem to be directly referred to natural numbers. Still, he also seems to consider that his views on these numbers extend to any other sorts of numbers, insofar as theories of the latter come from an extension of the theory of the former (or arithmetic, as usually intended). This is made clear by the parenthesis in the following claim: “ In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept [Zahlbegriff] entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought” (Dedekind 1901, p. 14).
- 30.
I thanks Emmylou Haffner for drawing my attention to this passage.
- 31.
Look at the two following quotes, from the Grundlagen (Frege 1884, § 14; 1953, p. 21), and from a coeval paper (“Über Formale Theorien der Arithmetik”, Frege 1967, pp. 103–111, especially p. 103; English translation in Frege 1984, pp. 112–121, especially p. 112), respectively:
The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even than that of geometry. The truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought.
As a matter of fact, we can count about everything that can be an object of thought: the ideal as well as the real, concepts as well as objects, temporal as well as spatial entities, events as well as bodies, methods as well as theorems; even numbers can in their turn be counted. What is required is really no more than a certain sharpness of delimitation, a certain logical completeness.
- 32.
To see the point, remark, firstly, that \( {\mathfrak{L}}^{{\text{L}_{2} }} \) includes no predicate constant, so that a restriction involving some predicate constant cannot be stated in \( {\mathfrak{L}}^{{\text{L}_{2} }} \). Remark, then, that in (13), the restricted quantifier ‘\( \forall_{{\mathcal{N}}} z \)’ can be equivalently replaced by an unrestricted one (its entering this principle is only motivated by easiness of understanding). Remark, finally the difference between the open formula ‘\( x = y \wedge \forall z \, \left[ {X\left( z \right) \Leftrightarrow Y\left( z \right)} \right] \)’, providing the right-hand side of (13), and the other open formulas ‘\( x + y^{\prime } = x^{\prime } + y \)’, ‘\( \left[ {y = 0_{{\mathcal{Z}}} \wedge y^{\prime } = 0_{{\mathcal{Z}}} } \right] \vee \left[ {y \ne 0_{{\mathcal{Z}}} \wedge y^{\prime } \ne 0_{{\mathcal{Z}}} \wedge x\cdot_{{\mathcal{Z}}} y^{\prime } = x^{\prime } \cdot_{{\mathcal{Z}}} y} \right] \)’, ‘\( \forall_{{\mathcal{Q}}} x\left( {F \trianglelefteq x \Leftrightarrow G \trianglelefteq x} \right) \)’, ‘\( \forall_{{{\mathcal{N}}^{ + } }} h,k\left[ {hx \lesseqgtr ky \Leftrightarrow hx^{ * } \lesseqgtr ky^{ * } } \right] \)’ and ‘\( \forall_{{{\mathcal{R}}^{{{\mathcal{N}}^{ + } }} }} \left[ {P\left( x \right) \Leftrightarrow Q\left( x \right)} \right] \)’, providing the right-hand sides of the abstraction principles involved in Shapiro’s definitions of real numbers, and in the adaptation of Hale’s to FA: whereas the first of these formulas is a formula of \( {\mathfrak{L}}^{{\text{L}_{2} }} \), this is so for none of the others.
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Panza, M. (2016). Abstraction and Epistemic Economy. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_17
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