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Naturalized Platonism
vs.
Platonized Naturalism
Bernard Linsky
Department of Philosophy
University of Alberta
and
Edward N. Zalta
Center for the Study of Language and Information
Stanford University
†
those objects required by the explanations of the natural sciences.with naturalism. Naturalism is the realist ontology that recognizes only In this paper, we argue that our knowledge of abstract objects is consistent
(^) But
properties and sets are empirical, discoveredboth) in the causal order, and to suggest that philosophical claims aboutentific laws. This has led some naturalists to locate properties or sets (orrequired for the proper philosophical account of scientific theories and sci-some abstract objects, such as mathematical objects and properties, are
(^) a posteriori
, and subject to
revision.
(^) We call this view
(^) Naturalized Platonism
, and in what follows,
Published in^ ∗
(^) The Journal of Philosophy
, (^) xcii /10 (October 1995): 525-555.
The authors would like to acknowledge support from the Social Sciences and Hu-^ †
Tawil, Mark Balaguer, and Gideon Rosen for their helpful comments on the paper.versity, and Monash University. We are especially indebted to Chris Swoyer, Nathansity of Otago, University of Queensland, University of Sydney, Australian National Uni-University of Auckland, Victoria University of Wellington, Massey University, Univer-at the following institutions, where the second author presented the penultimate draft:and Information. We would like to thank our New Zealand and Australian colleaguesmanities Research Council of Canada and from the Center for the Study of Language
Bernard Linsky and Edward N. Zalta
we contrast it with our own view, which we call
(^) Platonized Naturalism
Platonized Naturalism is the view that a more traditional kind of Pla-
tiotemporal and outside the causal order.ontology that recognizes abstract objects, i.e., objects that are nonspa-tonism is consistent with naturalism. Traditional Platonism is the realist
(^) The more traditional kind of
argue that such comprehension principles are synthetic and are knownhension principles that assert the existence of abstract objects. We shallPlatonism that we defend, however, is distinguished by general compre-
(^) a
priori
. Nevertheless, we claim they are consistent with naturalist stan-
principles for properties, relations, and propositions.we believe that similar arguments apply to corresponding comprehensionshall concentrate the argument of our paper on this particular principle,ematical truths is linked to our knowledge of this principle. Though weindividuals, and in what follows, we show that our knowledge of math-stracta, will be located. This is the comprehension principle for abstractthat governs the domain in which mathematical objects, among other ab-we shall demonstrate our claims with respect to a comprehension principleralized Platonism has gone wrong most clearly in the case of mathematics,dards of ontology, knowledge, and reference. Since we believe that Natu-
I. Naturalized Platonism
Meinong, philosophers have recognized that such naive theories are oftenever since Russell developed both his paradox of sets and his criticisms oforetical description of an abstract object is sufficient to identify it. Butevery predicate denotes a property (or picks out a class) or that a the-seem to rely on naive, often unstated, existence principles, such as thatlation of traditional Platonism. The problem is that traditional Platoniststraditional Platonism. One important problem concerns the very formu- Naturalized Platonism is an attempt to solve the problems inherent in Different philosophers use the term ‘naturalism’ in different ways. See, for example,^1
the anthology on naturalism by S. Wagner and R. Warner, eds.,
(^) Naturalism: A Critical
Appraisal
, (Notre Dame, Ind: University of Notre Dame Press, 1993). D. Armstrong,
in Universals and Scientific Realism
(^) (Cambridge: Cambridge University Press, 1978),
the literature. We shall say more about this ambiguity at the start of sectionsubtle ambiguity that reflects an ambiguity in the way in which the term is used inthat our definition of ‘naturalism’ is a serviceable one, in part because it contains auses ‘physicalism’ to label the view that the natural world is physical. But we thinkreserves the term for the denial that there are any objects outside of spacetime and
(^) vi .
Naturalized Platonism vs. Platonized Naturalism
arises from the mistake of reifying words into objects.for the logical positivists, talk of abstract objects is just empty talk thatof spatiotemporal objects, and logical knowledge is merely analytic. So,nonspatiotemporal abstracta. For we can have empirical knowledge onlyneither case could we have genuine, synthetic (ampliative) knowledge ofing that our knowledge is either empirical or logical in nature and that inknowledge of them. The logical positivists articulated this worry by argu-cognitive access to entities outside the causal order by which we obtainTraditional Platonism seems to require that we have a mystical kind ofsecond problem, namely, how we could ever know that the theory is true.lation of Platonism proves to be clear and consistent, it would still face a fraught with contradictions and inconsistencies. And even if some formu- However, Quine suggested that some abstract objects (namely, sets
theories quantify over both.par with the theoretical entities of natural science, for our best scientificand those mathematical entities thought to be reducible to sets) are on a
(^2) Quine formulated a limited and nontra-
continuousditional kind of Platonism by proposing that set theory and logic are
(^) with scientific theories, and that the theoretical framework as
a whole is subject to empirical confirmation.
(^3) Because set theory and
tify over mathematical entities but that mathematics ismodified Quine’s view by arguing not simply that our best theories quan-mediate revision by their distance from empirical observations. Putnamlogic stand in the center of the theoretical web, they are isolated from im-
(^) indispensable
(^) to
ories without quantifying over them).natural science (in the sense that there is no way to formulate such the-
(^4) Putnam also accepts properties
laws.on the grounds that they are needed in the proper formulation of natural (^5) Indeed, the appeal to properties also seems to provide a satisfying
and inter-theoretic reduction.account of physical measurement, causal relations, biological functions,
(^6) On this conception, the acceptance of ab-
“On What There Is,” reprinted in W. V. Quine,^2
(^) From a Logical Point of View
,
2nd rev. ed. (Cambridge, MA: Harvard University Press, 1980), pp. 1-19. Philosophy of Logic^3
(^) (Englewood Cliffs, NJ: Prentice Hall, 1970).
Philosophy of Logic^4
(^) (New York:
(^) Harper and Row, 1971); reprinted in H. Put-
nam, (^) Mathematics, Matter, and Method: Philosophical Papers I
(^) , 2nd ed. (Cambridge:
Cambridge University Press, 1979), pp. 323-357. “On Properties,” reprinted in^5
(^) Mathematics, Matter, and Method:
(^) Philosophical
Papers I
(^) , (^) op. cit.
, pp. 305-322.
See C. Swoyer, “The Metaphysics of Measurement,” in J. Forge, ed.,^6
(^) Measurement,
Realism, and Objectivity
(^) (Dordrecht: D. Reidel, 1987), for a description of some of the
ways in which an appeal to properties clarifies our understanding of natural science.
Bernard Linsky and Edward N. Zalta
to be reduced to sets (Quine) or sets and properties (Putnam).to the workings of science, and (2) other purported abstract entities aregroup of principles which are justified by the fact that they are essentiallogical commitment is to be kept to a minimum and governed by a smallstracta is constrained by principles of parsimony and reduction: (1) onto- Quine’s formulation of a limited Platonism was seen by many as in-
abstract objects. How do we obtaincomplete, however, for it did not provide an account of our access to
(^) knowledge
(^) of individual abstract ob-
jects? G¨
odel suggested that it was some perception-like intuition of those
objects that guides our choice of axioms.
(^7) But Benacerraf pointed out
reference.that this is still not compatible with a naturalist theory of knowledge and
(^8) Using the current causal theories of knowledge and reference
anisms for tracking and forming beliefs about such objects.or properties? It is not clear how there could be reliable cognitive mech-come to have reliable beliefs about nonspatiotemporal objects such as setsrecent externalist or reliabilist theories of knowledge, for how would oneties with the objects known. And the problem persists even for the moreas a guide, Benacerraf saw no natural way of linking our cognitive facul-
(^) Benacerraf
to sets.to arbitrate among equally acceptable reductions of other abstract objectsalso raised another question for Quine’s limited Platonism, namely, how
(^9) Benacerraf’s principal example was the fact that the von Neu-
von Neumann ordinals rather than the Zermelo ordinals, or vice versa.no principled reason, therefore, to say that the numbers “really are” theequally viable ways of identifying the natural numbers with sets. There ismann ordinals and the Zermelo ordinals are just two (of infinitely many) Three trends have developed in response to the first of the Benacer-
raf problems we discussed: (1) Field
10 and Mundy^
11 accept Benacerraf’s^
without an appeal to abstract individuals such as numbers or sets. Field,representing features of the world that can be essentially characterizeddispensable to natural science. Mathematics may be useful, but only forproblem as decisive and then challenge the idea that mathematics is in- “What is Cantor’s Continuum Problem,” reprinted in P. Benacerraf and H. Put-^7
nam, eds.,
(^) Philosophy of Mathematics
, 2nd ed.
(^) (Cambridge:
(^) Cambridge University
Press, 1983), pp. 470-485. “Mathematical Truth,”^8
(^) The Journal of Philosophy
, (^) LXX
/19 (November 1973):
661-679. “What Numbers Could Not Be,”^9
(^) Philosophical Review
, (^74) (^) (1965): 47-73.
10 Science Without Numbers
(^) (Princeton: Princeton University Press, 1980).
11 “Mathematical Physics and Elementary Logic,”
(^) Proceedings of the Philosophy of
Science Association
, (1990): 289-301.
Naturalized Platonism vs. Platonized Naturalism
On the other hand, let us assume that Burgess is correct and that the
ism. Quine’s view still faces certain otherBenacerraf problem has no force against Quine’s limited kind of Platon-
(^) prima facie
(^) obstacles, however.
The more serious ones are:
- There is no account of mathematics that is not applied in scientific do we account for the meaningfulness of that language?it is never applied, it is expressed in a meaningful language. Howtheories. Such mathematics certainly might be applied, and even if
- The mathematical portion of a scientific theory does
(^) not (^) seem to
the theory as a whole.receive confirmation from the empirical consequences derivable from
18 Sober points out that there is a core
portion.confers no incremental confirmation on the purely mathematicalsumed in every competing theory, evidence for the theory as a wholepotheses. Since this core group of mathematical principles are as-of mathematical principles common to all competing scientific hy-
19 Simply put, the evidence neither increases nor decreases^
not continuous with scientific theory.of every competing hypothesis. This suggests that mathematics isthe likelihood of those mathematical principles, since they are part
- If the overall scientific theory fails, scientists don’t
(^) revise
(^) the mathe-
matical portion but instead
(^) switch
(^) to a different mathematical the-
ory.
The revolutions in physics in the early part of this century
ometry.theories of non-Euclidean geometries, not by revising Euclidean ge-were accompanied by appeal to previously unapplied mathematical
(^) Even in those cases where the needs of physical theories
spurred the development of new mathematics, those needs never
of Mathematical Objects,” 111-135, and their followup article “A Reductio Ad Surdum? Field on the Contingency
(^) Mind
(^103) /410 (April 1994): 169-184. See also B. Linsky
ed.,and E. Zalta, “In Defense of the Simplest Quantified Modal Logic,” in J. Tomberlin, (^) Philosophical Perspectives 8:
(^) Logic and Language
(^) (Atascadero, CA: Ridgeview
Press, 1994), pp. 431-458.
(^) In that paper, the present authors introduce contingently
quantified modal logic (i.e., a logic that includes the Barcan formulas).nonconcrete objects in order to give an “actualistic” interpretation of the simplest
(^) But Field
18 ily, rather than contingently, nonconcrete.could not appeal to those objects to ground his conception, for numbers are necessar- Indeed, even for the scientific portion of the theory, different pieces of evidence
19 seem to bear on different parts of the theory. Confirmation doesn’t seem to be holistic. “Mathematics and Indispensability,”
(^) The Philosophical Review
, (^102) (^) (1993): 35-
Bernard Linsky and Edward N. Zalta
altered the normal
(^) a priori
(^) procedures of mathematical justifica-
doubt on the continuity of mathematics with natural science.tion by axiomatization, definition, and proof. This point also casts
- The account of logic doesn’t fit the facts.
(^) With the exception of
theory.native logics are not revisions of classical logic forced by empiricalcourse of arguing for alternative logics. The proliferation of alter-quantum logic, no empirical evidence has ever been adduced in the
(^) Quantum logics stand alone, rather than as the first of a
series of logics revised to suit the needs of physics.
- The other problem posed by Benacerraf, concerning the arbitrari- ness of reductions, still remains.
(^) And even if other mathematical
not some other (mathematical) thing.a strong intuition that every mathematical object is what it is andon set theory do not take themselves to be studying sets. There isfollow that they are just sets. Mathematicians who are not workingentities could be reduced to sets in a nonarbitrary way, it doesn’t
Finally, we consider those philosophers who meet Benacerraf’s chal-
properties (i.e., by locating them in the causal order).lenge by more thoroughly naturalizing Platonic entities such as sets or
By accepting
strong face further difficulties as well:scribed (except for the first part of the last problem). But she and Arm-Quine’s limited Platonism, Maddy inherits all of the problems just de-
- For Maddy, there seems to be no way to assess the rationality of maticians find most interesting.cardinal axioms. This is the very part of the discipline that mathe-arguments for the highly theoretical axioms of ZF, such as the large
- While Maddy solves the Benacerraf problem of arbitrary reductions practicing mathematicians) that numbers are (individual) objects.is that she denies the logical intuition (and common sense view ofby identifying numbers with structural properties of sets, the cost
- For Armstrong’s sparse conception of properties and states of affairs, accepting uninstantiated properties.both for natural science and the mathematics it requires, withoutthere is a problem of finding enough properties and states to account
Naturalized Platonism vs. Platonized Naturalism
- The combinatorial account of possibility Armstrong develops ap- don’t seem to be part of the causal order.peals to fictional entities (such as possible states of affairs), which
20
these worries surrounding the various responses to Benacerraf’s problem. In the remainder of the paper, we develop an alternative that is free of
II. Platonized Naturalism
shared by both the traditional and naturalized Platonists. We motivate our view by reexamining the conception of abstract objects
We believe
themselves.ent epistemological problems associated with Platonism quickly presentproper conception and theory of abstract objects, answers to the appar-about abstract objects. Once we are freed from these mistakes and get aobjects as physical objects, and (ii) the piecemeal approach to theorizingthat there are two mistakes in that conception: (i) the model of abstract Most Platonists conceive of abstract objects on the model of physical
ical objects:of abstract objects by analogy with the following three features of phys-objects. That is, they understand the objectivity and mind-independence
(^) (1) Physical objects are subject to an appearance/reality
inquiry.they appear, nor can those properties be known in advance of empiricalerties physical objects have can’t be immediately inferred from the waydistinction. This distinction can be unpacked in two ways: (a) the prop-
(^) Rather, they have to be discovered, and in the process of dis-
sides”.appearances; for example, we assume that physical objects have “back(b) There is more to a physical object than that presented to us by itsa physical object as having certain features is no guarantee that it does.covery we can be surprised by what we find. The fact that you think of
(^) (2) Physical objects are sparse.
(^) You can assert that they exist
So when we have aabout), and that they are determinate down to the last physical detail.may not know about (indeed, more properties than we could ever knowWe simply assume that physical objects have all sorts of properties wesometimes guided by theoretical need. (3) Physical objects are complete.a piecemeal fashion, and this is sometimes guided by direct observation,only after you discover them. This means they have to be discovered in
(^) bona fide
(^) physical object
(^) x , then for every property
20 See D. Armstrong,
(^) A Combinatorial Theory of Possibility
(^) (Cambridge: Cambridge
University Press, 1989), especially pages 45-50.
Bernard Linsky and Edward N. Zalta
F
, either
(^) x (^) has (^) F (^) or (^) x (^) has the negation of
F
(^). Features (1), (2), and (3)
ground the objectivity and mind-independence of physical objects. We call those Platonists who conceive of and theorize about abstract
objects on this model of physical objects
(^) Piecemeal Platonists
Histor-
and characterized by theories developed on a piecemeal basis.abstract objects are “out there in a sparse way” waiting to be discoveredPlatonism, for traditional Platonists typically assume that their preferredically, Piecemeal Platonism has been the dominant form of traditional
(^) Natural-
compatibility of these two realist ontologies begins to emerge.between Platonism and naturalism. By rejecting this model, the essentialand the resulting piecemeal theories as the root of the apparent conflictand that it is a mistake to conceive of them in this way. We see this modelthat abstract objects are fundamentally different from physical objects,jects, inheriting his conception from traditional Platonism. But we thinkPlatonist. He conceives of abstract objects on the model of physical ob-principled reason for accepting some abstract objects, he is a Piecemealtain particular abstracta exist while others don’t? Though Quine offers ato have reliable beliefs about them, and how can we explain why cer-some way, via some manifold analogous to spacetime, how could we comethan others. If we are not differentially connected to abstract objects inalso because there seems to be no principled reason to accept some ratherplanation of how we can come to have reliable beliefs about them, butabstract objects on a piecemeal basis, not simply because there is no ex-ists are quite right to be suspicious of postulating causally disconnected To explain the mind-independence and objectivity of causally inert
that yield aabstract objects, one must assert topic-neutral comprehension principles
(^) plenitude
(^) of abstract objects. Comprehension principles are
tute a plenum.these principles guarantee that the abstract objects in question consti-as there could possibly be (without logical inconsistency); i.e., some ofthat they assert that there are as many abstract objects of a certain sortof a certain sort. Some of these principles are distinguished by the factvery general existence claims stating which conditions specify an object
21 Any theory of abstract objects based on such compre-^
21 Some comprehension principles are unconditional; for example, a schema which
sponds in some way to the condition.requires, for every suitable condition, that there exists an abstract object that corre-
(^) Others are conditional; for example, a modal
thing satisfy the condition, then something exists that satisfies the condition.conditional which asserts, for every suitable condition, that if it is possible that some-
Naturalized Platonism vs. Platonized Naturalism
quasi-Kantian “transcendental” argument for the synthetic thus for the very meaningfulness of any such theory. So we shall present a
(^) a priori
(^) char-
acter of the comprehension principle.
29 Taken together, the material in^
sections
(^) iii (^) though
(^) v (^) constitutes Platonized Naturalism, which we put for-
argument in sectionward as an alternative to Naturalized Platonism. We then complete our
(^) vi (^) by showing that Platonized Naturalism is indeed
for ontology.a kind of naturalism and is compatible with all the naturalist standards
III. A Principled Platonism
any mathematical notions or axioms. The theory of abstract objects to which we now turn does not appeal to
(^) However, both mathematical ob-
jects (^) and (^) mathematical theories will be identified as abstract individuals
described by the theory.
30 The theory is expressed in terms of the pred-^
icate ‘
A! x’ (which asserts that
(^) x (^) is abstract
) and the primitive notion of
encoding
. Encoding is a mode of predication and, as such, contrasts with
the traditional
(^) exemplification
(^) mode of predication. That is, in addition
to the traditional reading of ‘
x is F ’ as (^) x (^) exemplifies
F
F x ’), we in-
troduce the reading that
(^) x (^) encodes
F
xF (^) ’). The three most important
principles that govern the notion of encoding are:
- For every condition on properties, there is an abstract individual that (^) encodes
(^) exactly the properties satisfying the condition.
∃x (A !x (^) & (^) ∀ F (^) ( xF (^) ≡ (^) φ )), where
(^) x (^) is not free in
(^) φ
- If (^) x (^) possibly encodes a property
F
(^) , it does so necessarily.
xF (^) → (^) xF
29 We say “quasi-Kantian” because, unlike Kant, we shall not ground the synthetic
(^) a
priori (^) character of the comprehension principles on facts about possible psychological
states of consciousness and experience. See I. Kant,
(^) Critique of Pure Reason
, Norman
Kemp Smith, trans. (New York: St. Martins, 1965).
(^) Whereas Kant argued that the
30 Platonism is a presupposition of any possible science.possible experience, we argue that the use of the abstract objects of our Principleduse of the categories of the understanding in judgements was a presupposition of any A typed version of the theory asserts not only the existence of abstract individuals,
relations (which are distinguished from ordinary properties and relations). Seebut also the existence of higher-order abstract objects, such as abstract properties and
(^) Abstract
Objects
, (^) op. cit
. Mathematical relations such as
(^) successor
(^) and (^) membership
(^) can then be
the sense of ‘individual’.the theory to higher-order abstracta. So, in what follows, we use the term ‘object’ inidentified as abstract relations. But we shall not spend time at this point generalizing
Bernard Linsky and Edward N. Zalta
- If (^) x (^) and (^) y (^) are abstract individuals, then they are identical iff they
A! encode the same properties. x & (^) A !y (^) → (^) (x (^) = (^) y ≡ ∀ F (^) ( xF (^) ≡ (^) yF (^) ))
encoded property.a given condition if distinct abstract objects have to differ by at least onebe two distinct abstract objects encoding exactly the properties satisfyingthat encodes just the properties satisfying the condition; there couldn’tthat for every condition on properties, there is a unique abstract objectstract objects. It is a simple consequence of the first and third principlesessentially encoded; the third principle is the identity principle for ab-second principle says that what an abstract object (possibly) encodes isThe first principle is the comprehension principle for abstract objects; the The comprehension principle asserts the existence of a wide variety of
because Clinton either exemplifiesencodes just the properties Clinton exemplifies. This object is completeone instance of comprehension asserts there exists an abstract object thatties they encode, while others are incomplete in this respect. For example,abstract objects, some of which are complete with respect to the proper-
F
(^) or exemplifies the negation of
F
(^) , for
every property
F
(^). Another instance of comprehension asserts that there
is an abstract object that encodes just the two properties:
(^) being blue
(^) and
being round
(^). This object is incomplete because for every
(^) other
(^) property
F
, it encodes neither
F
(^) nor the negation of
F
(^). But though abstract ob-
complete with respect to the properties theyjects may be partial with respect to their encoded properties, they are all
(^) exemplify
. In other words,
the following principle of classical logic is preserved: for every object
(^) x
and property
F
(^) , either
(^) x (^) exemplifies
F
(^) or (^) x (^) exemplifies the negation of
F
We can express this formally if we use
(^) λ -notation to define the negation
of (^) F (^) (‘ F ¯ F (^) ’) as follows:¯ = df [λy (^) ¬ F y ]
We may read the
(^) λ -predicate as: being an object
(^) y (^) such that
(^) y (^) fails to
exemplify
F
So we preserve the following formal principle of classical
logic:
(^) ∀ F (^) ∀ x( F x (^) ∨ (^) F x¯ ). 31
The comprehension principle can be formulated without restrictions
31 Note that encoding satisfies classical bivalence:
(^) ∀ F (^) ∀ x( xF (^) ∨ ¬ xF (^) ). But the in-
general true:completeness of abstract objects is captured by the fact that the following is not in
(^) xF (^) ∨ (^) x F (^) .¯
Naturalized Platonism vs. Platonized Naturalism
because
(^) xF (^) does not entail
(^) F x
. 32 It captures the idea that to describe^
hension principle asserts the existence of a plenitude of abstract objects.are as many abstract objects as there could possibly be. So the compre-that there is an abstract object for every group of properties, then thereof distinguishing among objects, and the comprehension principle assertsproperties involved in that conception. If properties are the possible waysto mind to conceive of a thing, there is something that encodes just thehension principle guarantees that no matter what properties one bringsused to specify it. Another way of thinking about this is that the compre-tify them completely. An abstract object encodes exactly the propertiesdistinctive about abstract objects is that this is all one has to do to iden-any abstract object, one must specify a group of properties. But what is Given the above, it should be clear that none of the elements of the
different in kind from ordinary spatiotemporal objects.model of physical objects apply to abstracta—abstract objects are simply
Ordinary spa-
spacetime.ties. Abstract objects are not the kind of thing that could be located intiotemporal objects are not the kind of thing that could encode proper-
(^) We assert that abstract objects
(^) necessarily fail to exemplify
certain ordinary properties.
(^) They necessarily fail to have a location in
eration and decay, etc.be material objects, they necessarily fail to be subject to the laws of gen-spacetime, they necessarily fail to have a shape, they necessarily fail to
Consequently, by the classical laws of complex
properties.properties, abstract objects necessarily exemplify the negations of these
But notice that the properties abstract objects encode are
tions to ordinary objects, such as being thought object byto mention that abstract objects may contingently exemplify certain rela-former are the ones by which we individuate them. And it is importantmore important than the properties they necessarily exemplify, since the
(^) y , being studied
by (^) z , inspiring
(^) u (^) to action, etc.
Our three principles of encoding are part of a larger system which in-
erties, relations, and propositions.cludes complementary existence and identity principles for ordinary prop-
33 The framework as a whole has been^
32 So abstract objects may encode incompatible properties without contradiction, for
the latter are defined as properties that couldn’t be
(^) exemplified
(^) by the same objects.
The following, for example, are jointly consistent:
(^) x (^) encodes roundness (
xR ), x en-
codes squareness (
xS ), and necessarily everything that
(^) exemplifies
(^) being round fails to
exemplify
(^) being square (
∀ y(Ry (^) → ¬ Sy )). Thus, the notorious “round square” may
33 simply be the abstract object that encodes just being round and being square. While the comprehension principle for abstract objects is not restricted, that for
Bernard Linsky and Edward N. Zalta
worlds, and intentional entities such as fictions, among other things.applied to the analysis of complex properties and propositions, possible
34 As^
be analyzed in the same terms we use to analyze mathematical theories.least, provides stories about mathematical objects and such stories cancovers informal mathematics as well. Informal mathematics, at the veryand set theory as typical examples of mathematical theories, our analysismatical theories and objects. Though we shall use formal number theoryanother application of the theory, we now develop an analysis of mathe- We begin by extending the notion of an object encoding a property
to that of an object encoding a proposition.
We do this by treating
propositions as 0-place properties.
(^) If we let ‘
p’ range over propositions,
then an object
(^) x (^) may encode the proposition
(^) p (^) in virtue of encoding the
complex propositional property
(^) being such that p
. We will symbolize such
a propositional property as: [
λy p ]. 35 These notions allow us to identify^
a mathematical theory
(^) T (^) with that abstract object that encodes just the
propositions asserted by
(^) T (^).
Next we define a technical notion of truth in a theory as follows: a
proposition
(^) p (^) is (^) true in
(^) a theory
(^) T (^) (‘ T |= (^) p’) iff
(^) T (^) encodes the property
being such that p
. Formally:
T |= (^) p = df T [λy p ]
We use this definition to analyze the ordinary claim ‘In theory
(^) T (^) , a is
F
’ as follows: the proposition that
(^) a (^) exemplifies
F
(^) is true in theory
(^) T (^) ,
i.e., (^) T (^) |= (^) F a . In the present context, the symbol ‘
|=’ does
(^) not (^) express
relations must be restricted to avoid paradox:of which have been analyzed elsewhere in terms of encoding.true-at-a-world, factual-in-a-situation, and true-according-to-a-fiction, allmodel-theoretic consequence, but rather the family of notions such as
∃F (^) n ∀x (^1)... (^) ∀x n( F (^) n 1 x
(^)... x n (^) ≡ (^) φ ), where
(^) φ (^) has no encoding subformulas and
no quantifiers binding relation variables
ditions for properties:Identity conditions for relations can be defined in terms of the following identity con-
(^) F (^) and (^) G (^) are identical iff necessarily, all and only the objects
that encode
(^) F (^) encode
(^) G .
34 See E. Zalta,
(^) Abstract Objects
, (^) op. (^) cit. , (^) Intensional Logic and the Metaphysics
of Intentionality
, (^) op. cit.
, and “Twenty-Five Basic Theorems in Situation and World
Theory,”
(^) Journal of Philosophical Logic
, (^22) (^) (1993): 385-428.
35 The propositional property [
λy p ] is logically well-behaved despite the vacuously
bound
(^) λ -variable
(^) y . It is constrained by the ordinary logic of complex predicates,
which has the following consequence:
(^) x (^) exemplifies
(^) [λy p
] iff (^) p, i.e., [
λy p ]x ≡ p.
Naturalized Platonism vs. Platonized Naturalism
Given this reading, we can explain the
(^) necessity
(^) of ordinary mathematical
which they are true are necessary.statements by the fact that the encoding claims that provide the sense in
(^) This is a consequence of the second
principle of encoding. We now have a conception of both abstract objects in general and
mathematical objects in particular.
This conception distinguishes the
properties these objects encode from the properties they exemplify.
(^) On
theory are not the properties that it exemplifiesthis conception, the properties attributed to a mathematical object in a
(^) simpliciter
(^) , for theories
about the properties encoded by abstract mathematical objects.are frequently incomplete and inconsistent. On our view, mathematics is
(^) Math-
which it serves to anchor.this conception as we more fully describe the philosophy of mathematicsis extra-mathematical. In the next section, we draw out consequences oftheir abstract nature, but the fact that they exemplify such propertiesematical objects certainly exemplify properties that are characteristic of
IV. A Philosophy of Mathematics
By analyzing mathematical objects as
(^) bona fide
(^) abstract objects in a
tional elements of Platonist philosophies of mathematics.realist ontology, our Principled Platonism preserves the following tradi-
39 Mathematical^
decay.They are not spatiotemporal and therefore not subject to generation andobjects are essentially different in kind from ordinary material objects.
(^) Mathematical truths are necessary, and moreover, mathematical
objects necessarily exist.
40 Like all abstract objects, they couldn’t possi-^
operator. a detective’, and argues that both are acceptable only if prefixed by an ‘In-the-story’Even though they encode only the properties attributed to them by theirdenoted by a given symbol, or being thought about by mathematicians.these properties, and they contingently exemplify such properties as beingbeing a building, etc. Indeed they necessarily exemplify the negations ofbly exemplify ordinary properties like having a shape, having a texture,
(^) But he offers neither truth conditions for the story operator nor a reading
of unprefixed
(^) sentences such as ‘2 is prime’ on which they turn out true.
39 For a summary of these traditional elements, see A. Irvine, “Introduction” to
Physicalism in Mathematics
, (^) op. cit.
, pp. xix–xx. Our theory also appears to preserve
many of the aspects of historical Platonism.
(^) See J. Moravcsik,
(^) Plato and Platonism
40 (Cambridge, MA: Blackwell, 1992). The necessary existence of abstract objects is a consequence of applying the Rule
of Necessitation to the comprehension principle.
Bernard Linsky and Edward N. Zalta
are complete with respect to the properties that they exemplify.respective theories, they are nevertheless determinate objects, for they
(^) Note
ner, and its semantics is therefore compositional.is analyzed, from a logical point of view, in the simplest possible man-of mathematical theories denote abstract objects. Mathematical languageindividuals), and we preserve the logical intuition that the singular termsthat we preserve the common sense view that numbers are objects (i.e.,
(^) The truth conditions
terms and the way in which they are arranged.of mathematical sentences are stated in terms of the denotations of their
41 This gives a sense in^
which the ontology is realist and in which its truths are objective. Our view of theories and objects is very fine-grained.
If the math-
ometries.Consider, for example, Euclidean, Riemannian, and Lobachevskian ge-ematical theories are different, the mathematical objects are different.
(^) It is natural for a Platonist to think that different geometri-
are not ZF sets of any kind.ferent theory from the theory of Zermelo-Fraenkel sets, Peano numbersordinals simply doesn’t apply. Since Peano’s theory of numbers is a dif-the Peano numbers “really are” the von Neumann ordinals or the Zermeloand not some other thing. So Benacerraf’s problem of explaining whetherjects of some foundational theory. Each mathematical object is what it isdoesn’t require us to reduce the various mathematical objects to the ob-cal theories are about different objects. Moreover, Principled Platonism It may be wondered whether our theory is too fine-grained, providing
too many objects.
(^) The worry is that the number 1 of Peano’s Number
41 not the same objects, and the reason they are not is that the number 1 ofof ZFC (i.e., ZF plus the Axiom of Choice). But we reply that these aretheory, and that the emptyset of ZF is the same object as the emptysetTheory seems to be the same object as the number 1 of real number For example, the sentence ‘In Peano Number Theory, 3 is greater than 2’ receives
the analysis:
(^) PNT
(^) |= 3 (^) > (^) 2 (dropping the subscripts on ‘3’ and ‘2’).
(^) The ordinary
traditional relational analysis of the formsentence ‘3 is greater than 2’, inside the operator ‘In Peano Number Theory’, receives a
(^) Rxy
. Since theories are closed under logical
consequence, it follows both that:
PNT (^) |= [ λy y >
(^) 2]
PNT (^) |= [ λy (^3)
y ]
described in ‘Mathematical Truth,’‘In Peano Number Theory, 3 is greater than 2’. This satisfies a desideratum Benacerrafnoted by ‘Peano Number Theory’, ‘2’, and ‘3’ in the compositional truth conditions forSo we may use the analysis of the previous section to identify the abstract objects de-
(^) op. cit.
Naturalized Platonism vs. Platonized Naturalism
fails to have (encode) this property.is nothing between 1 and 2, whereas the number 1 of real number theory Peano Number theory has (encodes) the property of being such that there
So they are different.
Similarly,
ZF^ ∅
(^) and (^) ∅ ZFC (^) are different because
ZFC (^) has (encodes) the property of
being such that every nonempty set of sets has a choice set, but
ZF (^) lacks
different because they are embedded in distinct theories.(fails to encode) this property. What this shows that that the objects are We don’t see mathematicians as searching for the unique true theory
to “CH” and “non-CH” sets. To suppose otherwise is to make thethis applies not only to well-founded and non-wellfounded sets, but alsohegemonic set-theory—there are many equally real universes of sets, andwhether a proposition is derivable from the axioms. But there is no single,sequences of their own theories—there is an objective matter concerningof reality for the whole of reality. They can each be looking for the con-real, but each party to the disagreement mistakes their limited portionplained by the common vocabulary. What each has in mind is perfectlysimply talking about different sets. The appearance of disagreement is ex-is “true”. It seems clear in this latter case that the mathematicians aretwo mathematicians “disagree” about whether the Axiom of FoundationAxiom of Foundation and later “rejects” it, or the situation in whichthe analogous situation of a mathematician who at one time accepts theare thinking about different objects—they just don’t realize it. Considermaticians who disagree about whether it is “true”. We claim that theyuum Hypothesis (CH), and then later rejects it, or consider two mathe-of sets. Consider a mathematician who at one time accepts the Contin-
(^) mistake
other mathematical theories?ing, and which set theory is so powerful that we could have done withoutquestions, which overall theory of sets is the most powerful and interest-one true theory. The real disagreements among set theorists concern thenot “out there in a sparse way” waiting to be discovered and described by of conceiving abstract objects on the model of physical objects—sets are Indeed, we extend our conclusion to claim that there is no single hege-
monic membership relation.
Our view is that not only should we not
or determinate.ance/reality distinction for mathematical relations; they are not completestract mathematical relations as ordinary relations. There is no appear-model abstract objects as physical objects, but we should not model ab-
(^) Rather, they are just the way we specify them to be—
they are creatures of theory just as much as mathematical objects, and as
Bernard Linsky and Edward N. Zalta
such, are indeterminate. So we treat mathematical relation
R
(^) of theory
T as that
(^) abstract relation
(^) that encodes just the properties of relations
F
that are attributed to
R
(^) in (^) T
(^). 42 Thus, the membership relation of ZF^
can be theoretically described as follows:
ZF^ ∈
(^) ıR (R F ≡ (^) ZF (^) |= (^) F ∈ ZF (^) )
ematical relations as ordinary relations.To suppose otherwise is to make the mistake of modeling abstract math-If the theories are different, so are the abstract mathematical relations.
43
Of course, by accepting an incorrect proof, a mathematician might
erroneously judge that in theory
(^) T (^) , x is F , for some
(^) T (^) , x and (^) F (^). The
about the objects of a theory iswhat properties follow from the theory. So we allow for error—a mistakeerties that a mathematical object encodes by making a mistake aboutfollow from that theory. It is possible to make a mistake about the prop-mathematical objects of a theory encode the properties that genuinely
(^) not (^) a successful discovery of a truth about
can form new judgements of the form ‘Insome different objects. Similarly, we allow for ignorance—mathematicians
(^) T (^) , (^) x (^) is (^) F (^) ’ without thereby think-
objects described earlier.tinction is rather different from the ones involved in the model of physicalwith our knowledge of mathematical objects. But this version of the dis-version of the appearance/reality distinction presents itself in connectioning of objects of a different theory. By allowing for error and ignorance, a Nor is our theory a version of “if-thenism”.
(^) If-thenism is the thesis
assertions are really just certain logically true conditionals.prime’ is derivable from axioms of PNT, i.e., that apparent categoricalthat a mathematical claim like 2 is prime is really the claim that ‘2 is
(^) But on our
notion ‘sis, ‘2 is prime’ is a simple categorical claim. Moreover, we distinguish theview, mathematical statements are categorical assertions. On our analy-
p is true in theory T’ from the notion ‘
p is derivable from some
that a mathematical statement has a meaning of its own.derivability in a theory, it would seem that if-thenists cannot maintainaxiomatization of T’. By collapsing the notions of truth in a theory and
(^) For were it
42 This requires the type-theoretic version of the comprehension principle for abstract
objects mentioned in footnote 30. See
(^) Abstract Objects
, (^) op. cit.
, Chapters V and VI.
43 Note that, on our view, mathematical individuals are, in some sense, even more
ematical individuals encode abstract properties.abstract than fictions. Whereas fictional individuals encode ordinary properties, math-
Naturalized Platonism vs. Platonized Naturalism
ensures that incomplete descriptions will successfully refer.properties that they encode, and this, together with the identity principle, introduces abstract objects that may be incomplete with respect to the Knowledge of particular abstract objects doesn’t require any causal
de reconnection to them, but we know them on a one-to-one basis because
(^) knowledge of abstracta is by description.
All one has to do to
become so acquainted
(^) de re
(^) with an abstract object is to understand its
encodes are precisely those expressed by their defining conditions.descriptive, defining condition, for the properties that an abstract object
48 So^
one we use to understand the comprehension principle.our cognitive faculty for acquiring knowledge of abstracta is simply the
We therefore
link between our cognitive faculty of understanding and abstract objects.The comprehension and identity axioms of Principled Platonism are thefaculties and abstract objects accounts for our knowledge of the latter.have an answer to Benacerraf’s worry that no link between our cognitive The comprehension principle as a whole, we argue, is
(^) synthetic
(^) and
known
(^) a priori
. It is synthetic because it asserts the existence of objects
encoding certain properties.
(^) It’s not part of the meaning of ‘abstract’,
used to express it.So the principle isn’t true in virtue of the very meanings of the wordsabstract object that encodes just the properties satisfying the condition.‘encodes’, and ‘property’ that for every condition on properties there is an
(^) Moreover, if it is known, it is known
(^) a priori
The
general, topic-neutral, and constitutes a simple extension of thefacts bear on the truth of the comprehension principle. It is completelyanalogy with the principles of logic—like such principles, no contingentempirical evidence. This can be seen, at the very least, by inspection andreason is that it is not subject to confirmation or refutation on the basis of
(^) a priori
truths of logic.
49 Its^ (^) a priori
(^) character can also be established, moreover,
48 This depends, of course, on the fact that we can refer to the properties involved
predicates and the ones denoted by complex predicates.however, the ordinary properties fall into two kinds, the ones denoted by primitiveproperties, the development of which would take us too far afield at this point. Briefly,in the description. To explain such reference, we would rely on the theory of ordinary
(^) With respect to the latter,
comprehension principle forreference to properties is by description, where the description is grounded by the
(^) properties
. With respect to the former, either reference
49 takes place in the context of a scientific theory (e.g., physical, primary qualities).to the property is by acquaintance (e.g., experienced, secondary qualities) or reference This is similar to the turn-of-the-century logicians’ view that the unrestricted
comprehension principle for sets was known
(^) a priori
. If one thinks that our (consistent)
comprehension principle for abstracta just is logical rather than metaphysical, then our
Bernard Linsky and Edward N. Zalta
by the
(^) a priori
(^) character of its theorems.
50
But the most important ingredient of Platonized Naturalism is the
argument that Principled Platonism is consistent with naturalism.
(^) In-
deed, from a naturalist perspective, how can there be synthetic
(^) a priori
truths like the comprehension principle?
The answer is that there can
categories ofthat the analysis of formal scientific theories will be based on the logicalterms of logical consequence. Here we have in mind not simply the factpart of natural language, and that scientific inferences are analyzable inpremise, we simply note that scientific language is a special (if systematic)required for the proper analysis of scientific theories. In support of thisanalysis of natural language and inference in general is the frameworkpremises. The first is that the logical framework required for the properin just this way, we offer a very general argument that begins with twoistic theories. To establish that our comprehension principle is requiredthat is, if they are required for our very understanding of those natural-be such truths if they are required to make sense of naturalistic theories,
(^) object
(^) ( individual
(^) property
quality
), and
(^) exemplification
(instantiation
), which play an essential role in the analysis of ordinary
everyday discourse.in scientific practice as a special case of the intensional language used inhave happened, the distinction between fact and fiction, etc.) shows upthe future, talk about possibilities, subjunctive talk about what mightlanguage, but also the fact that intensional language (such as talk about
51 So the logical framework required for the analysis^
50 project can be seen as a kind of logicism.logic of encoding are an essential part of the logical framework requiredNow the second premise is simply that the comprehension principle andentific thought will be a special case of the analysis of thought in general.by their very nature, universally applicable, and that the analysis of sci-first premise, then, simply expresses the idea that the laws of thought are,quired for the analysis of scientific theories and scientific reasoning. Thisof natural language and inference as a whole is the framework that is re- There is a wide range of
(^) a priori
(^) theorems that can be derived from the compre-
hension principle.
(^) See (^) Abstract Objects
, (^) op. cit.
, and “Twenty-Five Basic Theorems
in Situation and World Theory,”
(^) op. cit.
51 In further support of this premise, we note that such fictions as frictionless planes,
are distinct. Some charge that even the laws of science are useful fictions.generalizations such as ‘All ravens are black’ and ‘All non-black things are non-ravens’tific laws, and that the paradoxes of confirmation suggest that necessarily equivalentideal gases, centers of gravity and other point-sized bits of matter, are used in scien-
Naturalized Platonism vs. Platonized Naturalism
failures of important logical principles.course about fictions, puzzles about definite descriptions, and apparentconstructions such as propositional attitude reports, modal contexts, dis-best explanation of the logical form and consequences of such problematiccoding are the central components of an intensional logic that offers thethis premise, we claim that the comprehension principle and logic of en- for the proper analysis of natural language and inference. In support of
52 We also point to our analysis^
in section
(^) iii , which suggests that this logic also offers the best explana-
required for the proper analysis of scientific theories and inferences.conclude that the comprehension principle and the logic of encoding aretion of mathematical language and inference. From our two premises, we
(^) As
tent with naturalism because it is required by naturalism.understanding of any such theory. So our Principled Platonism is consis-to make sense of any possible scientific theory, i.e., required for our verysion that the comprehension principle and logic of encoding are requiredthe final step to our argument, we claim that it follows from this conclu- Note that the reason why Principled Platonism is consistent with nat-
as we distinguish the notion of best overallory. But we can accept Quine’s talk about the best overall theory as longtheory is stronger than the claim that it is part of the best scientific the-sion principle is required for our understanding of any possible scientificof the best scientific theory of the world. The claim that the comprehen-uralism is stronger than Quine’s claim that his limited Platonism is part
(^) scientific
(^) theory from the no-
tion of best overall logical account of scientific theories.
(^) Arguments for
believing it.most uniform understanding of scientific theories, then we are justified inparticular comprehension principle is part of the best logic and offers theprinciple. For if on the basis of rational argument we conclude that oursonably claim to know (i.e., to be justified in believing) the comprehension(among other things). Indeed, on the basis of such arguments, we can rea-offers a uniform understanding of the variety of different scientific theoriesderivable from one specific theory, but rather by the way the frameworkto be judged (i.e., justified or refuted) not by the empirical consequencesthe latter are different from the arguments for the former, for the latter is
(^) We emphasize that we may be wrong about which frame-
general, and of mathematics, in particular.work offers the most uniform understanding of thought and language, in
(^) Some other Principled Pla-
52 tonism may have the best analysis of logical form (and if so, the ideas in See (^) Intensional Logic and the Metaphysics of Intentionality
, (^) op. cit.
Bernard Linsky and Edward N. Zalta
than ours). Though we claim that our comprehension principle isthe previous paragraph would apply to its comprehension principle rather
(^) a pri-
ori , there is room for rational debate about its status as part of the best
overall framework. We can be fallibilists about the
(^) a priori
So we can claim to know the comprehension principle if we can ra-
the physical reality of some metaphysically possible world.bizarre or convoluted, might be needed to characterize some portion ofematical theory describes an abstract mathematical realm that, howeverpossible world in one of the natural sciences there. Every consistent math-ical theories. Each consistent such theory could have been used in someanalysis of mathematics covers the whole range of (possible) mathemat-uniform understanding of scientific theories. To this end, we note that ourtionally conclude that it is part of the best analysis and offers the most
(^) This is why
science in which that theory is employed.mathematical theory, our logic and ontology makes sense of any possiblemathematical theories. By offering a correct representation of any suchit is important to have an analysis of currently unapplied or dispensable
(^) Indeed, the full comprehen-
might prove useful).principle, has not made some arbitrary choices about which mathematicswhy their theory, which would have to be more complex than our single(anyone claiming that a weaker theory could do this job has to explainsis of every possible mathematical theory employed in a possible sciencecomprehension principle is sufficient to the task of providing an analy-sion principle must be accepted for this project. Only a maximally broad
(^) So some such maximal comprehension principle is
best explanation.alternatives that are sufficient to the task, we believe ours to offer theprinciple suffices for this task and that there is currently an absence ofrequired for the analysis of mathematics, and given both that our specific We conclude this section with a brief discussion of mathematical know-
ledge.
(^) The truths of mathematics, on our view, inherit the synthetic
(^) a
priori
(^) character of the comprehension principle. The basic truths of math-
ematics are not such unadorned sentences as ‘2 is prime’ and ‘
but rather have the form:
53
In Peano Number Theory, 2 is prime.
53 There are, however, some unadorned claims about the properties that the
(^) natural
cardinals and the
(^) natural
(^) sets encode.
(^) These are independent of any mathematical
sketch.theory and are derivable from our comprehension principle. See footnote 37 for a brief
Naturalized Platonism vs. Platonized Naturalism
a plenum is not an arbitrary selection from some larger class.acknowledges that a maximal ontology of abstracta is the simplest becauseentities to a minimum, theories are kept simple. Platonized Naturalisming Ockham’s Razor is that by keeping the number of kinds of theoretical non-arbitrary way is to add them all! The traditional justification for cit- Our comprehension principle is consistent, so it is contradiction and
paradox free.
56 But some Naturalized Platonists have objected to com-^
all and only those who do not shave themselves.of themselves as strictly analogous to the puzzle of the barber who shavesample, present Russell’s paradox of the set of all sets that are not membersprehension principles for abstract objects. Bigelow and Pargetter, for ex-
57 Comprehension princi-^
a slowly uncovered realm.describe a complete universe of sets, but is rather a partial description ofto describe sets as the latter are encountered. On this view, ZF does notone at a time according to theoretical need. Axioms are to be formulatedshould proceed in a piecemeal fashion, postulating objects and propertiesprinciples for barbers. The lesson they learn from this is that naturalistsples for sets or properties, they argue, are as mistaken as comprehension But we think this is to misunderstand the nature of mathematical the-
prehension principleories and abstract objects. The abandonment of Frege’s unrestricted com-
58 and its replacement by formal set theories (like ZF)^
is not a move from an
(^) a priori
(^) approach to sets to one that is
(^) a posteriori
and of a piece with natural science. The axioms of set theory are
(^) not (^) like
pairing axiom asserts the existence ofhypotheses about a newly discovered class of fundamental particle. The
(^) arbitrary
(^) pair sets using disjunction
as a guide to existence; the power set axiom asserts the existence of
(^) all
can revise their existence claims without abandoning thenot be the same as that to the Barber Paradox. Platonized Naturalistsempirical theories is inaccurate. The response to Russell’s Paradox should subsets of a given set. So we think Bigelow and Pargetter’s analogy with
(^) a priori
(^) status
56 Two models of the theory of abstract objects have been developed.
(^) The first is
suggested by Dana Scott, and is reported in
(^) Abstract Objects
, (^) op. cit.
, Appendix A.
Calculus and its Interpretation,” in M. de Rijke, ed.,The second is suggested by Peter Aczel and is reported in Zalta, “The Modal Object
(^) Advances in Intensional Logic
(Dordrecht:
(^) Kluwer, forthcoming).
(^) These models describe interpretations on which
57 for ordinary relations turn out true.both the comprehension principle for abstract objects and the comprehension principle Science and Necessity
(^) (Cambridge: Cambridge University Press, 1991).
58 G. Frege,
(^) Basic Laws of Arithmetic
, M. Furth, trans.
(^) (Berkeley:
(^) University of
California Press, 1967).
Bernard Linsky and Edward N. Zalta
a piecemeal one. To repeat, we can be fallibilists about theof those claims and without abandoning their systematic methodology for
(^) a priori
VII. Conclusion
vation about what we have tried to accomplish. We conclude with a suggestion for extending our ideas and a final obser-
(^) Platonized Naturalism
cal objects. Properly understood as ancan be extended to account for logical objects in addition to mathemati-
(^) a priori
(^) science, logic has always
sibilia (for modal logic).abstract structures (for model theory), or possible worlds and other pos-values, properties or pluralities (for the quantifiers of second order logic),required objects of some sort, whether propositions, extensions, truth
(^) Neither naturalism nor its predecessor, logical
matter of logic.empiricism, has ever supplied a satisfactory explanation of the subject
(^) The only account proffered was that logic consisted of
pirical truth.linguistic conventions. This however, as Pap showed, collapses into em-
(^60) There simply never was a naturalist account of logic as
an object-free
(^) a priori
(^) science.
A Platonized Naturalist, however, can treat some logical objects (such
sion principle.stract models) as abstract individuals already covered by the comprehen-as possible worlds, truth values, extensions, natural cardinals, and ab-
61 The remaining logical objects require other comprehen-^
59 sion principles, such as principles asserting a plenitude of properties, re- Given this interpretation of the development of set theory, one might wonder
whether ZF itself constitutes a Principled Platonism.
(^) We rule out ZF and other set
not piecemeal, they do not assert the existence of a plenitude of sets.theories for the following reasons: (1) While the comprehension principles of ZF are
(^) Just consider
60 the framework in which any possible natural science would be formulated.successfully argue that exemplification logic and the axioms of set theory constitutemathematics that would be used in any possible scientific theory. Hence, one cannnotmathematical theory is reducible to ZF, and so questionable whether ZF provides theof language and thought. (4) Finally, it is at least questionable whether every possibletory foundation for intensional logic nor a general framework for the logical analysisdistinguishing features of the former. So we believe that ZF offers neither a satisfac-sitions to extensional entities collapse important distinctions and so do not capture thein a nonarbitrary way, reductions of intensional entities such as properties and propo-properties, and propositions to sets. (3) Even if the reductions could be accomplishedof the kind described by Benacerraf (1965) in the reductions of entities like numbers,the many large cardinal axioms that are independent of ZF. (2) There is arbitrariness Semantics and Necessary Truth
(^) (New Haven: Yale University Press, 1958).
61 See, for example, “The Theory of Fregean Logical Objects” (
op. cit.
) and “Twenty-
Five Basic Theorems in Situation and World Theory” (
op. cit.
).
Naturalized Platonism vs. Platonized Naturalism
they are a part of the Principled Platonism we defend.describe these other plenitude axioms in any detail in the present paper, lations, and propositions or a plenitude of possibilia. Though we did not
62 They would
matter is central to the applicability of logic in the rest of natural science.logic has a legimitate subject matter of its own and that this subjectciple for abstract individuals. Platonized Naturalism acknowledges thatreceive the same epistemological justification as the comprehension prin- We have tried to address the traditional naturalist concerns about
abstract objects.
If we are right, then the bald thesis that there are
no abstract objects is no longer justifiable.
We have tried to develop
an insight that exists at the intersection of work by Kant, Frege,
63 and^
Russell.
64 We defend the Kantian idea that there are synthetic^
(^) a priori
with a robust sense of reality.objects; and we defend the Russellian idea that such objects are consistent truths; we defend the Fregean idea that logic and mathematics are about
We have employed rigorous logical and
to the study of abstract objects.epistemological standards to eliminate the arbitrary, piecemeal approach
(^) Belief in these objects is justified if it
62 complies with these standards. The plenum principle for possibilia is just the first-order Barcan Formula, under
the actualist interpretation formulated in B. Linsky and E. Zalta,
(^) op. cit
. Though we
do not have individual,
(^) de re (^) knowledge of these entities, our knowledge of the Barcan
formula would nevertheless be justified on the grounds outlined in sections
(^) v and (^) vi .
63 The Foundations of Arithmetic
, J. L. Austin, trans., 2nd rev. ed. (Oxford: Black-
well, 1974), and
(^) The Basic Laws of Arithmetic
, (^) op. cit.
64 The Problems of Philosophy
(^) (Oxford: Oxford University Press, 1964).
