Mark Balaguer:  Platonism and Anti-Platonism in Mathematics

 

Reviewed by Matthew McGrath

 

 

     Mark Balaguer has written a provocative and original book.  The book is as ambitious as a work of philosophy of mathematics could be.  It defends both of the dominant views concerning the ontology of mathematics, Platonism and Anti-Platonism, and then closes with an argument that there is no fact of the matter which is right. 

     Balaguer first defends a form of Platonism he calls “Full Blooded Platonism” (FBP).  This is the view that every possible mathematical object exists.  (Throughout, Balaguer assumes, reasonably, that if there are mathematical objects, they are abstract, i.e., non-spatio-temporal.)  Formulated more carefully, FBP holds that every consistent mathematical theory is true of some part of the mathematical realm.  FBP is full-blooded, since it does not distinguish between those consistent mathematical theories that manage to truly describe abstract objects and those that don’t; they all do.  Balaguer argues, persuasively, that FBP has virtues that other forms of Platonism lack, among which are these:  (1) FBP is consistent with, and would lead us to predict, the remarkable fact that mathematicians accept mathematical theories in many cases prior to, and independently of, learning of any applications of the theory; consistency and usefulness in mathematics is enough.  (2):  FBP does not force us to regard undecidable propositions, such as the continuum hypothesis (CH), as having a truth-value.  Both ZF + CH and ZF + ~CH are true of parts of the mathematical realm, just different parts.  Thus, neither theory is objectively better than the other.  (3): FBP avoids Benacerraf’s two well-known objections to Platonism, the epistemological objection, and the objection from multiple-reductions.

     I want to discuss in particular (2) and (3).  First, (2).  Truth, under FBP, is distinct from being true of part of the mathematical realm; it requires “jibing” with our intuitive conception of the entities in question.  For example, because our intuitive conception of sets builds in extensionality, no theory according to which two sets had the same members is true, though all consistent such theories are true of parts of the mathematical realm.  One is puzzled, however, how a sentence and its negation could be true of different parts of the mathematical realm.  The negation of C ought to say that C isn’t the case, but how can it do so if it has a different subject matter than C?  Indeed Balaguer seems to deny that not-C is a negation of C:  “while it does follow from FBP that both C and not-C truly describe parts of the mathematical realm, we can obtain this result only by interpreting C in two different ways in the two different cases…”  This is unacceptable.  We ought to be able to negate mathematical theories.[1]   ‘Not-(C is true of part of the mathematical realm)’ ought to be distinguishable from ‘not-C is true of part of the mathematical realm’.  One wonders whether Balaguer might solve this problem by denying that negations of mathematical theories are mathematical theories.  A mathematical theory ought to introduce objects and state axioms about them.  The negation of, say, a second-order formulation of Peano Arithmetic does neither of these.  Indeed, it could be, and would be, accepted by someone who rejected all of mathematics.

     How does FBP fare by Benacerraf’s two objections?  Consider the epistemological objection.  Balaguer claims that, to reply to the objection, the FBP-ist needs only to provide an externalist epistemology, i.e., an account of how mathematicians reliably get it right.  She can do this, he claims.  First, independently of FBP, she can explain why it is that mathematicians tend to accept a mathematical theory only if it is consistent (the notion of consistency is taken as primitive).  Then, using FBP, she can conclude that mathematicians tend to accept a mathematical theory only if it truly describes part of the mathematical realm. 

     Balaguer is aware that some will regard this as an explanation of lucky true belief, or lucky reliable belief, rather than knowledge:  it just happens that the mathematical realm is there to make our theories true.  Balaguer replies that it would be too much to demand that mathematicians know the truth of FBP, just as it would be too much to require that ordinary perceivers know the truth of an assumption about the reliability of sense perception.  Reliability is enough, and FBP gives us that.  However, some doubts remain.  For Balaguer’s FBP-ist, and for Balaguer (p. 132), the mathematical realm does not necessarily exist.  So the counterfactual ‘If there were no mathematical realm, then mathematician S would still accept ZF’ makes sense.  Later in the book, in motivating fictionalist Anti-Platonism, Balaguer claims that such counterfactuals are true (p. 132).  This makes it seem that mathematical belief does not track the truth, under FBP.  Indeed, FBP’s account of mathematical knowledge seems no better than the following account of knowledge of the instantiation of lengths.  Suppose a community believes ‘There is an object with length L’ for arbitrarily large positive real values of L.  And suppose they have these beliefs independently of any empirical investigation; perhaps they believe them because they are consistent.  Now, suppose it turns out that every length is instantiated by some object.  Does this mean that these people have knowledge of the instantiation of such lengths?  Surely not.  They would have believed just as they did had what they believed been false.

     A close relative of the epistemological objection is that even reference to mathematical objects is impossible under Platonism.  Balaguer’s response is to invoke a distinction between thick and thin aboutness.  ‘Santa Claus is jolly’ is thinly, but not thickly, about Santa.  The differences between the two notions seem to come to this:  (1) a statement that is thinly about an object can be true independently of whether it exists; not so for thick aboutness (p. 188n47); (2) thin aboutness is never unique; or at least, it is never de re, but is rather designation by description. (p. 189n7).  According to Balaguer, the FBP-ist is committed only to the claim that mathematical theories are thinly about mathematical objects, a thesis he believes that everyone must admit.  How can the FBP-ist avoid thick aboutness?  Because, for each mathematical theory, she admits the existence of many, many systems of objects that the theory truly describes.  Our full conception of natural numbers, unlike our full conception of sets may fix on a system of objects up to isomorphism, but this still leaves room for an infinite number of distinct isomorphic systems, all of which “jibe” with our conception of natural numbers. 

     The notion of thin aboutness is important for Balaguer.  He uses it to answer the “multiple-reductions” objection, arguing that the objection succeeds only against “Uniqueness Platonism,” i.e., only against a view that treats mathematical theories as describing unique domains of entities.  Thus, the axioms of Peano Arithmetic, under Uniqueness Platonism, describe a unique progression, with ‘0’ and function terms such as ‘SS0’ referring to unique elements in the progression.  FBP avoids the multiple-reductions objection because it is a form of Non-Uniqueness Platonism.

     Thin aboutness is puzzling, however, and not because of (1) or (2) taken separately, but because of their combination.  There ought to be room for a view according to which mathematical theories imply the existence of mathematical objects, but do not contain singular terms that fix uniquely on any particular such objects.  Indeed, the FBP-ist needs to invoke such a sense of aboutness.  For, working with Balaguer’s notion, the FBP-ist can’t claim that all our talk is thinly about mathematical objects and still count as a Platonist. For she would then have to treat FBP itself as only thinly about mathematical objects, and so as not requiring their existence for its truth.[2]  FBP, by its own lights, must be about mathematical objects in the sense of (2) and not in the sense of (1), and the same should hold, under FBP, for all consistent mathematical theories.

     However, even the invocation of thin aboutness in the sense of (2) seems to make FBP susceptible to Benacerraf’s dilemma.  Benacerraf claimed that in theorizing about mathematics, we ought to treat both the semantics and the epistemology of mathematics as continuous with semantics and epistemology concerning other domains.  We want reference in mathematics to be reference, knowledge of mathematics to be knowledge.  He then argued that we preserve continuity in one of these areas only at the cost of relinquishing it in the other.  FBP aims to preserve continuity in both areas.  But can it preserve continuity in semantics by taking apparent singular terms such as ‘0’ to refer to infinitely many objects?  It seems not. 

     Balaguer’s insistence on non-unique reference is strongly reminiscient of structuralism.  Balaguer himself regards the issue of structuralism as a red herring:  the structuralist Platonist must construe structures as entities of some kind; but then structures, too, will be non-unique.  However, there are non-ontological forms of structuralism, e.g., what Stewart Shapiro calls “eliminative structuralism”.  Talk of natural numbers, for this sort of structuralist, is best understood as talk about all progressions; so talk of “0” is talk of the minimal element of every progression.   It is difficult not to read FBP as a form of eliminative structuralism.

     Balaguer defends fictionalist Anti-Platonism in the second part of the book.  The fictionalist agrees with the FBP-ist on the semantics of mathematics, but claims that all mathematical theories are false.  Balaguer argues that fictionalists can answer the most serious traditional objections to their view:  the Fregean objection that mathematics is indispensable to empirical science and therefore ought to be regarded as true, and the objection that if mathematics were story-telling, it would not the applications to empirical science that it has.  Balaguer goes some significant distance toward nominalizing quantum mechanics.  But he argues that even if mathematics proves indispensable to empirical science, such indispensability is no reason to think it even partly true.  For the fictionalist is just as well off as the Platonist in explaining the applicability of mathematics; indeed, Balaguer thinks the two will explain such applicability in the same way.  Empirical science does not attribute to mathematical entities any causal-explanatory role; it uses them as mere descriptive aids.  (In defense of this point, Balaguer appeals to the “intuition” that, if the mathematical objects existed, their sudden disappearance would not alter anything in the concrete world.)  But descriptive aids need not exist; they may merely be elements in a consistent story. 

     Must the fictionalist who grants the indispensability of mathematics be a scientific anti-realist as well?  No.  He may distinguish the nominalistic from the Platonistic content of empirical theories and statements.  Thus, ‘System S is 40º C’ has a nominalistic content, about the system’s temperature state, and a Platonistic content, about the number 40.  The fictionalist denies the latter but accepts the former, while admitting that there are some nominalistic contents that elude nominalistic expression.  (One worries about whether fictionalism itself is Platonistic, since it invokes stories and items in stories.  Must the fictionalist be fictionalist about fictionalism itself?  Can the fictionalist in any way state what he literally believes?  It’s no good to say “What I accept is the nominalistic content of fictionalism,” since (if the fictionalist is right) there are no “contents” at all.)

     Finally, Balaguer argues that there is no fact of the matter which of FBP and fictionalism is correct.  The two disagree only on the truth of mathematical statements and the existence of mathematical objects.  Balaguer considers it no argument at all in favor of FBP that we say ‘1 + 1 = 2’ is true.  Nor does he think Ockham’s Razor favors fictionalism, for FBP purports to explain putative facts that fictionalism doesn’t, viz. that some mathematical theories are true. 

     Could this be a case of mere epistemic limitation?  Balaguer argues no.  We do not know what must hold of a world for it contain non-spatio-temporal entities, just as Malcolm said we do not know what it would be for a person’s head (as opposed to one’s leg or arm) to be asleep.  Thus, ‘There are non-spatio-temporal objects’ must be regarded as lacking a truth-value.  So, both FBP and fictionalism must be rejected, for both take stands on the truth-value of this statement.

     There are problems with this line of reasoning.  First, if there is no fact of the matter, doesn’t a close relative of fictionalism win?  Fictionalism becomes true (modulo the worry about “stories”) if slightly modified thus:  all objects are spatio-temporal, and since mathematical theories are true only if they this isn’t so, mathematical theories aren’t true (nor are they false); rather mathematical claims are parts of consistent stories.  Second, if ‘All objects are spatio-temporal’ is intelligible, which it must be if it is to be true, as Balaguer thinks it is, mustn’t ‘Some objects are not spatio-temporal’ also be intelligible?  If P has possible worlds truth-conditions, won’t ~P, too?  Third, Balaguer himself subscribes to the semantics common to FBP and fictionalism.  But this semantics treats mathematical statements as true just in case there are certain sorts of abstract objects.  However, we are told that ‘There are abstract objects’ lacks truth-conditions.  Doesn’t this mean that ‘1+1=2’ lacks truth-conditions?  This suggests we don’t know what we’re saying when we say that 1+1=2.  And that conclusion is hard to live with.

     Platonism and Anti-Platonism in Mathematics may not persuade many Platonists or Anti-Platonists, but it will give them a lot to think about.