Plenitudinous Platonism

Plenitudinous platonism, which Balaguer takes to be the most defensible form of platonism, is the view that all logically possible mathematical objects exist, or, a bit more carefully, the view that the ordinary, actually existing objects exhaust all of the logical possibilities for mathematical objects (5, 6).

Note that Balaguer (so far as I can see) nowhere defines "mathematical object." Why, for example, isn't it logically possible for there to be a causally efficacious mathematical object or, for that matter, a mathematical object that usually wears a bowler? If those are logical possibilities, it would seem that they are counterexamples to plenitudinous platonism. What Balaguer seems to have in mind is that all logically possible objects to which we ascribe only mathematical properties (I don't know what those are, but being causally efficacious and wearing a bowler are not among them) are ordinary, actually existing objects. I'm not convinced that even that will work.

The basic idea

The basic idea is that, while we cannot know that there is someone wearing saffron robes in a Nepalese village without some correlation (contact) between the formation of that belief and the village (Field), we could know that such things were true of some Nepalese village if we knew that all possible Nepalese villages existed.

Note that we don't get knowledge about a particular Nepalese village, only that there is some Nepalese village such that …. It is worth noting that, analogously, Balaguer does not claim that the plenitudinous platonist has (thick) knowledge about particular mathematical objects, only that some mathematical objects or other have some properties. He can only get away with that weak form of mathematical knowledge by endorsing some view closely related to structuralism: mathematical knowledge about, for example, numbers, is not knowledge about particular objects but about all progressions. (He might be able to finesse that, since, after all, knowledge about all progressions is, in particular, knowledge about the progression that is actually the natural numbers, but he doesn't make that move.)

So, the idea is that if we come to know that a certain mathematical statement is possibly true, then (by plenitudinous platonism) it is actually true. We gain mathematical knowledge on the basis of

• Knowledge of what is mathematically possible
  

and

• The outright assumption of plenitudinous platonism

How can we know what is mathematically possible? His answer is that it is enough to know that it is thinly possible. Even little children know that it is possible that Santa Claus lives at the North Pole, since to know that one needn't know whether Santa exists.

Actually, we can't, because of Gödel's second incompleteness theorem, know that it is possible that mathematical objects answering to various important mathematical theories exist. Therefore, we cannot know, however Balaguer finesses the definition of possibility, whether even the natural numbers exist. We do, however, ordinarily think that we have good reason to think that the natural number exist and that, for example, Peano arithmetic is consistent, though we don't have the mathematical certainty of a proof. Therefore it is important to notice that to meet Benacerraf's challenge all we need is that we have good reason to think that various mathematical statements are true, short of the kind of absolute certainty that skeptics always seem to want.

Having said that, here is Balaguer's formulation of the plenitudinous platonist's answer to the epistemological objection (51):

1. Plenitudinous platonists can account for the fact that we can formulate mathematical theories. They do so by noting that we can formulate such theories taking them thinly.
2. Plenitudinous platonists can account for the fact that we know that many of those theories are consistent. Big gap. He attempts to fill it in %$\S$%5.
3. Using 2, plenitudinous platonists can account for the fact that, as a general rule mathematical theories we accept are consistent.

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Therefore,

• If plenitudinous platonism is true, plenitudinous platonists can account for the fact that, as a general rule, the mathematical theories we accept are theories about mathematical objects that actually exist.

The conclusion is, Balaguer claims, what we need.

Actually, that conclusion is not what we need, what we need is that

• If plenitudinous platonism is true, plenitudinous platonists can account for the fact that, as a general rule, we can know that the mathematical theories we accept are true theories about mathematical objects that actually exist.

I have a different theory, trivial platonism that also accomplishes what Balaguer claims we need.

Trivial platonism

As a general rule, the mathematical theories we accept are theories about mathematical objects that actually exist.

I claim that the following is true of trivial platonism:

• If trivial platonism is true, then trivial platonists can account for the fact that, as a general rule, the mathematical theories we accept are theories about mathematical objects that actually exist.

I leave the argument for the claim as an exercise.

What's the point? We need some answer to the question, "What does plenitudinous platonism accomplish that trivial platonism does not?"

Externalism

The internalist-externalist debate is a debate within epistemology that is quite independent of philosophy of mathematics. It is often introduced using

The KK thesis

If you know that p, then you know that you know that p.

Many internalists accept the KK thesis; externalists deny it.

Skeptics claim that we do not know many things that we ordinarily take ourselves to know by arguing that we cannot explain how it is that we know them: at some point the explanation encounters a gap.

Analogously, even if plenitudinous platonism is correct, and we could know that, say, 2+2=4, on the basis of plenitudinous platonism, most people who know that 2+2=4 have no idea about anything like plenitudinous platonism, and that includes practically all mathematicians, who have mathematical knowledge if anyone does.

The two cases are analogous and suggest a similar reply: to know something, you must have come to believe it in an appropriate way, but you needn't know even that it is an appropriate way, let alone know how to show that it is an appropriate way.

Everyone knows that there are various medium-sized dry goods in their vicinity because they can see them. If anyone knows how it is that seeing results in such knowledge, it is a small group of philosophers, and it is not clear that anyone has an adequate account of that.

It is therefore, clearly possible to know things without knowing how you know them or being able to explain how you know them. An internalist claims that to know something, you must have an internal knowledge of how you know it. An externalist claims that to know something, you must have come to know it via a reliable method, but that you do not need to know that the method is reliable, let alone why the method is reliable. (I seem to be taking the only alternative to be reliablism, but I'm not.)

Alvin Goldman discussed barn façades.

So, how do we know that there is a book on the table next to me?

The externalist account that Balaguer endorses goes like this:

I see the book, and, seeing medium-sized dry goods is a reliable method of determining that they exist, whether I or anyone else knows that.

So, how do I know that 2+2=4, according to Balaguer?

It follows from my beliefs about arithmetic that 2+2=4 and I know that my arithmetical beliefs about arithmetic are consistent (logically possible), and, the consequences of consistent beliefs are true (that is plenitudinous platonism) whether I or anyone else knows that.

Balaguer takes himself to be able to just assume plenitudinous platonism (better, takes it that plenitudinous platonism is true) in answering Benacerraf's challenge, by analogy with externalist accounts of perceptual knowledge. He makes an ad hominem point that Benacerraf says that we know how we get knowledge of physical things: we get such knowledge through perception, and Benacerraf doesn't worry about how or whether we might know that perception is a reliable means of acquiring such knowledge.

Now, analysis. What is the problem Benacerraf's work suggests? It suggests (Benacerraf and Putnam, 409) that "an acceptable semantics for mathematics must fit an acceptable epistemology," and that means an acceptable overall epistemology.

Is the following part of an acceptable epistemology?

• We just believe that perception is reliable, and that it happens that that belief is true, though we have, and can have, no justification for that belief.

It seems to me, contra Balaguer, that the answer is no: to know something, one must come to believe it via a reliable process, and one need not know why the process is reliable, but, it seems to me, it must be possible to know that the process is reliable, whether or not one knows why. An individual knower need not know that the process is reliable, but we only take a belief to be knowledge when evaluating from the outside on the basis of evidence that the process actually is reliable. (Chicken sexing.)

We can check that perception is reliable—though not in a way that would make the skeptic happy. Nothing parallel is true of the case of plenitudinous platonism.

Various Objections to Plenitudinous Platonism

We don't think our knowledge about, say, numbers, is just knowledge that some mathematical system or other exists answering to our axioms, but rather knowledge of specific objects, the numbers. The axioms follow from our knowledge of numbers, not the other way around. Balaguer's answer is that all words, including mathematical words, have an intended interpretation, and our intentions constrain the use of, for example, number words, and hence what axioms "hold of numbers." Any consistent mathematical theory will, by Balaguer's lights, be true, but that allows that we may have a not fully articulate intention in our use of some mathematical words, and the resulting beliefs will be true if the intentions lead to consistent usage. That allows that there can be genuine disputes about whether, for example, something is true of the numbers even if it is independent of our current systems of axioms. However, on Balaguer's account, such phenomena are not mathematical but "sociological." That is a claim that goes against what a lot of people think about mathematics; it is a pretty big bullet to bite, but Balaguer does.

Consistency

Plenitudinous platonism does not work unless we can determine whether a system of mathematical beliefs is consistent in a thin way, that is, without assuming that the mathematical objects described exist. But consistency, as ordinarily construed, is itself a platonistic notion: all proofs from the axioms do not result in a contradiction; all the axioms are true of some model. Balaguer's way out is to take both of those to be ex post facto mathematical analyses of an intuitive notion. He just says that consistency is an intuitive idea that is basic. Even if that is correct, how can we come to have good reason to think that we know that a theory is consistent? He has two answers:

1. Even students in an elementary logic course can do it, before any technical training.
2. Consistency is a notion that is ineliminable from mathematics. If there is no such intuitive notion, then consistency is a platonistic notion that is ineliminable from mathematics, and hence platonism is true.

Balaguer's argument that we have a nonplatonic notion of consistency is unsatisfying because he gives no account of in what that notion consists. Were I him, I would introduce a notion of consistency that does what he wants as follows: for a theory to be consistent, it is enough that it could have a model—note, not does have a model, but merely could have a model. That means that we can have reason to believe that a theory is consistent if we can imagine a model, take the theory to have a model in a thin sense. But he is already committed to our having the capacity to do that, since he thinks we initially acquire mathematical beliefs on a thin basis, so, it seems to me, he loses nothing and gains a lot by introducing nonplatonic consistency in that way.

Trivial Platonism

Plenitudinous platonism

Ordinary, actually existing objects exhaust all of the logical possibilities for mathematical objects.

Trivial platonism

As a general rule, the mathematical theories we accept are theories about mathematical objects that actually exist.

As we have seen, the trivial platonist can answer the Benacerrafian epistemological challenge to platonism in precisely the same way that the plenitudinous platonist can. Moreover, on its face, it requires fewer mathematical objects to exist (only the ones we actually accept) than does plenitudinous platonism. It is, furthermore, a defensible view: many, I believe most, philosophers would today deny that there is a sense in which philosophy is prior to the sciences and that the sciences depend on philosophy for their legitimacy—in Quine's slogan, no first philosophy. Thus, philosophy is practiced as on a par with the sciences, and philosophers are entitled to make use of results from the sciences in their analyses. But then, philosophers should, it seems, be entitled to make use of results from mathematics in their analyses (as indeed they do, employing, for example, formal logic, in various ways). But, if that is right (and I am inclined to think that it is), then the testimony of mathematicians about mathematical objects is conclusive, and trivial platonism is justified. (Maddy gives an argument pretty much along those lines.) Even a philosopher who wants to reject the testimony of mathematicians is committed to trivial platonism in accepting the sciences, since the sciences are committed to a substantial bit of mathematics, certainly enough to establish platonism.

The considerations of the previous paragraph raise the question what advantage plenitudinous platonism might be supposed to have over trivial platonism? Balaguer does not discuss trivial platonism (it is my, unpublished, invention so he hardly could), but he discusses a not dissimilar example (56–57:

Zermelo-Fraenkel (ZF) platonism

Zermelo-Fraenkel set theory is true of part of the mathematical realm.

Balaguer claims that plenitudinous platonism "is not trivial in the way that" ZF platonism is, and he offers two reasons.

1. ZF platonism "does not describe a method of mathematical belief acquisition … that is reliable in general." Actually, since the rest of mathematics can be embedded in ZF, I'm not sure he's right about that, but, more important, the criticism does not apply to trivial platonism.
2. ZF platonism "is a mathematical theory, whereas [plenitudinous platonism] is an ontological theory." Thus, he says, ZF platonism presumes some mathematical knowledge while plenitudinous platonism does not. It strikes me that there may be a reason to prefer plenitudinous platonism along the lines suggested, but his formulation is so sloppy that it is hard to make out exactly what the argument is. After all, while ZF set theory is a mathematical theory, ZF platonism, despite what Balaguer says, is clearly an ontological theory—it mentions "the mathematical realm." So, at best, the criticism should have been that ZF platonism is an ontological theory that presumes the truth of a mathematical theory. But plenitudinous platonism presumes knowledge of consistency. Isn't that mathematical knowledge too? Taking consistency to be basic doesn't help: the ZF platonist can take set theory to be basic. Trivial platonism, it seems to me, fares better than either alternative.

-- ShaughanLavine - 29 Mar 2007 - 03 Apr 2007 - 05 Apr 2007