Introduction

Balaguer's argument has three parts.

1. Platonism is tenable.
2. Anti-Platonism is tenable.
3. There is no fact of the matter.

When he argues that a view is tenable, what that means is that he will argue that it can defeat the standard objections to it. It doesn't mean he has an argument to show that it is correct, only that he can show that the arguments taken to rule it out fail. The final argument is a lot like that of Benacerraf in "What Numbers Could Not Be": we have equally attractive alternatives with no means to decide between them, and so there is no fact of the matter. That is a bit ironic since the first two parts of the argument consist primarily in attempting to refute the two parts of Benacerraf's argument in "Mathematical Truth."

Balaguer agrees with Benacerraf that the arguments in "Mathematical Truth" pose serious problems for standard forms of Platonism and anti-Platonism, and so his arguments for the first two parts proceed by seeing what is wrong with standard forms of Platonism, proposing an alternative that he thinks meets Benacerraf's challenge, and then doing the same for anti-Platonism. Thus, we have the following organization:

1. Benacerraf's challenge to Platonists
2. What responses are possible
3. Contemporary versions of those responses, and why they fail
4. Balaguer's final answer

Next, the same for anti-Platonism.

Thus, we start with Benacerraf's epistemological challenge to Platonism, and then see how Balaguer thinks it wipes out all standard versions of Platonism before giving his answer. That is a bit of a rhetorical flourish on Balaguer's part: as he acknowledges in footnotes, the key ideas of his theory are not all original and some of his arguments come from the other theories, but it does make the exposition clearer.

---

Benacerraf's Epistemological Challenge

1. Human beings are objects of a certain kind (spacetime).
2. If Platonism is correct, then mathematical objects are of a different kind that has no contact with the kind that human beings are (outside of spacetime, no causal relations).
3. Contact is required for knowledge.
4. Human beings do have mathematical knowledge.
   

Therefore,

1. Platonism is not correct.

Balaguer seems to endorse a strong form of physicalism, the view that the mental is entirely reducible to the physical, especially in his formulation of the first premise. I reformulated the argument a bit to make it clear that he didn't need to do that: his argument goes through without any essential change even for the most unregenerate Cartesian dualist, since both matter and mind are of a different kind from the mathematical objects of the Platonist.

Karl asked about the notion of contact. That is the $64-dollar question. The way Benacerraf tried to resolve it was by saying that any satisfactory theory of knowledge must be a lot like a causal theory of knowledge, and on a causal theory of knowledge the requisite kind of contact is causal interaction. There is obviously no causal interaction with abstract, mathematical objects.

Aside about Balaguer's notion of abstract object

Balaguer says that he isn't going to pay much attention to the question what abstract objects are. He takes the marks of an abstract object to be that it is not located in space or time and that it has no causal powers. When those he is discussing have a different view, he just fudges and says they take mathematical objects to have different marks of abstractness.

In my view, that is the most serious deficiency of the book. The thesis of Platonism is that mathematical objects are abstract objects, and, often, that there are abstract objects. That means that the prime distinction of the work is between abstract and not abstract. (He seems to think, concrete.) But mental objects are not abstract, and so concrete is not the same as not abstract. Moreover, there are lots of abstract objects that are located in space and time (the Equator) and that do have causal powers (Microsoft). The problem isn't fatal to the interest of what Balaguer says because no one thinks that mathematical abstract objects have causal powers, and few (Maddy is the main example) think they are located in space and time.

Back to the Epistemological Challenge

Benacerraf's use of a causal theory of knowledge has led to an industry that has produced a library rack of books and articles opposing the causal theory of knowledge, with the motivation of defeating the Challenge to Platonism by making Benacerraf's argument irrelevant. Many authors of such works say that Benacerraf adopted or endorsed a causal theory of knowledge.

Balaguer bypasses that entire debate by pointing out that all that Benacerraf's argument requires is that a theory of knowledge require some form of contact with mathematical objects that we don't have. whether causal, mystical, or other. Criticizing various epistemological theories to show that Benacerraf's argument doesn't defeat Platonism can't work, because there is a prima facie plausibility to the argument that can only be defeated by giving a positive account of how we might obtain mathematical knowledge. That is, Balaguer takes the best form of Benacerraf's argument to be not one that defeats Platonism, but one that place a burden on the Platonist of showing how mathematical knowledge is possible.

Types of Response to Benacerraf's Epistemological Challenge to Platonism

Well, there are two basic types: Contact and Noncontact. Contact responses try to show that we can have contact with mathematical objects. Noncontact responses try to show that we can have knowledge about things with which we have no contact. Today, we discuss contact.

Contact Responses

There are two types of contact response:

1. Those that take human beings to be of a sort that can have contact with the usual sort of mathematical object.
2. Those that take mathematical objects to be of a sort that ordinary human beings can have contact with.

Type One: "Gödel"

The game here is that human beings have "something like a perception" of mathematical objects also. The relevant form of contact is often called mathematical intuition, and theories of this type take us to have a capacity for mathematical intuition that puts us in contact with mathematical objects. That would require, to say the least, some account of how we might do that, and no one has one. Such views are only defensible through a rather unpopular act of faith. (Katz is probably an example.)

Type Two: Early Maddy

Maddy's idea is that we see small sets of medium-sized dry goods. She gave the example, I'm baking a cake, I need 4 eggs. I check the fridge and see that I have 6. As Frege taught us, 6 is a property of something over and above the physical stuff in the fridge, of a "concept" or, Maddy's candidate, a "set." Set theory arises as a generalization of a theory that starts off being about such mundane objects as collections of eggs with which we have contact.

The view needs to avoid collapsing into the view that collections of eggs are physical objects. That might be a tenable view, but it wouldn't be platonism, our present topic, since then mathematics wouldn't be about abstract objects.

What we have perceptual contact with when we see the collection of eggs is the same stuff as what we have contact with when we see the aggregate of eggs. In fact, there is a proper class of sets one can build above the eggs. The reply is that the collection and the aggregate are structured differently, in our perception. Balaguer (33) denies that one can perceive the additional "abstract" component, the structure.

Maddy says we learn to see physical objects instead of just perceptual data. Babies can't yet do it. We grow special nerve assemblies for the purpose. Similarly, we learn to see collections of physical objects, and the collections are associated with different nerve assemblies.

Balaguer wants to say that all that we see is the "stuff," that the additional structure is in the head, not the world, and hence we don't get any additional knowledge of collections from the additional nerve assemblies. But Maddy has an argument against that that Balaguer seems to ignore: by parity of reasoning, Balaguer seems to be committed to the view that we don't see physical objects. Balaguer's argument "proves" too much.

Physicalist vs. hybrid platonism. How do we get knowledge of fully abstract sets, the ones mathematicians really use. Maddy has two replies:

• Hybrid platonism: The fully abstract ones are of the same kind as the ones with physical objects in their transitive closure. Once we know about the ones with physical objects in the transitive closure, we come to know general principles about object of that kind, including the fully abstract ones.
• Physicalist platonism: Only the sets with physical objects in their transitive closure do exist. But that is all we need.

Balaguer answers that the sets with physical objects in their transitive closure can exist whether or not the fully abstract ones do, and so no amount of knowledge about the first could give us knowledge about the second.

Extrinsic justification. The account needs an explanation of the reliability of extrinsic justification. Maddy says that that is a problem for all philosophies of mathematics. But it is, as Maddy and Balaguer fail to note, a problem for all philosophies of science as well.

-- ShaughanLavine - 22 Mar 2007