Intuition of Objects vs. Holism

In a previous post I wondered what the role of intuition of quasi-concrete objects like stroke-inscriptions really was in Parsons' overall epistemology of mathematics. After finally finishing the book, it seems that one clear role, at least, is Parsons' objections to holism of the sort familiar from Quine and championed in more detail for mathematics by Resnik. In the last chapter of the book Parsons makes this point:

Intuition does play a role in making arithmetic evident to the degree that it is, in that there is a ground level of arithmetic, not extending very far, that is intuitively evident. Furthermore, the objects that play the role of numbers in this low-level arithmetic can continue to do so in a more full-blooded arithmetic theory.

After noting that logical notions allow this further extension, he insists that

the role of intuition does not disappear, because it is central to our conception of a domain of objects satisfying the principles of arithmetic ... an intuitive domain witnesses the possibility of the structure of the numbers (336).

Here, then, we have a definite epistemic role for intuition of objects. It helps us to explain what is different about arithmetic, or at least the fragment of arithmetic that is closely related to these intuitions. (In chapter 7, this fragment is said to not even include exponentiation, so it fars fall short of PRA.)

While this objection to holism is quite persuasive, Parsons is at pains to emphasize how modest it really is. He offers some additional discussion of the implications for set theory, but the book seems primarily focused on what distinguishes arithmetic from other mathematical theories. It is an impressive achievement that I am sure will frame much of philosophy of mathematics for a long time.

Intuition of Quasi-Concrete Objects

In his intriguing discussion of our intuition of quasi-concrete objects Parsons focuses on a series of stroke-inscriptions (familiar from Hilbert) and their respective types. For Parsons, a perception or imagining of the inscription is not sufficient for an intuition of the type:

I do not want to say, however, that seeing a stroke-inscription necessarily counts as intuition of the type it instantiates. One has to approach it with the concept of the type; first of all to have the capacity to recognize other situations either as presenting the same type, or a different one, or as not presenting a string of this language at all. But for intuiting a type something more than mere capacity is involved, which, at least in the case of a real inscription, could be described as seeing something as the type (165).

Again, later Parons says that "intuition of an abstract object requires a certain conceptualization brought to the situation by the subject" (179).

Unfortunately, Parsons says little about what this concept is or what role it plays in the intuition of the type. The risk, to my mind, is that a clarification of the role for this concept might make the perception or imagination of the token irrelevant to the intuition. If, for example, we think of concepts along Peacocke's lines, then possessing the concept of a type is sufficient to think about the type. Some might think that acquiring the concept requires the perception or imagination of the token, but Parsons says nothing to suggest this. I would like to think this has something to do with his view that the token is an intrinsic representation of the type, but the connection is far from clear to me.

(Logic Matters has an extended discussion of the Parsons book as well.)

Quasi-Concrete Objects

One of the central claims of Charles Parsons' remarkable new book Mathematical Thought and Its Objects is that abstract objects come in two flavors: pure and quasi-concrete. Independently of the epistemological consequences of this division, we can ask how cogent and well-motivated it really is. Abstract objects are introduced using the standard negative tests: "an object is abstract if it is not located in space and time and does not stand in causal relations" (1). Quasi-concrete objects are abstract objects that have an additional feature:

What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete "representation," such that different objects of the kind in question are distinguishable by having different representations (34).

A central example of this sort of objects is expression-types. Each token represents a type and we individuate a given type by the tokens that represent it. It is clear how the token is concrete and also fairly clear how its representation of the type is intrinsic. One could say that the token stands in the relation it does to the type solely in virture of its intrinsic features.

Still, the situation is less clear with a second central case:

Although sets are in general not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership (35).

The first objection that Parsons notes to this proposal is that the representation relationship is too different because "one element can hardly represent the set as a whole". But it seems to me that a more serious objection focuses on the intrinsic nature of the membership relation. For concrete objects do not stand in any intrinsic relationship to sets. That is, a concrete object is not the member of a set solely in virtue of its intrinsic features. If we drop this intrinsic-ness test, then the motivation for carving out the quasi-concrete objects escapes me.

It is true that an impure set stands it an essential relation to its members, and so we might say that it also stands in an intrinsic relation to its members. But this it to reverse the direction of representation that Parsons originally invokes.