What Abstract Objects Are

Slide 1: Thanks

Thanks for inviting me.

Slide 2: Abstractness Is Vagueness

The thesis of this talk is that abstract objects and vague objects are but two species of a common genus: Incomplete Objects.

Slide 3: Abstractness Is Vagueness

The thesis of this talk is that abstract objects and vague objects are but two species of a common genus: Incomplete Objects.

The thesis is not intended to commit me to the existence of incomplete objects as a metaphysical thesis. I am loosely calling the referents of incompletely referring terms incomplete objects.

Slide 4: Abstractness Is Vagueness

The thesis of this talk is that abstract objects and vague objects are but two species of a common genus: Incomplete Objects.

The thesis is not intended to commit me to the existence of incomplete objects as a metaphysical thesis. I am loosely calling the referents of incompletely referring terms incomplete objects.

In many cases, vague reference is not a defect of a referring term, but a carefully crafted feature, witness Tappenden's example of the brownrate: the rate at which schools in the United States were to be desegregated in accordance with the Brown v Board of Ed Supreme Court Decision .

Slide 5: Abstractness Is Vagueness

In many cases, vague reference is not a defect of a referring term, but a carefully crafted feature, witness Tappenden's example of the brownrate: the rate at which schools in the United States were to be desegregated in accordance with the Brown v Board of Ed Supreme Court Decision .

I now turn to making this idea of a referring term that permits increases in precision more precise.

Slide 6: Completeable Reference

In the standard case, precise reference, one can simply give the referent of a term.

Slide 7: Completeable Reference

In the standard case, precise reference, one can simply give the referent of a term.

It is not much of a stretch to think of specifying the reference of an incompletely referring term, in the simplest case, by giving a collection of things any one of which might be taken to be, or, perhaps better, no one of which is clearly not, the referent of the term. One takes the reference to be incomplete in that it is only specified that the reference is to something in the collection. For example, the reference of "Mt. Everest" might be to a collection of lumps of earth and stone, all of which include the peak and a considerable amount of other stuff, but that differ on exactly where the mountain ends and the foothills begin.

Slide 8: Completeable Reference

The reference of "Mt. Everest" might be to a collection of lumps of earth and stone, all of which include the peak and a considerable amount of other stuff, but that differ on exactly where the mountain ends and the foothills begin. The simple picture I have just suggested is too simple because our rules for further stipulating the references of incompletely referring terms (or, for that matter, predicates with incompletely determined extension) constrain each other in nontrivial ways: If a piece of dirt is on Mt. Everest, it cannot also be on Mount Lhotse, and vice versa. We need something more sophisticated.

Slide 9: Holistic Completeable Reference

Given the potentially complex and intertwined interactions between our possible further stipulations on incompletely referring terms and predicates with incompletely specified extension, I see nothing for it but to regard such specifications as being made for the whole language at once. Thus, instead of associating each term and predicate with its possible further specifications, I shall take the whole language to be incomplete, and associated with a collection of possible future specifications. It is possible to give a formal model, and I shall do so in a talk to the Vagueness Seminar.

Slide 10: Types and tokens

The example I have been pursuing so far—Mount Everest, with its less than fully determined boundaries—is one of a vague object. I now want to turn to an example of a somewhat abstract object: a letter of the alphabet.

Slide 11: Types and tokens

The example I have been pursuing so far—Mount Everest, with its less than fully determined boundaries—is one of a vague object. I now want to turn to an example of a somewhat abstract object: a letter of the alphabet.

How many letters in the word " aardvark "? Well, eight tokens, but only five types.

Slide 12: Types and tokens

The example I have been pursuing so far—Mount Everest, with its less than fully determined boundaries—is one of a vague object. I now want to turn to an example of a somewhat abstract object: a letter of the alphabet.

How many letters in the word " aardvark "? Well, eight tokens, but only five types. That is the traditional fast and loose answer among philosophers, though it poses some problems: how can there be eight letter tokens, not in a particular token of the word "aardvark," but in the word type?

I'll come back to that later, if there is time, but for now, what sort of object is a letter type?

Slide 13: Types and tokens

The example I have been pursuing so far—Mount Everest, with its less than fully determined boundaries—is one of a vague object. I now want to turn to an example of a somewhat abstract object: a letter of the alphabet.

How many letters in the word " aardvark "? Well, eight tokens, but only five types. What sort of object is a letter type?

A letter is certainly an incomplete object, it has no size, no location in physical space, and so on. Moreover, and this is the main point for my purposes, it is incomplete in a way that permits further temporary stipulation: A first-grade teacher, teaching the alphabet, can say, holding up a suitable inscription, this is the letter A, and be perfectly correct in so saying.

Slide 14: Types and tokens

A letter is certainly an incomplete object, it has no size, no location in physical space, and so on. Moreover, and this is the main point for my purposes, it is incomplete in a way that permits further temporary stipulation.

Abstract objects like types and vague objects are incomplete in much the same way and for much the same reason: to permit further stipulation as needed in particular circumstances. The main difference is not metaphysical but pragmatic: If an object is vague in some respect, it may be possible to further stipulate what it is in that respect on a permanent basis; if an object is abstract in some respect, while it is possible to further stipulate what it is in that respect temporarily for some purpose or other, it is not possible to permanently stipulate what it is in that respect.

Slide 15: Abstract mathematical objects

The abstract objects of modern mathematics are exactly the incomplete objects that

1. admit of no further permanent stipulations—that is what makes them abstract,
   and
2. have no referential constraints on how they may be further specified, only what Kit Fine calls "penumbral" constraints.

Slide 16: Abstract mathematical objects

The abstract objects of modern mathematics are exactly the incomplete objects that admit of no further permanent stipulations—that is what makes them abstract, and have no referential constraints on how they may be further specified, only what Kit Fine calls "penumbral" constraints.

The value of such objects is precisely that they may be interpreted on any domain of objects as a way of imposing the penumbral constraints (that is, their "structure") on that domain. The lack of referential constraints makes such objects play a purely structural role in their applications, and so they provide an account of mathematical objects that meet the desiderata of the mathematical structuralists, an account that explicates the notion of structure instead of taking it to in some way be basic.

Slide 17: Abstract mathematical objects

The value of the abstract objects of modern mathematics is precisely that they may be interpreted on any domain of objects as a way of imposing the penumbral constraints (that is, their "structure") on that domain. The lack of referential constraints makes such objects play a purely structural role in their applications, and so they provide an account of mathematical objects that meet the desiderata of the mathematical structuralists, an account that explicates the notion of structure instead of taking it to in some way be basic.

It is not my purpose in this talk to develop this account further or to defend its correctness. I have a prior worry:

Slide 18: Motivation

What I am doing is attempting to argue that such comparatively clear and respectable entities as the natural numbers are relevantly similar to such problematic ones as heaps. It would not be uncharitable of you to think that
I must be out of my mind

Slide 19: Structuralism

What I am doing is attempting to argue that such comparatively clear and respectable entities as the natural numbers are relevantly similar to such problematic ones as heaps. Why?

There is a common view, to which I more or less subscribe, that the objects of modern mathematics are purely structural, that they are something like rôles in patterns or structures, determined only up to isomorphism.

Slide 20: Structuralism

What I am doing is attempting to argue that such comparatively clear and respectable entities as the natural numbers are relevantly similar to such problematic ones as heaps. Why?

There is a common view, to which I more or less subscribe, that the objects of modern mathematics are purely structural, that they are something like rôles in patterns or structures, determined only up to isomorphism.

That is all very well, but what are such objects? However one cashes out what a rôle or a position in a pattern is—and that is no small problem—one is apparently faced with relying on some notion of a structure or pattern.

Slide 21: Structuralism

That is all very well, but what are such objects? However one cashes out what a rôle or a position in a pattern is—and that is no small problem—one is apparently faced with relying on some notion of a structure or pattern. There are three main stratagems in the literature.

1. Structural accounts of structure. Such accounts may yield an argument for self-consistency, but are otherwise uninformative. One can bite the bullet on that: explanation must stop somewhere, mathematics is not in need of a metaphysical foundation any more than is any other subject of knowledge, and so forth.

Slide 22: Structuralism

That is all very well, but what are such objects? However one cashes out what a rôle or a position in a pattern is—and that is no small problem—one is apparently faced with relying on some notion of a structure or pattern.

There are three main stratagems in the literature.

1. Structural accounts of structure.
2. It's not my department. Mathematical structures are kinds of patterns, and the question what patterns are, while it is surely interesting and important, is beyond the scope of the philosophy of mathematics.

Slide 23: Structuralism

There are three main stratagems in the literature.

1. Structural accounts of structure.
2. It's not my department. Mathematical structures are kinds of patterns, and the question what patterns are, while it is surely interesting and important, is beyond the scope of the philosophy of mathematics.

This would be absolutely unexceptionable, if only we had an independent and complete philosophical account of what patterns are. Shifting a burden of explanation doesn't meet it.

Slide 24: Structuralism

1. Structural accounts of structure.
2. It's not my department.
3. Structures are metaphysically basic. They form a fundamental metaphysical category. Now stop pestering me with questions.

Slide 25: Structuralism

1. Structural accounts of structure.
2. It's not my department.
3. Structures are metaphysically basic.
   Though each of these views is internally coherent, my attitude, following Charles Parsons, is that structures are mathematical entities. Even if one can successfully reduce all questions about mathematical entities to questions about mathematical structures and rôles in them, structures are themselves mathematical entities, and structuralism leaves us with the residual problem of what structures are. Call that an additional view if you like:
4. Structures are mathematical entities.

Slide 26: Motivation

I hope what I just said about structuralism has made my attempt to assimilate mathematical objects to vague objects seem less ill-advised. Mathematical objects, if anything resembling any of the versions of structuralism I've mentioned is correct, are sui generis. If they are incomplete objects much like heaps and mountains they are not.

Slide 27: Motivation

I hope what I just said about structuralism has made my attempt to assimilate mathematical objects to vague objects seem less ill-advised. Mathematical objects, if anything resembling any of the versions of structuralism I've mentioned is correct, are sui generis. If they are incomplete objects much like heaps and mountains they are not. My account is intended to preserve a central intuition behind structuralism—mathematical objects aren't anything in particular other than themselves (thus, numbers are not particular sets or rulers of ancient Rome) without relegating them to sui generis status.

Slide 28: Motivation

I hope what I just said about structuralism has made my attempt to assimilate mathematical objects to vague objects seem less ill-advised. Mathematical objects, if anything resembling any of the versions of structuralism I've mentioned is correct, are sui generis. If they are incomplete objects much like heaps and mountains they are not.

All well and good, but why all the fuss? To answer a challenge of Benacerraf: Any acceptable philosophy of mathematics must provide both coherent semantics—an account of why mathematical truths are true in some ordinary sense of the word true— and a coherent epistemology—an account of how our reasons for holding mathematical beliefs are connected with the truth of those beliefs.

Slide 29: Motivation

If mathematical objects are nothing more than ordinary incomplete objects, then statements about them are true for the same kinds of reasons that statements about other incomplete objects are true:
Compare

• 2 is even

to

• Mountains are tall

Slide 30: Motivation

If mathematical objects are nothing more than ordinary incomplete objects, then statements about them are true for the same kinds of reasons that statements about other incomplete objects are true
and we know that statements about them are true in the same way that we know that statements about other incomplete objects are true. Compare how you know that

• 2 is even

to how you know that

• Mountains are tall

Slide 31: Conclusion

There is, of course, a substantial burden for this program to meet: I need an acceptable semantics for incomplete objects. That will be the topic of my talk to the vagueness seminar.

-- ShaughanLavine - 09 Mar 2005