Knowledge of Mathematical Objects without Contact

Wright and Hale and Frege

D

The direction of line a is identical to the direction of line b.

L

Line a is parallel to line b.

(From p. 36.)

The argument rests on the claim that knowledge of L is unproblematic, and the D is equivalent to L. It follows that knowledge of D is unproblematic, and hence that there are abstract objects, namely, directions. Note that for the purposes of the argument lines are not taken to be abstract; they may as well be unproblematic concrete objects.

Balaguer is even willing to concede that knowledge of D is unproblematic. His challenge is that it would be problematic if it were knowledge about mathematical objects. So one needs to explain why D is unproblematic.

Wright takes L to be unproblematic because it is not about mathematical objects, and Hale adds that it is a priori. I'll just discuss Wright. Balaguer must agree about L that there is no further problem about abstract objects, since he is prepared to agree that L mentions no abstract objects. Thus, Wright will have provided the needed justification of D if he can show that D is equivalent to L without presuming the existence of abstract objects. Thus, Balaguer's argument that Wright has no argument for the existence of directions is wrong. If he wants to establish that Wright has not shown that there are directions, he must argue against Wright's claim that D follows from L on grounds that do not presume the existence of abstract objects. He doesn't even mention the argument.

What I just said is entirely ad hominem. Lots of other people have argued that Wright's (and Hale's) argument from L to D smuggles in abstract objects. I'm not sure how much that really helps Balaguer, since, as we'll see below, Balaguer thinks we are entitled to just assume that abstract objects exist; that the problem is determining what properties they have given that they exist.

No-Contact Mathematical Intuition

Parsons (types), Katz, Steiner (abstraction), Resnik (abstraction), Shapiro (abstraction), Gödel.

Balaguer claims that no version provides an explanation of why the intuition is reliable. "It may as well be a process of creative writing or dreaming up, instead of anything so respectable-sounding as abstraction" (39).

Intuition does yield knowledge about mathematical objects in a thin (Santa Claus) sense.

These theories, even if true, merely postpone the epistemological question.

By his own lights, Balaguer's objection is merely skeptical. What he would need to object is that we have no reason to believe that, given that mathematical objects exist, intuition reliably informs us about their nature.

Holism

Quine, Putnam, Steiner, Resnik.

Balaguer presents the argument from confirmational holism: Our best theories of the physical world, which are well confirmed theories, include mathematics. Theories are confirmed as a whole, and therefore the mathematics in our physical theories is as well confirmed as the physics.

He then argues against confirmational holism

But the real indispensability argument makes no use of confirmational holism. (Dieveney.) The argument is rather that if one is committed to a theory, one is committed to everything the theory says. To say, I believe theory T, and theory T entails that there are mathematical objects, but I don't believe that there are mathematical objects is intellectually dishonest. Confirmation, holistic or not, is a red herring. To believe a theory must be to believe everything it entails. If you believe some parts of a theory but not others, you need to express what you believe without claiming to believe theory T.

Now, the existence of mathematical objects follows from the claim that we are committed to physical theories that entail the existence of mathematical objects, and that the mathematical objects are ineliminable from those theories. But that seems to be the case.

Balaguer also claims that "mathematicians know that their theories are true when they first construct them, that is, before they're used in applications" (40). But all mathematical theories are applied when they are first constructed: pure mathematics is mathematics that is, when it is first constructed, only applied to other mathematics. If the other mathematics is indispensable for physical theories, or is applied to mathematics that is, then the new mathematics inherits, at least to some extent, the warrant of the other mathematics. At the very least, that shows that Balaguer's argument is too fast.

Necessity

Katz and Lewis.

Field's objection: "'there is still a problem of explaining the actual correlation between our' mathematical beliefs and the mathematical facts" (42). That is most interesting in the case of mathematical truths that are independent of the axioms.

what anti-platonists will demand is an explanation of how we could know that there is an object in the mathematical realm that answers to this description ['successor of 3']. (43)

Balaguer's answer: FBP.

The game rests on giving a "well-motivated account of what metaphysical necessity consists in" (44). Balaguer doubts that that can be done.

Structuralism

Resnik and Shapiro.

Full-Blooded Platonism

A problem: the relevant background notion of consistency or coherence is itself platonistic. (46, posed by Balaguer for structuralism, not FBP.)

-- ShaughanLavine - 27 Mar 2007