Nature of mathematical objects...

Hey guys!

I'm writing a paper on the nature of mathematical objects that was inspired by some of the philosophy of science literature (mainly by this paper by Godfrey-Smith: http://www.people.fas.harvard.edu/~pgs/PGS_ModelsFictions-09Final.pdf). I was wondering if you guys could comment on it. At any rate, here's the argument:

The Physical Objects of this World are Mathematical Abstracta in Some Other Possible World and Vice Versa

The Argument

1. Any object x that counterfactually engages in causal relations is a physical object in some possible world w, such that x does engage in those causal relations in w.

2. The objects in physics models are some species of mathematical abstracta.

3. The objects in physics models counterfactually engage in causal relations (e.g., had frictionless inclined planes existed physically, they would have engaged in causal relations with other physical objects.)

4. If (1) and (3) then the objects in physics models are physical objects in some possible world.

5. Therefore, the objects in physics models are physical objects in some possible world (MP from (1), (3), and (4).)

6. If (2) and (5) then there is some possible world where there are physical object(s) that are also mathematical abstracta.

7. Therefore, there is some possible world where there are physical object(s) that are also mathematical abstracta (MP from (2), (5), and (6).)

8. There exists some possible world w* where some physical object x* that exists physically in this world fails to exist physically, but does exist in the physics models of that world.

9. If x* fails to exist in w* but could have existed there, x* is said to counterfactually exist in w*.

10. Therefore, x* counterfactually exists in w* (MP from (8) and (9).)

11. Therefore, there exists a possible world where the physical objects of this world are mathematical abstracta.

Possible Rejections

One could try to reject (1). However, (1) cannot be rejected because it follows from the definition of "possible world" and "counterfactual" -- i.e. any proposition phi that is true counterfactually is a contingent truth and to say that phi is true contingently is just to say that diamond-phi is true. Therefore, if it is counterfactually true that engages in causal relations then it is true that there exists a possible world such that engages in causal relations.

One could also try to reject (2) by merely claiming that things like inclined planes and ideal gases are not species of mathematical abstracta but rather another kind of object altogether. But on what basis would one make that claim? Consider a mathematical model of a frictionless inclined plane. Such an object is a model in the semantic sense in that it makes true some set of propositions in some formal system. Further, such a model will satisfy certain syntactic relations (i.e. it is axiomatizable) and will be fully describable in terms of geometry, functions, and so forth (with additional axioms from Newtonian physics.) It is difficult to imagine a property that a frictionless inclined plane has that any given mathematical object could fail to have. Furthermore, it would be question begging to merely define mathematical objects as those kinds of objects which fail to engage in causal relations. But suppose one argued that we need to decouple objects from their descriptions and that it is insufficient to merely note that frictionless inclined planes are fully mathematically describable. One can generate numerous isomorphic descriptions of the same object, and one might feel that these descriptions (and the propositions employed therein) are what is properly called mathematics. However, I argue that this does not actually solve the problem, but pushes it back a step farther. Few would deny that objects like triangles and squares are, at least, abstracta of some kind. Most would probably argue that these are to be understood as mathematical abstracta. However, there are numerous isomorphic axiomatizations of Euclidean geometry. Due to the isomorphism involved, we feel that all of these axiomatizations (and the formal systems that result) refer to the same abstract objects (i.e., the triangles, squares, and so forth.) I merely argue that frictionless inclined planes have the same status as the abstract objects of Euclidean geometry and other branches of mathematics. Thus, this worry about what is to be called mathematics is merely definitional and need not warrant criticism of premise (2).

Consider premise (3). It is difficult to see how any one could reject this premise. In any standard physics problem, one speaks of the objects in the associated models as if they engaged in causal relations:

[CENTER] "An 8.0kg box is released on a incline and accellerates down the incline at 0.30 m/s^2. Find the frictional force impeding the motion. How large is the coefficient of friction in this situation?" (page 58, problem 4.44, 3000 Solved Problems in Physics.)


[/CENTER]
In this problem, the inclined plane impedes the motion of the box (so that it doesn't just fall straight down and so that its descent down the incline is slowed from what it would have been had there been no friction acting.) The interaction between the box and the inclined plane seems to be some kind of causal relation born out between two objects. Consider, for example, a counterfactual conception of causation. Had the inclined plane not been there, the box would have fallen straight down. Further, had the inclined plane been frictionless, its motion down the incline would not have been impeded by friction. But since the inclined plane was present, and it had the property of having friction, the motion of the box was altered accordingly. Thus, had the inclined plane and the box existed physically (and presumably in the presence of gravity) then there would have been a causal relation between the box and the inclined plane. In addition, if one believed in a "conserved quantities" conception of causation, one would be happy to know that there is a conserved quantity (namely, momentum) being exchanged between the inclined plane and the box (for otherwise there would have been no force acting between the two.) Therefore, there are atleast two conceptions of causation compatible with premise (3).