Monday, May 19, 2008

Mathematics and causal theories of knowledge and content

Note: I am on a new faculty retreat until Friday. I will have no Internet access. My home server is going to automatically make posts every morning (unless something goes wrong), but I will only be able to respond to comments when I return. But, dear readers, feel free to engage in discussion among yourselves. And please pray for me while I am on retreat, that the Lord will give me a listening heart and a humble mind.

A standard problem for causal theories of knowledge or content is the case of mathematical knowledge. It is widely held that Platonic entities like the number seven are not causally efficacious, and hence if knowledge requires a causal relation, or if it is required for having content in a particular subject area that some beliefs in that subject area have been caused by entities in that subject area, then it seems mathematical knowledge or event contentful mathematical belief is impossible. One way out is to follow St. Augustine and make mathematical entities exist in the mind of God. Then as mental entities, they may enter into causal explanations.

But I want to offer a second suggestion. Causal priority is a special case of a more general relation: explanatory priority. If A causes B, then A's occurrence is explanatorily prior to B's. But not all instances of causal priority are instances of explanatory priority. For instance, why is it wrong to tell lies? On a Kantian story, because it fails to respect the other as an end. Explanatory relations yield the same kind of counterfactuals as causal ones, though the counterfactuals may be counterpossible: if lying respected the interlocutor as an end, it wouldn't be wrong.

But let us suppose we generalize the causal requirements for knowledge or content to corresponding explanatory requirements. Thus, if one required for knowledge that the object known should partly cause the belief, now one requires merely that the object known should partly explain the belief. If the causal requirement was more complicated, the explanatory requirement would build in this additional complexity.

This helps. Mathematical facts clearly enter into explanations. Central Limit Theorems explain the bell-shaped distributions of many natural features. Mathematical facts about algorithms explain features of the behavior of computers. And so on. Likewise, mathematical facts can explain features of our mental behavior, and in particular of why we hold certain beliefs. Now whether in fact the right mathematical facts explain our mathematical beliefs in the right way is a difficult (and partly empirical) question, but to explore that would go beyond this sketch.

Generalizing from causation to explanation also damages the argument from mathematical knowledge against naturalism. One way to put that argument would be that our mind would have to be non-natural to be "affected" by mathematical facts. But a purely physical mind would be explanatorily, though maybe not causally, "affected" by mathematical facts.

Of course a purely physical mind is a problematic concept for other reasons.


Anonymous said...

"Mathematical facts clearly enter into explanations."

They may provide useful models, that doesn't mean they provide explanations. Just because you can model something mathematically doesn't mean that the mathematics explains something.

The question of why mathematics is so helpful in modeling the physical world is an interesting one. However, it's different to say that mathematics explains the world, or causes? the world.

Clark Goble said...

Is this necessary for constructivists about mathematics? Or even formalists? Admittedly both may reject as mathematical certain proofs and even certain claims. But they can explain the causation issue.

Alexander R Pruss said...


The epistemological problem may not come up for constructivists or formalists, but of course they face other problems.


Surely Newtonian physics provided an apparent (apparent, because the physics turned out to be false) explanation of why the planets have approximately elliptical orbits. Moreover, a part of this explanation was a bunch of mathematical facts.

Anonymous said...

"Surely Newtonian physics provided an apparent (apparent, because the physics turned out to be false) explanation of why the planets have approximately elliptical orbits. Moreover, a part of this explanation was a bunch of mathematical facts."

Newtonian physics provided a way to model the orbits of the planets. They provided a system that allowed us to predict the orbits, etc. But they did not explain the orbits. Science produces models, which we can check by observation. It doesn't explain.

Planets don't follow Newton's laws. Newton's laws are just tools for describing planetary motion.

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