Applied Mathematics and Scrutability

Also via Leiter’s blog I was perusing the Philosopher’s Annual list of the ten best papers of 2008. The paper on Mill is very interesting and I have heard a lot about belief and alief lately but what really caught my attention is Penelope Maddy’s How Applied Mathematics Became Pure.

The whole paper is really very interesting and I would highly recommend that you read the whole thing but I want to quickly discuss one of the morals that she draws from the story she tells. She says,

This story has morals, it seems to me, about how mathematics functions both in application and in its pure pursuit. One clear moral for our understanding of mathematics in application is that we are not in fact uncovering the underlying mathematical structures realized in the world; rather, we are constructing abstract mathematical models and trying our best to make true assertions about the ways in which they do and do not correspond to the physical facts. There are rare cases where this correspondence is something like isomorphism – we have touched on elementary arithmetic and the simple combinatorics of beginning statistical mechanics, and there are probably others, like the use of finite group theory to describe simple symmetries – but most of the time, the correspondence is something more complex, and all too often, it is something we simply do not yet understand: we do not know the small-scale structure of space-time or the physical structures that underlie quantum mechanics. And even this leaves out the additional approximations and accommodations required to move from the initial mathematical model to actual predictions.

I wonder if this is right if it causes problems for the kinds of scrutability claims that David Chalmers wants to defend, and which for the most part I am highly sympathetic to (of course where we differ is over whether we need to include phenomenal truths in the base truths or not…I think probably not since they can be derived just as easily as other ordinary macroscopic truths).

The problem, it seems to me, is that if this is right (i.e. if at the limit we do not end up with a unified mathematical model of the world but rather patchwork models that apply only in various respects) then which mathematical model we apply or assumption we make will crucially depend on empirical knowledge (for instance knowing that the equations for a harmonic oscillator  are a good model of a molecule’s vibration only in the region of the minimum (see page 35)). Am I missing an easy response?

I’ll have to think about it later because now I’m off to Jared Blank’s cogsci talk

The Philosophical Method

It seems to me that philosophy is distinguished from other endeavors by the method that it adopts. This is not unusual, as science is usually identified by the scientific method. But what is the philosophical method? This question is obviously controversial but I think a good case can be made that the philosophical method involves a commitment to reason and argument as a source of knowledge.

In its earliest form it was often argued that reason could discern facts about reality that were in opposition to the way that the senses revealed reality to be. This was taken as evidence that only reason was a source of knowledge (this is rationalism). So Parmenides argued that though reality appeared as a plurality that was in constant change in actuality it was a static unity that never changed. The reason that we are supposed to adopt this radical position is that positing the reality of many changing objects leads to a contradiction (that of something coming from nothing or opposites existing in the same place at the same time).

This may make it seem as though empiricists who see philosophy as continuous with the sciences (or as I prefer, see science as natural philosophy) are not really doing philosophy anymore. They are doing science, or at least advocating that they should be doing science. But this is wrong. The empiricist is using the philosophical method because their belief in empiricism is based on reasoned argument. Hume’s arguments are just as good as any rationalists; perhaps better!

The philosophical method then involves a commitment to the following:

A good argument with the conclusion that p is a reason to believe p

What counts as a good argument (or even an argument at all) will be debated but everyone agrees that if there is a good argument with the conclusion that p then there is a reason to believe that p. This also lets us see how it is that science is a type of philosophy. The scientific method presupposes the philosophical method with the restriction that good arguments come from empirical testing of theory. So though Einstein used thought experiments to come up with relativity no one believed it until there was empirical confirmation.

Even this doesn’t preclude the rationalist from agreeing that the scientific method presupposes the philosophical method. They may hold that we have to do science because we are not omnicient. But a purely rational being that new every physical fact (i.e. the position of every fundamental unit of physics and the laws that govern them) could deduce what was possible and actual a priori.

So the identification of the philosophical method with a commitment to reason and argument as a source of knowledge (or at least justification for people to believe) seems reasonably viable.