PSA Symposium: Applied Mathematics and the Philosophy of Science

As the final version of the PSA program is finally online, it is about time for me to promote the symposium that I will be in. Here are the details:

Applied Mathematics and the Philosophy of Science
PSA 2008 Symposium
Parallel Session 6: Saturday, November 8, 9-11:45 am
Room CCA (Conference Center A)
Chair: Paul Teller

Proposed schedule:
9:00-9:30 Christopher Pincock, “The Value of Mathematics for Scientific Confirmation”
9:30-10:00 Stathis Psillos, “What If There Are No Mathematical Entities? Lessons for Scientific Realism”
10:00-10:20 discussion
10:20-10:25 break
10:25-10:55 Mark Wilson, “Leibniz’ ‘Possibilities’ and Our Own”
10:55-11:25 Robert Batterman, “Essential Models and Explanatory Mathematics”
11:25-11:45 discussion

Abstract: This symposium will explore the relevance of philosophical reflection on the details of applied mathematics for current debates in the philosophy of science along four dimensions: (i) scientific representation, (ii) confirmation of scientific theories, (iii) idealization and scientific explanation, (iv) scientific realism. In all four cases the participants aim to show that a clear focus on the contribution that mathematics makes to science sheds new light on traditional positions in the philosophy of science. In some cases the viability of a philosophical view is called into question, while in others a standard thesis receives new support. The symposium is motivated by the realization that the philosophy of mathematics has changed considerably in the last twenty years and the hope that philosophers of science can benefit from this transformation.

For those of you who can't be there, here is a link to a rough draft of my paper. Constructive comments appreciated! Update (April 11, 2009): I have removed this old draft and hope to repost a final version sometime this spring.

Downward Causation in Fluids?

Bishop claims to have found a case of downward causation in physics based on the existence of what is known as Rayleigh-Benard convection in fluids. In the simplest case we have a fluid like water that is heated from below. What can result, as this image from Wikipedia shows, is a series of cells, known as Benard cells, where the dominant large-scale structure is fluid flowing in interlocking circular patterns.

The claim is that these patterns require new causal powers over and above what can be ascribed to the smaller scale fluid elements: "although the fluid elements are necessary to the existence and dynamics of Benard cells, they are not sufficient to determine the dynamics, nor are they sufficient to fully determine their own motions. Rather, the large-scale structure supplies a governing influence constraining the local dynamics of the fluid elements" (p. 239).

There is no doubt that this is an interesting case that should receive more scrutiny. As with McGivern's article, the tricky interpretative question is how closely we should link the workings of the mathematical model to the genuine causes operating in the system. Bishop's conclusion seems based on taking the representation of fluid elements very seriously, but I am not sure that the link between the representation and reality at this level is well enough understood. Still, I would concede his point that many features of downward causation from philosophical accounts appear in this example.

Two Papers on Modeling

I have recently posted two preprint versions of papers that approach modeling from (hopefully) complementary directions. The first, "Modeling Reality", argues that model autonomy and model pluralism are consistent with a limited form of scientific realism. The second, "Towards a Philosophy of Applied Mathematics", argues that applied mathematics is a distinct area of mathematics that deserves further scrutiny by philosophers of mathematics.

Comments are welcome as both papers are part of my larger project!

Math Education Humor

The Onion offers some math education humor which can probably apply equally well to some introductory logic courses out there.

Maddy on Applied Mathematics

In her recent Review of Symbolic Logic article “How Applied Mathematics Became Pure” Maddy offers a rich discussion of the various changes that have occurred in mathematics, science and their relationship. While I am generally sympathetic to her main conclusion that mathematics as it is practiced today has its own science-independent standards of success, I am surprised by her pessimistic conclusions concerning the sort of project that I am engaged in for applications.

Maddy first summarizes the history of scientists who took a modest perspective on the degree to which their mathematical representations were capturing ultimate causal mechanisms:

we have seen how our best mathematical accounts of physical phenomena are not the literal truths Newton took them for but freestanding abstract models that resemble the world in ways that are complex and sometimes not fully understood (p. 33).

She continues that

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 (p. 33).

After surveying some successful accounts of particular cases where we can make these distinctions, she concludes

Given the diversity of the considerations raised to delimit and defend these various mathematizations, anything other than a patient case-by-case approach would appear singularly unpromising (p. 35).

But nothing in the article precludes a useful classification of these sorts of successes into kinds. Of course, such a classification must start with individual cases. This would be just the beginning, especially if we could find patterns across cases. Indeed, it seems like this is just what applied mathematicians are trained to do, as a review of any applied mathematics textbook would reveal.

I grant that this is just a promissory note at this stage, but the attempt to understand and classify successful cases of mathematical modeling is really just another instance of the naturalistic methods that Maddy has applied to set theory.

McGivern on Multiscale Structure

Back in July I made a brief post on multiscale modeling from the perspective of recent debates on modeling and representation. So I was very happy to come across a recent excellent article by McGivern on “Reductive levels and multi-scale structure”. McGivern gives a very accessible summary of a successful representation of a system involving two time scales, and then goes on to use this to question some of the central steps in Kim’s influential argument against nonreductive physicalism.

To appreciate the central worry, we need the basics of his example. McGivern discusses the case of a damped harmonic oscillator, like a spring suspended in a fluid, where the damping is given as a constant factor of the velocity. So, instead of the simple linear harmonic oscillator

my’’ + ky = 0

we have

my’’ + cy’ + ky = 0

Now this sort of system can be solved exactly, so a multiscale analysis is not required. Still, it is required in other cases, and McGivern shows how it can lead to not only accurate representations of the evolution of the system but also genuine explanatory insight into its features. In this case, we think of the spring evolving according to two time scales, t_s and t_f, where t_f = t and t_s = εt and ε is small. Mathematical operations on the original equation then lead to

y(t) ~ exp(-t_s/2)cos(t_f)

where ~ indicates that this representation of y is an approximation (essentially because we have dropped terms that are higher-order in ε). McGivern then plots the results of this multiscale analysis against the exact analysis and shows how closely they agree.

McGivern’s argument, then, is that the t_s and the t_f components represent distinct multiscale structural properties of the oscillator, but that they are not readily identified with the “micro-based properties” championed by Kim. McGivern goes on to consider the reply that these are not genuine properties of the system, but merely products of mathematical manipulation. This seems to me to be the most serious challenge to his argument, but the important point is that we need to work through the details to see how to interpret the mathematics here. I would expect that different applications of multiscale methods would result in different implications for our metaphysics. I hope that this paper will be studied not only by the philosophy of mind community, but also by people working on modeling. If we can move both debates closer to actual scientific practice, then surely that will be a good thing!

Norton on Pincock

Over at The Last Donut, John Norton offers a very generous summary of my recent lunchtime talk at the Pittsburgh Center for the Philosophy of Science. I hope to have a revised draft online soon!

Intuition of Objects vs. Holism

In a previous post I wondered what the role of intuition of quasi-concrete objects like stroke-inscriptions really was in Parsons' overall epistemology of mathematics. After finally finishing the book, it seems that one clear role, at least, is Parsons' objections to holism of the sort familiar from Quine and championed in more detail for mathematics by Resnik. In the last chapter of the book Parsons makes this point:

Intuition does play a role in making arithmetic evident to the degree that it is, in that there is a ground level of arithmetic, not extending very far, that is intuitively evident. Furthermore, the objects that play the role of numbers in this low-level arithmetic can continue to do so in a more full-blooded arithmetic theory.

After noting that logical notions allow this further extension, he insists that

the role of intuition does not disappear, because it is central to our conception of a domain of objects satisfying the principles of arithmetic ... an intuitive domain witnesses the possibility of the structure of the numbers (336).

Here, then, we have a definite epistemic role for intuition of objects. It helps us to explain what is different about arithmetic, or at least the fragment of arithmetic that is closely related to these intuitions. (In chapter 7, this fragment is said to not even include exponentiation, so it fars fall short of PRA.)

While this objection to holism is quite persuasive, Parsons is at pains to emphasize how modest it really is. He offers some additional discussion of the implications for set theory, but the book seems primarily focused on what distinguishes arithmetic from other mathematical theories. It is an impressive achievement that I am sure will frame much of philosophy of mathematics for a long time.