Mathematics and Scientific Representation

by Christopher Pincock
Oxford University Press, 2012 / 2014, 330 pages
reviewed by T. Nelson

It's gratifying that philosophers are so interested in science that they now have not one, but several branches of philosophy dedicated to it. Instead of philosophy of science, we now have philosophy of biology, philosophy of physics, philosophy of psychology, philosophy of math, and several more.

A recurring criticism has been that philosophers' ideas have little connection to how science is actually practiced. Scientists themselves generally pay little attention to it other than to praise Popper's idea that science can never prove a theory, and can only disprove it by falsifi­ca­tion. This is a restatement of modus tollens and is a round­about way of saying that the goal of science is to establish causal relationships. But philosophers of science, starting with Thomas “Paradigm Shift” Kuhn, routinely dismiss Popper in favor of their own theory.

This book makes the claim that mathematics makes an epistemic contribution to scientific representation. Math does this, says Christopher Pincock, by confirming a theory's accuracy, calibrating its content, giving insight, and making problems more tractable.

This is a ‘realist’ approach: science depicts the natural world through connections or ‘representations’ between science and math. Realism has mostly displaced the earlier social conception of science, which claimed that the content of science is meaningless and the goal of science is to obtain a consensus or “agreement” among scientists. Pincock says this view is now ignored by philosophers of science.

Pincock says that for mathematical representations, content is exclu­sively structural. His scheme is S↔M↔S*, where S is a concrete system, S* is a math system, and M is the structural representation between them. A scientific representation is anything that has content. Content means conditions where the representation is correct and where it is incorrect.

After nicely and logically stating these ideas, in the next three chapters Pincock presents examples of equations that make use of partial derivatives, including the Navier-Stokes equation, perturbation theory, and Prandtl's 1904 boundary layer theory in aerodynamics. Pincock likes these because they are abstract varying representations, which he says is one of two types of abstraction, the other being acausal representation, which also uses math.

Then in Chapter 7 Pincock describes cases where math failed, including Galileo's failed theory of stone columns, which broke due to faulty assumptions; the famous 1940 Tacoma Narrows bridge disaster, where equations based on small models didn't account for wind resonance; and the Black-Scholes model of options pricing, based on a heat diffusion equation, which didn't account for large-scale changes in stock prices. Long Term Capital Management used this simple formula to give investors 40% yields from 1996 to 1998, when the equation suddenly zigged when the market zagged and LTCM's solvency went to zero.

Another good point is that he is one of the few writers who spells ‘Snel's law’ correctly.

This chapter should have come before the chapters on the math so that readers could understand why making connections between science and math is not trivial—there are many ways it could fail. For example, Navier-Stokes is famously difficult to solve either analytically or numerically. Physicists in the late 19th century calculated that powered flight was impossible until Prandtl derived a simplified equation—about a year after the Wright brothers' flight. I recommend reading Chapter 7 before starting Chapter 4.

In Part II, Pincock reviews what other philosophers have said about math and science, starting with Quine and Hilary Putnam, who said that the fact that math entities are indispensable for science commits us to accepting that they exist. Pincock argues that the descriptive success of science supports his idea of platonism of math. By this Pincock means that math is a collection of abstract objects that are causally isolated from the world. He disagrees with Putnam, who thought that alternatives like modal logic could be equivalent to abstract objects. On page 194 he says Quine's argument that science needs math and therefore math must exist is also unsound because math need not be true to aid science.

This sort of thing, where people debate about whether something everybody uses really exists or not, happens a lot in philosophy. Pincock is unable to find an argument from indispensability to prove his platonism over the nominalist approach proposed by others, which says that abstract concepts are just names, so he settles for a compromise position.

This isn't a big failing. We don't study philosophy to get answers; we study it to learn how to think clearly about questions. Pincock, for instance, was a co-editor on Philosophy of Science: The Central Issues, an excellent tutorial on the field up to about 2013. (Reviewed at right.)

In Chapter 12, Pincock argues against fictionalism, which says that math is just a story. This should have been like shooting fish in a barrel (and it was), but Pincock fails to engage the “shut up and calculate” crowd which is so influential in quantum mechanics.

One difference between this book and The Central Issues is that this book is stuffed to the gills with contrived sentences using hypothetical shes and hers. Unfortunately, there are now at least 78 such candidate pronouns, none of which have any semantic content. Writers who use a she or he or she now risk being accused of heteronormative bias; somebody who insists on ze, xem, and pnkyself is bound to get offended. To be truly inclusive, he'd have to write sentences like this:

This point seems especially difficult for the objective Bayesian as pnkyself insists that degrees of belief track objective chances.

Then he'd have to count up the instances of each of the other 77 pronouns to make them all equal. Sounds like a lot of trouble to me.

jun 12, 2024

Philosophy of Science: The Central Issues, 2e

by Martin Curd, JA Cover, and Christopher Pincock, eds.
Norton, 2013, 1393 pages
reviewed by T. Nelson

This is a collection of excerpts from the important books and papers from major philosophers of science. Each section begins and ends with a discussion of what the main topics were and what they mean. The sections are:

1. Science and Pseudoscience

2. Rationality, Objectivity, and Values in Science

3. The Duhem-Quine Thesis and Underdetermination

4. Induction, Prediction, and Evidence

5. Bayesian Approaches

6. Models of Explanation

7. Laws of Nature

8. Intertheoretic Reduction

9. Empiricism and Scientific Realism

The order in which the 52 excerpts are presented is mostly chronological: one philosopher's views are presented, and each subsequent one refutes and slags off the previous one. That makes it different from science, where people focus on the discoveries instead of the individuals, and it lets us see how their culture works: unlike science, philosophy makes progress by arguments, not experimentation.

That makes it a very refined and exacting discipline. The argumen­ta­tion approach sometimes leads the authors to see things as all or none—what they would probably call conceptual monism—but what you will gain from this book, thanks to the skillful way the chapters are organized, is a solid understanding of what these philosophers were arguing about.

That doesn't always help with more specialized topics, like those discussed in The Foundations of Spacetime Physics: Philosophical Perspectives, where the authors discuss things unfamiliar even to many scientists. The focus here is on well-known theories created by big famous scientists: Newton, Einstein, Heisenberg, Copernicus, Maxwell, etc.; and topics like phlogiston, particle/wave dualism of light, charge of the electron, the caloric theory of heat, the gene theory, and Ptolemaic astronomy, rather than the daily workings of Dr Joe Shmoe scientist in the corner lab. Some of them recognize that this could distort their conclusions.

The footnotes are also very helpful. For example, they mention that early Feyerabend made solid contributions to the dialogue before he went off into crazy left-wing-uncle territory. Most of the authors reject the relativist, social construc­tionist, feminist, and post-modern­ist ideas that intruded into the field as elsewhere in academia.

One problem, of course, is that solid conclusions are hard to come by. In philosophy, unlike science, there is no empirical grounding, only argument. Thus, some philosophers can claim to have “disproved” Popper's ‘falsification’ theory, while others defend it or make subtle changes—which is exactly what they claim scientists do. The reader needs to keep this in mind: philosophy's purpose is really just to clarify how people think.

For the reader, the main benefit is that you don't have to read 52 long-winded philosophy books.

Students beware: the version that I have incorrectly states that the Second Law of Thermodynamics has been proven false.

jun 24 2024. updated jul 31 2024