Monday, October 27, 2014

Two new papers on abstract (mathematical) explanation

There has not been much activity here lately, but I wanted to link to two new papers of mine that tackle the vexing issue of mathematical explanation in math and in science. I try to isolate a kind of "abstract" explanation using two cases, and explore their significance.

The Unsolvability of the Quintic: A Case Study in Abstract Mathematical Explanation
Philosophers' Imprint, forthcoming.
Abstract: This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between distinct mathematical domains. This case study highlights the importance of the conceptual, as opposed to computational, turn of much of modern mathematics, as recently emphasized by Tappenden and Avigad. The approach adopted here is contrasted with alternative proposals by Steiner and Kitcher.

Abstract Explanations in Science
British Journal for the Philosophy of Science, forthcoming.
A previous version of this paper is online here.
Abstract: This paper focuses on a case that expert practitioners count as an explanation: a mathematical account of Plateau's laws for soap films. I argue that this example falls into a class of explanations that I call abstract explanations. Abstract explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need not be causally relevant to the explanandum for its features to be explanatorily relevant. However, it remains unclear how to unify abstract and causal explanations as instances of a single sort of thing. I conclude by examining the implications of the claim that explanations require objective dependence relations. If this claim is accepted, then there are several kinds of objective dependence relations.

It remains to be seen if this "ontic" approach is the best way to go, but I believe it is a promising avenue to explore.

Saturday, September 7, 2013

New Symposium on Glock's What is Analytic Philosophy?

The most recent issue of the Journal for the History of Analytical Philosophy has just appeared with a long-awaited symposium on Glock's book on the nature of analytic philosophy. Discussants include me, Leila Haaparanta, Panu Raatikainen and Graham Stevens. Glock also offers an extended and helpful reply. This issue marks the end of the term of our first editor in chief, Mark Textor. I would like to thank him for all his work in getting this new open-access journal going. I would also like to welcome our new editor in chief, Sandra Lapointe!

Wednesday, April 24, 2013

Six Papers in Mind About Mathematical Fictionalism

The October 2012 issue of Mind (posted today here) has an extended discussion section where mathematical fictionalists of various stripes respond to Colyvan's earlier article "There is No Easy Road to Nominalism". The discussion concludes with a detailed reply by Colyvan. While I am a fan of neither Colyvan's explanatory indispensability argument nor its fictionalist critics, I look forward to reading this discussion and engaging with it soon!

The contents:

Jody Azzouni Taking the Easy Road Out of Dodge Mind (2012) 121(484): 951-965

Otávio Bueno An Easy Road to Nominalism Mind (2012) 121(484): 967-982

Mary Leng Taking it Easy: A Response to Colyvan Mind (2012) 121(484): 983-995

David Liggins Weaseling and the Content of Science Mind (2012) 121(484): 997-1005

Stephen Yablo Explanation, Extrapolation, and Existence Mind (2012) 121(484): 1007-1029

Mark Colyvan Road Work Ahead: Heavy Machinery on the Easy Road Mind (2012) 121(484): 1031-1046

Thursday, April 11, 2013

E. O. Wilson on Science and Math

Prominent biologist and science writer E. O. Wilson has a provocative Wall Street Journal opinion piece about the link between mathematical ability and scientific achievement. Perhaps the central ambiguity of his argument is illustrated by the two different titles the article seems to have. The browser heading is "Great Scientists Don't Need Math", while the actual title is "Great Scientist Does not Equal Good at Math". While the latter claim is almost trivial, the former claim seems very contentious. Of course, I am biased on this issue, having written a book arguing that mathematics makes several crucial contributions to the formulation and justification of our scientific knowledge. But setting that philosophical discussion aside, it is somewhat disturbing to find such a simplistic view of the way mathematics helps in science being presented by such a distinguished scientist.

Wilson's basic idea is that great scientists don't need to be good at math because they can always call on specialists in the relevant areas of mathematics. On Wilson's picture, the great scientists come up with great ideas, and these ideas are then implemented and tested via mathematical models. But the ideas themselves are completely non-mathematical:

Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition.
Everyone sometimes daydreams like a scientist. Ramped up and disciplined, fantasies are the fountainhead of all creative thinking. Newton dreamed, Darwin dreamed, you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper, and they take on life as real examples are sought and found.
Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics. Most of the stereotypical photographs of scientists studying rows of equations on a blackboard are instructors explaining discoveries already made. Real progress comes in the field writing notes, at the office amid a litter of doodled paper, in the hallway struggling to explain something to a friend, or eating lunch alone. Eureka moments require hard work. And focus.
Ideas in science emerge most readily when some part of the world is studied for its own sake. They follow from thorough, well-organized knowledge of all that is known or can be imagined of real entities and processes within that fragment of existence. When something new is encountered, the follow-up steps usually require mathematical and statistical methods to move the analysis forward. If that step proves too technically difficult for the person who made the discovery, a mathematician or statistician can be added as a collaborator.
Now, it is clear that some ideas that drive scientific discoveries are non-mathematical. But I do not see much evidence that most of these ideas are like that or that scientists should trust non-scientists to implement their ideas in mathematical terms. It is precisely at this stage that some of the most important and innovative work is done, and it is not clear to me how collaborations can work if one side, the scientist, doesn't understand what the other side, the mathematician, is doing.

See here for another critique of Wilson.

Friday, January 25, 2013

Two upcoming conferences on the history of analytic philosophy

This spring will see two conferences in the midwest on the history of analytic philosophy.
First, there is Early Analytic Philosophy 7, hosted by Indiana-Purdue University, Fort Wayne during the weekend of March 15th. The keynote speaker is Michael Mi of Soochow University (Taiwan). The call for papers for this conference closes on Feb. 15th. See here for more details.
Later in the spring, Indiana University will host the Society for the Study of the History of Analytic Philosophy 2 conference. It is scheduled for the weekend of May 9th. The keynote speakers are Warren Goldfarb (Harvard), Joan Weiner (Indiana) and Peter Sullivan (Stirling). The call for papers for this conference closes on March 1st. See here for more details.

Thursday, January 24, 2013

Philosophy of Science: The Central Issues, Second Edition

I was lucky enough to be involved in the new edition of the popular philosophy of science anthology Philosophy of Science: The Central Issues. It appeared late last year and, judging from its Amazon ranking, it is selling well! Martin Curd and Jan Cover did an excellent job on the first edition, but they wanted to update it by changing a few of the selections and augmenting the already considerable commentaries. Norton, the publisher, has been very generous with its review copies, so please feel free to request one if you think you might use it for a course!

Monday, January 7, 2013

Models and Simulations 4 (Special Issue of Synthese)

The whole issue is now online. Thanks again to my co-editor Marion Vorms, the authors and of course the referees.