Department Chair, Philosophy
My main research interests lie at the intersection between philosophy of mathematics and philosophy of science. By focusing on how mathematics is applied in science, I aim to gain insights both into traditional metaphysical and epistemological questions concerning the nature of mathematics, and into methodological questions about science.
PHILOSOPHY OF MATHEMATICS
(I) METAPHYSICS of mathematics
What is mathematics about? I defend the position known as platonism, which holds that mathematical entities such as numbers, sets, and functions really exist, and that the aim of mathematics is to truly describe this realm of abstract mathematical objects.
I think that the best argument for platonism is the indispensability argument, whose original version was developed by W. V. O. Quine and Hilary Putnam in the 1950’s and 1960’s. According to this argument, we ought to believe in the existence of mathematical objects because mathematics plays an indispensable role in science. We have the same kind of reason for believing in numbers as we do for believing in theoretical physical posits such as electrons and black holes.
Over the past twenty years I have developed and defended a version of the above argument that has come to be known as the enhanced indispensability argument (or EIA, for short). According to EIA, the crucial facet of indispensability involves explanation. In other words, we ought to believe in the existence of a given entity if it is indispensable for explaining other facts that we believe. I argue that mathematics sometimes plays an indispensable explanatory role in science, and hence that we ought to add mathematical objects to our ontology. A presentation of the enhanced indispensability argument can be found in my 2023 book Indispensability (co-authored with A. C. Paseau).
One of my main case studies involves periodical cicadas. Cicadas have life-cycles of 13 or 17 years, and after spending this long period of time underground they all emerge at the same time and in massive numbers. This strategy is known as “predator satiation.” Basically the idea is to overwhelm potential predators by sheer numbers, ensuring that many cicadas survive uneaten. There is a mathematical explanation of the period lengths of cicadas that focuses on the fact that 13 and 17 are both prime numbers. It can be shown that emerging after a prime number of years minimizes the frequency of intersection with predators that are themselves periodical. Discussion of the periodical cicada example can be found in my 2005 paper Are There Genuine Mathematical Explanations of Physical Phenomena? [pdf].
(II) METHODOLOGY of mathematics
The above work has also led to a broader engagement with the topic of mathematical explanation, including whether (and, if so, how) a distinction might be drawn between explanatory and non-explanatory proofs of mathematical theorems.
- Mathematical Accidents and the End of Explanation [pdf]
- Mathematical Induction and Explanation [pdf]
Another interest of mine concerns methodological issues within mathematics, and in particular the use of 'experimental' methods in mathematics.
Philosophy of Science
(I) metaphysics of science
What is it for a scientific phenomenon to be simple? How does the simplicity of a phenomenon relate to the simplicity of the theories that we use to describe and explain it? I discuss this and related issues in my Stanford Encyclopedia of Philosophy article Simplicity [web link].
Following the award of a 'New Directions' Fellowship from the Andrew W. Mellon Foundation, I spent the 2008 - 09 academic year in the UK investigating the newly emerging field of complexity science. I am interested in several conceptual issues connected to complex systems, including definitions of complexity, the concept of emergence and its relation to complexity, the rise of the 'network approach' to the study and analysis of complex systems, and explanation in complex systems.
(II) methodology of science
I have pursued research on theory choice in science, especially appeals to simplicity and to simplicity-related principles such as Occam's Razor.
I am also interested in issues related to scientific models, especially the role of idealization and issues involving fictionalism and explanation, and the role of simulation as a distinctive methodological approach that lies between theory and observation.
At Swarthmore, I regularly teach classes in logic (Phil 12: Logic; Phil 31: Advanced Logic), in philosophy of science (Phil 18; Phil 119), and in philosophy of mind (Phil 86; Phil 118).
In addition to my permanent position at Swarthmore College, I have lived, worked, and held research or teaching positions in Philosophy at a number of different institutions around the world including in Africa (at the University of Ghana), in Britain (at the University of Cambridge and at the University of Oxford), in continental Europe (at Tartu University in Estonia), and in Asia (at Kyoto University in Japan).
Since 2006, I have been a resident scholar (along with my wife, historian of science Shelley Costa) at the Whitehall Museum House, former home of the 18th-century philosopher George Berkeley, which is situated in Middletown, Rhode Island. We have spent 8 summers residing at Whitehall, giving tours of the house, and explaining Berkeley’s philosophy of idealism (or immaterialism) to the general public.
Article: "Philosopher in Residence" on my 2006 stay at Whitehall.
My main non-academic interests center around abstract strategy games, classical music, racket sports, and cycling.
I grew up playing a lot of chess. In college, I represented Cambridge twice in the Varsity Match against Oxford, the oldest annual chess fixture in the world, and in 1998 I won the New York State Amateur Chess Championship.
Shogi is the Japanese version of chess. After discovering shogi in the mid-1990’s I mostly stopped playing chess and switched to shogi instead. In 2004 I founded the Swarthmore College Shogi Club, which is now the longest running college-based shogi club outside of Japan. I became U.S. shogi champion in 2008 and represented the U.S. at the 3rd World Shogi Forum in Japan, winning the bronze medal. Swarthmore College has twice hosted the U.S. Shogi Championship, in 2011 and 2016, and has also received many visits from professional shogi players from Japan. Since 2017 I have been the captain of the USA team in the World Shogi League, and in 2023 the USA won the League for the first time, ahead of 26 other countries, including Japan.
I play squash at the Arlen Specter National Squash Center in Philadelphia, and I am a former assistant coach of the Swarthmore College squash team. I grew up playing the more esoteric game of court tennis (also known as real tennis), the historical precursor of modern tennis.
I regularly play violin in the Swarthmore College Orchestra, and will be participating in my 25th Orchestra concert in Spring 2024. I also play piano for Swarthmore Friends Meeting, and organ on the College’s newly renovated pipe organ in Lang Concert Hall.
In a burst of adventurousness during my college years, I undertook three extended cycling trips over three consecutive summers for a combined total distance of around 10,000 miles. The first trip started in England and ended at the northern edge of the Sahara (via France, Spain, and Morocco). The second trip was around Lake Victoria in East Africa (via Kenya, Tanzania, Burundi, Zaire, and Uganda). The third trip started in France and ended in Istanbul (via Switzerland, Germany, Austria, Czechoslovakia, Hungary, Romania, Bulgaria, Greece, and Turkey). Since that time, I have continued to make international cycling trips, albeit on a much more modest scale. My main cycling these days is my daily commute to and from the Swarthmore College campus.