In mathematics, axioms are statements that don鈥檛 need to be proved; they are truths one can assume, such as the axioms 鈥渇or any number x, x + 0 = x鈥 or 鈥淏etween any two points is a line.鈥
Working in the field of logic, , a Klarman Postdoctoral Fellow in philosophy, studies the axiomatic method, a central methodology in mathematics whereby claims are proven from axioms.
Based in the Sage School of Philosophy in the 麻豆视频 and 麻豆视频, Walsh is tapping into Cornell鈥檚 robust logic resources in philosophy, mathematics and linguistics to accomplish three years of study under the Klarman Postdoctoral Fellowship, working closely with , assistant professor of philosophy.
Walsh is trying to get to the heart of a discrepancy between the natural theories, which arise in mathematical practice, and unnatural ones that do not.
鈥淥ne of the things we do in logic is measure and compare the strengths of theories,鈥 Walsh said. 鈥淲hen we look at all axiomatic theories鈥攊ncluding both the natural and unnatural ones鈥攁nd we compare them according to their strength, the result is very messy. We get lots of pathology. It turns out that it鈥檚 not generally possible to compare the strength of axiomatic theories.鈥
However, if the focus is shifted to just the natural theories, which are the ones that arise in practice, those pathologies disappear, Walsh said. 鈥淎ny two natural theories can be compared according to their strength, and it鈥檚 possible to rank the natural theories,鈥 he said. 鈥淭hat contrast 鈥 between the axiomatic theories in general and the natural theories 鈥 is mysterious.鈥
Walsh stands out for his ability to bridge the divide between mathematics and philosophy, Kocurek said. 鈥淥ften, researchers in logic are either really just mathematicians or really just philosophers. But James is really both: He publishes serious mathematical work while also being able to isolate the philosophically important aspects of that work.鈥
Logic is fundamentally interdisciplinary, Walsh said. At Cornell, he has been participating in a seminar in mathematics and a joint research group between linguistics and philosophy including faculty members and graduate students. He said that scholars from different disciplines can talk with him about different facets of his central project.
Walsh first heard of the mysterious discrepancy between natural theories and unnatural theories in a philosophy seminar during a year between his undergraduate and graduate studies spent as a visiting student at Harvard.
鈥淲hat was interesting to me is that people evoke this observation that the natural theories have this nice structure in the course of certain philosophical arguments,鈥 Walsh said. 鈥淚 wondered: How do we know this is true and that there isn鈥檛 some undiscovered counterexample? Has anyone proved a theorem that indicates that the claim is true?鈥
Walsh said it is widely claimed that the problem cannot be approached mathematically since 鈥渘aturalness鈥 is not a precise notion.
The research community at the University of California, Berkeley, where he completed his Ph.D., encouraged Walsh to continue thinking about the contrast between natural and unnatural theories. The Klarman Fellowship, Walsh said, allows him to focus on research he had just begun in his doctoral studies.
鈥淗e鈥檚 tackling deep and important issues at the philosophical foundations of linguistics and philosophy of language,鈥 Kocurek said. 鈥淗e鈥檚 also incredibly productive. James鈥 philosophical work makes real progress on longstanding philosophical problems.鈥
This month, Walsh received the , recognizing his thesis, 鈥淩eflection Principles and Ordinal Analysis,鈥 in which he鈥檚 made progress towards explaining 鈥渨hy proof-theoretic results tend to be robust with respect to the kinds of theories that occur naturally in mathematics.鈥
Logic appeals to Walsh because the field combines the precision and methods of math with the interests of philosophy. 鈥淵ou prove things that say something important about the scope of what we can know and what we can prove,鈥 he said. 鈥淏ut you do so with mathematical techniques and a lot of precision.鈥
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