The motivation of my research is to understand what is truth, in particular, mathematical truth, in what sense we say a statement is true, and how we decide it. Although no restriction should be put on the methodology when pursuing the truth, modern mathematical logic is no doubt providing the most reliable and yet irreplaceable framework for us to address those fundamental problems. In my view, doing mathematical logic is doing philosophy, and is a beautiful style to do philosophy.
It is natural to ask if a mathematical statement must be true or false, and, if so, how we can decide it. The first question is evident for realists such as Gödel, but the second remains a problem. Upset by the difficulties, many mathematicians “rush to the shelter of formalism” (Cohen, 1971). However, for formalists, it is still inevitable to facing the problem if the arithmetic statement Con(PA) or even Con(ZF) is true or false. Hence, there is actually no shelter from philosophy, and one is either a realist or a partial realist in the philosophy of mathematics. Gödel’s hierarchy provides standard landmarks to address one’s positions as a (partial) realist. Lower in the hierarchy, it is bounded arithmetic for strict finitists, higher in the hierarchy, it is set theory and even large cardinals for who are much more confident in classical mathematics. For the latter ones, the philosophy of mathematics is somehow reduced to the philosophy of set theory. Still, there are realists who maintain that every set theory statement is either true of false about the universe of sets, and partial realists or set theory pluralists think that some statements do not has a decidable truth value, and they can be true under this concept of sets or false under that one. It is apparently not a provable mathematical statement whether the realism or some sort of pluralism holds. However, modern development in set theory provides substantial evidences arguably both for realism and pluralism. Among those are the decidability of second-order arithmetic from large cardinals, the undecidability of third-order arithmetic, especially the continuum hypothesis from large cardinals, and more recent, Woodin’s argument that the inner model program could be achieved once for all, and Hamkins’s arguments for the set theory potentialism and multiverse view. My research in philosophy of set theory is trying to understand how Woodin’s program (or more generally, Gödel’s program) might success or fail mathematically, philosophically or sociologically, and how pluralism in set theory is supported by newly proofed results.
I am also interested in the historical issue on how analytic philosophy and mathematical logic emerged together and departed from each other afterward as we see now. I am trying to find out whether it is a historical incident or there is some serious reason behind it. This also leads me to the study of the history of ordinary language philosophy and naturalism tendency in analytic philosophy, especially the impact of late Wittgenstein and Quine. I am also holding a wide interest in Metaphysics (say, the philosophy of time) and even political philosophy (how the theories of political philosophy would look like if we eliminate the hypothesis that no man is immortal). Recently, I have been attracted by the idea of blockchain. I believe it is more about social progress than technology one. I am mostly interested in whether it can bring revolution to the scientific industry and how it might affect the sociology and philosophy of science.
Please visit Logic Fudan for more information about our logic group.