How do mathematicians think, and what does that tell us about how we both discover, and come to understand, universal and eternal truths? We propose to create an interdisciplinary hub for the study of how we explain and make sense of universal truths. Our project will be housed at Carnegie Mellon University, and will draw on the intellectual and organizational resources of the Santa Fe Institute, the Institute for Advanced Study at Princeton, and a network of collaborators across the cognitive, mathematical, and philosophical sciences.
We will address three foundational questions of how meaning emerges from the interaction of contemplation and thought about transcendental objects with human experience. A core group of three postdoctoral fellows, one in cognitive science, one in mathematics, logic, or computer science, and one in philosophy or philosophy of science will work on these questions. The research will be overseen by a project leader, and guided by a team of mentor-hosts, established and influential scholars in the cognitive, mathematical, computational, and philosophical sciences.
Proofs in Practice: how do human mathematicians discover and make sense of mathematical proofs, both as individuals and collectives? How do they explain the proofs they find, and how is that explanation influenced by the search process? How do the objects of mathematics co-evolve with the logical structure of the proofs they participate in?
Transcendental Structures: how does this psychological process interact with basic metamathematical constraints, such as those from computational complexity, and with the more abstract structures revealed to us by foundational proof-first formalisms such as dependent type theory?
Cyborg Proofs: how are these processes affected by computer-aided proof assistants and the proof-search tools of artificial intelligence? How will machines change the ways mathematicians understand mathematics and direct their attention?
