My main research interest lies in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. I have worked particularly with forcing and large cardinals, those strong axioms of infinity, and have been particularly interested in the interaction of these two central set-theoretic concepts.  I have worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess.  My work on the automorphism tower problem lies at the intersection of group theory and set theory.  Recently, I am preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as my work on the modal logic of forcing and set-theoretic geology.  I was recently interviewed by Richard Marshall at 3:AM Magazine about my work.

My permanent position is Professor at The City University of New York, at the Graduate Center of CUNY and the College of Staten Island of CUNY.  I have held academic faculty positions at various other universities and institutions around the world:

Appointments

• The City University of New York, since 1995
• College of Staten Island, CUNY
• Professor of Mathematics
• The Graduate Center of CUNY
• Professor of Mathematics, doctoral faculty
• Professor of Philosophy, doctoral faculty
• Professor of Computer Science, doctoral faculty
• Fields Institute, Toronto, Scientific Researcher, August 2012
• Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., Visiting Fellow, March–April, June, 2012.
• New York University, Visiting Professor of Philosophy, July-December, 2011.
• University of Vienna, Kurt Gödel Research Center, Guest Professor, June, 2009.
• Universiteit van Amsterdam, Institute for Logic, Language & Computation
• NWO Bezoekersbeurs Visiting Researcher, June–August 2005, June 2006.
• Visiting Professor, April–August 2007.
• Universität Münster, Institut für mathematische Logik, Germany, Mercator-Gastprofessor, DFG, May–August 2004.
• Georgia State University, Associate Professor of Mathematics and Statistics, 2002–2003.
• Carnegie Mellon University, Visiting Associate Professor of Mathematics, 2000–2001.
• Kobe University, Japan, JSPS Research Fellow, Jan–Dec 1998.
• Univ. California at Berkeley, Visiting Assistant Professor of Mathematics, 1994–1995.

Education

• Ph.D. in mathematics, 1994, University of California at Berkeley
• C.Phil., 1991, University of California at Berkeley
• B.S. in mathematics (with honor), 1988, California Institute of Technology

1. Dear Prof Dr Joel David Hamkins,
My name is Ellena Caudwell and I am a 3rd year Undergraduate Mathematics student from the University of Exeter, UK.
I am currently involved in a group project based around D.E.Knuth’s book ‘Surreal Numbers’, and we were hoping to find examples of where this book has been used as a tool for teaching mathematics. I found an old page on the University of Amsterdam’s website explaining a course you ran in the 2nd Semester 2004/05: Surreal Numbers (http://www.illc.uva.nl/MScLogic/courses/Projects-0405-IIc/Hamkins.html) and it appears this is exactly what we were looking for.
I understand this was a long time ago, and you must be very busy, but we would be very grateful if you could answer a couple of questions to help us in our work.

Firstly, what level were the students who took part in the course, and in what way were they assessed?
Second, D.E. Knuth states in the postscript of his book that he wasn’t really trying to teach the theory of surreal numbers, but to ‘provide some material that would help to overcome… the lack of training for research work’. Therefore it is questionable how well this book helps teach surreal numbers.
1. Do you feel this course showed this book can be used successfully to teach surreal numbers? Was this your aim?
2. Do you feel it effectively shows the process of exploration and discovery of mathematical proof?

Any other thoughts or ideas on the topic would be most useful.
Thank you for your time and I look forward to hearing from you.
Ellena Caudwell

• The course was a lot of fun for all of us, and I count it as a success. I would definitely be interested in running such a course again, and I think the book works very well for this kind of course. Shorter than a regular semester course, the course was filled mostly with masters degree students, with a few PhD students, but there would be no problem running a similar course for much longer. It was a small class, with about 8 students, which I think was relevant for its success. Following the idea of Knuth that you mention, we used the book not only to learn about the surreals but also to illustrate the practice of mathematical research. In particular, the students themselves presented much of the material on a rotating schedule, filling out the ideas of the book. Thus, the practice was that they would read the next part of the book, master the material, including whatever proofs needed filling in (which is the nature of the book), and make a presentation on that topic, at which we all would ask further questions and figure things out together. In addition, as is my general practice, the students wrote term papers, on a topic chosen in discussion with me. Assessment was based on the presentations and the paper.

To answer your specific questions: (1) The book can definitely be used successfully to teach surreal numbers, and yes, this was a major part of my aim. But this book requires more work from the students than a regular textbook, laying out all the material, since it is more open-ended. (2) As a result, yes, I do feel that the book and the way we used it in that course gave a good introduction to the process of mathematical research, particularly at the masters degree level. I think it could also work well with advanced undergraduates, if they were very motivated.

2. Dear professor Hamkins,
I am looking for a reviewer to my paper in progress “$\mathcal{L}_{\omega_1\omega}$ -transfer principle in algebraic geometry”. Are you or someone you know interested in reviewing my work?
Sincerely yours,
Ali Bleybel