Joel David Hamkins

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Joel David Hamkins
Nationality American
Fields Mathematics, Philosophy
Institutions City University of New York
Alma mater University of California, Berkeley California Institute of Technology
Doctoral advisor W. Hugh Woodin
Doctoral students George Leibman

Jonas Reitz
Victoria Gitman
Thomas Johnstone
Jason Schanker
Brent Cody
Norman Perlmutter

Erin Carmody

Joel David Hamkins is an American mathematician and philosopher based at the City University of New York. He has made contributions in mathematical and philosophical logic, particularly set theory, as well as in group theory, computability theory and in the philosophy of set theory.

Biography

After earning a B.S. in mathematics at the California Institute of Technology, Hamkins earned his Ph.D. in mathematics in 1994 at the University of California, Berkeley under the supervision of W. Hugh Woodin, with a dissertation entitled Lifting and Extending Measures by Forcing; Fragile Measurability. He joined the faculty of the City University of New York in 1995, where he is a member of the doctoral faculties in Mathematics, in Philosophy and in Computer Science at the CUNY Graduate Center and professor of mathematics at the College of Staten Island. He has also held various faculty or visiting fellow positions at University of California at Berkeley, Kobe University, Carnegie Mellon University, University of Muenster, Georgia State University, University of Amsterdam, the Fields Institute, New York University and the Isaac Newton Institute.

Research Contributions

Hamkins research work is cited,[1] and he gives talks,[2] including events for the general public.[3][4][5][6] Hamkins was interviewed on his research by Richard Marshall in 2013 for 3:AM Magazine, as part of an on-going interview series for that magazine of prominent philosophers and public intellectuals,[7] and he is occasionally interviewed by the popular science media about issues in the philosophy of mathematics.[8][9]

Set Theory

In set theory, Hamkins has investigated the indestructibility phenomenon of large cardinals, proving that small forcing necessarily ruins the indestructibility of supercompact and other large cardinals[10] and introducing the lottery preparation as a general method of forcing indestructibility.[11] Hamkins introduced the modal logic of forcing and proved with Benedikt Löwe that if ZFC is consistent, then the ZFC-provably valid principles of forcing are exactly those in S4.2.[12] Hamkins, Linetsky and Reitz proved that every countable model of Gödel-Bernays set theory has a class forcing extension to a pointwise definable model, in which every set and class is definable without parameters.[13] Hamkins and Reitz introduced the Ground axiom, which asserts that the set-theoretic universe is not a forcing extension of any inner model by set forcing. Hamkins proved that any two countable models of set theory are comparable by embeddability, and in particular that every countable model of set theory embeds into its own constructible universe.[14]

Philosophy of Set Theory

In his philosophical work, Hamkins has defended a multiverse perspective of mathematical truth,[15][16] arguing that diverse concepts of set give rise to different set-theoretic universes with different theories of mathematical truth. He argues that the Continuum Hypothesis question, for example, "is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for." (Hamkins 2012) Elliott Mendelson writes of Hamkins's work on the set-theoretic multiverse that, "the resulting study is an array of new fantastic, and sometimes bewildering, concepts and results that already have yielded a flowering of what amounts to a new branch of set theory. This ground-breaking paper gives us a glimpse of the amazingly fecund developments spearheaded by the author and...others..."[17]

Infinitary Computability

Hamkins introduced with Jeff Kidder and Andy Lewis the theory of infinite time Turing machines, a part of the subject of hypercomputation, with connections to descriptive set theory.[18]

In other computability work, Hamkins and Miasnikov proved that the classical halting problem for Turing machines, although undecidable, is nevertheless decidable on a set of asymptotic probability one, one of several results in generic-case complexity showing that a difficult or unsolvable problem can be easy on average.[19]

Group Theory

In group theory, Hamkins proved that every group has a terminating transfinite automorphism tower.[20] With Simon Thomas, he proved that the height of the automorphism tower of a group can be modified by forcing.

Infinite Chess

On the topic of infinite chess, Hamkins, Brumleve and Schlicht proved that the mate-in-n problem of infinite chess is decidable.[21] Hamkins and Evans investigated transfinite game values in infinite chess, proving that every countable ordinal arises as the game value of a position in infinite three-dimensional chess.[22]

MathOverflow

Hamkins is the top-rated[23] user by reputation score on MathOverflow.[24][25][26] Gil Kalai describes him as "one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else."[27]

References

  1. ↑ J. D. Hamkins Google Scholar profile.
  2. ↑ List of talks, from Hamkins's web page.
  3. ↑ The Span of Infinity, Helix Center roundtable, October 25, 2014. (Hamkins was a panelist.)
  4. ↑ J. D. Hamkins, plenary General Public Lecture, Higher infinity and the Foundations of Mathematics, American Association for the Advancement of Science, Pacific Division, June, 2014.
  5. ↑ A Meeting at the Crossroads - Science, Performance and the Art of Possibility, The Intrinsic Value Project, Underground Zero, New York City, July 9 & 10, 2014. (Hamkins was a panelist.)
  6. ↑ The Future of Infinity: Solving Math's Most Notorious Problem, World Science Festival, New York City, June 1, 2013. (Hamkins was a panelist.)
  7. ↑ Richard Marshall, Playing Infinite Chess, 3AM Magazine, March 25, 2013.
  8. ↑ Jacob Aron, Mathematicians Think Like Machines for Perfect Proofs New Scientist, 26 June 2013.
  9. ↑ Erica Klarreich, Infinite Wisdom, Science News, Volume 164, No. 9, August 30, 2003, p. 139.
  10. ↑ Joel David Hamkins, Small Forcing Makes any Cardinal Superdestructible, The Journal of Symbolic Logic, Vol. 63, No. 1 (Mar., 1998), pp. 51-58, DOI: 10.2307/2586586
  11. ↑ Joel David Hamkins, The Lottery Preparation, Annals of Pure and Applied Logic, Volume 101, Issues 2–3, 3 February 2000, Pages 103–146, DOI:10.1016/S0168-0072(99)00010-X
  12. ↑ Joel David Hamkins and Benedikt Löwe, The modal logic of forcing, Transactions of the American Mathematical Society, Volume 360, Number 4, April 2008, Pages 1793–1817, DOI:S 0002-9947(07)04297-3
  13. ↑ Joel David Hamkins, David Linetsky and Jonas Reitz, Pointwise definable models of set theory, The Journal of Symbolic Logic, 78, pp 139-156, 2013, doi:10.2178/jsl.7801090.
  14. ↑ Joel David Hamkins, Every countable model of set theory embeds into its own constructible universe, J. Math. Log., 13, 1350006 (2013) [27 pages] DOI: 10.1142/S0219061313500062
  15. ↑ The set-theoretic multiverse, The Review of Symbolic Logic, Volume 5, Issue 03, September 2012, pp 416-449, DOI: http://dx.doi.org/10.1017/S1755020311000359
  16. ↑ J. D. Hamkins, The multiverse perspective on determinateness in set theory, talk at the Exploring the Frontiers of Incompleteness, Harvard University, October 19, 2011. video
  17. ↑ Elliott Mendelson, Zentralblatt review Zbl 1260.03103 of J. D. Hamkins, The set-theoretic multiverse, Review of Symbolic Logic, 5, No. 3, 416-449 (2012).
  18. ↑ Joel David Hamkins and Andy Lewis, Infinite time Turing machines, The Journal of Symbolic Logic, Volume 65, Issue 02, June 2000, pp 567-604, DOI:10.2307/2586556
  19. ↑ Joel David Hamkins and Alexei Miasnikov, The Halting Problem Is Decidable on a Set of Asymptotic Probability One, Notre Dame J. Formal Logic, Volume 47, Number 4 (2006), 515-524, DOI:doi:10.1305/ndjfl/1168352664
  20. ↑ Every group has a terminating automorphism tower, Proceedings of the American Mathematical Society, Volume 126, Number 11, November 1998, Pages 3223-3226, DOI:S 0002-9939(98)04797-2
  21. ↑ Dan Brumleve, Joel David Hamkins and Philipp Schlicht, The mate-in-n problem of infinite chess is decidable, in How the World Computes, Lecture Notes in Computer Science Volume 7318, 2012, pp 78-88, DOI:10.1007/978-3-642-30870-3_9
  22. ↑ C. D. A. Evans and J. D. Hamkins, “Transfinite game values in infinite chess,” Integers, volume 14, Paper No. G2, 36, 2014.
  23. ↑ MathOverflow users, by reputation score.
  24. ↑ MathOverflow announcement of Hamkins breaking 100,000 reputation score, September 17, 2014.
  25. ↑ MathOverflow announcement of Hamkins posting 1000th answer, January 30, 2014.
  26. ↑ Erica Klarreich, The Global Math Commons, Simons Foundation Science News, May 18, 2011.
  27. ↑ Gil Kalai on Hamkins's MathOverflow achievements, January 29, 2014.

External links

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