Joel David Hamkins is an American mathematician and philosopher who is O'Hara Professor of Philosophy and Mathematics at the
University of Notre Dame
The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic university, Catholic research university in Notre Dame, Indiana, outside the city of South Bend, Indiana, South Bend. French priest Edward Sorin fo ...
. He has made contributions in
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
philosophical logic,
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and philosophy of set theory (particularly the idea of the
set-theoretic multiverse), in
computability theory, and in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.
Biography
After earning a
B.S.
A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years.
The first university to admit a student to the degree of Bachelor of Science was the University ...
in mathematics at the
California Institute of Technology
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
, Hamkins earned his
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
in mathematics in 1994 at the
University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
under the
supervision of
W. Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
, 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 was a member of the doctoral faculties in Mathematics, in Philosophy and in Computer Science at the
CUNY Graduate Center
The Graduate School and University Center of the City University of New York (CUNY Graduate Center) is a public research institution and post-graduate university in New York City. Serving as the principal doctorate-granting institution of the C ...
and professor of mathematics at the
College of Staten Island
The College of Staten Island (CSI) is a public university in Staten Island, New York. It is one of the 11 four-year senior colleges within the City University of New York system. Programs in the liberal arts and sciences and professional studie ...
. He has also held various faculty or visiting fellow positions at
University of California at Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant uni ...
,
Kobe University
, also known in the Kansai region as , is a leading Japanese national university located in the city of Kobe, in Hyōgo. It was established in 1949, but the academic origins of Kobe University trace back to the establishment of Kobe Higher Comme ...
,
Carnegie Mellon University,
University of Münster
The University of Münster (german: Westfälische Wilhelms-Universität Münster, WWU) is a public research university located in the city of Münster, North Rhine-Westphalia in Germany.
With more than 43,000 students and over 120 fields of stud ...
,
Georgia State University
Georgia State University (Georgia State, State, or GSU) is a public research university in Atlanta, Georgia. Founded in 1913, it is one of the University System of Georgia's four research universities. It is also the largest institution of hig ...
,
University of Amsterdam
The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being ...
, the
Fields Institute
The Fields Institute for Research in Mathematical Sciences, commonly known simply as the Fields Institute, is an international centre for scientific research in mathematical sciences. It is an independent non-profit with strong ties to 20 Ontar ...
,
New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then- Secretary of the Treasury Albert Gallatin.
In 1832, th ...
and the
Isaac Newton Institute
The Isaac Newton Institute for Mathematical Sciences is an international research institute for mathematics and its many applications at the University of Cambridge. It is named after one of the university's most illustrious figures, the mathema ...
.
In September 2018, Hamkins moved to the
University of Oxford
, mottoeng = The Lord is my light
, established =
, endowment = £6.1 billion (including colleges) (2019)
, budget = £2.145 billion (2019–20)
, chancellor ...
to become Professor of Logic in the Faculty of Philosophy and Sir Peter Strawson Fellow in Philosophy in
University College, Oxford
University College (in full The College of the Great Hall of the University of Oxford, colloquially referred to as "Univ") is a constituent college of the University of Oxford in England. It has a claim to being the oldest college of the unive ...
. In January 2022 he moved to the
University of Notre Dame
The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic university, Catholic research university in Notre Dame, Indiana, outside the city of South Bend, Indiana, South Bend. French priest Edward Sorin fo ...
as the O'Hara Professor of Philosophy and Mathematics.
Research contributions
Hamkins research work is cited, and he gives talks, including events for the general public. Hamkins was interviewed on his research by Richard Marshall in 2013 for ''
3:AM Magazine'', as part of an ongoing interview series for that magazine of prominent philosophers and public intellectuals, and he is occasionally interviewed by the popular science media about issues in the philosophy of mathematics.
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 and introducing the lottery preparation as a general method of forcing indestructibility. 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 the modal theory known as S4.2. 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. 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.
Philosophy of set theory
In his philosophical work, Hamkins has defended a
multiverse perspective of mathematical truth, 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
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
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
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard Univ ...
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..."
Infinitary computability
Hamkins introduced with Jeff Kidder and Andy Lewis the theory of
infinite-time Turing machines, a part of the subject of
hypercomputation
Hypercomputation or super-Turing computation refers to models of computation that can provide outputs that are not Turing-computable. Super-Turing computing, introduced at the early 1990's by Hava Siegelmann, refers to such neurological inspired, b ...
, with connections to
descriptive set theory.
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.
Group theory
In group theory, Hamkins proved that every group has a terminating transfinite automorphism tower. 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
Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a mo ...
is
decidable. 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.
MathOverflow
Hamkins is the top-rated user by reputation score on
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a ...
.
Gil Kalai
Gil Kalai (born 1955) is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics ...
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."
Gil Kalai
on Hamkins's MathOverflow achievements, January 29, 2014.
References
External links
*
*Hamkins's blog
Mathematics and philosophy of the infinite
Joel David Hamkins
on MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a ...
.
*Interview at 3AM Magazine
Playing infinite chess
{{DEFAULTSORT:Hamkins, Joel David
21st-century American mathematicians
University of California, Berkeley alumni
American logicians
20th-century American mathematicians
Living people
Set theorists
Year of birth missing (living people)
Fellows of University College, Oxford
California Institute of Technology alumni
University of Notre Dame faculty
City University of New York faculty