Editing Paul Cohen

{{Short description|American mathematician (1934–2007)}}
{{hatnote|For other people named Paul Cohen, see [[Paul Cohen (disambiguation)]]. Not to be confused with [[Paul Cohn]].}}
{{Infobox scientist
| name              = Paul J. Cohen
| image             = 
| image_size        = 
| caption           = Cohen in 1966
| birth_date        = {{birth date|1934|4|2|mf=y}}
| birth_place       = [[Long Branch, New Jersey]], U.S.
| death_date        = {{death date and age|2007|3|23|1934|4|2|mf=y}}
| death_place       = [[Stanford, California]], U.S.
| field             = [[Mathematics]]
| work_institutions = [[Stanford University]]
| alma_mater        = [[University of Chicago]] ([[Master of Science|MS]], [[PhD]])
| thesis_title      = Topics in the Theory of Uniqueness of Trigonometrical Series
| thesis_year       = 1958
| thesis_url        = http://pi.lib.uchicago.edu/1001/cat/bib/4173372
| doctoral_advisor  = [[Antoni Zygmund]]
| doctoral_students = [[Peter Sarnak]]
| known_for         = [[List of forcing notions#Cohen forcing|Cohen forcing]]<br>[[Continuum hypothesis]]
| prizes            = [[Bôcher Memorial Prize|Bôcher Prize]] (1964)<br>[[Fields Medal]] (1966)<br />[[National Medal of Science]] (1967)
| footnotes         = 
}}

'''Paul Joseph Cohen''' (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the [[continuum hypothesis]] and the [[axiom of choice]] are [[independence (mathematical logic)|independent]] from [[Zermelo–Fraenkel set theory]], for which he was awarded a [[Fields Medal]].<ref name="Stanford_obit">{{cite news |last=Levy |first=Dawn |date=2007-03-28 |title=Paul Cohen, winner of world's top mathematics prize, dies at 72 |url=http://news-service.stanford.edu/news/2007/april4/cohen-040407.html |access-date=2007-10-31 |publisher=Stanford Report}}</ref><ref>{{cite news|url=https://www.nytimes.com/2007/04/02/us/02cohen.html?_r=0|author=Pearce, Jeremy|title=Paul J. Cohen, Mathematics Trailblazer, Dies at 72|newspaper=NY Times|date=2 April 2007}}</ref>

==Early life and education==
Cohen was born in [[Long Branch, New Jersey]] in 1934, into a [[Jews|Jewish]] family that had immigrated to the United States from what is now [[Poland]]; he grew up in [[Brooklyn]].<ref>Macintyre, A.J. [http://old.lms.ac.uk/newsletter/360/360_09.html "Paul Joseph Cohen"] {{webarchive|url=https://web.archive.org/web/20101225053150/http://old.lms.ac.uk/newsletter/360/360_09.html |date=2010-12-25 }}, [[London Mathematical Society]]. Accessed March 3, 2011. "Cohen's origins were humble. He was born in Long Branch, New Jersey on 2 April 1934, into a Polish immigrant family."</ref><ref name="mmp">{{citation|contribution=Paul Cohen|title=More Mathematical People|editor1-first=Donald J.|editor1-last=Albers|editor2-first=Gerald L.|editor2-last=Alexanderson|editor2-link=Gerald L. Alexanderson|editor3-first=Constance|editor3-last=Reid|editor3-link=Constance Reid|publisher=Harcourt Brace Jovanovich|year=1990|pages=42–58}}.</ref> He graduated in 1950, at age 16, from [[Stuyvesant High School]] in [[New York City]].<ref name="Stanford_obit"/><ref name="mmp"/>

Cohen next studied at the [[Brooklyn College]] from 1950 to 1953, but he left without earning his [[bachelor's degree]] when he learned that he could start his graduate studies at the [[University of Chicago]] with just two years of college. At [[Chicago]], Cohen completed his master's degree in mathematics in 1954 and his [[Doctor of Philosophy]] degree in 1958, under supervision of [[Antoni Zygmund]]. The title of his doctoral thesis was ''Topics in the Theory of Uniqueness of Trigonometrical Series''.{{sfn|Cohen|1958}}<ref name="pro">{{cite web |url=https://www.proquest.com/openview/710f92e193e41d7b2bc082f6df8f09a8/1?pq-origsite=gscholar&cbl=18750&diss=y |title=Topics in the Theory of Uniqueness of Trigonometrical Series |website=ProQuest |access-date=2024-11-02}}</ref>

In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at the [[University of Rochester]] for a year. He then spent the academic year 1958–59 at the [[Massachusetts Institute of Technology]] before spending 1959–61 as a fellow at the [[Institute for Advanced Study]] at Princeton. These were years in which Cohen made a number of significant mathematical breakthroughs. In ''Factorization in group algebras'' (1959) he showed that any integrable function on a locally compact group is the convolution of two such functions, solving a problem posed by [[Walter Rudin]]. In {{harvtxt|Cohen|1960}} he made a significant breakthrough in solving the Littlewood conjecture.<ref>{{MacTutor Biography|id=Cohen|title=Paul Joseph Cohen}}</ref>

Cohen was a member of the [[American Academy of Arts and Sciences]],<ref>{{Cite web |title=Paul Joseph Cohen |url=https://www.amacad.org/person/paul-joseph-cohen |access-date=2022-08-22 |website=American Academy of Arts & Sciences |language=en}}</ref> the United States [[National Academy of Sciences]],<ref>{{Cite web |title=Paul J. Cohen |url=http://www.nasonline.org/member-directory/deceased-members/56837.html |access-date=2022-08-22 |website=www.nasonline.org}}</ref> and the [[American Philosophical Society]].<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Paul+Cohen&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-08-22 |website=search.amphilsoc.org}}</ref> On June 2, 1995, Cohen received an [[Honorary degree|honorary doctorate]] from the Faculty of Science and Technology at [[Uppsala University]], [[Sweden]].<ref>{{cite web|url=http://www.uu.se/en/about-uu/traditions/prizes/honorary-doctorates/|title=Honorary doctorates - Uppsala University, Sweden|website=www.uu.se|access-date=21 March 2018}}</ref>

==Career==
Cohen is noted for developing a mathematical technique called [[forcing (mathematics)|forcing]], which he used to prove that neither the [[continuum hypothesis]] (CH) nor the [[axiom of choice]] can be proved from the standard [[Zermelo–Fraenkel axioms]] (ZF) of [[set theory]]. In conjunction with the earlier work of [[Kurt Gödel|Gödel]], this showed that both of these statements are [[logical independence|logically independent]] of the ZF axioms: these statements can be neither proved nor disproved from these axioms. In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory.

For his result on the continuum hypothesis, Cohen won the [[Fields Medal]] in mathematics in 1966, and also the [[National Medal of Science]] in 1967.<ref>{{cite web|url=https://www.nsf.gov/od/nms/recip_details.cfm?recip_id=80|title=The President's National Medal of Science: Recipient Details - NSF - National Science Foundation|website=www.nsf.gov|access-date=21 March 2018}}</ref> The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic, as of 2022.

Apart from his work in set theory, Cohen also made many valuable contributions to analysis. He was awarded the [[Bôcher Memorial Prize]] in [[mathematical analysis]] in 1964 for his paper "On a conjecture by [[John Edensor Littlewood|Littlewood]] and [[idempotent measure]]s",{{sfn|Cohen|1960}} and lends his name to the [[Cohen–Hewitt factorization theorem]].

Cohen was a full professor of mathematics at [[Stanford University]]. He was an Invited Speaker at the [[International Congress of Mathematicians|ICM]] in 1962 in Stockholm and in 1966 in Moscow.

[[Angus MacIntyre]] of the [[Queen Mary University of London]] stated about Cohen: "He was dauntingly clever, and one would have had to be naive or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s." He went on to compare Cohen to [[Kurt Gödel]], saying: "Nothing more dramatic than their work has happened in the history of the subject."<ref name="chronicle">{{cite news |newspaper=[[San Francisco Chronicle]] |url=http://www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2007/03/30/BAG8DOUKEG1.DTL |title=Paul Cohen -- Stanford professor, acclaimed mathematician |first=Keay |last=Davidson |date=2007-03-30 |access-date=2007-10-31}}</ref> Gödel himself wrote a letter to Cohen in 1963, a draft of which stated, "Let me repeat that it is really a delight to read your proof of the ind[ependence] of the cont[inuum] hyp[othesis]. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play."<ref>[[Solomon Feferman]], The Gödel Editorial Project: A synopsis [http://math.stanford.edu/~feferman/papers/Goedel-Project-Synopsis.pdf] p. 11.</ref>

===Continuum hypothesis===
While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he had "had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory. Indeed, they thought you had to be slightly crazy even to think about the problem."<ref name="nytimes">{{cite news |work=[[The New York Times]] |title=Paul J. Cohen, Mathematics Trailblazer, Dies at 72 |first=Jeremy |last=Pearce |date=2007-04-02 |url=https://www.nytimes.com/2007/04/02/us/02cohen.html?_r=1&oref=slogin |access-date=2007-10-31}}</ref>

{{blockquote|1=A point of view which the author [Cohen] feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the [[axiom of infinity]] is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now <math>\aleph_1</math> is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set <math>C</math> [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the [[axiom of power set|power set axiom]]. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the [[axiom schema of replacement|replacement axiom]] can ever reach <math>C</math>.<p>Thus <math>C</math> is greater than <math>\aleph_n, \aleph_\omega, \aleph_a</math>, where <math>a = \aleph_\omega</math>, etc. This point of view regards <math>C</math> as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.</p>|2={{harvtxt|Cohen|2008}}}}

An "enduring and powerful product" of Cohen's work on the continuum hypothesis, and one that has been used by "countless mathematicians"<ref name="nytimes"/> is known as [[forcing (mathematics)|"forcing"]], and it is used to construct mathematical models to test a given hypothesis for truth or falsehood.

Shortly before his death, Cohen gave a lecture describing his solution to the problem of the continuum hypothesis at the 2006 Gödel centennial conference in [[Vienna]].<ref>{{YouTube|VBFLWk7k1Zo|Paul Cohen lecture video, six parts, Gödel Centennial, Vienna 2006}}</ref>

== Death ==
Cohen and his wife, Christina (née Karls), had three sons. Cohen died on March 23, 2007, in [[Stanford, California]], after suffering from [[lung disease]].<ref>{{Cite news|last=Pearce|first=Jeremy|date=2007-04-02|title=Paul J. Cohen, Mathematics Trailblazer, Dies at 72|language=en-US|work=The New York Times|url=https://www.nytimes.com/2007/04/02/us/02cohen.html|access-date=2020-06-13|issn=0362-4331}}</ref>

==Selected publications==
*{{cite web|last=Cohen|first=Paul Joseph|date=1958|url=http://www.chronomaitre.org/cohen.pdf|title=Topics in the theory of uniqueness of trigonometrical series|access-date=2010-02-19|archive-date=2011-07-25|archive-url=https://web.archive.org/web/20110725165254/http://www.chronomaitre.org/cohen.pdf|url-status=dead}}
*{{cite journal|last=Cohen|first=Paul Joseph|title=On a conjecture of Littlewood and idempotent measures|journal=Amer. J. Math.|year=1960|volume=82|issue=2|pages=191–212|mr=0133397|doi=10.2307/2372731|jstor=2372731}}
*{{cite journal|last=Cohen|first=Paul Joseph|title=The independence of the continuum hypothesis|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=50|issue=6|pages=1143–1148|date=December 1963|doi = 10.1073/pnas.50.6.1143 |pmid=16578557|pmc=221287|bibcode=1963PNAS...50.1143C|doi-access=free}}
*{{cite journal|last=Cohen|first=Paul Joseph|title=The independence of the continuum hypothesis, II|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=51|issue=1|pages=105–110|date=January 1964|doi=10.1073/pnas.51.1.105 |pmid=16591132|pmc=300611|bibcode=1964PNAS...51..105C|doi-access=free}}
*{{Cite book|last=Cohen|first=Paul Joseph|date=2008|orig-date=1966|title=Set theory and the continuum hypothesis|location=Mineola, New York City |publisher=Dover Publications|isbn=978-0-486-46921-8|pages=151}}

==See also==
{{Portal|Biographies|Mathematics}}
*[[Cohen algebra]]
{{clear}}

==References==
{{Reflist|30em}}

==Further reading==
*[[Akihiro Kanamori]], "[http://math.bu.edu/people/aki/14.pdf Cohen and Set Theory]", ''The Bulletin of Symbolic Logic'', Volume 14, Number 3, Sept. 2008.
* {{cite journal |last=Sarnak |first=Peter |author-link=Peter Sarnak |date=December 2007 |title=Remembering Paul Cohen |journal=MAA Focus |volume=27 |issue=9 |pages=21–22 |publisher=Mathematical Association of America |location=Washington, DC |issn=0731-2040 |url=https://web.math.princeton.edu/sarnak/RememberingPaulCohen.pdf |access-date=2009-05-31}}

==External links==
{{wikiquote}}
* {{MacTutor Biography|id=Cohen|title=Paul Joseph Cohen}}
* {{MathGenealogy |id=6479|title=Paul Joseph Cohen}}
* [https://web.archive.org/web/20070613110312/http://paulcohen.org/ paulcohen.org] - a commemorative website celebrating the life of Paul Cohen
* [http://news-service.stanford.edu/news/2007/april4/cohen-040407.html Stanford obituary]

{{Set theory}}

{{Fields medalists}}
{{Winners of the National Medal of Science|math-stat-comp}}

{{Authority control}}

{{DEFAULTSORT:Cohen, Paul}}
[[Category:1934 births]]
[[Category:2007 deaths]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:American people of Polish-Jewish descent]]
[[Category:Brooklyn College alumni]]
[[Category:Fields Medalists]]
[[Category:Institute for Advanced Study visiting scholars]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:National Medal of Science laureates]]
[[Category:People from Long Branch, New Jersey]]
[[Category:Set theorists]]
[[Category:Stuyvesant High School alumni]]
[[Category:University of Chicago alumni]]
[[Category:Mathematicians from New York (state)]]
[[Category:Members of the American Philosophical Society]]