{{short description|American mathematician}}{{Infobox academic | name = Stanley Tennenbaum | image = Stanley Tennenbaum.jpg | birth_date = 1927 or 1928 | death_date = May 4, 2006 | death_place = Princeton, New Jersey, U.S. | alma_mater = University of Chicago | discipline = Mathematics | sub_discipline = Mathematical logic | workplaces = University of Pennsylvania<br>University of Rochester<br>New Mexico State University }}
'''Stanley Tennenbaum''' (1927/28 – May 4, 2005)<ref name=":1">{{Cite web |title=Memorial – Stanley Tennenbaum American Original |url=https://stanleytennenbaumamericanoriginal.com/memorial/ |access-date=2026-05-01 |language=en}}</ref> was an American mathematical logician. Though best remembered for Tennenbaum's theorem, despite having never published a proof of it, Tennenbaum had also studied Suslin's problem and computably enumerable sets.
Over his academic career in the 1960s and 1970s, he switched between several U.S. institutions; namely, in order, the University of Michigan, the University of Pennsylvania, the University of Rochester, the Institute for Advanced Study, and New Mexico State University.
== Biography ==
=== Early life and education (1927–1956) === Tennenbaum grew up in Cincinnati, Ohio.<ref name=":2" /> Though he rarely spoke about his childhood later in life, he had had one sister, born about the same time as he was, and he had played football in high school.<ref name=":2" />
At age 16, Tennenbaum entered the University of Chicago as an undergrad;<ref name=":2">{{Cite web |title=Bill Howard – Stanley Tennenbaum American Original |url=https://stanleytennenbaumamericanoriginal.com/bill-howard/ |access-date=2026-05-01 |language=en}}</ref> he received his PhD from the same university in 1956.<ref name=":3">{{Cite web |last=Roberts |first=Siobhan |date=2019-12-09 |title=Stanley Tennenbaum {{!}} Scholars {{!}} Institute for Advanced Study |url=https://www.ias.edu/scholars/stanley-tennenbaum |access-date=2026-05-01 |website=www.ias.edu |language=en}}</ref> He also spent a few months as one of the counselors for "Bettelheim's Orthogenic School", a school, also located in the city of Chicago, Illinois, which served autistic and schizophrenic children.<ref name=":2" /> In the fall of 1953, Tennenbaum briefly squared off with Paul Halmos, who was teaching at time about the connection between Godel's completeness theorem and cylindrical algebras, accusing the latter of "destroy[ing] logic by subsuming it into ordinary mathematics" and "being egged on by Saunders Mac Lane".<ref name=":2" />
=== Research career (1956–1972) === As a mathematician, Tennenbaum lived an itinerant lifestyle.<ref name=":4">{{Cite web |last=Landsburg |first=Steve |date=2012-04-26 |title=That Does Not Compute |url=https://www.thebigquestions.com/2012/04/26/that-does-not-compute/ |access-date=2026-05-01 |website=Steven Landsburg {{!}} The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics |language=en-US}}</ref> As of 1961, he had a research scholarship from the University of Michigan Department of Mathematics.<ref>{{Cite web |last=Erlewine |first=Michael |date=December 14, 2012 |title=LIFE TEACHERS: STANLEY TENNENBAUM |url=https://www.startypes.com/pdf/articles/Rites%20of%20Passage/LIFE%20TEACHERS%20-%20STANLEY%20TENNENBAUM.pdf |url-status=live |archive-url=https://web.archive.org/web/20160411153717/https://www.startypes.com/pdf/articles/Rites%20of%20Passage/LIFE%20TEACHERS%20-%20STANLEY%20TENNENBAUM.pdf |archive-date=April 11, 2016 |access-date=May 1, 2026}}</ref> Later on, he became a visiting professor of philosophy at the University of Pennsylvania as of 1964.<ref name=":5">{{Cite web |last=Rochester |first=Melvyn B. |date=April 7, 2006 |title=TENNENBAUM AT PENN AND ROCHESTER |url=https://www.theoryofnumbers.com/melnathanson/pdfs/nath2008-125.pdf |url-status=live |archive-url=https://web.archive.org/web/20260501043946/https://www.theoryofnumbers.com/melnathanson/pdfs/nath2008-125.pdf |archive-date=May 1, 2026 |access-date=May 1, 2026 |website=www.theoryofnumbers.com}}</ref> He then went on to teach at the University of Rochester,<ref name=":5" /> his only ever permanent position.<ref name=":4" /> Resigning after he spat on the shoes of the university's president during a faculty meeting,<ref name=":4" /> he continued to live in the same city of Rochester, New York thereafter.<ref name=":5" /> At both universities, he became a staunch friend of fellow mathematician Melvyn B. Nathanson. He was a member of the Institute for Advanced Study School of Mathematics from January to June 1967, and five years later he was a visitor to the same school from May to August 1972.<ref name=":3" /> Tennenbaum had also spent much of his life thinking about possibly solving the Riemann hypothesis.<ref name=":2" /> He was good friends with André Weil, jokingly referring to Weil (whose surname was pronounced "Vey") as "Mr. Oy Vey".<ref name=":2" />
=== Decline and death (1972–2005) === By 1972, Tennenbaum's health was in decline.<ref name=":2" /> His finances were worsening, he was showing signs of schizophrenia, and he had been using pot.<ref name=":2" /> In the 1970s, Tennenbaum was affiliated with New Mexico State University, and he had taught a popular seminar on constructive mathematics there.<ref>{{Cite web |title=Newcomb Greenleaf – Stanley Tennenbaum American Original |url=https://stanleytennenbaumamericanoriginal.com/newcomb-greenleaf/ |access-date=2026-05-01 |language=en}}</ref>
Around the spring of 1991, Tennenbaum had showed up at the University of Illinois Chicago campus while Bill Howard, one of Tennenbaum's friends, had been teaching a class titled "Finite Math for Business Students", during which some students left and one had dropped a bottle. After the class, Tennenbaum told Howard that the class was bad for the latter's mental health.<ref name=":2" />
Tennenbaum died in 2005, aged 78, sitting on a chair<ref name=":1" /> in Princeton, New Jersey<ref name=":5" /> with a phone in his hand.<ref name=":1" />
== Research == Sometime in the 1950s, Tennenbaum discovered a short proof of the irrationality of the square root of 2, which was later republished in the 2006 book ''Power''.<ref name=":7">{{Citation |last=Miller |first=Steven J. |title=Irrationality From The Book |date=2018-06-29 |url=http://arxiv.org/abs/0909.4913 |access-date=2026-05-17 |publisher=arXiv |doi=10.48550/arXiv.0909.4913 |id=arXiv:0909.4913 |last2=Montague |first2=David}}</ref> Namely, if two red squares of minimal integer sidelength could equal a larger white square of integer sidelength, by putting both red squares inside the white square and double counting, the doubly counted part (a square) must be equal in area to the uncounted part (two even smaller squares). This contradicts the minimality assumption.<ref name=":7" />
In the June 1959 issue of ''Notices of the American Mathematical Society'', Tennenbaum wrote in an abstract, without proof, that no countable nonstandard model of Peano arithmetic (PA) can be recursive.<ref name=":6">{{cite journal | author=Tennenbaum | title=Non-archimedean models for arithmetic | journal=Notices of the American Mathematical Society | volume=6 | pages=270 |url=https://www.ams.org/journals/notices/195906/195906FullIssue.pdf?adat=June%201959&trk=&cat=none&type=.pdf|issue=38|first=Stanley|date=4 March 1959}}</ref> Although Tennenbaum himself never gave a proof,<ref>{{Citation |last=Schmerl |first=James H. |title=Tennenbaum's theorem and recursive reducts |date=2011 |work=Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies |pages=112–149 |editor-last=Kennedy |editor-first=Juliette |url=https://www.cambridge.org/core/books/set-theory-arithmetic-and-foundations-of-mathematics/tennenbaums-theorem-and-recursive-reducts/1E85D80DCF61BF68846D519E5F7AC768 |access-date=2026-05-17 |series=Lecture Notes in Logic |place=Cambridge |publisher=Cambridge University Press |isbn=978-1-107-00804-5 |editor2-last=Kossak |editor2-first=Roman}}</ref> this theorem would later become known as Tennenbaum's theorem.<ref>{{Cite web |last=Smith |first=Peter |date=16 March 2011 |title=Tennenbaum's Theorem |url=https://www.logicmatters.net/resources/pdfs/TennenbaumTheorem.pdf |access-date=17 May 2026 |website=www.logicmatters.net}}</ref> In the same abstract, Tennenbaum also wrote, without proof, that if a set is representable by a first-order formula in a countable nonstandard model of PA, then the Turing degree of that set is at most the Turing degree of the operations of said model.<ref name=":6" /> As a corollary, citing work of Feferman, Tennenbaum also wrote that there is no arithmetically definable model for all true sentences of arithmetic.<ref name=":6" />
In 1962, Tennenbaum proved that the function which enumerates the complement of any maximal recursively enumerable set must grow faster than any general recursive function.<ref>{{Cite web |title=Document Zbl 0199.02504 - zbMATH Open |url=http://zbmath.org/0199.02504 |archive-url=http://web.archive.org/web/20221120035517/http://zbmath.org/0199.02504 |archive-date=2022-11-20 |access-date=2026-05-17 |website=zbmath.org |language=en}}</ref>
In 1963, Tennenbaum proved the relative consistency of the negation of Suslin's hypothesis by applying an early version of forcing;<ref name=":0">{{Cite web |last=Kanamori |first=Akihiro |title=Historical Remarks on Suslin’s Problem |url=https://math.bu.edu/people/aki/18.pdf}}</ref> the same result was independently proven by Thomas Jech in 1967 using the method of nabla-models introduced by Petr Vopěnka.<ref name=":0" /> Later, in 1965, Tennenbaum and Robert Solovay proved the relative consistency of Suslin's hypothesis.<ref name=":0" />
== Legacy == A conference was held in his honor at the City University of New York on April 7, 2006.<ref>{{Cite web |title=News and Events: Conferences {{!}} Institute for Logic, Language and Computation |url=https://www.illc.uva.nl/NewsandEvents/Events/Conferences?displayMode=month&month=04&year=2006 |access-date=2026-05-01 |website=www.illc.uva.nl}}</ref> He had also had one daughter, Susan.<ref name=":2" />
== References == <!--- See Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically --> {{Reflist}}
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{{DEFAULTSORT:Tennenbaum, Stanley}} Category:2005 deaths Category:20th-century American mathematicians Category:21st-century American mathematicians
Category:1920s births Category:University of Pennsylvania faculty Category:University of Rochester faculty Category:New Mexico State University faculty