Editing Emil Leon Post

{{Short description|American mathematician and logician (1897 – 1954)}}
{{Infobox scientist
| name = Emil Leon Post
| image = Emil Leon Post.jpg
| birth_date = February 11, 1897
| birth_place = [[Augustów]], [[Suwałki Governorate]], [[Congress Poland]], [[Russian Empire]]<br>(now Poland)
| death_date = {{death date and age|1954|4|21|1897|2|11}}
| death_place = New York City, U.S.
| residence = 
| nationality =
| field = [[Mathematics]], [[logic]]
| work_institution = [[Princeton University]]
| alma_mater        = [[City College of New York]] (B.S., 1917)<ref name="Urquhart">Urquhart (2008)</ref><br>[[Columbia University]] (A.M. 1918, Ph.D. 1920)<ref name="MacTutor"/>
| thesis_title      = Introduction to a General Theory of Elementary Propositions
| thesis_url        = <!--(or  | thesis1_url  =   and  | thesis2_url  =  )-->
| thesis_year       = 1920
| doctoral_advisor  = [[Cassius Jackson Keyser]]
| doctoral_students =
| known_for  = [[Post–Turing machine|Formulation 1]]<br>[[Post correspondence problem]]<br>Completeness-proof of ''[[Principia Mathematica|Principia]]''{{'s}} [[propositional calculus]]<br>[[Post's inversion formula]]<br>[[Post's lattice]]<br>[[Post's theorem]]
}}

'''Emil Leon Post''' ({{IPAc-en|p|oʊ|s|t}}; February 11, 1897 – April 21, 1954) was an American [[mathematician]] and [[logician]]. He is best known for his work in the field that eventually became known as [[computability theory]].

==Life==
Post was born in [[Augustów]], [[Suwałki Governorate]], [[Congress Poland]], [[Russian Empire]] (now Poland) into a [[Polish Jews|Polish-Jewish]] family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post.<ref name="MacTutor">{{MacTutor Biography|id=Post}}</ref>

Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer, leading to his decision to pursue mathematics rather than astronomy.<ref>Urquhart (2008), p. 429.</ref>

Post attended the [[Townsend Harris High School]] and continued on to graduate from [[City College of New York]] in 1917 with a B.S. in mathematics.<ref name="Urquhart"/>

After completing his [[Doctor of philosophy|Ph.D.]] in mathematics in 1920 at [[Columbia University]], supervised by [[Cassius Jackson Keyser]], he did a post-doctorate at [[Princeton University]] in the 1920–1921 academic year. Post then became a high school mathematics teacher in New York City.

Post married Gertrude Singer (1900–1956) in 1929, with whom he had a daughter, Phyllis Post Goodman (1932–1995).<ref>{{cite web|title=Phyllis Post Goodman Park|website=NYC Parks|url=https://www.nycgovparks.org/parks/phyliss-post-goodman-park/history}}</ref> Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton.<ref name="Urquhart 2008, p. 430">Urquhart (2008), p. 430.</ref>

In 1936, he was appointed to the mathematics department at the City College of New York. He died in April 1954 of a [[myocardial infarction|heart attack]] following [[Electroconvulsive therapy|electroshock treatment]] for [[Depression (mood)|depression]].<ref name="Urquhart 2008, p. 430"/><ref>{{cite book |editor=Baaz, Matthias |year=2011 |title=Kurt Gödel and the Foundations of Mathematics: Horizons of Truth |edition=1st |publisher=Cambridge University Press |isbn=9781139498432}}</ref>

==Early work==
In his doctoral thesis, later shortened and published as "Introduction to a General Theory of Elementary Propositions" (1921), Post proved, among other things, that the [[propositional calculus]] of ''[[Principia Mathematica]]'' was complete: all [[Tautology (logic)|tautologies]] are [[theorem]]s, given the ''Principia'' axioms and the rules of [[substitution (logic)|substitution]] and [[modus ponens]]. Post also devised [[truth table]]s independently of [[Charles Sanders Peirce|C. S. Peirce]] and [[Ludwig Wittgenstein]] and put them to good mathematical use. [[Jean van Heijenoort]]'s well-known source book on mathematical logic (1966) reprinted Post's classic 1921 article setting out these results.

While at Princeton, Post came very close to discovering the incompleteness of ''Principia Mathematica'', which [[Kurt Gödel]] proved in 1931. Post initially failed to publish his ideas as he believed he needed a 'complete analysis' for them to be accepted.<ref name="MacTutor"/> As Post said in a postcard to Gödel in 1938:

:I would have discovered Gödel's theorem in 1921—if I had been Gödel.<ref>{{cite journal |last1=Stillwell |first1=John |title=Emil Post and His Anticipation of Gödel and Turing |journal=Mathematics Magazine |date=2004 |volume=77 |issue=1 |pages=3–14 |doi=10.2307/3219226 |jstor=3219226 |url=https://www.jstor.org/stable/3219226 |issn=0025-570X|url-access=subscription }}</ref>

==Recursion theory==
In 1936, Post developed, independently of [[Alan Turing]], a mathematical model of computation that was essentially equivalent to the [[Turing machine]] model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper [[Formulation 1]]. This model is sometimes called "Post's machine" or a [[Post–Turing machine]], but is not to be confused with [[Tag system|Post's tag machines]] or other special kinds of [[Post canonical system]], a computational model using [[string rewriting]] and developed by Post in the 1920s but first published in 1943.  Post's rewrite technique is now ubiquitous in programming language specification and design, and so with [[Alonzo Church|Church]]'s [[lambda calculus]] is a salient influence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all.

Correspondence systems were introduced by Post in 1946 to give simple examples of [[undecidable problem|undecidability]].<ref name="Post46">{{cite journal|year = 1946|author = E. L. Post|author-link = Emil Post| title= A variant of a recursively unsolvable problem |journal = [[Bull. Amer. Math. Soc.]]| volume =  52|issue = 4|url=https://www.ams.org/bull/1946-52-04/S0002-9904-1946-08555-9/S0002-9904-1946-08555-9.pdf|doi=10.1090/s0002-9904-1946-08555-9|pages=264–269|doi-access = free}}</ref> He showed that the [[Post correspondence problem]] (PCP) of satisfying their constraints is, in general, undecidable. The undecidability of the correspondence problem turned out to be exactly what was needed to obtain undecidability results in the theory of [[formal languages]].

In an influential address to the [[American Mathematical Society]] in 1944, he raised the question of the existence of an uncomputable [[recursively enumerable set]] whose [[Turing degree]] is less than that of the [[halting problem]]. This question, which became known as [[Post's problem]], stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful [[Turing degree#Post's problem and the priority method|priority method]] in [[computability theory]].

==Polyadic groups==
Post made a fundamental and still influential contribution to the theory of [[n-ary group|polyadic, or ''n''-ary, groups]] in a long paper published in 1940.  His major theorem showed that a polyadic group is the iterated product of elements of a [[normal subgroup]] of a [[group (mathematics)|group]], such that the [[quotient group]] is [[cyclic group|cyclic]] of order ''n'' &minus; 1.  He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set.  The paper contains many other important results.

==Selected papers==
* {{cite journal |first=Emil L. |last=Post |year=1919 |title= The generalized gamma functions |journal=[[Annals of Mathematics]] |series=Second Series |volume=20 |issue=3 |pages=202–217 |doi=10.2307/1967871 |jstor=1967871 |doi-access=free }}
* {{cite journal |first=Emil L. |last=Post |year=1921 |title=Introduction to a general theory of elementary propositions |journal= [[American Journal of Mathematics]] |volume=43 |issue=3 |pages=163–185 |doi=10.2307/2370324|jstor=2370324 |hdl=2027/uiuo.ark:/13960/t9j450f7q |hdl-access=free }}
* {{cite journal |first=Emil L. |last=Post |year=1936 |title=Finite combinatory processes – Formulation 1 |journal=[[Journal of Symbolic Logic]] |volume=1 |issue=3 |pages=103–105 |doi=10.2307/2269031|jstor=2269031 |s2cid=40284503 }}
* {{cite journal |first=Emil L. |last=Post |year=1940 |url=https://www.ams.org/journals/tran/1940-048-02/S0002-9947-1940-0002894-7/home.html|title=Polyadic groups |journal=[[Transactions of the American Mathematical Society]] |volume=48 |issue=2 |pages=208–350 |doi=10.2307/1990085|jstor=1990085 |doi-access=free }}
* {{cite journal |first=Emil L. |last=Post |year=1943 |title=Formal reductions of the general combinatorial decision problem |journal=American Journal of Mathematics |volume=65 |issue=2 |pages=197–215 |doi=10.2307/2371809|jstor=2371809 }}
* {{cite journal |first=Emil L. |last=Post |year=1944 |url=https://www.ams.org/journals/bull/1944-50-05/S0002-9904-1944-08111-1/home.html|title=Recursively enumerable sets of positive integers and their decision problems |journal=Bulletin of the American Mathematical Society |volume=50 |issue=5 |pages=284–316 |doi=10.1090/s0002-9904-1944-08111-1|doi-access=free }} Introduces the important concept of [[many-one reduction]].

==See also==
*[[Arithmetical hierarchy]]
*[[Functional completeness]]
*[[List of multiple discoveries#20th century|List of multiple discoveries]]
*[[List of pioneers in computer science]]

==Notes==
{{Reflist|30em}}

==References==
*{{citation|last = Stillwell|first = John|author-link = John Stillwell|title = Emil Post and His Anticipation of Gödel and Turing|journal = [[Mathematics Magazine]]|volume = 77|issue = 1|year = 2004|pages = 3–14|doi = 10.2307/3219226|jstor = 3219226|url = https://www.maa.org/sites/default/files/321922608817.pdf.bannered.pdf}}
*{{cite book |chapter-url=https://www.ualberta.ca/~francisp/papers/UrquhartPost.pdf |chapter=Emil Post |first=Alasdair |last=Urquhart |author-link=Alasdair Urquhart |series=Handbook of the History of Logic |volume=5 |title=Logic from Russell to Church |editor1=Gabbay, Dov M. |editor2=Woods, John Woods |year=2008 |publisher=Elsevier BV}}
*Neary, Turlough (2015), "Undecidability in binary tag systems and the post correspondence problem for five pairs of words", International Symposium on Theoretical Aspects of Computer Science, Leibniz International Proceedings in Informatics (LIPIcs), pages 649–661, 2015.

== Further reading ==
* {{cite journal |last1=Anshel |first1=Iris Lee |last2=Anshel |first2=Michael |jstor=2324657  |title=From the Post–Markov Theorem Through Decision Problems to Public-Key Cryptography |journal=The American Mathematical Monthly |volume=100 |number=9 |date=November 1993 |pages=835–844 |publisher=Mathematical Association of America |doi=10.2307/2324657}} 
*:Dedicated to Emil Post and contains special material on Post. This includes "Post's Relation to the Cryptology and Cryptographists of his Era: ... Steven Brams, the noted game theorist and political scientist, has remarked to us that the life and legacy of Emil Post represents one aspect of New York intellectual life during the first half of the twentieth century that is very much in need of deeper exploration. The authors hope that this paper serves to further this pursuit". (pp.&nbsp;842–843)
*{{cite book |editor=Davis, Martin |year=1993 |title=The Undecidable |url=https://archive.org/details/undecidablebasic00davi_950 |url-access=limited |pages=[https://archive.org/details/undecidablebasic00davi_950/page/n291 288]–406 |publisher=Dover |isbn=0-486-43228-9}} 
*:Reprints several papers by Post.
*{{cite book |last=Davis |first=Martin |year=1994 |chapter=Emil L. Post: His Life and Work |title=Solvability, Provability, Definability: The Collected Works of Emil L. Post |publisher=Birkhäuser |pages=xi–xxviii}}
*:A biographical essay.
* {{cite journal  |last=Jackson |first=Allyn |url=https://www.ams.org/notices/200805/ |title=An interview with Martin Davis |journal=Notices of the AMS |volume=55 |number=5 |date=May 2008 |pages=560–571}}
*:Much material on Emil Post from his first-hand recollections.
*{{cite journal |last= Jackson |first=Allyn
|url=https://inference-review.com/article/psychological-fidelity/ |title=Emil Post: Psychological Fidelity
|journal=Inference: International Review of Science |date=October 2018|doi=10.37282/991819.18.48
|s2cid=240012225
|url-access=subscription}}
*:A biographical article.

==External links==
* [https://web.archive.org/web/20100820215747/http://www.amphilsoc.org/mole/view?docId=ead%2FMss.Ms.Coll.45-ead.xml Emil Leon Post Papers 1927-1991], [[American Philosophical Society]], Philadelphia, Pennsylvania.
* {{cite web|title=Celebrating Emil Post & His "Intractable Problem" of Tag: 100 Years Later|date=May 19, 2021|website=YouTube|publisher=Wolfram|url=https://www.youtube.com/watch?v=ultMxODJE7o |archive-url=https://ghostarchive.org/varchive/youtube/20211221/ultMxODJE7o |archive-date=2021-12-21 |url-status=live}}{{cbignore}}

{{Authority control}}

{{DEFAULTSORT:Post, Emil Leon}}
[[Category:1897 births]]
[[Category:1954 deaths]]
[[Category:20th-century American mathematicians]]
[[Category:American amputees]]
[[Category:American logicians]]
[[Category:American people of Polish-Jewish descent]]
[[Category:City College of New York faculty]]
[[Category:Columbia Graduate School of Arts and Sciences alumni]]
[[Category:Computability theorists]]
[[Category:Emigrants from Congress Poland to the United States]]
[[Category:Mathematicians from New York (state)]]
[[Category:People from Augustów]]
[[Category:People with bipolar disorder]]
[[Category:Townsend Harris High School alumni]]