Disproving the Myth that many early logicians were a few axioms short of a complete set

While I was working on this post another blogger posted on the same topic here and I found a book review of Logicomix that touched on some of the same issues here. (For MY review of Logicomix see here.) They are very good sources and I will refer to them in this post.

There is a notion that logicians who work in foundations early on in the field were crazy. I give examples of where this is said and then I look at the real evidence.

1. In Rudy Rucker's post about Turing he writes

   ... it really does seem possible that Turing killed himself. Like the other logicians Godel and Cantor, he seems to have been somewhat nuts. Funny how many logicians are crazy and irrational. A paradox.

2. In Logicomix, a great comic book about the foundations of logic, there is an allusion to Logicians being crazy.
3. In Gian-Carlo Rota Indiscrete Thoughts he writes: it cannot be a complete coincidence that several outstanding logicians of the 20th century found shelter in asylums at some point in their lives: Cantor, Zermelo, Godel, and Post are some.

I've also seen an explanation for this alleged phenomena: Logicians were searching for absolute certainly and either it can drive you crazy or thinking you can find absolute certainly means you were crazy ahead of time.

So the people above, and others, give some examples of logicians being crazy and then claim that many logicians are crazy. I am reminded of people who say It was cold the other day, looks like Global warming is wrong.

Let us look at the actual record. I will look at all of the logicians in Wikipedia's list of logicians who

1. were born between 1845 and 1912. (1845 is when Cantor was born, 1912 is when Turing was born.)
2. I ruled out a few people who were really philosophers, and also Banach who I don't think would call himself a logician.

For each logician on the list who meets my criteria I say if I think they are sane or a few axioms short of a complete set. I am not a historian--- corrections are more than welcome. I also freely admit that crazy is not well defined.

You may well disagree with what years I pick and my opinions. The point is to get an intelligent discussion going.

1. Wilhelm Ackerman (1896-1962): He defined the function that bares his name. He also worked on the epsilon-calculus which formed the basis for Bourbaki's logic. Reading Bourbaki might drive one crazy; however, forming the basis for it does not. He was quite sane (Ackerman that is-- Bourbaki had multiple personality disorder.)
2. Alice Ambrose (1906-2001): She had the longest lifespan of anyone on this list. She studied with Moore and Wittgenstein and got two PhD's. (In those days a women had to do twice as much as a man to get a job.) She was more on the philosophy side of logic, but certainly had math training. She wrote a textbook with her husband, known as Ambrose and Lazerowitz. Sane!
3. Paul Bernays (1888-1977): He worked with Hilbert on alternative set theories. Sane!
4. Evert Willem Beth (1908-1964): He helped to establish Logic as a discipline. Sane!
5. L.E.J. Brouwer (1881-1966): He thought that all math should be constructive. This point of view lost the battle if ideas; however, that does not make him crazy. The Wikipedia article quotes Martin Davis as saying: he felt more and more isolated, and spend his last years under the spell of totally unfounded financial worries and a paranoid fear of bankruptcy, persecution, and illness. However, Dirk van Dal en wrote a scholarly two-volume biography of Brouwer that indicates that Brouwer was not crazy. And I agree. Sane!
6. Georg Cantor (1845-1918): He had a new way of looking at infinity that was brilliant and is now accepted. That does not make him crazy. He was also convinced that Bacon wrote the plays of Shakespeare and that Joseph of Arimathea was the father of Jesus Christ. That does not make him crazy. However, he was obsessed with these views and was in and out of sanitariums. A few axioms short of a complete set.
7. Rudolph Carnap (1891-1970): I originally thought he was more of a philosopher; however, he published in thermodynamics and the foundations of probability. He fled Hitler's regime and later refused to sign a loyalty oath in America (during the McCarthy Era). His second wife committed suicide. He led an interesting life but was sane.
8. Alonzo Church (1903-1995): He invented (discovered?) The Lambda Calculus, proved that Peano Arithmetic was undecidable, and articulated what is now called the Church-Turing Thesis. These are all sane things to do. (Bob Soare distinguishes Church's Thesis from Turing's Thesis here.)
9. Haskell Curry (1900-1982): He worked in combinatory logic. There is a programming logic named after his first name! (see here). Sane!
10. Adolf Fraenkel (1891-1965): The F in ZF-set-theory. Provably Sane!
11. Gottlob Frege (1848-1925) He hated Jews, Catholics, and the French. That might make him unpleasant to hang around, especially if you are a French Jew who converts to Catholicism. However, that does not make him crazy. He is often given as an example of someone who was crazy, though the links ( here and here) argues for Frege being sane. I defer to the two links. Sane!
12. Gerhard Gentzen (1909-1945): He made the cut- Sane!
13. Kurt Godel (1906-1978): He stopped eating because he thought people were trying to poison his food. They weren't. A few axioms short of a complete set.
14. Jean Van Heijenoort (1912-1986): Best known in Logic for writing From Frege to Godel, a history of Logic from ... Frege to Godel (duh). Best known outside of logic for being Trotsky's secretary and later a historian of that movement. He was killed by his estranged fourth spouse. An interesting life, an interesting death, but he was sane.
15. Jacques Herbrand (1908-1931) Has the shortest lifespan (died at 23 in a mountaineering accident) of anyone on this list. He worked in proof theory. Sane!
16. Arend Heyting (1898-1980) He continued Brouwer's work on intuitionism. Sane!
17. David Hilbert (1862-1943): In Logiccomix they claim that Hilbert's son Franz had a mental illness and Hilbert cut off all contact with him. However, this refutes this and claims that Hilbert's son was only put away for 3 years and then re-joined his family. One may question if David Hilbert deserves a World's Greatest Father mug, but one cannot question his sanity.
18. Clarence Irving (1883-1964): He took exception to Principia's use of material implication. I'm impressed that he read and understood Principia enough to have objections. Sane!
19. Stanislaw Jaskowski (1906-1965): He worked in Intuitionistic Logics. Since I can't prove that he was crazy I assume he was sane.
20. William Ernest Johnson (1858-1931): He wrote three volumes on logic which showed technical expertise but was superseded by Principia Mathematica. This did NOT drive him crazy. Sane!
21. Philip Jourdain (1879-1919): He was interested in paradoxes and formed the card version of the liar's paradox. He also worked on algebraic logic. Quite sane. His sister Eleanor Jourdain claimed to have traveled through time and seen ghosts, but was not a logician.
22. Stephen Kleene (1909-1994): Kleene hierarchy, Kleene star, Kleene algebras are all named after him. He also proved the recursion theorem. Did this go to his head and make him insane? NO- he was totally sane.
23. Christine Ladd-Franklin (1847-1930): Her PhD was on Algebra and Logic. She faced problems being a women in a man's field but kept her sanity.
24. Stanislaw Lesniewski (1886-1939): He rejected axiomatic set theory (because of Russell's paradox) and tried to obtain other formal systems to replace it. A noble effort that failed. Still, he kept his sanity.
25. Adolf Lindenbaum (1904-1941): He proved Lindenbaum's Lemma- every consistent theory of predicate logic can be extended to a complete consistent theory. Like many major advances, profound at the time, easy to prove now. Certainly sane.
26. Leopold Lowenheim (1878-1957): The Lowenheim of Lowenheim-Skolem. See Skolem for more on that. A model of sanity.
27. Jan Lukasiewicz (1978-1956) Wikipedia says He thought innovatively about traditional propositional logic. Is innovatively a word? My spell checker does not think so but whoever wrote his Wikipedia entry thinks so. Sane.
28. Saunders Mac Lane (1909-2005) (He preferred the space between Mac and Lane.) His PhD thesis was on Logic and he also worked in Category theory. But he also did lots of Algebra. Sane.
29. Carew Arthur Meredith (1904-1976): He worked on obtaining short axiom basis for logic systems. Sane.
30. John von Neumann (1903-1957): Calling him a logician seems odd since he contributed to so many fields. Sane.
31. Jean Nicod (1893-1924): Co-discovered the Sheffer Stroke from which you can do everything in prop logic. Sane.
32. Pyotr Novikov (1901-1975): He proved the word problem for groups undecidable. His son Sergei Novikov won a Fields Medal in 1970 and, more importantly, is a professor at The University of Maryland! Sane.
33. Giuseppe Peano (1858-1932): His Wikipedia entry calls him the founder of Mathematical Logic and Set Theory. That seems over-the-top, but not by much. His axiom system is still the standard. Sane.
34. Emil Post (1897-1954) He introduced Turing Degrees. In the mid 1940's he posed Post's Problem which is to find a r.e. set (now called c.e.) that is neither decidable nor complete. This was solved in 1956 by Friedberg and Munhnik independently. He suffered from mental illness. A few axioms short of a complete set.
35. Mojzesz Presburger (1904-1943): Presburger proved Presburger Arithmetic was decidable. What are the odds of that!? Sane!
36. William Quine (1908-2000): He was more of a philosopher; however he did do some math. At Harvard he taught Symbolic Logic every fall for 50 years. That might drive some crazy; however, he was quite sane.
37. Frank Ramsey (1903-1930): The paper where he proved what is now known as Ramsey Theory was titled A Problem in Formal Logic and solved a case of the Decision Problem. He regarded himself as a logician so we shall too. Speculation: He would be surprised at where his work lead to (combinatorics) and then pleased that it lead back to logic again : The Large Ramsey Theorem (see also here) and much work in the reverse mathematics of Ramsey's theorem".
38. Raphael Robinson (1911-1995): He worked in Logic and Number Theory. He is probably best known for his work on tiling the plane. He married Julia Bowman (who changed her name to Julia Robinson) who was also a logician but born in 1919--- a little too late to be on this list. Having two academics in the same area get married might drive some crazy, but not them. Sane!
39. J. Barkley Rosser (1907-1989): He strengthened Godel's incompleteness theorem. Sane!
40. Bertrand Russell (1872-1970): He was obsessed with the quest for certainty; however, that does not make him crazy. He had several wives (not at the same time) and believed in open marriage. He was not crazy, just ahead of his time. Sane.
41. Moses Schonfinkel (1889-1942): He worked in Combinatory Logic. By 1927 he was in a sanitarium. The only non-famous logician on my list who was a few axioms short of a complete set.
42. Thoralf Skolem (1887-1963): He is best known for the Lowenheim-Skolem theorem: The notion that any consistent set of axioms has a countable model is very interesting--- One corollary: there is a countable model of the reals. Thinking about that might drive some crazy, but not him. Sane!
43. Alfred Tarski (1901-1983): The Banach-Tarski paradox is crazy; however, Tarski was not. Sane.
44. Alan Turing (1912-1954): He defined Turing Machines, though he didn't call them that. The story I had assumed was true is that the British Government made him take hormones (or something) to cure him of his homosexuality, and this drove him to suicide. But the story doesn't quite work with the timeline. He committed suicide a few years after he was forced to take drugs. Delayed reaction? Suicide for some other reason? Really was an accident? In any case, since his possible suicide is the only evidence that he was crazy I say Sane!
45. Nicolai Vasilev (1880-1940): The originator of non-Aristotelian logics. Sane.
46. Alfred North Whitehead (1861-1947): In Russell-Whitehead's Principia Mathematica they spend 300 pages proving that 1+1=2. This might drive some insane but not him. Whitehead was stark raving sane.
47. Ludwig Wittgenstein (1889-1951): He gave away all his money and seemed to be a self-hating Jew. Odd yes, but he was sane. (NOTE- Scott Aaronson left a comment that argues that Wittgenstein should be classified as a few axioms short of a complete set. I believe his arguments (they are backed up by facts) and may later redo the stats at the end of this post.)
48. Ernest Zermelo (1871-1953): The Z in ZF set theory. He disapproved of Hitler's Regime. Hardly crazy. Rota says that Zermelo was crazy but neither I nor this post have been able to find any evidence of this. Zermelo did spend time in a hospital for lung problems, which may have confused Rota.

So what to make of all of this?

1. Cantor, Godel, Post and Schonfinkel were crazy. So we have 4 out of 48 were crazy. That's around 8%. This website claims that 6% of all people are crazy. So 8 seems high, but the sample space is pretty small. Conclusion: Same as the posts on the same topic referenced at the beginning: the notion that people in logic are crazy is not well founded. In addition, this post argues that the problems Cantor, Godel, and Post had were unrelated to their study of logic. (There was no comment on Schonfinkel.)
2. AH- but Rota said that so many outstanding logicians were crazy. Since three of the four who I say were crazy were outstanding there may be a point here. One could look at who on my list was outstanding and see what percent of them logicians were crazy. However, determining who was outstanding is even harder than determining who was crazy, so I leave it to others to continue this work.
3. There were some on the list that in my opinion were sane but others think were crazy: Brouwer, Frege, Turing, Zermelo. Perhaps more. If enough of them turn out to be crazy then there may be something to this logicians are crazy theme; however, I doubt this will happen.
4. Was it crazy to spend so much time and effort on this one post? I am not on the logic list, nor was I born between 1845 and 1912 so the answer is not relevant to the study.
5. This blog posting has a crazy number of links: 71. That breaks the record for this blog which was held by this entry which had around 37.