Rurweg 3 synopses of G6del's 1931 incomplete- a few; he also quickly found for him- D-41844 Wegberg ness theorems, the Erkenntnis piece of self a society of like-minded and, judg- Germany 1931 due to G6del himself, and the text ing from the photographs, high-living, e-mail: [email protected] of a lecture delivered in in 1932 university friends, who, as was cus- by . And although the au- tomary then in Vienna, spent their days thors write in the introduction that the in a series of roving discussions in cafes, Kurt GOdel. Das book is meant as an "easily digestible some of which are pictured. Many of introduction" to G6del's life, work, and these friends were to go on to promi- the Viennese culture in which he lived nent positions in post-war academia: Album-The Album and flourished, it will undoubtedly be , for example, a student of Karl Sigmund, John Dawson, and of interest to readers on every level, and one of G6del's best Kurt Mfihlberger, with a Foreword by from newcomers to this area of the his- friends, was to become president of the Hans Magnus Enzensberger tory of mathematics to seasoned logi- American Philosophical Association-- cians and mathematicians. but he is seen here as a young man ca- WIESBADEN, VIEWEG, 2006, 225 PP., WITH 200 vorting at the beach with the Schlick PICTURES, (~29,90, ISBN 3-8348-0173-9 G6del's Life family. Other friends from this period "GOdel in a good mood and brilliant as who would later emigrate to the United REVIEWED BY JULIETrE KENNEDY usual .... I like him infinitely much and States include Karl Menger, Olga no one, no one of my friends can stim- Taussky, and Oskar Morgenstern. he Austrian logician Kurt G6del, ulate me as he does." So wrote G6del's Schlick, a central figure in the Vien- a brilliant man beset by misfor- friend Oskar Morgenstern after visiting nese philosophical culture at the time, T tunes of all kinds, has drawn the with G6del, presumably sometime in was, together with Hans Hahn, the attention of a number of biographers-- the mid-1970s. leader of the so-called , with mixed results. One wonders about A number of interesting documents also called the Schlick Circle, a discus- the rights of the deceased in such mat- are included in this first section. For ex- sion group G6del attended that quickly ters. These lines from Othello's final so- ample, a few pages from one of G6del's became identified with the doctrine of liloquy (from Act 5 of Shakespeare's schoolbooks are reproduced on page "" (a term coined by play) seem quite apt: 18. They are undated, but GOdel must Feigl and Blumberg in their 1931 "Log- I have done the state some service have been a very young child when he ical positivism: A new movement in Eu- and they know't. wrote these lines. In a typical example ropean philosophy" [1]). G6del was No more of that. I pray you, in your of the old-fashioned rote method of never drawn to logical positivism him- letters, teaching, in the workbook GOdel had self, but the exposure to the discussions When you shall these unlucky deeds to write the symbols "+" and the num- that took place at those meetings must relate, bers "2" and "Y' a hundred (or more) have been crucial to his development Speak of me as I am. Nothing ex- times, presumably for practice, and then as a logician. , Karl tenuate, finally the equations "1 + 1 = 2" and Menger, Gustav Bergmann, Rudolf Car- Nor aught set down in malice. "2 + 1 = 3." But then his mind wan- nap, and Friedrich Waismann were just Such being the case, Kurt GOdel. The dered and he wrote "1+ = 21." His re- a few of the regular members; Ludwig Album comes as a welcome addition port card of February 10, 1917 is here Wittgenstein visited periodically, as did to the literature on G6del. As the cat- too, on page 19, and there we see that outsiders like Frank Ramsey and W.V.O. alogue of the exhibition which took the I0-year-old boy rated "sehr gut" in Quine. place at the this every subject--except mathematics!, in That Hahn and Schlick would come year in conjunction with the celebra- which he only rated a "gut." together to lead this historic seminar tion there on the occasion of G6del's A fascinating set of pictures and cap- was characteristic of the hybrid nature centennial, it is to a large extent made tions deal with G6del's first few years of logic at the time. G6del commented up of reproductions of documents and in Vienna, where he became a student on this to Hao Wang in the late 1970s photographs with accompanying com- at the University of Vienna in 1924. It ([4], p. 82): mentary. And while far too many of the seems to have been a golden time for When I entered the field of logic, letters and other documents are un- G6del, intellectually and in other ways there were fifty percent philosophy dated, compromising the historical too. One could hardly walk down the and fifty percent mathematics. There value of the volume, it is nevertheless street in Vienna in the 1920s without are now ninety percent mathematics one of the most fascinating and also bumping into a major cultural or scien- and only one percent philosophy... one of the most corrective books to ap- tific luminary of the twentieth century-- G6del began his studies in Vienna pear about G6del so far. and the University of Vienna was no in 1924 in physics with Hans Thirring; The book is divided into three sec- less blessed in this respect. G6del at- and although he switched to mathe- tions: a biographical section comprising tended lectures by Philipp Furtw~ngler roughly half of the book, a section de- (cousin of the great conductor), Hein- The voted to GOdel's work, a brief section rich Gomperz, Hans Hahn, Moritz sun was" setting on this titled "GOdel's Vienna." These are fol- Schlick, and (all of golden period of G6del's life lowed by an appendix consisting of two whom are pictured here), to name just

2007 Springer Science+Business Media, Inc., Volume 29, Number 3, 2007 73 matics two years later, studying first lectures. Such an occasion is pictured tion involving the interrogation of number theory under Furtw~ingler and in a chilling photograph on page 55 of G6del's friends and colleagues. Never- then logic under Hahn and, informally, the book. theless, G6del was eventually granted Carnap, he continued to take physics Finally, in a tragic symptom of the the Dozentur in 1940; but by that time courses up to the time he graduated in times, Schlick was assassinated at the he and his wife Adele had already 1929. The background in physics G6del university in 1936 by an ex-student moved to the United States. The G6dels obtained as a student may explain his named Nelb6ck, whom GOdel knew were never to return to Europe. In fact being able to make a significant con- from his mathematics and philosophy G6del would write to his mother (in a tribution to relativity in 1947. (See [2].) classes. After mounting a defense based letter reproduced on page 87) that for In fact, G6del wrote this about his work on the idea that Schlick's atheism had a time he was plagued by nightmares in relativity in a 1955 letter to Carl caused him to become deranged, Nel- of being trapped in Vienna and not be- Seelig (see [3], p. 252): b6ck got off after serving only two years ing able to leave. I have, however, in connection with for the crime. In a post-war coda to the As for G6del's life in the United certain philosophical problems, de- Nelb6ck episode, we learn (on page States, one letter stands out particularly voted myself for some time to a less 186) that Viktor Kraft, who wrote the as a footnote to the GOdel-Thirring con- difficult complex of questions from first book about the Vienna Circle, was nection. Before G6del left Vienna for general relativity theory, namely cos- successfully sued by Nelb6ck in 1950, the United States in 1940, Thirring asked mology. The fact that here I, as a for Kraft's use of the term "persecution him to warn Einstein that newcomer to the field of relativity mania" to describe Nelb6ck at the time might be in a position to develop a nu- theory, could immediately obtain es- of the assassinationithat is to say, Kraft clear weapon (the possibility of atomic sentially new results seems to me had to withdraw the expression. Need- fission having been discovered in Berlin sufficient to demonstrate the unfin- less to say, deprived of one of its found- in late December 1938, when for the ished state of the theory. ing members, the Vienna Circle broke first time fission was actually produced That the sun was setting on this up soon afterwards. in the laboratory). Many years later (in golden period of GOdel's, if not Vi- The thought of G6del in the midst of 1972) Thirring wrote to G6del and enna's, if not indeed Europe's exis- all this gives one pause, although of asked him if he had ever passed on the tence, is first alluded to on page 26, course, not being Jewish, his difficulties warning. In this letter (reproduced on where it is mentioned that when G6del paled besides those of his Jewish friends page 143) G6del replied that he had not obtained his Austrian citizenship in and teachers. Nevertheless G6del's dif- passed it on. He gave a number of rea- 1929, the year he turned in his thesis ficulties during the 1930s--his most pro- sons: at the time he had been out of containing the completeness theorem ductive decade mathematically--were contact with physics for a decade, and for first-order logic, the political situa- substantial, and are documented here. hadn't known about the development tion in Vienna was becoming increas- He was hospitalized a number of times of fission; but then when he did finally ingly violent, with riots in 1927 leaving with nervous breakdowns, due perhaps hear about "these things," he was skep- 89 people dead and the conservatives not only to a natural inclination in this tical-not for any scientific reason, but in essentially open war with the so- direction, but perhaps in part also due for what he called a "sociological rea- cialist majority. The political situation to outside circumstances. Hahn as well son": he didn't think the culture at the manifested itself with particular vehe- as Schlick had died, and his Dozentur, time was at such a point in its evolu- mence at the University of Vienna, which he had gotten in 1932, was with- tion for such a development to ensue. which, being considered particularly drawn and could only be restored as a Rather this should come at the end of "red," was increasingly targeted by the so-called "Dozent neuer Ordnung" upon our "Kulturperiode." authorities--with, of course, disastrous his being found ethnically and politically Although it is not difficult to imag- results as the 1930s wore on and Na- acceptable to the Nazi regime. At the ine G6del's real reasons for not con- tional Socialism eventually saturated same time his problems with the au- tacting Einstein about Thirring's warn- the university. By May of 1938, out of thorities were compounded by his three ing--certainly the reasons G6del gives 258 emeritus and full professors, asso- trips back and forth to the United States here do not seem very convincing---one ciate professors, and lecturers, 94 had during that decade, each of which re- may nevertheless be strongly inclined "retired" from the philosophical faculty; quired a travel visa. Finally, he was to take G6del to task over the matter. and those salaried employees who roughed up in the street one day by As it happens the physicist Leo Szilard wished to remain had to endure a so- some young men, reportedly members had already visited Einstein in 1939 to called dejudification procedure in or- of the Hitler Youth. convey the same warning. In any case, der to keep their positions. Regarding GOdel's university posi- after the war G6del's and Einstein's The oath to the F/ihrer that univer- tion, in a letter to the Rector (repro- views about the stockpiling of nuclear sity personnel had to sign is reproduced duced here on page 68), the Ministry of weapons and the pursuit of the ensu- here on page 57, as is a letter from the Interior and Cultural Affairs recom- ing arms race between the USSR and Studentenfi_ihrer from that month on mended against G6del being granted the United States, and about matters of page 56, apprising students of the new the neue Dozentur, on the grounds of war and peace in general, were nearly post- decorum, which re- his "doubtful" political stance. This was indistinguishable. GOdel had no hesita- quired them to stand and give the Hitler not just an idle guess on the part of the tion about expressing those views--at salute at the beginning and end of all Ministry, but the result of an investiga- least in the 1950s:

74 THE MATHEMATICAL~NTELLIGENCER Einstein warned the world not to try To show how G6del's work in reminiscent of the project of informal to attain peace by rearmament and physics was seen by contemporary rigor due to G6del and also to Georg intimidating the adversaries. He said physicists, the authors include a note Kreisel. that this procedure would lead to from the then Director of the Institute The book ends with two letters the war and not to peace, and he was for Advanced Study, Harry Woolf (to poet Hans Magnus Enzensberger wrote quite right. And the fact is well himself, presumably) listing the topics to GOdel in 1957 and in 1974, asking known that the other procedure (try- to be spoken about at GOdel's funeral, him for an interview. I was delighted ing to come to an agreement in an which took place in 1978: to notice that the return address on amiable way) wasn't even attempted 1. set theory + the continuum hypoth- the 1974 letter is 15 Commerce Street, by America, but refused from the esis , which was then the first. It isn't the one and only ques- 2. logic = incompleteness + consis- home of Christiane Zimmer, n#e von tion as to who started matters, and tency Hofmannstahl, daughter of the famous for the most part it would be diffi- 3. (Minor): relativity--not worth a poet and Strauss librettist Hugo von cult to establish. But one thing is cer- talk X. Hofmannstahl, and herself a wartime tain: under the slogan "democracy," On the opposite end of the spec- emigre to the United States with her hus- America is waging a war for an ab- trum, so to speak, the subsection called band the Indianologist Heinrich Zim- solutely unpopular regime and un- "Theology" begins somewhat inauspi- mer. Most likely Enzensberger would der the name of a "police action" for ciously with an undated notebook from have been staying with Mrs. Zimmer, the UN and does things to which the G6del Nachlass titled: "Errors in the who by 1974 had become the sine qua even the UN does not agree... Bible." But then it goes on to discuss non of emigre German and Austrian life GOdel's correspondence with his G6del's ontological proof and his wider in New York City: she offered material mother was being monitored by the views on exact theology in a very sen- and other support to untold numbers of FBI, presumably because of GOdel's sible way. artists--for example, the food served at friendship with Einstein. Accordingly One of the most interesting docu- her Sunday night salons, which I had this passage of GOdel's 1950 letter was ments in the book appears in this sec- the pleasure of attending in its last years, included in a report written by one tion (page 159): the receipt for the copy was often the only square meal an artist General Cornelius Moynihan of the U.S. of the papal encyclical called "Mit bren- could have gotten in New York that armed forces, to none other than J. nender Sorge" (in English "With burn- week. She also offered hospitality to any Edgar Hoover (see page 145). ing worry") issued by Pope Pius XI. 1 and all visiting German and Austrian The encyclical, published (unusually) in writers, such as Max Frisch, Gtinter G6del's Work German in March of 1937 but distrib- Grass, and Siegfried Lenz. In this section of the book one sees uted secretly, condemns Nazism. The Her picture should perhaps have most clearly that it is aimed at a gen- receipt indicates that G6del purchased been included here; not necessarily for eral audience; the expert may take is- his copy on December 20, 1937. her very faint connection to GOdel, but sue with the finer points of the de- Unfortunately, very short shrift is because she demonstrated so beauti- scriptions of GOdel's theorems or of his given to G6del's philosophical work. fully the idea that one could create a philosophical work, or find those ac- For example, in a section called "Plato's new and vibrant life in a new country, counts too sparse. But the mathemat- Shadow," subtitled "An unadulterated even after being forced out of one's ics here is in all particulars completely Platonist,'2 we are only told that GOdel's own country--a lesson GOdel was correct. Also the historical emphasis Platonist views "appear strange in the never to learn, scarred as he was by the is right: Hilbert, Russell, Tarski, von twentieth century." events of the 1930s, and perhaps by Neumann, Turing--all are included, events from his childhood too. with no small amount of text devoted G6del's Vienna to their contributions to logic. As is Much to their credit, the authors cast a REFERENCES Cantor--indeed a page from a letter wide net in this section, including not [1] Herbert Feigl and Albert Blumberg, Logical he wrote to Hilbert is included here, only photographs and text devoted to positivism, a new movement in European written in Cantor's extravagantly florid GOdel's scientific milieu, but to figures handwriting. We also meet up with from the wider culture such as Robert philosophy. Journal of Philosophy 28:281 - Husserl, and Leibniz, and even Gold- Musil, whom G6del never met, and Her- 296, 1931. bach. In pages headed "Time travel mann Broch, whom he did. Interest- [2] Kurt GOdel, "An example of a new type of with GOdel," GOdel's work in relativity ingly, both wrote novels whose heroes cosmological solutions of Einstein's field is touched upon, with a long quote are mathematicians; in fact, the authors equations of gravitation," Reviews of mod- from Palle Yourgrau summarizing remind us on page 194 that Broch's ern physics 21:447-450, 1949. GOdel's work and why it implies that 1933 novel called The Unknown Quan- [3] Kurt G6del, Collected Works. V." Corre- "one can travel to any region of the tity features a hero who dreams of find- spondence H-Z. S. Feferman, et al., eds., past." ing a logic without axioms--not a little Oxford University Press, Oxford, 2003.

~ln a rare slip, the authors get the German title wrong, misidentify it as a bull, and attribute it to Pius XII. 2This is actually Russell's description of GOdel in the second volume of Russell's Autobiography, on the occasion of his meeting G6del at Einstein's house in Prince- ton in 1943.

2007 Springer Science+Business Media, inc., Volume 29, Number 3, 2007 75 [4] Hao Wang, A logical journey: From G6del study of parts of the continuum, the set -the resistance from the older gen- to philosophy. Final edition and with an ad- of all real numbers. Cantor used the eration of mathematicians to the dition to the preface by Palle Yourgrau and language of sets and functions, and new mathematics, Leigh Cauman. MIT Press, Cambridge, MA, with his proof that the set of real num- -the resistance from younger ele- 1996. bers is non-denumerable he set the ments like Cantor's friend Schwarz, stage for the new mathematical theory -the efforts of Cantor to prove the Juliette Kennedy of infinity. Then he introduced the main Continuum Hypothesis (CH), one of Department of Mathematics and Statistics actors: transfinite numbers, cardinals the central questions of set theory University of Helsinki and ordinals, and the Alephs. But Can- even now, and Helsinki, Finland tor was also hoping to achieve a sub- -Cantor's mental problems and e-mail: [email protected] lime goal--understanding the infinity problems connected with his wife of infinities. Not only was he aware of and daughter. the religious dimension of his work, On stage Cantor is euphoric when he it was a strong stimulus, as it had thinks he has proven CH, and goes been for other mathematicians, such as ahead with the hierarchy of the Alephs. Die Vermessung des Pythagoras, or Luzin and the Name- During depressive periods he sits by a worshippers, or G6del. Cantor consid- river--a feature of much German ro- Unendlichen ered his theory of sets a revelation of mantic literature (one cannot help truth inspired by God. thinking of H61derlin's poems). (Measuring Infinity) Much later, set theory became the The Continuum Hypothesis is more an opera by Ingomar Granauer lingua franca of mathematics, and al- than a leitmotif: it is the symbol of the though some important difficulties had opera. The formula 2~o = c is written at REVIEWED BY JEAN-MICHEL KANTOR been apparent from the beginning, the the center of the stage curtain, and is new math (as they called it) was taught often quickly flashed onto the wall, a art of mathematics consists in giv- in most schools from the 1960s on. hundred times during some scenes. It is p ing precise meanings to specific Cantor, who created set theory, was even chalked by Cantor on his violin. words. Take 'infinity': it first ap- born in Saint Petersburg, but moved Axel KOhler plays Cantor, a perfect peared two thousand years ago when with his family to Germany when he role for this very gifted singer. He sings Anaximander of Miletus, a pre-Socratic was twelve. He was a professor at Halle as a baritone when remembering his philosopher, coined the word 'Ape- University from 1869 to 1913. 4 Cantor youth or the happy periods of his life, iron'--an obscure notion of vagueness suffered early from manic depression, and as a counter-tenor when thinking and unboundedness. 1 It took centuries which increased after 1899, and he later about set theory. When Cantor opposes to give a mathematical meaning to in- spent periods in various psychiatric in- his enemies, like Kronecker, the singer finity. And it is still not over! 2 The first, stitutions. He is buried in Halle. speaks, while his depressive moods are crucial step was Aristotle's claim that The city of Halle was founded 1200 represented by the sound of the violin there was no 'actual infinity', just years ago, but it has recently fallen on he plays. The old mathematics is rep- the 'potential infinity' represented by hard times: due to the economic crisis resented by old mandarins in wigs, one, two, three, . . . In fact, until re- in the former German Democratic Re- sometimes even carrying plaster statues. cent surprising discoveries concerning public, the city lost a third of its popu- The new mathematics is represented by Archimedes, 3 general opinion held that lation in five years. The city center has four charming Alephs, dancing women for the Ancient Greeks there was no real many old and well-preserved buildings, who come and take off the wigs of the infinity. among them the house where Georg old teachers. The music is an oratorio The mathematics of infinity in the Friedrich H~indel was born, but the with most of the themes inspired by the modern sense began in the middle of outer parts of the city resemble a Russ- famous Fugue in B Minor from Bach's the nineteenth century with the priest ian provincial town. The Opernhaus of 'Well-tempered clavier'. This fugue is Bernard Bolzano (1781-1848). Bolzano Halle is a nice building of classical eigh- also well-known to musicians because wrote a book, Paradoxes of the Infi- teenth-century style. it uses all twelve half-tones, and it in- nite, in which he tried to define a cal- For its Jubilaeum, the city council, to spired Schoenberg. Grgnauer's post-se- culus with all sorts of infinities (infi- honor Cantor, the other of its two great rial style is still quite popular among nitely small and large). When Georg men, commissioned an opera from In- musicians in Germany. The orchestra, Cantor (1845-1918), studying the sets gomar GrOnauer, a composer born in led by Roger Epple, a popular director of unicity (now called exceptional sets) Vienna in 1938. Grunauer did not try to there, received strong applause. of trigonometric series, extended the give a precise mathematical account of The staging of the opera is quite work of his colleague Heinrich Heine Cantor's work, but rather was loosely elaborate, with part of the orchestra at (1821-1881), he was led naturally to the inspired by the rear, sometimes visible, with a part

~Simplicius quoting Anaximander in "Commentaries of Aristotle's Physics," 24,13. 2Woodin, Hugh W., The continuum hypothesis, Parts 1 and 2, Notices of the AMS, vol. 48, 2001. 3cf. Reviel Netz and William Noel, The Archimedes Codex, May 2007, Weidenfeld & Nicoison. 4Stern, Manfred, Memorial Places of Georg Cantor in Halle, The Mathematical Intelligencer, v. 10, n.3, 1988, 48-49.

76 THE MATHEMATICALINTELLIGENCER