The "Here" and the "There" of Mathematics: Kurt Gödel

Review of
Logical Dilemmas: The Life and Work of Kurt Gödel
By John W. Dawson, Jr.
A.K. Peters, Wellesley, Massachusetts, 1997, 320 pages, $49.95

The name of Kurt Gödel (1906-1978), a singular genius and a man of extraordinary mathematical insight, is known to the general reading public, if only through Douglas Hofstadter's prize-winning Gödel, Escher, Bach (1979).

In the scientific world there are Gödel numberings, Gödel programming languages, Gödel societies that hold colloquia; there are explications and simplifications by the hundreds of the famous Gödel theorems. More popularly, there are speculations on the implications of Gödel's theorems for art, music, the mind-body problem, on whether cognition can be computational, on the limitations of human intelligence, on immortality, the existence of God --- you name it.

Less well known are his contributions to set theory, even among the mathematical community, which nonetheless may be ready to say that Gödel was the most original, most insightful mathematical mind of the 20th century, the greatest logician since Aristotle. His cosmological discoveries ("Gödel's rotating worlds" based on the Einstein field equations), I gather, are not much studied, and his philosophical and theological ideas, along with the facts of his personal life, have been diffused largely through community gossip.

From an earlier book by Ed Regis, Who Got Einstein's Office (Addison Wesley, 1987), I learned that John Dawson, the author of the biography under review, has had access to Gödel's two tall file cabinets plus 60 or so cardboard boxes piled six feet high in the archives of the Institute for Advanced Study in Princeton. Dawson has sifted through these leftovers --- the Nachlass --- which contained not only mathematical material, but also cosmological material, philosophical material, notes on everything under the sun in a German shorthand known as Gabelsberger, library slips, house leases, clothing bills, counter checks for the occasional beer, in short, both the pearls and the detritus of a remarkable life on two continents.

Two years at the very least of Herculean labor were required to go through and to straighten out and interpret the contents. And after than came the writeup. How does one squeeze the contents of some 60 boxes into a book of convenient size? What aspects to emphasize? What to forget or --- even more strongly --- what to censor? (Consider the Harvard philosophy department's decades-long restriction of access to the residue of Charles Sanders Peirce, an earlier logician.)

I suspect that Dawson's notes on this material, when added to his outside knowledge, would have allowed him to expand each of the following aspects of Gödel's life and work into a book in its own right:

  1. The day-to-day life, birth to death. The comings, the goings, the relatives, the friends, the interaction with the historic events of his day, where and how Gödel lived. His appointments, honors. The stuff of Who's Who.
  2. Gödel's accomplishments in logics, set theory, and cosmology. Their background, their evolution, and their post-Gödelian sequels.
  3. The Institut for Advanced Study during the first quarter century of its existence and the interaction between Gödel and Einstein and other luminaries in residence.
  4. Gödel's philosophical and theological views.
  5. Gödel as a case study for Oliver Sacks. The role of mental instability, paranoia, hypochondria, depression in creativity. The old idea (and indeed the frequent observation of the general public) of the close relationship between genius and mental instability.
  6. The important role played by Adele Porkert Gödel in keeping her husband alive and on track.
The reader of Dawson's splendid biography will find discussions of these point, some in more depth than others, adding up to an entirely satisfactory blend. The remainder of this review touches on some of these points.

•       •       •

Kurt Gödel was born in 1906, in what is now Brno, Czech Republic, and what was then part of the Austro-Hungarian Empire. He was baptized in the German Lutheran church. His father worked his way up to a partnership in a famous textile plant; his mother's father had a similar background in textile. In 1924 Gödel entered the University of Vienna, hoping for a degree in physics. Impressed by a brilliant course of lectures in number theory by Philipp Furtwängler, he switched to mathematics. His main teacher, Hans Hahn, introduced him to the regular meetings of the "Vienna Circle," in which a new philosophy of science was being hammered out. Gödel did not go along with the varieties of positivism espoused by the participants in the Wienerkreis.

Was there anything particular in Austrian (or Austro-Hungarian) education or society, one may ask, in the half century beginning, say, in 1888 that would account for its giving birth to so many logicians and philosophers of science: Mach, the whole Vienna Circle, Wittgenstein, Gödel, Popper, Lakatos, Feyerabend? Making connections between art, science, and society is an iffy project, but one might point to the absolute rigidity in the rites and the protocols of the Habsburg court and army that diffused all the way down into the educational system. There was an "invisible barrier of authority" that stood between teacher ad pupil and an "unfeeling and soulless method of ... education" (Stefan Zweig). What is mathematical logic if not absolute rigidity and protocol? What human feelings are attributed by the average person to the page after page of naked symbols that constitute a mathematical text?

Gödel's Incompleteness Theorem came in 1931, and the shock waves left in its wake can still be felt in some quarters. Mathematical logic had been developing since the mid-1800's. De Morgan, Boole, Frege are some of the early names. By the time of the publication of their Principia Mathematica, (1910-1913), Bertrand Russell and Alfred North Whitehead "saw logic as a universal system within which all mathematics could be derived." Mathematicians, joining their opinion to that of David Hilbert, had gone so far as to assert that all sensible mathematical claims could either be proved or disproved.

Gödel showed that this was not the case. Ambiguity is possible. The views of Russell and Whitehead and of many others lay shattered. Initially Gödel's theorems were misunderstood and considered bewildering by such logicians and philosophers as Zermelo and Wittgenstein. Russell (Literature Prize Nobelist), who had early on turned from mathematics to social activism was "glad he was no longer working at mathematical logic."

Gödel's theorem can be stated as follows: Given a consistent formalization of arithmetic, there exist arithmetic truths that are not provable in that system. To put it another way: Arithmetic cannot be completely formalized. Or still more popularly: You can't always get there from here. The "there" is some perfectly good and obviously existing condition or true statement, and the "here" consists of the axioms or the starting positions from which, after a finite number of rigidly described and allowed logical steps, one has been asked to arrive at the goal called "there".

As a crude illustration: Almost everyone has played with the 4 x 4 square plastic frame that has little movable squares numbered 1 to 15 and one vacant square. From an initial distribution of the numbered squares, the object of the puzzle is to move them around until they are in the order 1, 2, 3, 4, 5, ..., 13, 14, 15, blank.

It is well known and demonstrable that for half of the possible initial dispositions of the little squares, the desired end condition can be achieved. For the other half, it cannot. In such cases,you simply can't get there from here. You can always solve the puzzle, however, if you allow yourself to lift the squares up into three-dimensional space and rearrange them. If you were frustrated by the stated conditions of the puzzle and if your life depended on achieving the goal, that is exactly what common sense would tell you to do.

On the strength of his stunning achievement, Gödel was forthwith invited by Oswald Veblen to join the newly formed Institute for Advanced Study in Princeton. He was there briefly in the fall of 1933. The following May he returned to Austria.

He met Adele Porkert, the woman he would marry, in the Lokale (pub/dance hall) "Koralle" in the Porzellangasse (still there!), where she worked --- so my Viennese informants tel me --- as a taxi-dancer. "Aber ganz harmlos" they add. She claimed also to have been a ballet dancer. She was crude, course, loud, uneducated, "a Viennese washerwoman type: garrulous, uncultured, strong will." So wrote Gödel's friend Oskar Morgenstern, the economist and game theorist. She was six years older than Gödel and had been married briefly and divorced, but without having children. Gödel's family was scandalized by the match: In those days, any dancer, even a ballet dancer, was considered to be a whore. "A ballet dancer ... was available for any man ... in Vienna for two hundred crowns" (Stefan Zweig). Kurt and Adele were married in 1938 and remained together for the rest of their lives.

In March 1939, he was out of a job and was facing a draft call into the Nazi army. World War II began on September 2, 1939. Gödel and Adele decided that it was time to get out of Europe. Somehow, they made it across Russia and the Pacific to America, and in March 1940, Gödel was again at the Institute for Advanced Study. He remained there for the rest of his life, rising extraordinarily slowly to a professorship --- the inmates of the Institute protected their titles with the fine discrimination of central European monarchs who distinguished very carefully between high aristocracy and the blood royal.

Adele was a big city girl who hated the small town life of Princeton; she was lonesome there. And her relationship with the university wives in Princeton was one of mutual abhorrence. She was proud of her husband; she appreciated that he was famous in scientific circles.

•       •       •

Shy and reclusive, Gödel appeared to some to be a man/child, to others "a precocious youth who became old before his time". Gödel was hospitalized twice for serious depression, He suffered delusions and personality disturbances. He became excessively paranoid, the paranoia deriving, some have conjectured, from his super-logicality and overly intense introspection. He tended to believe in secret intrigues and conspiracies. He feared bodily harm from specific individuals. He feared poisoning, but at the same time told Morgenstern that if Morgenstern were a true friend, he would bring him some cyanide. As he grew older Gödel ate less and less. He argued with himself as to whether he should commit himself to a mental institution, deciding ultimately against it. In the end, he died of self-starvation.

Adele took good care of her man/child "Kurtele". She had saved him from an attack by Nazi thugs outside the University of Vienna. She nursed him through his phobias, depressions, and other mental difficulties. She acted as his food taster and fed him his meals, spoonful by spoonful. It was she, thought Morgenstern, who preserved his life.

The old question of the relation between genius and insanity arises here. Would any parent enter into a Faustian bargain that granted genius to a child at the price of severe abnormality? And yet, I have heard from the mouth of one of Europe's leading thinkers that a little craziness is a good thing, that aberrations keep the world from stagnating.

If medication had been available to clear up Gödel's condition (such pills as are now around are only partially alleviatory), one might reasonably ask what mathematics would have lost. I wonder how his Platonic personality would have answered the question. According to this philosophy, Gödel's theorems must surely exist in a Gödel-free Universe.

Gödel was a Platonist, an "unadulterated" Platonist in the opinion of Bertrand Russell. I use the term in the sense of one who believes that the whole mathematical enterprise, its objects, its conclusions, exist independently of humans. They are "out there" somewhere, everywhere, independent of time. As Gödel put it, "concepts form an objective reality of their own, which we cannot create or change, but only perceive and describe." He believed that conceptual understanding derived from introspection. There was and is nothing unusual about these opinions.

Most professional mathematicians, in fact, are Platonists, believing that 2+2=4 expresses an eternal truth, valid in all possible worlds, believing that the set of all integers 1,2,3 ... has objective reality, as does the set of all sets of integers, as does the set of all sets of all sets of integers, and so forth.

From Platonism to the belief that change and time are illusions is an "easy" step (if you have in mind to take that step). From Platonism to the existence of God is an easy step: Who or what shall guarantee that mathematical objects and constructions are "out there" and are meaningful? God, obviously.

From Platonism to determinism (in which Gödel shared a belief with Einstein) is an easy step: If mathematics is out there, then the cosmos itself is a rationally ordered harmonic entity in which all is, of necessity, foreordained.

From Platonism to neo-Platonism (the belief in mysticism) is an easy step: Gödel's interest in parapsychological phenomena (the occult, ghosts, telepathy) "went beyond mere open-mindedness." The step to belief in an afterlife in which all is revealed is also an easy one, as is that to the death of this world and the rebirth of new worlds.

Gödel believed all of these things --- or at least he believed that scientific theory did not logically preclude them. He deplored the positivists, who denied validity to religious concepts even as they espoused their own philosophy with the intensity of tent-meeting missionaries. His view should be contrasted with that of Karl Popper, who said that a theory should be considered scientific only if it can be tested in a way that risks its refutation.

•       •       •

A famous statement of theoretical computer science, the undecidability of the halting problem, can be traced through the work of Alan Turing back to Gödel. (The life of Turing (1912-1954) has been the basis of a play, Breaking the Code, and a later movie, both with Derek Jacobi. Briefly, this statement is that it is impossible to write a program P that can monitor all other programs Q and tell whether, when Q is run, the computer will stop or grind on forever. Numerous other computer impossibilities can be derived from this single impossibility.

Despite this development, it is an important fact that the brilliant, earth-shaking theorems of Gödel are of absolute unimportant to 99.5% of research mathematicians in their professional work. This is a paradox that is rarely discussed. I've asked a number of informed mathematicians for their comments and have received a wide spread of answers.

One response, which I consider a bit flip, is "Well, the discipline of mathematics is so large and so fragmented that most professionals hardly know or care about what their colleagues at the next desk are doing. We all do out own thing." A prominent member of the British Royal Society (of Science) speaking in public, recently put it this way: "If I took time to read what other people say, I wouldn't have time to do what I want to do."

While all this is true enough, considering that Gödel's work and that of his followers lie at what are supposed to be the foundations of mathematics, where does that leave the material at the higher levels of the structure?

Another response is that Gödel's Incompleteness Theorem is like the Bible or perhaps the Ten Commandments. It states certain limitations. We all know what they are, honor them in principle, and then assume that they do not pertain to our own work.

Although Gödel's theorems are now a significant part of the unstated metamathematical assumptions of research, they are relegated to a far back burner. IF a number theorist is working, say, on the famous and as yet unsolved problem of whether there are an unlimited number of twin primes (e.g. 11,13; 17,19), then the strong assumption underlying the work, even in our post-Gödelian period, is that the answer is either yes or no. It is not assumed that on the basis of the traditional axioms of arithmetic we cannot decide for true or for false.

Some have answered the paradox of irrelevance in this way: "Wait, the impact is yet to be felt." Thus philosopher of science David Berlinski's, "We have just begun --- begun! ---- to assess the full importance of Gödel's Incompleteness Theorem for philosophy, mathematics, and computer science.

To this one might respond that if the truth of a large fraction of what mathematicians have wanted traditionally to prove turns out to be unprovable, this would mean that the mathematical enterprise is totally different from the way it has been represented for centuries.

This leads me to my own explanation, which I cannot develop fully in this review and which is not far from the views of the Hungarian philosopher Imre Lakatos. It is that the role of deductive proof in mathematics has been misappraised. We have witnessed and apotheosis of Euclid's method --- definitions, axioms, theorems, proofs --- that has increased in the past 150 years of mathematical history, and that embodies a false description of how mathematicians operate, how they think, what, in fact, they do. I believe that a notion of mathematical "evidence," one that includes logical deduction as one of its components, dominates the process of mathematizing. And as regards deduction itself, it can have only a local, informal character. The "here" and the "there" are subject to constant historic renegotiation, modulated by the process of new concept formation.

•       •       •

One of the side pleasures I find in reading books is in following up on interesting tangential material that is often unimportant to the main story. Dawson's book suggested two tangents to me, and he was even kind enough to deepen one of them for me by mail.

Gödel was not one of your great intellectuals (a word I hate, and one that is used much more often in Europe than in the U.S.). He was a middle-brow as regards cultural matters. His wife was a low-brow. He liked operettas and pleasant plays. He visited museums, he watched variety shows on TV.

Classical music? Only with great difficulty did he bring himself to sit through two hours of Bach and Handel at an Einstein memorial. Literature? His favorite novel was Effie Briest by Theodor Fontane. Fontane? Who's he? I had never heard of him and I doubt whether my American readers have either. Yet the Brown University Library stacks have 130 books by and about him.

Effie Briest (1895) is Fontane's most popular novel and has been made into a movie four times, once by the well-known German filmmaker Rainer Werner Fassbinder. It tells of a young, lonesome married woman who falls into a brief adultery. The discovery of the affair by her inflexible army husband, who adheres rigidly to standardized officer corps behavior, leads to a dual, and then to the ruin of all the principals. Standardized operating procedure? Isn't this what mathematics is all about? Now see how rigidity can lead to disaster.

But Effie Briest was probably a double bill for Gödel. There is a ghost story build into the plot, and Gödel, who was soft on ghosts, was undoubtedly attracted to it.

A rather more suggestive piece of tangentialism is this: Both Gödel and his mother took an unusual interest in the historic (i.e. not fictional) Mayerling case. Mayerling is a small village outside Vienna where Crown Prince Rudolf of Austria had a hunting lodge. The case, briefly, involves the alleged murder by the prince of the young Countess Mary Vetsera followed by his suicide (1889). This tragedy was immediately subjected to cover-ups, suppression and destruction of evidence, creation of misinformation. The prince's history is sufficiently complex and is surrounded by sufficiently many ambiguities that it still bears discussion in Austria today. (See for example Kriminfall Mayerling: Level und Sterben der Mary Vetsera, Georg Markus, Amaltheaa, Vienna, 1993.)

Gödel did not believe in the romantic love affair between Rudolf and Mary implies by the thirties movie Mayerline (Charles Boyer and Danielle Darrieux.) Many Mayerling buffs would agree. In a letter to his mother on June 2, 1964 Gödel wrote, "I think a political conspiracy in which Rudolph himself took part is much more probable."

If you are an unadulterated Platonist, you will say that there is an answer to the question of what really and truly happened at Mayerling. Someone may even claim to know the answer. But how can we now validate the claim? What do the philosophers of legal evidence say is sufficient. What, speaking quite generally, is so and how do we know it? Gödel it seems, was attracted to the tension between the provable and the unprovable.

The discovery of the jarring Incompleteness Theorem notwithstanding, Dawson asserts that Gödel always believed in his heart of hearts that the questions thrown up by mathematics are always decidable by mathematics, that the ambiguities asserted by his undecidability theorems represented false views in a logically ordered, harmonious universe. He believed that we can always get "there" from "here" if only we are clever enough to discover and to declare where the "here" is from which we must start.

The French philosopher Jean-Marie Guyau (1854-88) anticipated the artistic freedom believed by all mathematicians to accompany the rigidities of their subject when he remarked that hypotheses were the poetry of the savant.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at

Reprinted from SIAM News Volume 30-8 October 1997. (C) 1997 by Society for Industrial and Applied Mathematics All rights reserved.