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by Brig Klyce | 20 March 2008
Albert Einstein (1879-1955) spent the last twenty years of his life at the Institute for Advanced Study in Princeton New Jersey. He was already famous, and he was careful about the company he kept. Nonetheless, from about 1942 until he died, Einstein often sought the company of a mathematician also at the Institute. Freeman Dyson, who knew them both, observed that the mathematician was the only person alive who could discourse with Einstein as an intellectual equal. The mathematicians name was Kurt Gdel.
I first became interested in Kurt Gdel in connection with my research about panspermia. I mentioned Gdels Incompleteness Theorem on my website, and one of my readers challenged me. He said that my own understanding was the thing that was incomplete! So I undertook a more thorough study of his life and work.
In the course of that study I came to admire Gdel immensely. He was able to do something that I would like to do. That is, he saw something wrong with the prevailing philosophy of his time, and he exposed the error in a way that soon convinced his opponents. He did all this by the age of 26. After learning about his contribution to philosophy, I have come to believe that some familiarity with Kurt Gdel would be enriching for anyone.
General Philosophical Background
Let me prepare for the essence of this story by giving a very brief synopsis of the historical background for the prevailing philosophy of Gdels time.
Rene Descartes (1596-1650) whom most consider to be the founder of modern Western philosophy, began by doubting virtually everything. The one thing he could not doubt was his own mental activity his thoughts. To think, he must exist. Cogito ergo sum he wrote, in 1637.
Doubt was also the chosen stance by influential British philosophers who succeeded him. John Locke (1632-1704) held that there is no a priori knowledge. In 1690, in his Essay Concerning Human Understanding, he explained that knowledge comes only from experience. But even our experience has a limited range: shapes may be real, but colors and sounds depend on the perceiver. All we can know about them is our perceptions, he wrote.
Following him, George Berkeley (1685-1753) denied even the existence of matter. In A Treatise concerning the Principles of Human Knowledge, Part I, published in 1710, and Three Dialogues between Hylas and Philonous, published in 1713, he said, basically, Its all in your mind. Only minds and mental events exist. Nothing else can be confirmed.
David Hume (1711-1776), in writings mostly published after his death in 1776, claimed to doubt even his own selfhood. He wrote that we cannot be sure that cause and effect are related, and that inductive reasoning is not logically certain, but merely habitual.
With Hume, skepticism had, apparently, reached its limit. These developments by the British empiricists have been caricatured as, 1) You can only know what you perceive with your senses; 2) You cant be sure that perceptions pertain to anything real; 3) In fact, you cant really be sure of anything.
This skepticism was disturbing to Immanuel Kant (1724-1804). In 1781, he published his 800-page Critique of Pure Reason (English translation 1887), which included an attempt to refute the established skepticism. Kant believed that a priori knowledge was possible, in geometry for example; and that intuition is, we must admit, the way we understand concepts like space and time. But his hope for a priori knowledge in geometry was not widely accepted, and anyway it was later overturned by the development of non-Euclidian geometries. And his advocacy of intuition did not win many converts among philosophers.
A partial solution to the problem of sure knowledge was introduced by Auguste Comte (1798-1857), the founder of the philosophy called positivism. In his Course of Positive Philosophy, published between 1830 and 1842, he held that the object or goal of knowledge is to explain only what we experience, without taking a position on the existence of an external reality to which experience may pertain. In this way, truth could be measured by the consistency and coherence of the body of descriptions.
Positivism was adopted by physicists of the late 19th century, notably Ernst Mach (1838-1916), who never accepted the existence of atoms, because they were unseen. Machs positivism influenced Albert Einstein. When Einstein said that simultaneity was a term that should be discarded, this was an example of positivism as a stance or a style. But ultimately, Einstein was not as interested in the philosophy underlying science and math as he was in the usefulness of the tools they supplied. If logic seemed to contradict reality, as he thought was the case with quantum theory, he favored reality. In other words, he was willing to rely on his own intuition.
Mathematical Philosophy
In mathematics, the most exact of sciences, the issue of certain knowledge is especially pressing. Between roughly 1875 and 1925, partly influenced by the new positivist philosophy, a movement developed within mathematics to make the subject completely rigorous to eliminate paradox, contradiction, uncertainty and doubt from it. To do this, mathematicians would start at the beginning.
One who did so is Giuseppe Peano (1858-1932), an Italian who began to ask, after about 1880, What is a number? and What counts as a proper definition? He laid down five simple axioms that were considered sufficient to serve as the foundation of arithmetic. And he pioneered the use of symbolic logic.
For an example of a mathematical system that begins with very few axioms, think of Euclidean Geometry. Euclid, who lived in Alexandria c. 300 BC, also began his study of plane geometry with 4 or 5 axioms. For example, the first one was that two points uniquely determine a straight line. From these axioms, by straightforward logic, a very rich set of postulates and relationships can be derived. Euclids scheme was so successful that his textbook was still in use in geometry classes in the 20th century!
Another mathematician who participated in this effort was Gottlob Frege (1848-1925), of the University of Jena. He made the first use of what we now call predicate calculus, in which proof is more formal and systematic than before. His work led to what we now call set theory. He hoped to show that mathematics was simply a branch of logic. Logic, of course, must be logical. Therefore, if Frege is right, mathematics must also be logical. This made the project to certify mathematics seem promising.
Peano and Frege influenced Bertrand Russell (1872-1970), the one true celebrity among mathematicians of this era. (Young Russell, by the way, was one who studied from Euclids text.) Soon after he met Peano at a conference, Russell discovered, in 1901, Russells Paradox. The paradox, pertaining to set theory, revealed a serious flaw in Freges plan to make mathematics merely a branch of logic. Russells paradox was similar to the so-called Liars Paradox attributed to the Cretan, Epimides: All Cretans are liars. Other paradoxes, like Richards paradox, added to the list. Soon these were recognized serious problems demanding attention. Frege was fairly devastated by the issue that Russell had raised.
In view of the paradoxes, was Freges plan unrealistic? If so, could mathematics be underwritten in another way? Russell committed himself to resolving these questions. After a decade of collaborative work, he and Alfred North Whitehead (1861-1947), published the 3 volumes of Principia Mathematica, between 1910-1913. In it they laid down symbols and rules for the symbolic logic that Peano had worked on. But their system was more thorough than any before, and probably any since. Their intention was to devise a system of tools that would enable mathematics to achieve its goals. They hoped that, by using the symbolic logic laid down in Principia Mathematica, it should be possible to eventually eradicate all contradiction and paradox from the field of mathematics.
A German mathematician, David Hilbert (1862-1943), was also a prominent actor in the program to place mathematics on a firm foundation. His Foundations of Geometry, published in 1899, introduced the requirements that every set of axioms be (1) complete, (2) independent, and (3) consistent. By 1920, Hilbert had expanded this into a research project that became known as Hilberts program. It called for all of mathematics to be restated, beginning with formal axioms that are consistent with each other and are sufficient to serve as the premises for proofs that would follow them. The proofs would ultimately certify the rest of mathematics.
Kurt Gdel
Now onto the scene comes Kurt Gdel (1906-1978), born in Brno, a city now in the Czech Republic. He was an exceptional student, reading all of Kant and mastering several languages by the age of 16. At 18 (1924) he moved to Vienna to study physics. Of course he became familiar with the works of Mach and Einstein. But soon he turned his attention to mathematics, under the influence of a lecturer named Philip Furtwngler, who gave his lectures from a wheelchair, without notes, while an assistant wrote equations on the blackboard. In 1929 Gdels PhD thesis on the completeness of elementary logic was accepted.
In Vienna, Gdel had also been attending weekly meetings of a group known as the Vienna Circle. This group included Gdels academic advisor the mathematician Hans Hahn (1879-1934), and Freges former student Rudolph Carnap (1891-1970). The group was keenly interested in the positivist movement, which came to be called logical positivism in mathematics. The group was in general agreement with Hilberts program. They were also heavily influenced by Ludwig Wittgenstein (1889-1951), who hoped to eradicate confused metaphysical thinking with careful analysis of language. You could say that Wittgenstein hoped to do for language what Russell and Hilbert wanted to do for mathematics.
Gdel listened carefully. He became very familiar with the tools established in Principia Mathematica, and with the details of Hilberts program. But apparently he did not agree with the essence of the program.
Hilberts Program
Hilberts program requires consistency among the axioms and completeness in the scope of proofs which they lead to. Consistency means the axioms cannot lead to contradictions, and completeness means that everything that is true will logically follow from the axioms, by proofs that are already known, or ones that will be found.
The requirements mean, among other things, that proof and truth are synonymous. In mathematics, anything that is proven is true, and under Hilberts program, anything that is true must be provable, at least eventually.
Hilberts program got its most famous explication on 3 September 1928, at a conference in Bologna. Hilberts address there, entitled Problems in laying the foundations of mathematics, received stormy applause at the beginning, and again at the end. In it he reviewed recent progress on the topic, and then he divided the problem of laying the foundations of mathematics into four related parts, or four separate problems.
Gdels Answer
Within about two years, Gdel settled all [four] of them in a definitive and surprising way. So wrote Hao Wang, one of Gdels biographers. He added, In many ways this was probably a unique occurrence in the history of science.
Gdel announced his theorem in 1930, at The Conference on the Epistemology of the Exact Sciences, in Knigsberg (now named Kaliningrad, a seaport in the Russian enclave between Poland and Lithuania on the Baltic Sea.) On October 7, Gdel delivered a short paper entitled, On formally undecidable propositions of Principia Mathematica and related systems I. It showed that Hilberts program could not succeed.
Gdels theorem can be restated in different ways, and it is very easy to state it wrongly. This is where language must be perfectly precise. If its slightly wrong, its just wrong. One certified statement of it is
For every consistent formalization of arithmetic, there exist arithmetic truths that are not provable within that formal system.
This is the very opposite of the prevailing philosophy at the time, which held that truth and proof were ultimately synonymous. Here we learn that some truths are impossible of proof within a given formal system. Furthermore, if one tries to solve the problem in a given instance by expanding the formal system, more unprovable truths will follow.
For an example of an unprovable truth, consider Euclids parallel postulate. It is stated various ways, but it is usually taken as the rule that parallel lines never meet. This apparent truth was never proven by Euclid, nor by anyone subsequently. Under Hilberts program, if the parallel postulate is true (and it probably is), a proof will eventually be found. But after the advent of Gdels theorem we know that Hilberts faith is unfounded. Within any sufficiently rich mathematical system, like Euclidean Geometry, some truths are unprovable. Perhaps the parallel postulate is one of them.
One of the first to notice the importance of Gdels theorem was John von Neuman (1903-1957), then 27 and among the attendees at Knigsberg. He quickly saw a way to expand Gdels proof and wrote to him about it. Gdel replied that he had seen the same thing and already had written it up as a followup to the original paper. The followup stated that no consistent system can be proven to be consistent from within. Its sometimes called Gdels second incompleteness theorem. Usually it is treated as a corollary of the main theorem just stated above.
How did he do it?
Gdels famous paper ran about twenty-five pages, and much of it consists of unfamiliar logical symbols. But most laymens explanations of the proof are longer than that. The proof is not easy to understand, even for people in the field. We will try simply to get the texture and flavor of it. One writer, Palle Yourgrau, says, To appreciate Gdels theorem is your birthright; let no one, including the mathematical police, deprive you of what you have a right to enjoy.
I. Gdel created a numbering scheme such that every term, statement, formula or proof in the terms of Principia Mathematica can be assigned a unique number called its Gdel number. Gdel numbers can be translated back into the terms, statements, formulae or proofs that they represent. For example, Gdel assigned simple numbers to the short list of logical operators like not, or, If...then, equals, and zero. Other numbers stand for x, y and z, and higher number stand for other algebraic symbols. These numbers are used as exponents to be applied to prime numbers used in sequence.
The important point is that every expression in the system (outlined in Principia Mathematica) whether an elementary sign, a sequence of signs, or a sequence of sequences can be assigned a unique Gdel number. Any given expression can be easily translated into its Gdel number. And, after only a few terms, the numbers get large. And they are multiplied together, so the resulting Gdel number of any expression is a very large number. For example, under one convention, the expression 0=0 would have the Gdel number of 243,000,000.
Not every number is a Gdel number. This is not unlike the fact that not every number is a prime number. There is a straightforward way to examine a number to see if it is a Gdel number. If it is, it can immediately be translated into the expression it came from. For example, under the same convention as used just above, 243,000,000 is the Gdel number for the expression 0=0.
II. Gdel showed that any operation in Principia Mathematica can be perfectly mirrored by an arithmetical operation on the corresponding Gdel numbers. For example, a proof in Principia Mathematica corresponds to a numerical relationship between the Gdel numbers that stand for the thing-to-be-proven, and the proof-argument.
To arithmetize something is not new. Think of what Descartes did in geometry. For Euclid, geometry was always about figures like triangles and circles. Descartes arithmetized the subject by showing that algebraic equations could stand for circles, parabolas, and almost any other figure. The equations could be manipulated to produce perfectly accurate solutions to geometry problems without any drawn figures just algebra. Now Gdel has arithmetized the syntax of Principia Mathematica.
III. Gdel wrote, The formula that happens to have Gdel number g is not provable via the rules of Principia Mathematica, and this formula translates to the Gdel number g! (I wondered if this is possible Can a metamathematical expression be fully represented by a Gdel number which is contained within the expression? But apparently it is possible. (Douglas Hofstadter devotes 7 pages to this issue.))
IV. On the one hand, Gdels expression can be treated arithmetically. By those means, the sentence is a true arithmetical formula. On the other hand, Gdels expression is a metamathematical one which says of itself that it is unprovable with the logic of Principia Mathematica. And so we have an example of something true (the arithmetic shows) which cannot be proven (because it says so)! This demonstrates the existence of an arithmetic truth that is not provable within the formal system containing it.
Reaction
Gdels theorem went almost unnoticed initially, John von Neuman being a rare exception. David Hilbert at first was angered, and tried to find flaws in the theorem. But he was soon convinced, and quietly shifted his focus away from the broad program he had launched to more specific issues. Bertrand Russell tried to devise a way around Gdels theorem, but his method was not sound. Later in life he asked, Are we to think that two plus two equals 4.001? Ludwig Wittgenstein seems to have never really understood the importance of Gdels theorem either.
Opinions about the impact of Gdels theorem seem to fall into two different categories. Some stress the importance of confining his result to the narrow realm of high-order mathematical logic. They emphasize what Gdels theorem does not say. They remind us that many things can still be proven.
Others are more effusive. Hao Wang writes, It is not exaggeration to regard [Gdels paper of Oct 7, 1930] as the greatest single piece of work in the whole history of mathematical logic.
For another example of an enthusiastic reaction, here is a passage from the play by Hugh Whitemore about Alan Turing. Turing (1912-1954) was the computer genius who figured out how to decipher the secret code used by the Germans in World War II. The play, Breaking the Code, opened at Haymarket Theatre in November, 1986.
Act One, Scene Five; Alan Turing is speaking --
[Bertrand Russell]s written an immense book about [telling right from wrong]: Principia Mathematica. His idea was to break down all mathematical concepts and arguments into little bits and then show that they could be derived from pure logic, but it didnt quite work out that way. After many years of intensive work, all he was able to do was to show that its terribly difficult to do anything of the kind. But it was an important book. Important and influential. It influenced both David Hilbert and Kurt Gdel.
[a brief digression] You could, I suppose make a comparison between these preoccupations and what physicists call splitting the atom. As analyzing the physical atom has led to a new kind of physics, so the attempt to analyze these mathematical atoms has led to a new kind of mathematics.
[resuming the main thread of his explanation] Hilbert took the whole thing a stage further. I dont suppose his name means m-m-much to you if anything but there we are, thats the way of the world; people never seem to hear of the really great mathematicians. Hilbert looked at the problem from a completely different angle, and he said, if we are going to have any fundamental system for mathematics like the one Russell was trying to work out it must satisfy three basic requirements: consistency, completeness and decidability. Consistency means that you wont ever get a contradiction in your own system; in other words youll never be able to f-f-follow the rules of your system and end up by showing that two and two make five. Completeness means that if any statement is true, there must be some way of proving it by using the rules of your system. And decidability means, uh decidability means that there must exist some method, some d-d-definite procedure or test, which can be applied to any given assertion and which will decide whether or not the assertion is provable. Hilbert thought that this was a very reasonable set of requirements to impose; but within a few years Kurt Gdel showed that no system for mathematics could be both consistent and complete. He did this by constructing a mathematical assertion that said in effect: This assertion cannot be proved. A classic paradox. This assertion cannot be proved. Well either it can be proved or it cant. If it can be proved we have a contradiction, and the system is inconsistent. If it cannot be proved then the assertion is true but it cant be proved; which means that the system is incomplete. Its a beautiful theorem, quite b-b-beautiful. Gdels theorem is the most beautiful thing I know.
Later life
Gdel himself continued to work on related issues in mathematics, after 1940 in Princeton. He also took an interest in Einsteins theory of relativity, and found implications in it that Einstein himself had not foreseen.
When he wasnt with Einstein or his own wife, Adele, he was sometimes seen walking alone, leaning forward, looking down, with his hands behind his back. Probably he was pondering issues in mathematics. But during WWII, people who saw him walking along the Maine coast sometimes wondered if he was a German spy!
In 1947, Gdel went to Trenton NJ to appear before Judge Phillip Foreman to be examined for US citizenship. Two witnesses accompanied him, Oskar Morgenstern and Albert Einstein. In anticipation of the examination, Gdel had been studying the US Constitution. Before the judge, he first took issue on the matter of his citizenship. His passport was German, because he had remained in Vienna for two months after the German invasion. But he insisted that he was an Austrian citizen. Judge Foreman made a comment about the wicked dictator, Hitler, adding, Fortunately, that is not possible here. At this point Gdel tried to launch into an explanation of how it could happen the US Constitution was flawed! Apparently Einstein had anticipated such a moment and had warned the others. All three aligned themselves to change the subject, and Gdel passed the exam.
In later life, Gdel suffered from hypochondria and paranoia. He was afraid that gasses from the heating system in his Princeton apartment were ruining his health. He insisted that his wife, Adele, taste his food first, in case someone was trying to poison him. When he died, he weighed only about 70 lbs. He had, in effect, starved himself to death.
Gdels legacy
The irrational aspects of Gdels personal life reinforce a negative opinion of his legacy for mathematics and philosophy. One biographer called him a spy in the house of logic. The prevailing way of assessing him seems to be this The skepticism and uncertainty that came down from Descartes, Locke, Berkeley and Hume, and that defeated Kant and many who followed, is actually worse than even they suspected. Gdels theorem leaves us absolutely no hope of establishing sure knowledge. He has left us in a hopeless situation.
But Gdel himself believed in a real world. He believed in objective truth that was not merely invention, but was here to be discovered. Furthermore, he believed in God, and ghosts, and in disembodied souls and life after death.
He had strong convictions. He knew that some things are true, but cannot be proven within any given system. To establish the veracity of such truths, one must leave the bounds of that system and view the situation as from outside or above it.
One of the implications of Gdels theorem is that the capabilities of a human mind can never be adequately matched by a machine.
Gdel knew that logic is an excellent tool. No one was a better logician than he. But what he proved with his Incompleteness Theorem was that truth must sometimes be discovered by means other than logic. He established, therefore, that intuition can be a valid way of knowing. I find this thought liberating.
Bibliography
John L. Casti and Werner DePauli, Gdel: A Life of Logic, Perseus Publishing, 2000.
John W.Dawson, Jr., Logical Dilemmas: The Life and Work of Kurt Gdel, A K Peters, Ltd., 1997.
Jaakko Hintikka, On Gdel, Belmont CA, Wadsworth/Thompson Learning, Inc., 2000.
Douglas Hofstadter, Gdel, Escher Bach: An Eternal Golden Braid, Basic Books, 1979.
Douglas Hofstadter, I Am a Strange Loop, Basic Books, 2007.
Jim Holt, What were Einstein and Gdel talking about? The New Yorker, 28 February 2005.
Ernest Nagel and James R. Newman, Gdels Proof, New York University Press, 1958.
Hao Wang, Reflections on Kurt Gdel, The MIT Press, 1987.
Palle Yourgrau, A World Without Time: The Forgotten Legacy of Gdel and Einstein, Basic Books, 2005.
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