Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (27 page)
2. For any consistent formal system
S
within which a certain amount of elementary arithmetic can be carried out, the consistency of
S
cannot be proved in
S
itself.
The words may appear to be benign, but the implications for the formalists’ program were far-reaching. Put somewhat simplistically, the incompleteness theorems proved that Hilbert’s formalist program was essentially doomed from the start. Gödel showed that any formal system that is powerful enough to be of any interest is inherently
either incomplete or inconsistent. That is, in the best case, there will always be assertions that the formal system can neither prove nor disprove. In the worst, the system would yield contradictions. Since it is always the case that for any statement
T,
either
T
or not-
T
has to be true, the fact that a finite formal system can neither prove nor disprove certain assertions means that true statements will always exist that are not provable within the system. In other words, Gödel demonstrated that no formal system composed of a finite set of axioms and rules of inference can
ever
capture the entire body of truths of mathematics. The most one can hope for is that the commonly accepted axiomatizations are only incomplete and not inconsistent.
Gödel himself believed that an independent, Platonic notion of mathematical truth did exist. In an article published in 1947 he wrote:
But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.
By an ironic twist of fate, just as the formalists were getting ready for their victory march, Kurt Gödel—an avowed Platonist—came and rained on the parade of the formalist program.
The famous mathematician John von Neumann (1903–57), who was lecturing on Hilbert’s work at the time, canceled the rest of his planned course and devoted the remaining time to Gödel’s findings.
Gödel the man was every bit as complex as his theorems. In 1940, he and his wife Adele fled Nazi Austria so he could take up a position at the Institute for Advanced Study in Princeton, New Jersey. There he became a good friend and walking partner of Albert Einstein. When Gödel applied for naturalization as an American citizen in 1948, it was Einstein who, together with Princeton University mathematician and economist Oskar Morgenstern (1902–77), accompanied Gödel to his interview at the Immigration and Naturalization Service office. The events surrounding this interview are generally
known, but they are so revealing about Gödel’s personality that I will give them now in full,
precisely
as they were recorded from memory by Oskar Morgenstern on September 13, 1971. I am grateful to Ms. Dorothy Morgenstern Thomas, Morgenstern’s widow, and to the Institute for Advanced Study for providing me with a copy of the document:
It was in 1946 that Gödel was to become an American citizen. He asked me to be his witness and as the other witness, he proposed Albert Einstein who also gladly consented. Einstein and I occasionally met and were full of anticipation as to what would happen during this time prior to the naturalization proceedings themselves and even during those.
Gödel whom I have seen of course time and again in the months before this event began to go in a thorough manner to prepare himself properly. Since he is a very thorough man, he started informing himself about the history of the settlement of North America by human beings. That led gradually to the study of the History of American Indians, their various tribes, etc. He called me many times on the phone to get literature which he diligently perused. There were many questions raised gradually and of course many doubts brought forth as to whether these histories really were correct and what peculiar circumstances were revealed in them. From that, Gödel gradually over the next weeks proceeded to study American history, concentrating in particular on matters of constitutional law. That also led him into the study of Princeton, and he wanted to know from me in particular where the borderline was between the borough and the township. I tried to explain that all this was totally unnecessary, of course, but with no avail. He persisted in finding out all the facts he wanted to know about and so I provided him with the proper information, also about Princeton. Then he wanted to know how the Borough Council was elected, the Township Council, and who the Mayor was, and how the Township Council functioned. He thought he might be asked about such matters. If he were to show that
he did not know the town in which he lived, it would make a bad impression.
I tried to convince him that such questions never were asked, that most questions were truly formal and that he would easily answer them; that at most they might ask what sort of government we have in this country or what the highest court is called, and questions of this kind. At any rate, he continued with the study of the Constitution.
Now came an interesting development. He rather excitedly told me that in looking at the Constitution, to his distress, he had found some inner contradictions and that he could show how in a perfectly legal manner it would be possible for somebody to become a dictator and set up a Fascist regime, never intended by those who drew up the Constitution. I told him that it was most unlikely that such events would ever occur, even assuming that he was right, which of course I doubted. But he was persistent and so we had many talks about this particular point. I tried to persuade him that he should avoid bringing up such matters at the examination before the court in Trenton, and I also told Einstein about it: he was horrified that such an idea had occurred to Gödel, and he also told him he should not worry about these things nor discuss that matter.
Many months went by and finally the date for the examination in Trenton came. On that particular day, I picked up Gödel in my car. He sat in the back and then we went to pick up Einstein at his house on Mercer Street, and from there we drove to Trenton. While we were driving, Einstein turned around a little and said, “Now, Gödel, are you
really
well prepared for this examination?” Of course, this remark upset Gödel tremendously, which was exactly what Einstein intended and he was greatly amused when he saw the worry on Gödel’s face. When we came to Trenton, we were ushered into a big room, and while normally the witnesses are questioned separately from the candidate, because of Einstein’s appearance, an exception was made and all three of us were invited to
sit down together, Gödel, in the center. The examinor [
sic
] first asked Einstein and then me whether we thought Gödel would make a good citizen. We assured him that this would certainly be the case, that he was a distinguished man, etc. And then he turned to Gödel and said, “Now, Mr. Gödel, where do you come from?”
Gödel: Where I come from? Austria.
The Examinor: What kind of government did you have in Austria?
Gödel: It was a republic, but the constitution was such that it finally was changed into a dictatorship.
The Examinor: Oh! This is very bad. This could not happen in this country.
Gödel: Oh, yes, I can
prove
it.
So of all the possible questions, just that critical one was asked by the Examinor. Einstein and I were horrified during this exchange; the Examinor was intelligent enough to quickly quieten Gödel and say, “Oh God, let’s not go into this,” and broke off the examination at this point, greatly to our relief. We finally left, and as we were walking out towards the elevators, a man came running after us with a piece of paper and a pen and approached Einstein and asked him for his autograph. Einstein obliged. When we went down in the elevator, I turned to Einstein and said, “It must be dreadful to be persecuted in this fashion by so many people.” Einstein said to me, “You know, this is just the last remnant of cannibalism.” I was puzzled and said, “How is that?” He said: “Yes, formerly they wanted your blood, now they want your ink.”
Then we left, drove back to Princeton, and as we came to the corner of Mercer Street, I asked Einstein whether he wanted to go to the Institute or home. He said, “Take me home, my work is not worth anything anyway anymore.” Then he quoted from an American political song (I unfortunately do not recall the
words, I may have it in my notes and I would certainly recognize it if somebody would suggest the particular phrase). Then off to Einstein’s home again, and then he turned back once more toward Gödel, and said, “Now, Gödel, this was your one but last examination.” Gödel: “Goodness, is there still another one to come?” and he was already worried. And then Einstein said, “Gödel, the next examination is when you step into your grave.” Gödel: “But Einstein, I don’t step into my grave,” and then Einstein said, “Gödel, that’s just the joke of it!” and with that he departed. I drove Gödel home. Everybody was relieved that this formidable affair was over; Gödel had his head free again to go about problems of philosophy and logic.
Late in life Gödel suffered from periods of serious mental disorder, which resulted in his refusal to eat. He died on January 14, 1978, of malnutrition and exhaustion.
Contrary to some popular misconceptions, Gödel’s incompleteness theorems do not imply that some truths will never become known. We also cannot infer from the theorems that the human capacity for understanding is somehow limited. Rather, the theorems only demonstrate the weaknesses and shortcomings of formal systems. It may therefore come as a surprise that in spite of the theorems’ broad import for the philosophy of mathematics, their impact on the effectiveness of mathematics as a theory-building machinery has been rather minimal. In fact, during the decades surrounding the publication of Gödel’s proof, mathematics was reaching some of its most spectacular successes in physical theories of the universe. Far from being abandoned as unreliable, mathematics and its logical conclusions were becoming increasingly essential for the understanding of the cosmos.
What this meant, however, was that the puzzle of the “unreasonable effectiveness” of mathematics became even thornier. Think about this for a moment. Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic—literally from the laws of thought. But how could such a deductive science so marvelously fit
natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Gödel. Now all that existed was an incomplete formal “game,” expressed in mathematical language. How could models based on such an “unreliable” system produce deep insights about the universe and its workings? Before I even attempt to address these questions, I want to sharpen them a bit further by examining a few case studies that demonstrate the subtleties of the effectiveness of mathematics.
8
UNREASONABLE EFFECTIVENESS?
In chapter 1, I noted that the success of mathematics in physical theories has two aspects: one I called “active” and one “passive.” The “active” side reflects the fact that scientists formulate the laws of nature in manifestly applicable mathematical terms. That is, they use mathematical entities, relations, and equations that were developed with an application in mind, often for the very topic under discussion. In those cases the researchers tend to rely on the perceived similarity between the properties of the mathematical concepts and the observed phenomena or experimental results. The effectiveness of mathematics may not appear to be so surprising in these cases, since one could argue that the theories were tailored to fit the observations. There is still, however, an astonishing part of the “active” use related to accuracy, which I will discuss later in this chapter. The “passive” effectiveness refers to cases in which entirely abstract mathematical theories had been developed, with no intended application, only to metamorphose later into powerfully predictive physical models.
Knot theory
provides a spectacular example of the interplay between active and passive effectiveness.
Knots
Knots are the stuff that even legends are made of. You may recall the Greek legend of the Gordian knot. An oracle decreed to the citizens of Phrygia that their next king would be the first man to enter the
capital in an oxcart. Gordius, an unsuspecting peasant who happened to ride an oxcart into town, thus became king. Overwhelmed with gratitude, Gordius dedicated his wagon to the gods, and he tied it to a pole with an intricate knot that defied all attempts to untie it. A later prophecy pronounced that the person to untie the knot would become king of Asia. As fate would have it, the man who eventually untied the knot (in the year 333 BC) was Alexander the Great, and he indeed subsequently became ruler of Asia. Alexander’s solution to the Gordian knot, however, was not exactly one we would call subtle or even fair—he apparently sliced through the knot with his sword!
But we don’t have to go all the way back to ancient Greece to encounter knots. A child tying his shoelaces, a girl braiding her hair, a grandma knitting a sweater, or a sailor mooring a boat are all using knots of some sort. Various knots were even given imaginative names, such as “fisherman’s bend,” “Englishman’s tie,” “cat’s paw,” “truelover’s knot,” “granny,” and “hangman’s knot.” Maritime knots in particular were historically considered sufficiently important to have inspired an entire collection of books about them in seventeenth century England. One of those books, incidentally, was written by none other than the English adventurer John Smith (1580–1631), better known for his romantic relationship with the native American princess Pocahontas.
The mathematical theory of knots was born in 1771 in a paper written by the French mathematician Alexandre-Théophile Vandermonde (1735–96). Vandermonde was the first to recognize that knots could be studied as part of the subject of
geometry of position,
which deals with relations depending on position alone, ignoring sizes and calculation with quantities. Next in line, in terms of his role in the development of knot theory, was the German “Prince of Mathematics,” Carl Friedrich Gauss. Several of Gauss’s notes contain drawings and detailed descriptions of knots, along with some analytic examinations of their properties. As important as the works of Vandermonde, Gauss, and a few other nineteenth century mathematicians were, however, the main driving force behind the modern mathematical knot theory came from an unexpected source—an attempt to explain the structure of matter. The idea originated in the mind of the famous
English physicist William Thomson, better known today as Lord Kelvin (1824–1907). Thomson’s efforts concentrated on formulating a theory of atoms, the basic building blocks of matter. According to his truly imaginative conjecture, atoms were really knotted tubes of ether—that mysterious substance that was supposed to permeate all space. The variety of chemical elements could, in the context of this model, be accounted for by the rich diversity of knots.
If Thomson’s speculation sounds almost crazy today, it is only because we have had an entire century to get used to and test experimentally the correct model of the atom, in which electrons orbit the atomic nucleus. But this was England of the 1860s, and Thomson was deeply impressed with the stability of complex smoke rings and their ability to vibrate—two properties considered essential for modeling atoms at the time. In order to develop the knot equivalent of a periodic table of the elements, Thomson had to be able to classify knots—find out which different knots are possible—and it was this need for knot tabulation that sparked a serious interest in the mathematics of knots.
As I explained already in chapter 1, a mathematical knot looks like a familiar knot in a string, only with the string’s ends spliced. In other words, a mathematical knot is portrayed by a closed curve with no loose ends. A few examples are presented in figure 54, where the three-dimensional knots are represented by their projections, or shadows, in the plane. The position in space of any two strands that cross each other is indicated in the figure by interrupting the line that depicts the lower strand. The simplest knot—the one called the
unknot
—is just a closed circular curve (as in figure 54a). The
trefoil knot
(shown in figure 54b) has three crossings of the strands, and the
figure eight knot
(figure 54c) has four crossings. In Thomson’s theory, these three knots could, in principle, be models of three atoms of increasing complexity, such as the hydrogen, carbon, and oxygen atoms, respectively. Still, a complete knot classification was badly needed, and the person who set out to sort the knots was Thomson’s friend the Scottish mathematical physicist Peter Guthrie Tait (1831–1901).
The types of questions mathematicians ask about knots are really not very different from those one might ask about an ordinary knotted string or a tangled ball of yarn. Is it really knotted? Is one knot
equivalent to another? What the latter question means is simply: Can one knot be deformed into the shape of the other without breaking the strands or pushing one strand through the other like a magician’s linking rings? The importance of this question is demonstrated in figure 55, which shows that by certain manipulations one can obtain two very different representations of what is actually the same knot. Ultimately, knot theory searches for some precise way of proving that certain knots (such as the trefoil knot and the figure eight knot; figures 54b and 54c) are really different, while ignoring the superficial differences of other knots, such as the two knots in figure 55.
Figure 54
Tait started his classification work the hard way. Without any rigorous mathematical principle to guide him, he compiled lists of curves with one crossing, two crossings, three crossings, and so on. In collaboration with the Reverend Thomas Penyngton Kirkman
(1806–95), who was also an amateur mathematician, he started sifting through the curves to eliminate duplications by equivalent knots. This was not a trivial task. You must realize that at every crossing, there are two ways to choose which strand would be uppermost. This means that if a curve contains, say, seven crossings, there are 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 knots to consider. In other words, human life is too short to complete in this intuitive way the classification of knots with tens of crossings or more. Nevertheless, Tait’s labor did not go unappreciated. The great James Clerk Maxwell, who formulated the classical theory of electricity and magnetism, treated Thomson’s atomic theory with respect, stating that “it satisfies more of the conditions than any atom hitherto considered.” Being at the same time well aware of Tait’s contribution, Maxwell offered the following rhyme:
Figure 55
Clear your coil of kinkings
Into perfect plaiting,
Locking loops and linkings
Interpenetrating.
By 1877, Tait had classified alternating knots with up to seven crossings. Alternating knots are those in which the crossings go alternately over and under, like the thread in a woven carpet. Tait also made a few more pragmatic discoveries, in the form of basic principles that were later christened
Tait’s conjectures.
These conjectures were so
substantial, by the way, that they resisted all attempts to prove them rigorously until the late 1980s. In 1885, Tait published tables of knots with up to ten crossings, and he decided to stop there. Independently, University of Nebraska professor Charles Newton Little (1858–1923) also published (in 1899) tables of nonalternating knots with ten or fewer crossings.
Lord Kelvin always thought fondly of Tait. At a ceremony at Peter-house College in Cambridge, where a portrait of Tait was presented, Lord Kelvin said:
I remember Tait once remarking that nothing but science is worth living for. It was sincerely said, but Tait himself proved it to be not true. Tait was a great reader. He would get Shakespeare, Dickens, and Thackeray off by heart. His memory was wonderful. What he once read sympathetically he ever after remembered.
Unfortunately, by the time Tait and Little completed their heroic work on knot tabulation, Kelvin’s theory had already been totally discarded as a potential atomic theory. Still, interest in knots continued for its own sake, the difference being that, as the mathematician Michael Atiyah has put it, “the study of knots became an esoteric branch of pure mathematics.”
The general area of mathematics where qualities such as size, smoothness, and in some sense even shape are ignored is called
topology
. Topology—the rubber-sheet geometry—examines those properties that remain unchanged when space is stretched or deformed in any fashion (without tearing off pieces or poking holes). By their very nature, knots belong in topology. Incidentally, mathematicians distinguish between
knots,
which are single knotted loops,
links,
which are sets of knotted loops all tangled together, and
braids,
which are sets of vertical strings attached to a horizontal bar at the top and bottom ends.
If you were not impressed with the difficulty of classifying knots, consider the following very telling fact. Charles Little’s table, published in 1899 after six years of work, contained forty-three nonal
ternating knots of ten crossings. This table was scrutinized by many mathematicians and believed to be correct for seventy-five years. Then in 1974, the New York lawyer and mathematician Kenneth Perko was experimenting with ropes on his living room floor. To his surprise, he discovered that two of the knots in Little’s table were in fact the same. We now believe that there are only forty-two distinct nonalternating knots of ten crossings.
While the twentieth century witnessed great strides in topology, progress in knot theory was relatively slow. One of the key goals of the mathematicians studying knots has been to identify properties that truly distinguish knots. Such properties are called
invariants of knots
—they represent quantities for which any two different projections of the same knot yield precisely the same value. In other words, an ideal invariant is literally a “fingerprint” of the knot—a characteristic property of the knot that does not change by deformations of the knot. Perhaps the simplest invariant one can think of is the minimum number of crossings in a drawing of the knot. For instance, no matter how hard you try to disentangle the trefoil knot (figure 54b), you will never reduce the number of crossings to fewer than three. Unfortunately, there are a number of reasons why the minimal number of crossings is not the most useful invariant. First, as figure 55 demonstrates, it is not always easy to determine whether a knot has been drawn with the minimum number of crossings. Second and more important, many knots that are actually different have the same number of crossings. In figure 54, for instance, there are three different knots with six crossings, and no fewer than seven different knots with seven crossings. The minimum number of crossings, therefore, does not distinguish most knots. Finally, the minimum number of crossings, by its very simplistic nature, does not provide much insight into the properties of knots in general.