4.1 Alice in Gödel-Land
The little girl was exhausted due to the long journey she had completed from her home to Gödel-land. However, because of the excitement of being in such a new and strange place, she found she was unable to settle down.
“Mom”, she said, “I can’t sleep.” To emphasise the point, she sat up, fluffed her pillow and tossed her head back down on the bed with a dramatic thump.
Her mother hid a smile; she knew where Alice was heading. “Do they have bedtime stories in Gödel-land?” Alice asked, as though the idea had just occurred to her.
“Of course, they do. Which one would you like?”.
“Little Red Riding Hood.” It was her favourite – for the moment.
“All right, but you really must try to sleep. We’ve got a busy day tomorrow.” It was Alice’s first visit to Gödel-land, and her mother had many treats planned for her. She settled back and her mother began the story. “Once upon a time, a little girl, whose name was Cinderella, lived with her two older step-sisters …”.
“No!” interrupted Alice. Her mother paused, and regarded her daughter, who was struggling to sit up again. “Mom, I don’t want Cinderella, I want the story about Little Red Riding Hood!”.
“Ah, Alice, but this is Gödel-land. Things are different here. We can’t just restrict ourselves to the fairyland of Little Red Riding Hood. While we’re here, we have to consider a wider fairyland, where Cinderella, Snow-white, Pinocchio and all those other characters live. Otherwise story-telling here isn’t possible.”
“But … but …” Alice was sure this could not be right. She had never heard of anything so silly, but her mother was very clever, and knew so much more than she did.
“Now let me go on, and do try to sleep.” Alice settled back down to listen. “‘I have some cookies here,’ said Cindy’s step-mother, ‘and I would like you to deliver them to your Aunt, to her house deep in the woods.’”.
“Mom, no!”.
“Now what?”.
“It was to her grandma, not her aunt!”.
“I know, Alice, I know. But as I told you before, here in Gödel-land we can’t just limit ourselves to the world of grandmas. We have to consider a wider world with all sorts of other relatives, and sometimes even friends. Now, if you don’t let me continue – without interruption – we’ll never finish.” Alice pouted, but restrained herself with an effort, and her mother carried on. “So, Cindy set out through the woods, and halfway to her aunt’s, what do you think she saw, in the middle of the path, barring her way, but a –”.
“Wolf! A big, bad wolf.” shouted Alice.
“No. Actually, it was a cat.”
“A cat?”.
“That’s right. A Cheshire cat.”
“It was a wolf. It’s always been a wolf. You’re spoiling the story. And it’s Little Red Riding Hood, not Cinderella.”
“Now, Alice,” said her mother, beginning to lose her patience, “You’re not listening. In Gödel-land, we can’t limit ourselves to the world of wolves –”.
“I know, I know. I suppose we have to consider the wider world of all animals. But I don’t like it. I don’t like it one little bit! It’s so confusing. Why do we have to do silly things like that? Why?”.
“Do you really want to know?”.
“Yes!”.
“All right then, I’ll try to explain.”
Her mother pulled her armchair closer to Alice’s bed, rearranged the cushions, and settled to begin what she knew would be a very long and complicated narrative. After just a few words, however, when she next glanced down at her daughter, she noticed that the little girl’s eyes were closed. Alice was fast asleep.
4.2 The Price of Cakes in Gödel-Land
Our next goal is to try to explain what Alice’s mother could not, i.e. the odd customs and rules of Gödel-land. However, first we need to abandon the land of fairy stories, and enter into an altogether different abstract dominion, the mathematical world of numbers. To begin, let us clarify what we mean by a number. The reader may think this is unnecessary. Surely, a number is just one of those quantities, like time and temperature, that everybody understands, but nobody seems quite able to define clearly (try asking around).
Nevertheless, the concept is not really so difficult: if you have a collection (or group) of any objects whatsoever and you disregard the nature of the objects, the number is the only characteristic that remains of the group. For instance, if you collect 2 pebbles, 3 cookies and one mango, and if you do not care what the objects are (i.e., pebbles, cookies or mangoes), the only thing that describes your collection is the number 6 (6 objects) …but don’t eat the pebbles! The numbers, thus defined, are called integers (or natural numbers), and are contained in one small portion of what we may call the number world.
As trade began to flourish between humans, it became clear that the world of integers was insufficient to encompass transactions when debt was incurred by some individuals. In this case instead of owning 3 cookies in their collection, the debtors might owe 3 cookies to their neighbour’s collection, with the two parties agreeing that the debt would be paid off some time in the future. The number world was therefore extended by including negative numbers to describe the debt. Much later, the number zero was included in the number world.1
Problems of the type: what is the price of one cookie, if it costs $12 to buy four? are common in commerce. The answer is, of course, $3. Suppose now that we replace “cookie” in the above question with some other object, e.g. liquorice stick. The answer is still the same, i.e. $3, irrespective of what object we choose. We can show this independence of the nature of the object by writing the question in the form: what is the price of x, if it costs $12 to buy 4x?, or even more succinctly by: Find x if 4x = 12. The answer is x = 3 (i.e. 12/4) dollars, if our unit of currency happens to be the dollar. What we have introduced above is the concept of an equation (where two quantities, one of which is unknown, are located on opposing sides of an equals sign. The rules for solving the equation (i.e. finding the value of the unknown x) are contained in elementary algebra.
So far, so good, and for the moment our integer number world seems adequate for our purposes. However, what happens if four cookies cost $10? (i.e. 4x = 10). We easily see that there is no integer in our number world to tell us how much one cookie costs. We do not need a calculator to find that x = 10/4 = $2.50. But 2.50 is no longer an integer. In other words, the number world of integers is too limited to include the solution of our new equation. To solve it we must go into a much wider number world, the world of “rational” numbers, which includes integers and fractions, such as 10/4, or 2.50.
Let us now assume that our problem is even more complicated, and we are asked to find a number (again let us call it x) such that when it is multiplied by itself (or squared) we obtain the value 4. We can write this in the form: x*x = 4, or more concisely: x2 = 4. Solving this equation presents no great problem, and we see that x = 2, because the square of 2 is 4, or alternatively, the square root of 4 is 2. (Another possible solution to this equation is x = −2).
A right-angled triangle: the length of its hypotenuse is expressed as an irrational number
In other words, to find the solution of the equation x2 = 2, one must go into a still wider number world, the one of “real” numbers, which includes both the rational and the irrational numbers. By the way, the word “irrational” conveys a deep apprehension of these numbers, which were not wanted by their discoverers (the school of Pythagoras) because they contradicted their faith in a numerological “purity”. According to legend, the discoverer, a student of Pythagoras, was forced to commit suicide, lest his sacrilegious discovery be propagated to others.
Surely, now at last, our number world is sufficiently broad to allow us to carry out all the numerical operations we are ever likely to encounter. Not quite. For instance, take up your calculator and ask it to calculate the square root of −1 (or any other negative number). You will obtain a fairly dismissive error message from your machine, as if you had done something rather stupid. However, negative numbers are a legitimate part of our number world, and taking the square root is a legitimate mathematical operation. Surely one might expect a valid result from such an endeavour.
Mathematicians certainly thought so, and introduced into the number world a quantity i which they defined to be the square root of −1(so that i * i, or i2 = −1). The square root of −4 then became 2i. Numbers of this type were deemed imaginary, because mathematicians thought that i was only a “trick” to solve the equations. The number world would now seem to have two separate divisions, one comprising real numbers and the other imaginary numbers. However, there are also complex numbers, which have one component that is real and another component that is imaginary. Complex numbers were for many years regarded with the same suspicion that Pythagoras had accorded irrational numbers. However, in the 19th Century it was realized that they were essential tools for the solution of many problems in science and engineering.
So, at last, our number world is complete. We can perform any mathematical operation we like on complex numbers and still find a result that is not outside the domain of complex numbers.3 If you replace “telling a story” with “solving an equation”, you can now see the analogy between Alice’s story and the contents of the current section: both cases illustrate the need of going to a wider system. In the next section, we’ll see what all this has to do with Gödel, and with some of the current developments of Physics.
4.3 Gödel and Completeness
Kurt Gödel, one of the most significant logicians of the twentieth century, as a student in Vienna in 1925. Image: Wikimedia Commons (Public domain - Kurt Gödel Papers, the Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ. https://commons.wikimedia.org/wiki/File:Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg)
It is curious that something similar had been said thirty years earlier about Einstein himself, when the Nobel Prize judges hesitated before assigning the prize to him, since their policy had always been to reward only experimental discoveries or experimentally verified theories, which was not yet the case with Einstein’s General Relativity. Eventually a way around the impasse was found by giving the Nobel Prize to Einstein for his work on the photoelectric effect. On that occasion one of the judges remarked that assigning the prize to him was an honour more for the prize than for Einstein himself. However, although Einstein is still something of a popular cult figure more than 60 years after his death, Gödel’s name and work are largely unknown, even among many scientists and engineers.
Kurt Gödel was born in 1906 in Brno, Slovak Republic, which at that time belonged to the Austro-Hungarian Empire. He studied in Vienna, where in 1929 he completed his Ph.D. in mathematical logic. His dissertation led him to his completeness and, later, incompleteness theorems, which he published two years later and which represent his most relevant scientific achievements. In 1933 he became lecturer at the University of Vienna, where he remained up to 1940, when he lost his position because he had many Jewish friends, and moved to the USA, where he became Professor at the Institute of Advanced Studies (IAS) in Princeton in 1953. He remained in Princeton until 1976, two years before his death (1978).
Perhaps, from a human point of view, the cause of his death is almost as puzzling as his beloved paradoxes. Late in his life, he developed severe paranoia, and would eat only food prepared for him by his wife, Adele, for fear of being poisoned. When she became hospitalised for six months in 1977, he refused to eat, starving himself to death.
Let us leave to psychologists the task of fathoming Gödel’s complex personality. His introvert nature was doubtless much tested by the murder of a friend during the Nazi period and by the upheaval, both enthusiastic and acrimonious, that followed his discoveries. Instead, let us see if we can obtain an inkling of the nature of the work that so impressed Einstein (and others), and the implications that still resound through mathematics and physics.
As we have discussed in Chap. 2, the aim of formal mathematics was (and still largely is) to begin with a small number of axioms and definitions, and by using the rules of logic, to deduce ever more complex truths, or theorems. Probably the most extreme example of this approach is the Principia Mathematica of Russell and Whitehead, which we cited in the previous Chapter.
One of the banes that can occur in logic is the discovery of a paradox, which may indicate either that we have made a mistake in our reasoning, or that our logical framework is insufficient for our purposes, as we saw in the last Section when attempting to find the square root of 2 while limiting ourselves to rational numbers.
In Chap. 1, we encountered Zeno’s Paradox of Achilles and the Tortoise. Another famous example, according to tradition, is due to Epimenides of Crete, who came up with the statement “All Cretans are liars”.4 Now this statement presents an intrinsic self-contradiction, since, if it is true, it cannot be asserted by a Cretan, because a Cretan would lie, and vice versa if it is false. Since Cretans, like everybody else, are not necessarily always lying or always truthful, the so-called “Liar paradox” is usually stated more succinctly as “This statement is false”. If the statement is true, its content indicates it must be false, and vice versa. Now, since language is essentially a tool of communication, rather than of logic, it is not surprising that it contains contradictions or inconsistencies. What is really surprising is that the same thing also happens in mathematics.
While working on Principia Mathematica, Bertrand Russell uncovered a paradox in Set Theory, which he was using as the formal mathematical framework for his ambitious opus. Normally first encountered by mathematics students in advanced University courses, Set Theory enjoyed a brief period of popularity in the 1960s when it formed the basis of “New Math” and was taught to primary school students to help them learn arithmetic. As satirist/mathematician, Tom Lehrer, stressed: “the important thing is to understand what you’re doing, rather than to get the right answer.” New Math has long been confined to the dustbin of history as another failed experiment by misguided educationalists. However, Set Theory is still an important component of the foundations of mathematics.
Russell’s Paradox (see Appendix 4.1) is of a self-referential nature, similar to the Liar’s paradox above. Obviously it is disconcerting to base a formal analysis of the structure of mathematics on a theory with a paradox at its heart. Various solutions have been proposed for the paradox. However, what is of interest to us here is that it inspired a young student by the name of Kurt Gödel to investigate the logical foundations of mathematics and the nature of mathematical proofs. His discoveries have shown that the formal derivation of all of mathematics from a few basic axioms, as attempted by Whitehead and Russell, is actually impossible. Let us try to be a little more specific, without going into technical details.
Two things we require from any branch of mathematics are consistency and completeness. In the case of Whitehead and Russell, it was arithmetic that they attempted to put on a rigorous formal basis. By consistency, we mean that it should not be possible, by following two different chains of reasoning from the same axioms, to prove that something is both true and false. By completeness, we mean that if something is true, it should be possible to prove that it is true, starting from our axioms. There should not be any theorems that cannot be proved. We may not be smart enough to prove them at the moment, but we should have the hope, sometime in the future, of discovering a proof. It should certainly not be the case that they can never be proved.
However, what Gödel succeeded in proving is that, if we have a system that is consistent (i.e. does not contain any contradictions), then it must contain at least one statement that is true, but which can never be proved. (See Appendix 4.2).
Trying to add an axiom to a branch of mathematics to “prove” an unprovable theorem is like trying to repair a leaky rainwater tank
We see from the above why Gödel’s work has severe implications for mathematics. Indeed, debate still rages in philosophical circles as to whether it demonstrates that there are “ineluctable limits to human reason” [2]. Leaving such lofty considerations to one side, in the next Section we shall explore the relevance of Gödel’s work to physics.
4.4 Hawking and His Epiphany
If there is incompleteness in mathematics, can we also expect to find an analogous kind of incompleteness in physics? The hope of many eminent physicists at the end of the twentieth century, following a flurry of far-reaching advances in the fields of Quantum Mechanics, General Relativity and High Energy Physics (amongst others), was to discover an all-encompassing theory that would unite these disparate fields. They called such a theory a TOE, or Theory of Everything. In analogy with what Gödel has shown to be the case in mathematics, are we now to suspect that there may be contradictions in physics that can never be resolved, or observed phenomena that we can never explain within the laws of physics? In other words, are there fundamental limits to our knowledge, i.e., some prohibition that prevents us from ever arriving at a TOE?
Certainly there are areas of physics, and we will encounter some of these in Part 2 of this book, where there are very definite limits to our knowledge. One example is a black hole. We shall discuss these strange entities in more detail in Chap. 7. Suffice it to say here that they are extremely dense objects, with gravitational fields so strong that nothing (not even light) can escape from their clutches, if it chances to venture too close. The central region of a black hole thus seems always destined to remain out of reach of scientific observation, and therefore beyond the limits of physics.
However, even if black holes represent an instance of incompleteness of our knowledge of the universe, they were certainly not predicted, or discovered as a consequence of Gödel’s theorem. (For that matter, neither were the irrational and complex numbers, which appeared, as uninvited guests, long before Gödel’s time). The eventual consequences of Gödel’s theorem in physics stem from the realization that physical theories are essentially mathematical models. Hence, if some mathematical results cannot be proved, then surely there must also be some physical theories that cannot be proved.
“Up to now, most people have implicitly assumed that there is an ultimate theory that we will eventually discover. Indeed, I myself have suggested we might find it quite soon. However, M-theory5 has made me wonder if this is true. Maybe it is not possible to formulate the theory of the universe in a finite number of statements. This is very reminiscent of Gödel's theorem. This says that any finite system of axioms is not sufficient to prove every result in mathematics.”
Stephen Hawking (1942–2018). (Image public domain, courtesy of NASA StarChild Learning Center https://commons.wikimedia.org/wiki/File:Stephen_Hawking.StarChild.jpg (accessed 2020/9/1))
“Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel’s theorem ensured there would always be a job for mathematicians. I think M-theory will do the same for physicists.”
In other words, Hawking concludes, on a note of optimism for the new generations of young physicists, that a TOE would be bad news for physicists looking for a job: much better if there is no TOE and, as a consequence, always something new to discover.
Hawking was not the first to question the impact of Gödel's theorem on physics. Stanley Jaki in his book “The Relevance of Physics” [5] states that a TOE must be a consistent non-trivial mathematical theory, and hence, following Gödel, incomplete. Freeman Dyson maintains: the laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them.” [6] On the other hand, many physicists find the use of Gödel's theorem in this context to be unconvincing, and that in an infinite universe, there will always be some things that cannot be proved.
If Hawking’s conjecture is correct, how do we begin the search for evidence of Gödel's incompleteness in physics? We cannot simply seek out some area that is at the moment poorly understood, since it is more likely that our ignorance depends only on the fact that we have not yet discovered the right theory. This has always been the case in the past, before any of the great discoveries of science.
More interesting is when we are faced with a theory, such as Quantum Mechanics (see Chap. 5), that seems to work extremely well, since it fits all the observations or experimental data, and yields accurate predictions, but which defies our common sense. Here the question arises: is this evidence of some incompleteness of the theory, or is it only a consequence of the fact that our common sense has evolved in an environment markedly different from the realm of Quantum Mechanics? Indeed, in our everyday life we are in contact only with macroscopic objects and phenomena, while Quantum Mechanics helps us deal with a quite different microscopic world.
In the previous chapters we have learned that both our senses and our reason can be very deceitful, and that our deeply-held intuitions of concepts such as truth and reality may be dubious, or even fallacious.
Hence, if there exists indeed some incompleteness in the sense of Gödel, where should we turn to look for it? In 2015, three mathematicians and computer scientists, applying an admittedly simplified model, found that the problem of determining whether there is an energy gap between the lowest energy levels in a material is “undecidable” [7]. We shall explore what we mean by energy levels in Chap. 8. However, for our purposes here, suffice it to say that this issue is of major importance in the understanding of the physics of superconductors. When we say that the question is undecidable, we mean that it is impossible to find an answer, no matter how hard we try. This is a direct consequence of Gödel’s theorem.
So, is there any chance that Gödel’s theorem might account for some of the great puzzles of modern physics, that we are presently compelled to sweep under the carpet, such as the inconsistency between Quantum Mechanics and Relativity? We have now reached the point where we can proceed no further towards the answers to these questions without a deeper understanding of modern physics, and the limits to knowledge that it has revealed.
In Part 2 of this book, we make a brief visit to the various coalfaces of current physics research, where thousands of dedicated workers are chipping away in search of the occasional diamond. However, it is not our goal to take part in this venture ourselves here, since we have insufficient time and resources available for that purpose. Instead, our role will be to guide the interested spectator.
To enjoy the beauty of a Beethoven sonata, one does not require a degree in the theory of music, even if such a qualification may add a deeper insight to our appreciation. Nor are all the audience members, on their feet at Stratford-on-Avon after a stirring performance of Hamlet, graduates from a Stanislavsky school of acting. Similarly, it should be possible for non-specialists to obtain an appreciation of the important problems being addressed by physicists today, and have a spot of fun for themselves along the way. This is what we shall attempt in Part 2. In Part 3 we shall return to tackle once more the questions raised above, taking advantage of what we will learn in Part 2.