In the years since 1985, we have realized that both supergravity and string theory, belong to a larger structure, known as M theory.why it should be called M Theory, is completely obscure. M theory, is not a theory in the usual sense.Rather it is a collection of theories, that look very different, but which describe the same physical situation. These theories are related by mappings, or correspondences, called dualities, which imply that they are all reflections of the same underlying theory. Each theory in the collection, works well in the limit, like low energy, or low dilaton, in which its effective coupling is small, but breaks down when the coupling is large. This means that none of the theories, can predict the future of the universe, to arbitrary accuracy.for that, one would need a single formulation of M-theory, that would work in all situations.
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-theory 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 Goedel's theorem.This says that any finite system of axyoms, is not sufficient to prove every result in mathematics.
Goedel's theorem is proved using statements that refer to themselves.sUch statements can lead to paradoxes.aN example is, this statement is false. If the statement is true, it is false.and if the statement is false, it is true. Another example is, the barber of Corfoo shaves every man who does not shave himself. Who shaves the barber?if he shaves himself, then he doesn't, and if he doesn't, then he does. Goedel went to great lengths to avoid such paradoxes, by carefully distinguishing between mathematics, like 2+2 =4,and meta mathematics, or statements about mathematics, such as mathematics is cool, or mathematics is consistent. that is why his paper is so difficult to read.but the idea is quite simple. First Goedel showed that each mathematical formula, like 2+2=4, can be given a unique number, the Goedel number.the Goedel number of 2+2=4, is *. Second, the meta mathematical statement, the sequence of formulas A, is a proof of the formula B, can be expressed as an arithmetical relation between the Goedel numbers for A- and B. Thus meta mathematics can be mapped into arithmetic, though I'm not sure how you translate the meta mathematical statement, 'mathematics is cool'. Third and last, consider the self referring Goedel statement, G.tHis is, the statement G can not be demonstrated from the axyoms of mathematics. Suppose that G could be demonstrated.tHen the axyoms must be inconsistent, because one could both demonstrate G, and show that it can not be demonstrated. On the other hand, if G can't be demonstrated, then G is true.bY the mapping into numbers, it corresponds to a true relation between numbers, but one which can not be deduced from the axyoms. Thus mathematics is either inconsistent, or imcomplete.tHe smart money, is on imcomplete.
What is the relation between Goedels theorem, and whether we can formulate the theory of the universe, in terms of a finite number of principles. One connection is obvious.aCcording to the positivist philosophy of science, a physical theory, is a mathematical model.sO if there are mathematical results that can not be proved, there are physical problems that can not be predicted. One example might be the Golbach conjecture.gIven an even number of wood blocks, can you always divide them into two piles, each of which can not be arranged in a rectangle.tHat is, it contains a prime number of blocks.
Although this is incompleteness of sort, it is not the kind of unpredictability I mean.gIven a specific number of blocks, one can determine with a finite number of trials, whether they can be divided into two primes. But I think that quantum theory and gravity together, introduces a new element into the discussion, that wasn't present with classical Newtonian theory. In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed, and can contain an arbitrary amount of information, without affecting the universes they describe. But we are not angels, who view the universe from the outside.iNstead, we and our models, are both part of the universe we are describing. Thus a physical theory, is self referencing, like in Goedels theorem.oNe might therefore expect it to be either inconsistent, or imcomplete.tHe theories we have so far, are ~both inconsistent, and imcomplete.
Quantum gravity is essential to the argument..THe information in the model, can be represented by an arrangement of particles.aCcording to quantum theory, a particle in a region of a given size, has a certain minimum amount of energy. Thus, as I said earlier, models don't live rent free.tHey cost energy.By Einsteins famous equation, E = mc squared, energy is equivalent to mass.aNd mass causes systems to collapse under gravity. It is like getting too many books together in a library.tHe floor would give way, and create a black hole that would swallow the information. Remarkably enough, Jacob Bekenstein and I, found that the amount of information in a black hole, is proportional to the area of the boundary of the hole, rather than the volume of the hole, as one might have expected. The black hole limit on the concentration of information, is fundamental, but it has not been properly incorporated into any of the formulations of M theory that we have so far. They all assume that one can define the wave function at each point of space.bUt that would be an infinite density of information, which is not allowed. On the other hand, if one can't define the wave function point wise, one can't predict the future to arbitrary accuracy, even in the reduced determinism of quantum theory. What we need, is a formulation of M theory, that takes account of the black hole information limit.bUt then our experience with supergravity and string theory, and the analogy of Goedels theorem, suggest that even this formulation, will be imcomplete.
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. Goedels theorem ensured there would always be a job for mathematicians.I think M theory will do the same for physicists. I'm sure Dirac would have approved.