The Road to Reality: A Complete Guide to the Laws of the Universe

by Roger Penrose

What they had to do first was to discover how to disentangle the true from the suppositional in mathematics.

The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers.

But the main thrust of this book has to do with the first of these mysteries: the remarkable relationship between mathematics and the actual behaviour of the physical world. No proper appreciation of the extraordinary power of modern science can be achieved without at least some acquaintance with these mathematical ideas.

In particular, it is not at all an obvious matter whether physical ‘squares’ exist on a cosmological scale in the actual universe. This is a matter for observation, and the evidence at the moment appears to be conflicting

Now, it turns out that the hyperbolic notion of an angle between any two curves, at their point of intersection, is precisely the same as the Euclidean measure of angle between the two curves at the intersection point. A representation of this nature is called conformal. For this reason, the particular representation of hyperbolic geometry that Escher used is sometimes referred to as the conformal model of the hyperbolic plane.

However, there is actually something particularly elegant and remarkable about what does happen when we add up the angles of a hyperbolic triangle: the shortfall is always proportional to the area of the triangle.

We shall find that logarithms are really very beautiful and mysterious entities (as is the number e), as well as being important in many different contexts.

Hyperbolic geometry, with this definition of distance, turns out to have all the properties of Euclidean geometry apart from those which need the parallel postulate. We can construct triangles and other plane figures of different shapes and sizes, and we can move them around ‘rigidly’ (keeping their hyperbolic shapes and sizes from changing) with as much freedom as we can in Euclidean geometry, so that a natural notion of when two shapes are ‘congruent’ arises, just as in Euclidean geometry, where ‘congruent’ means ‘can be moved around rigidly until they come into coincidence’. All the white fish in Escher’s woodcut are indeed congruent to each other, according to this hyperbolic geometry, and so also are all the black fish.

There are indeed other representations of hyperbolic geometry in terms of Euclidean geometry, which are distinct from the conformal one that Escher employed. One of these is that known as the projective model. Here, the entire hyperbolic plane is again depicted as the interior of a circle in a Euclidean plane, but the hyperbolic straight lines are now represented as straight Euclidean lines (rather than as circular arcs). There is, however, a price to pay for this apparent simplification, because the hyperbolic angles are now not the same as the Euclidean angles, and many people would regard this price as too high.

To end this section, let us return to the question of the existence of squares in hyperbolic geometry. Although squares whose angles are right angles do not exist in hyperbolic geometry, there are ‘squares’ of a more general type, whose angles are less than right angles. The easiest way to construct a square of this kind is to draw two straight lines intersecting at right angles at a point O. Our ‘square’ is now the quadrilateral whose four vertices are the intersections A, B, C, D (taken cyclicly) of these two lines with some circle with centre O. See Fig. 2.18. Because of the symmetry of the figure, the four sides of the resulting quadrilateral ABCD are all equal and all of its four angles must also be equal. But are these angles right angles? Not in hyperbolic geometry.

From the assumption that Euclid’s fifth postulate was false, he derived, instead of an actual contradiction, a host of strange-looking, barely believable, but interesting theorems. However, strange as these results appeared to be, none of them was actually a contradiction. As we now know, there was no chance that Saccheri would find a genuine contradiction in this way, for the reason that hyperbolic geometry does actually exist, in the mathematical sense that there is such a consistent structure. In the terminology of §1.3, hyperbolic geometry inhabits Plato’s world of mathematical forms. (The issue of hyperbolic geometry’s physical reality will be touched upon in §2.7 and §28.10.)

Beltrami’s pseudo-sphere model seems to be the one that persuaded mathematicians of the consistency of plane hyperbolic geometry, since the measure of hyperbolic distance agrees with the Euclidean distance along the surface. However, it is a somewhat awkward model, because it represents hyperbolic geometry only locally, rather than presenting the entire geometry all at once, as do Beltrami’s other models.

Hyperbolic geometry also works perfectly well in higher dimensions.

What about our actual universe on cosmological scales? Do we expect that its spatial geometry is Euclidean, or might it accord more closely with some other geometry, such as the remarkable hyperbolic geometry (but in three dimensions) that we have been examining in §§2.4–6. This is indeed a serious question. We know from Einstein’s general relativity (which we shall come to in §17.9 and §19.6) that Euclid’s geometry is only an (extraordinarily accurate) approximation to the actual geometry of physical space.

Fortunately for those, such as myself, who are attracted to the beauties of hyperbolic geometry, and also to the magnificence of modern physics, there is another role for this superb geometry that is undisputedly fundamental to our modern understanding of the physical universe. For the space of velocities, according to modern relativity theory, is certainly a three-dimensional hyperbolic geometry (see §18.4), rather than the Euclidean one that would hold in the older Newtonian theory.

Rational numbers are rather simple quantities, being describable and understood in simple finite terms; yet they can be used to specify distances that are as small as we please or as large as we please. If all geometry could be done with rationals, then this would make things relatively simple and easily comprehensible. The notion of an ‘irrational’ number, on the other hand, requires infinite processes, and this had presented considerable difficulties for the ancients (and with good reason).

One reason that it can be important to identify the precise assumptions that go into a proof—even though some of these assumptions could be perfectly ‘obvious’ things—is that mathematicians are frequently interested in other kinds of entity than those with which the proof might be originally concerned. If these other entities satisfy the same assumptions, then the proof will still go through and the assertion that had been proved will be seen to have a greater generality than originally perceived, since it will apply to these other entities also. On the other hand, if some of the needed assumptions fail to hold for these alternative entities, then the assertion that may turn out to be false for these entities.

By the 19th and 20th centuries, however, the view had emerged that the mathematical notion of number should stand separately from the nature of physical space.

Accordingly, Richard Dedekind and Georg Cantor developed their ideas of what real numbers ‘are’ by use of notions that do not directly refer to geometry.

The ability to take limits is fundamental for many mathematical notions, and it is this that gives the real numbers their particular strengths.

It is as though Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought. An example of this is the real-number system itself. We have no direct evidence from Nature that there is a physical notion of ‘distance’ that extends to arbitrarily large scales; still less is there evidence that such a notion can be applied on the indefinitely tiny level.

a crucial role for the real-number system has been that it provides a necessary framework for the standard formulation of the calculus (see Chapter 6). All successful dynamical theories have required notions of the calculus for their formulations. Now, the conventional approach to calculus requires the infinitesimal nature of the reals to be what it is. That is to say, on the small end of the scale, it is the entire range of the real numbers that is in principle being made use of. The ideas of calculus underlie other physical notions, such as velocity, momentum, and energy. Consequently, the real-number system enters our successful physical theories in a fundamental way for our description of all these quantities also.

However, as we now understand quantum mechanics, that theory does not force us (nor even lead us) to the view that there is a discrete or granular nature to space, time, or energy at its tiniest levels (see Chapters 21 and 22, particularly the last sentence of §22.13). Nevertheless, the idea has remained with us that there may indeed be, at root, such a fundamental discreteness to Nature, despite the fact that quantum mechanics, in its standard formulation, certainly does not imply this.

There are various ways in which natural numbers can be introduced in pure mathematics, and these do not seem to depend upon the actual nature of the physical universe at all. Basically, it is the notion of a ‘set’ which needs to be brought into play, this being an abstraction that does not appear to be concerned, in any essential way, with the specific structure of the physical universe.

It is a remarkable fact that only in about the last hundred years has it become apparent that the system of integers does indeed seem to have such direct physical relevance. The first example of a physical quantity which seems to be appropriately quantified by integers is electric charge.19 As far as is known (although there is as yet no complete theoretical justification of this fact), the electric charge of any discrete isolated body is indeed quantified in terms of integral multiples, positive, negative, or zero, of one particular value, namely the charge on the proton (or on the electron, which is the negative of that of the proton).

Recall that all we have done is to demand that −1 have a square root, together with demanding that the normal laws of arithmetic be retained, and we have ascertained that these demands can be satisfied consistently. This seems like a fairly simple thing to have done. But now for the magic!

However, it is hardly possible to convey serious mathematical ideas without the use of a certain amount of notation. I appreciate that there will be many readers who are uncomfortable with these things. My advice to such readers is basically just to read the words and not to bother too much about trying to understand the equations. At least, just skim over the various formulae and press on. There will, indeed, be quite a number of serious mathematical expressions scattered about this book, particularly in some of the later chapters. My guess is that certain aspects of understanding will eventually begin to come through even if you make little attempt to understand what all the expressions actually mean in detail.

But this is just the very beginning. We can ask about cube roots, fifth roots, 999th roots, πth roots—or even i-th roots. We find, miraculously, that whatever complex root we choose and whatever complex number we apply it to (excluding 0), there is always a complex-number solution to this problem.

We are now thinking of our power series as functions of the complex variable z, rather than the real variable x, and we can ask for those locations of z in the complex plane for which the series converges and those for which it diverges. The remarkable general answer,9 for any power series whatever a0 + a1z + a2z2 + a3z3 + ··· , is that there is some circle in the complex plane, centred at 0, called the circle of convergence, with the property that if the complex number z lies strictly inside the circle then the series converges for that value of z, whereas if z lies strictly outside the circle then the series diverges for that value of z.

when we multiply two complex numbers we add their arguments and multiply their moduli.[5.4] Note the remarkable fact here that, as far as the rule for the arguments is concerned, we have converted multiplication into addition. This fact is the basis of the use of logarithms

We shall see that logarithms have a profound role to play in relation to complex numbers. Indeed, the argument of a complex number really is a logarithm, in a certain clear sense. We shall try to understand how this comes about.

For this, we need to find the right definition of ‘b raised to the power z’, for complex z, and we want the same formula as the above, namely bw+z = bw × bz, to hold when the exponents w and z are complex. In fact, the procedure for doing this mirrors, to some extent, the very history of generalizing, step by step, from the positive integers to the complex numbers, as was done, starting from Pythagoras, via the work of Eudoxos, through Brahmagupta, until the time of Cardano and Bombelli (and later), as was indicated in §3.4, §4.1. First, the notion of ‘bz’ is initially understood, when z is a positive integer, as simply b × b × ··· × b, with z b’s multiplied together; in particular, b1 = b. Then (following the lead of Brahmagupta) we allow z to be zero, realizing that to preserve bw+z = bw × bz we need to define b0 = 1. Next we allow z to be negative, and realize, for the same reason, that for the case z = – 1 we must define b−1 to be the reciprocal of b (i.e. 1/b), and that b−n, for a natural number n, must be the nth power of b−1. We then try to generalize to the situations when z is a fraction, starting with the case z = 1/n, where n is a positive integer. Repeated application of bw × bz = bw+z leads us to conclude that (bz)n = bzn; thus, putting z = 1/n, we derive the fact that b1/n is an nth root of b.

More generally, we have n distinct complex answers for b1/n, when n is a positive integer: 1, 2, 3, 4, 5, . . . . In fact, we have some finite number of answers whenever n is a (non-zero) rational number. If n is irrational, then we have an infinite number of answers, as we shall be seeing shortly.

It is not immediately obvious that such an inverse to ez will necessarily exist. However, it turns out in fact that, for any complex number w, apart from 0, there always does exist z such that w = ez, so we can define log w = z. But there is a catch here: there is more than one answer.

z = log r + iθ, where log r is the ordinary natural logarithm of a positive real number—the inverse of the real exponential. Why? It is intuitively clear from Fig. 5.7 that such a real logarithm function exists. In Fig. 5.7a we have the graph of r = ex. We just flip the axes over to get the graph of the inverse function x = log r, as in Fig. 5.7b. It is not so surprising that the real part of z = log w is just an ordinary real logarithm. What is somewhat more remarkable4 is that the imaginary part of z is just the angle θ that is the argument of the complex number w. This fact makes explicit my earlier comment that the argument of a complex number is really just a form of logarithm.

Recall that there is an ambiguity in the definition of the argument of a complex number. We can add any integer multiple of 2π to θ, and this will do just as well (recall Fig. 5.4b). Accordingly, there are many different solutions z for a given choice of w in the relation w = ez. If we take one such z, then z + 2πin is another possible solution, where n is any integer that we care to choose. Thus, the logarithm of w is ambiguous up to the addition of any integer multiple of 2πi.

This feature of the complex logarithm seems, at this stage, to be just an awkward irritation. However, we shall be seeing in §7.2 that it is absolutely central to some of the most powerful, useful, and magical properties of complex numbers. Complex analysis depends crucially upon it.

There is indeed a magic about the direct way that such somewhat complicated formulae spring from simple complex-number expressions.

This ambiguity leads us into all sorts of problems if we are not careful.[5.10] The best way of avoiding these problems appears to be to adopt the rule that the notation wz is used only when a particular choice of log w has been specified. (In the special case of ez, the tacit convention is always to take the particular choice log e = 1. Then the standard notation ez is consistent with our more general wz.) Once this choice of log w is specified, then wz is unambiguously defined for all values of z.

More generally, we can return to the question of zth roots of a non-zero complex number w, where z is any non-zero complex number, that was alluded to in §4.2. We can express such a zth root as the expression w1/z, and we generally get an infinite number of alternative values for this, depending upon which choice of log w is specified.

When z = n is a positive integer, things are much simpler, and we get just n roots. A situation of particular interest occurs, in this case, when w = 1. Then, specifying some possible values of log 1 successively, namely 0, 2πi, 4πi, 6πi, . . . , we get 1 = e0, e2πi/n, e4πi/n, e6πi/n, . . . for the possible values of 11/n. We can write these as 1, є, є2, є3, . . . , where є = e2πi/n. In terms of the complex plane, we get n points equally spaced around the unit circle, called nth roots of unity. These points constitute the vertices of a regular n-gon (see Fig. 5.10). (Note that the choices, −2πi, − 4πi, − 6πi, etc., for log 1 would merely yield the same nth roots, in the reverse order.)