ART & ORGANISM
READING
Chapter 3 of The Emperors New Mind :
Concerning Computers, Minds, and the Laws of Physics
by Roger Penrose
Oxford University Press, Incorporated, 2002.
3. MATHEMATICS AND REALITY
THE LAND OF TOR’BLED-NAM
LET US IMAGINE that we have been travelling on a great journey to some far-off
world. We shall call this world Tor’Bled-Nam. Our remote sensing device has
picked up a signal which is now displayed on a screen in front of us. The image
comes into focus and we see (Fig. 3.1):
Fig. 3.1. A first glimpse of a strange world.
What can it be? Is it some strange-looking insect? Perhaps, instead, it is a darkcoloured
lake, with many mountain streams entering it. Or could it be some vast
and oddly shaped alien city, with roads going off in various directions to small
towns and villages nearby? Maybe it is an island – and then let us try to find
whether there is a nearby continent with which it is associated. This we can do by
‘backing away’, reducing the magnification of our sensing device by a linear
factor of about fifteen. Lo and behold, the entire world springs into view (Fig.
3.2):
Fig. 3.2. Tor ‘Bled-Nam’ in its entirety. The locations of the magnifications shown
in Figs 3.1, 3.3, and 3.4 are indicated beneath the arrows.
Our ‘island’ is seen as a small dot indicated below ‘Fig. 3.1’ in Fig. 3.2. The
filaments (streams, roads, bridges?), from the original island all come to an end,
with the exception of the one attached at the inside of its right-hand crevice, which
finally joins on to the very much larger object that we see depicted in Fig. 3.2.
This larger object is clearly similar to the island that we saw first – though it is not
precisely the same. If we focus more closely on what appears to be this object’s
coastline we see innumerable protuberances – roundish, but themselves possessing
similar protuberances of their own. Each small protuberance seems to be attached
to a larger one at some minute place, producing many warts upon warts. As the
picture becomes clearer, we see myriads of tiny filaments emanating from the
structure. The filaments themselves are forked at various places and often
meander wildly. At certain spots on the filaments we seem to see little knots of
complication which our sensing device, with its present magnification, cannot
resolve. Clearly the object is no actual island or continent, nor a landscape of any
kind. Perhaps, after all, we are viewing some monstrous beetle, and the first that
we saw was one of its offspring, attached to it still, by some kind of filamentary
umbilical cord.
Let us try to examine the nature of one of our creature’s warts, by turning up
the magnification of our sensing device by a linear factor of about ten (Fig. 3.3 –
the location being indicated under ‘Fig. 3.3’ in Fig. 3.2). The wart itself bears a
strong resemblance to the creature as a whole – except just at the point of
attachment. Notice that there are various places in Fig. 3.3 where five filaments
come together. There is perhaps a certain ‘fiveness’ about this particular wart (as
there would be a ‘threeness’ about the uppermost wart). Indeed, if we were to
examine the next reasonable-sized wart, a little down on the left on Fig. 3.2, we
should find a ‘sevenness’ about it; and for the next, a ‘nineness’, and so on. As we
enter the crevice between the two largest regions of Fig. 3.2, we find warts on the
right characterized by odd numbers, increasing by two each time. Let us peer deep
down into this crevice, turning up the magnification from that of Fig. 3.2 by a
factor of about ten (Fig. 3.4). We see numerous other tiny warts and also much
swirling activity. On the right, we can just discern some tiny spiral ‘seahorse tails’
– in an area we shall know as ‘seahorse valley’. Here we shall find, if the
magnification is turned up enough, various ‘sea anemones’ or regions with a
distinctly floral appearance. Perhaps, after all, this is indeed some exotic coastline
– maybe some coral reef, teeming with life of all kinds. What might have seemed
to be a flower would reveal itself, on further magnification, to be composed of
myriads of tiny, but incredibly complicated structures, each with numerous
filaments and swirling spiral tails. Let us examine one of the larger seahorse tails
in some detail, namely the one just discernible where indicated as ‘Fig. 3.5’ in Fig.
3.4 (which is attached to a wart with a ‘29-ness’ about it!). With a further
approximate 250-fold magnification, we are presented with the spiral depicted in
Fig. 3.5. We find that this is no ordinary tail, but is itself made up of the most
complicated swirlings back and forth, with innumerable tiny spirals, and regions
like octopuses and seahorses.
At many places, the structure is attached just where two spirals come together.
Let us examine one of these places (indicated below ‘Fig. 3.6’ in Fig. 3.5),
increasing our magnification by a factor of about thirty. Behold: do we discern a
strange but now familiar object in the middle? A further increase of magnification
by a factor of about six (Fig. 3.7) reveals a tiny baby creature – almost identical to
the entire structure we have been examining! If we look closely, we see that the
filaments emanating from it differ a little from those of the main structure, and
they swirl about and extend to relatively much greater distances. Yet the tiny
creature itself seems to differ hardly at all from its parent, even to the extent of
possessing offspring of its own, in closely corresponding positions. These we could
again examine if we turned up the magnification still further. The grandchildren
would also resemble their common ancestor – and one readily believes that this
continues indefinitely. We may explore this extraordinary world of Tor’Bled-Nam
as long as we wish, tuning our sensing device to higher and higher degrees of
magnification. We find an endless variety: no two regions are precisely alike – yet
there is a general flavour that we soon become accustomed to. The now familiar
beetle-like creatures emerge at yet tinier and tinier scales. Every time, the
neighbouring filamentary structures differ from what we had seen before, and
present us with fantastic new scenes of unbelievable complication.
Fig. 3.4. The main crevice. ‘Seahorse valley’ is just discernible on the lower right.
What is this strange, varied and most wonderfully intricate land that we have
stumbled upon? No doubt many readers will already know. But some will not. This
world is nothing but a piece of abstract mathematics – the set known as the
Mandelbrot set.1 Complicated it undoubtedly is; yet it is generated by a rule of
remarkable simplicity! To explain the rule properly, I shall first need to explain
what a complex number is. It is as well that I do so here. We shall need complex
numbers later. They are absolutely fundamental to the structure of quantum
mechanics, and are therefore basic to the workings of the very world in which we
live. They also constitute one of the Great Miracles of Mathematics. In order to
explain what a complex number is, I shall need, first, to remind the reader what is
meant by the term ‘real number’. It will be helpful, also, to indicate the
relationship between that concept and the very reality of the ‘real world’!
Fig. 3.5. A close-up of a seahorse tail.
Fig. 3.6. A further magnification of a joining point where two spirals come
together. A tiny baby is just visible at the central point.
Fig. 3.7. On magnification, the baby is seen closely to resemble the entire world.
REAL NUMBERS
Recall that the natural numbers are the whole quantities:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, . . .
These are the most elementary and basic amongst the different kinds of number.
Any type of discrete entity can be quantified by the use of natural numbers: we
may speak of twenty-seven sheep in a field, of two lightning flashes, twelve
nights, one thousand words, four conversations, zero new ideas, one mistake, six
absentees, two changes of direction, etc. Natural numbers can be added or
multiplied together to produce new natural numbers. They were the objects of our
general discussion of algorithms, as given in the last chapter.
However some important operations may take us outside the realm of the
natural numbers – the simplest being subtraction. For subtraction to be defined in
a systematic way, we need negative numbers; we can set out the whole system of
integers
. . ., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7, . . .
for this purpose. Certain things, such as electric charge, bank balances, or dates*
are quantified by numbers of this kind. These numbers are still too limited in their
scope, however, since we can come unstuck when we try to divide one such
number by another. Accordingly, we shall need the fractions, or rational numbers as
they are called
0, 1, –1, 1/2, –1/2, 2, –2, 3/2, –3/2, 1/3, . . .
These suffice for the operations of finite arithmetic, but for a good many
purposes we need to go further than this and include infinite or limiting
operations. The familiar – and mathematically highly important – quantity π, for
example, arises in many such infinite expressions. In particular, we have:
π = 2{(2/1)(2/3)(4/3)(4/5)(6/5)(6/7)(8/7)(8/9). . .}
and
π = 4(1 – 1/3 + 1/5 – + 1/9 – 1/11 + . . .).
These are famous expressions, the first having been found by the English
mathematician, grammarian, and cipher expert John Wallis, in 1655; and the
second, in effect, by the Scottish mathematician and astronomer (and inventor of
the first reflecting telescope) James Gregory, in 1671. As with π, numbers defined
in this way need not be rational (i.e. not of the form n/m, where n and m are
integers with m non-zero). The number system needs to be extended in order that
such quantities can be included.
This extended number system is referred to as the system of ‘real’ numbers –
those familiar numbers which can be represented as infinite decimal expansions,
such as:
–583.70264439121009538. . .
In terms of such a representation we have the well-known expression for π:
π = 3.14159265358979323846. . .
Among the types of number that can also be represented in this way are the
square roots (or cube roots or fourth roots, etc.) of positive rational numbers, such
as:
√2 = 1.41421356237309504. . .;
or, indeed, the square root (or cube root etc.) of any positive real number, as with
the expression for π found by the great Swiss mathematician Leonhard Euler:
π = √ {6(1 + 1/4 + 1/9 + 1/25 + 1/36 +. . .)}
Real numbers are, in effect, the familiar kinds of number that we have to deal
with in everyday life, although normally we are concerned merely with
approximations to such numbers, and are happy to work with expansions
involving only a small number of decimal places. In mathematical statements,
however, real numbers may need to be specified exactly, and we require some sort
of infinite description such as an entire infinite decimal expansion, or perhaps
some other infinite mathematical expression such as the above formulae for π
given by Wallis, Gregory, and Euler. (I shall normally use decimal expansions in
my descriptions here, but only because these are most familiar. To a
mathematician, there are various rather more satisfactory ways of presenting real
numbers, but we shall not need to worry about this here.)
It might be felt that it is impossible to contemplate an entire infinite expansion,
but this is not so. A simple example where one clearly can contemplate the entire
sequence is
1/3 = 0.333333333333333. . .,
where the dots indicate to us that the succession of 3s carries on indefinitely. To
contemplate this expansion, all we need to know is that the expansion does indeed
continue in the same way indefinitely with 3s. Every rational number has a
repeated (or finite) decimal expansion, such as
93/74 = 1.2567567567567567. . .,
where the sequence 567 is repeated indefinitely, and this can also be
contemplated in its entirety. Also, the expression
0.220002222000002222220000000222222220. . .,
which defines an irrational number, can certainly be contemplated in its entirety
(the string of 0s or 2s simply increasing in length by one each time), and many
similar examples can be given. In each case, we shall be satisfied when we know a
rule according to which the expansion is constructed. If there is some algorithm
which generates the successive digits, then knowledge of that algorithm provides
us with a way of contemplating the entire infinite decimal expansion. Real
numbers whose expansions can be generated by algorithms are called computable
numbers (see also p. 66). (The use of a denary rather than, say, a binary
expansion here has no significance. The numbers which are ‘computable’ in this
sense are just the same numbers whichever base for an expansion is used.) The
real numbers π and √2 that we have just been considering are examples of
computable numbers. In each case the rule would be a little complicated to state in
detail, but not hard in principle.
However, there are also many real numbers which are not computable in this
sense. We have seen in the last chapter that there are non-computable sequences
which are nevertheless perfectly well defined. For example, we could take the
decimal expansion whose nth digit is 1 or 0 according to whether or not the nth
Turing machine acting on the number n stops or does not stop. Generally, for a
real number, we just ask that there should be some infinite decimal expansion. We
do not ask that there should be an algorithm for generating the nth digit, nor even
that we should be aware of any kind of rule which in principle defines what the
nth digit actually is.2 Computable numbers are awkward things to work with. One
cannot keep all one’s operations computable, even when one works just with
computable numbers. For example, it is not even a computable matter to decide,
in general, whether two computable numbers are equal to one another or not! For
this kind of reason, we prefer to work, instead, with all real numbers, where the
decimal expansion can be anything at all, and need not just be, say, a computable
sequence.
Finally, I should point out that there is an identification between a real number
whose decimal expansion ends with an infinite succession of 9s and one whose
expansion ends with an infinite succession of Os; for example
–27.1860999999. . . = –27.1861000000. . .
HOW MANY REAL NUMBERS ARE THERE?
Let us pause for a moment to appreciate the vastness of the generalization that
has been achieved in moving from the rational numbers to the real numbers.
One might think, at first, that the number of integers is already greater than
the number of natural numbers; since every natural number is an integer whereas
some integers (namely the negative ones) are not natural numbers, and similarly
one might think that the number of fractions is greater than the number of
integers. However, this is not the case. According to the powerful and beautiful
theory of infinite numbers put forward in the late 1800s by the highly original
Russian-German mathematician Georg Cantor, the total number of fractions, the
total number of integers and the total number of natural numbers are all the same
infinite number, denoted 0 (‘aleph nought’). (Remarkably, this kind of idea had
been partly anticipated some 250 years before, in the early 1600s, by the great
Italian physicist and astronomer Galileo Galilei. We shall be reminded of some of
Galileo’s other achievements in Chapter 5.) One may see that the number of
integers is the same as the number of natural numbers by setting up a ‘one-to-one
correspondence’ as follows:
Note that each integer (in the left-hand column) and each natural number (in the
right-hand column) occurs once and once only in the list. The existence of a oneto-
one correspondence like this is what, in Cantor’s theory, establishes that the
number of objects in the left-hand column is the same as the number of objects in
the right-hand column. Thus, the number of integers is, indeed, the same as the
number of natural numbers. In this case the number is infinite, but no matter.
(The only peculiarity that occurs with infinite numbers is that we can leave out
some of the members of the list and still find a one-to-one correspondence between
the two lists!) In a similar, but somewhat more complicated way, we can set up a
one-to-one correspondence between the fractions and the integers. (For this we
can adapt one of the ways of representing pairs of natural numbers, the
numerators and denominators, as single natural numbers; see Chapter 2, p. 56.)
Sets that can be put into one-to-one correspondence with the natural numbers are
called countable, so the countable infinite sets are those with So elements. We have
now seen that the integers are countable, and so also are all the fractions.
Are there sets which are not countable? Although we have extended the system,
in passing from the natural numbers to first the integers and then the rational
numbers, we have not actually increased the total number of objects that we have
to work with. We have seen that the number of objects is actually countable in
each case. Perhaps the reader has indeed got the impression by now that all
infinite sets are countable. Not so; for the situation is very different in passing to
the real numbers. It was one of Cantor’s remarkable achievements to show that
there are actually more real numbers than rationals. The argument that Cantor
used is the ‘diagonal slash’ that was referred to in Chapter 2 and that Turing
adapted in his argument to show that the halting problem for Turing machines is
insoluble. Cantor’s argument, like Turing’s later one, proceeds by reductio ad
absurdum. Suppose that the result we are trying to establish is false, i.e. that the
set of all real numbers is countable. Then the real numbers between 0 and 1 are
certainly countable, and we shall have some list providing a one-to-one pairing of
all such numbers with the natural numbers, such as:
I have marked out the diagonal digits in bold type. These digits are, for this
particular listing,
1, 4, 1, 0, 0, 3, 1, 4, 8, 5, 1,. . .
and the diagonal slash procedure is to construct a real number (between 0 and 1)
whose decimal expansion (after the decimal point) differs from these digits in each
corresponding place. For definiteness, let us say that the digit is to be 1 whenever
the diagonal digit is different from 1 and it is 2 whenever the diagonal digit is 1.
Thus, in this case we get the real number
0.21211121112 . . .
This real number cannot appear in our listing since it differs from the first number
in the first decimal place (after the decimal point), from the second number in the
second place, from the third number in the third place, etc. This is a contradiction
because our list was supposed to contain all real numbers between 0 and 1. This
contradiction establishes what we are trying to prove, namely that there is no oneto-
one correspondence between the real numbers and the natural numbers and,
accordingly, the number of real numbers is actually greater than the number of
rational numbers and is not countable.
The number of real numbers is the infinite number labelled C. (C stands for
continuum, another name for the system of real numbers.) One might ask why this
number is not called, 1 say. In fact the symbol 1 stands for the next infinite
number greater than 0 and it is a famous unsolved problem to decide whether in
fact C = 1, the so-called continuum hypothesis.
It may be remarked that the computable numbers, on the other hand, are
countable. To count them we just list, in numerical order, those Turing machines
which generate real numbers (i.e. which produce the successive digits of real
numbers). We may wish to strike from the list any Turing machine which
generates a real number that has already appeared earlier in the list. Since the
Turing machines are countable, it must certainly be the case that the computable
real numbers are countable. Why can we not use the diagonal slash on that list
and produce a new computable number which is not in the list? The answer lies in
the fact that we cannot computably decide, in general, whether or not a Turing
machine should actually be in the list. For to do so would, in effect, involve our
being able to solve the halting problem. Some Turing machines may start to
produce the digits of a real number, and then get stuck and never again produce
another digit (because it ‘doesn’t stop’). There is no computable means of deciding
which Turing machines will get stuck in this way. This is basically the halting
problem. Thus, while our diagonal procedure will produce some real number, that
number will not be a computable number. In fact, this argument could have been
used to show the existence of non-computable numbers. Turing’s argument to show
the existence of classes of problems which cannot be solved algorithmically, as
was recounted in the last chapter, follows precisely this line of reasoning. We shall
see other applications of the diagonal slash later.
‘REALITY’ OF REAL NUMBERS
Setting aside the notion of computability, real numbers are called ‘real’ because
they seem to provide the magnitudes needed for the measurement of distance,
angle, time, energy, temperature, or of numerous other geometrical and physical
quantities. However, the relationship between the abstractly defined ‘real’
numbers and physical quantities is not as clear-cut as one might imagine. Real
numbers refer to a mathematical idealization rather than to any actual physically
objective quantity. The system of real numbers has the property, for example, that
between any two of them, no matter how close, there lies a third. It is not at all
clear that physical distances or times can realistically be said to have this
property. If we continue to divide up the physical distance between two points,
we would eventually reach scales so small that the very concept of distance, in the
ordinary sense, could cease to have meaning. It is anticipated that at the ‘quantum
gravity’ scale of 1020th of the size* of a subatomic particle, this would indeed be
the case. But to mirror the real numbers, we would have to go to scales
indefinitely smaller than this: 10200th, 102OOOth, or 1010 200th of a particle size, for
example. It is not at all clear that such absurdly tiny scales have any physical
meaning whatever. A similar remark would hold for correspondingly tiny intervals
of time.
The real number system is chosen in physics for its mathematical utility,
simplicity, and elegance, together with the fact that it accords, over a very wide
range, with the physical concepts of distance and time. It is not chosen because it
is known to agree with these physical concepts over all ranges. One might well
anticipate that there is indeed no such accord at very tiny scales of distance or
time. It is commonplace to use rulers for the measurement of simple distances, but
such rulers will themselves take on a granular nature when we get down to the
scale of their own atoms. This does not, in itself, prevent us from continuing to use
real numbers in an accurate way, but a good deal more sophistication is needed
for the measurement of yet smaller distances. We should at least be a little
suspicious that there might eventually be a difficulty of fundamental principle for
distances on the tiniest scale. As it turns out, Nature is remarkably kind to us, and
it appears that the same real numbers that we have grown used to for the
description of things at an everyday scale or larger retain their usefulness on
scales much smaller than atoms – certainly down to less than one-hundredth of the
‘classical’ diameter of a sub-atomic particle, say an electron or proton – and
seemingly down to the ‘quantum gravity scale’, twenty orders of magnitude
smaller than such a particle! This is a quite extraordinary extrapolation from
experience. The familiar concept of real-number distance seems to hold also out to
the most distant quasar and beyond, giving an overall range of at least 1042, and
perhaps 1060 or more. The appropriateness of the real number system is not often
questioned, in fact. Why is there so much confidence in these numbers for the
accurate description of physics, when our initial experience of the relevance of
such numbers lies in a comparatively limited range? This confidence – perhaps
misplaced – must rest (although this fact is not often recognized) on the logical
elegance, consistency, and mathematical power of the real number system,
together with a belief in the profound mathematical harmony of Nature.
COMPLEX NUMBERS
The real number system does not, as it turns out, have a monopoly with regard to
mathematical power and elegance. There is still a certain awkwardness in that,
for example, square roots can be taken only of positive numbers (or zero) and not
of negative ones. From the mathematical point of view – and leaving aside, for
the moment, any question of direct connection with the physical world – it turns
out to be extremely convenient to be able to extract square roots of negative as
well as positive numbers. Let us simply postulate, or ‘invent’ a square root for the
number –1. We shall denote this by the symbol ‘i’, so we have:
i2 = –1.
The quantity i cannot, of course, be a real number since the product of a real
number with itself is always positive (or zero, if the number is itself zero). For this
reason the term imaginary has been conventionally applied to numbers whose
squares are negative. However, it is important to stress the fact that these
‘imaginary’ numbers are no less real than the ‘real’ numbers that we have become
accustomed to. As I have emphasized earlier, the relationship between such ‘real’
numbers and physical reality is not as direct or compelling as it may at first seem
to be, involving, as it does, a mathematical idealization of infinite refinement for
which there is no clear a priori justification from Nature.
Having a square root for – 1, it is now no great effort to provide square roots
for all the real numbers. For if a is a positive real number, then the quantity
i × √a
is a square root of the negative real number –a. (There is also one other square
root, namely – i × √a.) What about i itself? Does this have a square root? It surely
does. For it is easily checked that the quantity
(1 + i) / √2
(and also the negative of this quantity) squares to i. Does this number have a
square root? Once again, the answer is yes; the square of
or its negative is indeed (1 + i)/√2.
Notice that in forming such quantities we have allowed ourselves to add
together real and imaginary numbers, as well as to multiply our numbers by
arbitrary real numbers (or divide by non-zero real numbers, which is the same
thing as multiplying by their reciprocals). The resulting objects are what are
referred to as complex numbers. A complex number is a number of the form
a + ib
where a and b are real numbers, called the real part and the imaginary part
respectively, of the complex number. The rules for adding and multiplying two
such numbers follow the ordinary rules of (school) algebra, with the added rule
that i2 = –1:
(a + ib) + (c + id) = (a + c) + i (b + d)
(a + ib) × (c + id) =(ac – bd) + i (ad + bc).
A remarkable thing now happens! Our motivation for this system of numbers
had been to provide the possibility that square roots can always be taken. It
achieves this task, though this is itself not yet obvious. But it also does a great deal
more: cube roots, fifth roots, ninety-ninth roots, nth roots, (1 + i)th roots, etc. can
all be taken with impunity (as the great eighteenth century mathematician
Leonhard Euler was able to show). As another example of the magic of complex
numbers, let us examine the somewhat complicated-looking formulae of
trigonometry that one has to learn in school; the sines and cosines of the sum of
two angles
sin (A + B) = sin A cos B + cos A sin B,
cos (A + B) = cos A cos B – sin A sin B,
are simply the imaginary and real parts, respectively, of the much simpler (and
much more memorable!) complex equation*
eiA + iB = eiA eiB.
Here all we need to know is ‘Euler’s formula’ (apparently also obtained many
years before Euler by the remarkable 16th century English mathematician Roger
Cotes)
eiA = cos A + i sin A,
which we substitute into the equation above. The resulting expression is
cos(A + B) + i sin(A + B) = (cos A + i sin A) (cos B + i sin B),
and multiplying out the right-hand side we obtain the required trigonometrical
relations.
What is more, any algebraic equation
a0 + a1z + a2z2 + a3z3 + . . . + anzn = 0
(for which a0, a1, a2 . . . an are complex numbers, with an ≠ 0) can always be
solved for some complex number z. For example,
there is a complex number z satisfying the relation
z102 + 999z33 – πz2 = – 417 +i,
though this is by no means obvious! The general fact is sometimes referred to as
‘the fundamental theorem of algebra’. Various eighteenth century mathematicians
had struggled to prove this result. Even Euler had not found a satisfactory general
argument. Then, in 1831, the great mathematician and scientist Carl Friedrich
Gauss gave a startlingly original line of argument and provided the first general
proof. A key ingredient of this proof was to represent the complex numbers
geometrically, and then to use a topological* argument.
Actually, Gauss was not really the first to use a geometrical description of
complex numbers. Wallis had done this, crudely, about two hundred years earlier,
though he had not used it to nearly such powerful effect as had Gauss. The name
normally attached to this geometrical representation of complex numbers belongs
to Jean Robert Argand, a Swiss bookkeeper, who described it in 1806, although
the Norwegian surveyor Caspar Wessel had, in fact, given a very complete
description nine years earlier. In accordance with this conventional (but not
altogether historically accurate) terminology I shall refer to the standard
geometrical representation of complex numbers as the Argand plane.
The Argand plane is an ordinary Euclidean plane with standard Cartesian
coordinates x and y, where x marks off horizontal distance (positive to the right
and negative to the left) and where y marks off vertical distance (positive
upwards and negative downwards). The complex number
z = x + iy
is then represented by the point of the Argand plane whose coordinates are
(x, y)
(see Fig. 3.8).
Fig. 3.8. The Argand plane, depicting a complex number z = x + iy.
Note that 0 (regarded as a complex number) is represented by the origin of
coordinates, and 1 is represented as a particular point on the x-axis.
The Argand plane simply provides us with a way of organizing our family of
complex numbers into a geometrically useful picture. This kind of thing is not
really something new to us. We are already familiar with the way that real
numbers can be organized into a geometrical picture, namely the picture of a
straight line that extends indefinitely in both directions. One particular point of
the line is labelled 0 and another is labelled 1. The point 2 is placed so that its
displacement from 1 is the same as the displacement of 1 from 0; the point 1/2 is
the mid-point of 0 and 1; the point –1 is situated so that 0 lies mid-way between it
and 1, etc., etc. The set of real numbers displayed in this way is referred to as the
real line. For complex numbers we have, in effect, two real numbers to use as
coordinates, namely a and b, for the complex number a + ib. These two numbers
give us coordinates for points on a plane – the Argand plane. As an example, I
have indicated in Fig. 3.9 approximately where the complex numbers
u = 1 + i 1.3, v = – 2 + i, w = – 1.5 – i 0.4,
should be placed.
Fig. 3.9. Locations in the Argand plane of u – 1 + i 1.3, v = – 2 + i, and w = –1.5
– i 0.4.
The basic algebraic operations of addition and multiplication of complex
numbers now find a clear geometrical form. Let us consider addition first. Suppose
u and ν are two complex numbers, represented on the Argand plane in accordance
with the above scheme. Then their sum u + ν is represented as the ‘vector sum’ of
the two points; that is to say, the point u + ν occurs at the place which completes
the parallelogram formed by u, ν, and the origin 0. That this construction (see Fig.
3.10) actually gives us the sum is not very hard to see, but I omit the argument
here.
Fig. 3.10. The sum u + of two complex numbers u and ν is obtained by the
parallelogram law.
Fig. 3.11. The product uv of two complex numbers u and v is such that the triangle
formed by 0, v, and uv is similar to that formed by 0, 1, and u. Equivalently: the
distance of uv from 0 is the product of the distances of u and ν from 0, and the
angle that uv makes with the real (horizontal) axis is the sum of the angles that u
and υ make with this axis.
The product uv also has a clear geometrical interpretation (see Fig. 3.11),
which is perhaps a little harder to see. (Again I omit the argument.) The angle,
subtended at the origin, between 1 and uv is the sum of the angles between 1 and
u and between 1 and ν (all angles being measured in an anticlockwise sense), and
the distance of uv from the origin is the product of the distances from the origin of
u and v. This is equivalent to saying that the triangle formed by 0, v, and uv is
similar (and similarly oriented) to the triangle formed by 0, 1, and u. (The
energetic reader who is not familiar with these constructions may care to verify
that they follow directly from the algebraic rules for adding and multiplying
complex numbers that were given earlier, together with the above trigonometric
identities.)
CONSTRUCTION OF THE MANDELBROT SET
We are now in a position to see how the Mandelbrot set is defined. Let z be some
arbitrarily chosen complex number. Whatever this complex number is, it will be
represented as some point on the Argand plane. Now consider the mapping
whereby z is replaced by a new complex number, given by
z →z2 + c
where c is another fixed (i.e. given) complex number. The number z2 + c will be
represented by some new point in the Argand plane. For example, if c happened
to be given as the number 1.63 – i4.2, then z would be mapped according to
z → z2 + 1.63 – i 4.2
so that, in particular, 3 would be replaced by
32 + 1.63 – i 4.2 = 9 + 1.63 – i 4.2 = 10.63 – i 4.2
and –2.7 + i 0.3 would be replaced by
(–2.7 + i 0.3)2 + 1.63 – i 4.2
= (–2.7)2 – (0.3)2 + 1.63 + i{2( .7)(0.3) – 4.2}
= 8.83 – i 5.82.
When such numbers get complicated, the calculations are best carried out by an
electronic computer.
Now, whatever c may be, the particular number 0 is replaced, under this
scheme, by the actual given number c. What about c itself? This must be replaced
by the number c2 + c. Suppose we continue this process and apply the
replacement to the number c2 + c; then we obtain
(c2 + c)2 + c = c4 + 2c3 + c2 + c.
Let us iterate the replacement again, applying it next to the above number to
obtain
(c4 + 2c3 + c2 + c)2 + c = c8 + 4c7 + 6c6 + 6c5 + 5c4 + 2c3 + c2 + c
and then again to this number, and so on. We obtain a sequence of complex
numbers, starting with 0:
0, c, c2 + c, c4 + 2c3 + c2 + c, . . .
Fig. 3.12. A sequence of points in the Argand plane is bounded if there is some
fixed circle that contains all the points. (This particular iteration starts with zero
and has .)
Now if we do this with certain choices of the given complex number c, the
sequence of numbers that we get in this way never wanders very far from the
origin in the Argand plane; more precisely, the sequence remains bounded for such
choices of c which is to say that every member of the sequence lies within some
fixed circle centred at the origin (see Fig. 3.12). A good example where this occurs
is the case c – 0, since in this case, every member of the sequence is in fact 0.
Another example of bounded behaviour occurs with c = –1, for then the sequence
is: 0, –1, 0, –1, 0, –1,. . . ; and yet another example occurs with c = i, the sequence
being 0, i, i –1, – i, i –1, –i, i –1, – i, . . . However, for various other complex
numbers c the sequence wanders farther and farther from the origin to indefinite
distance; i.e. the sequence is unbounded, and cannot be contained within any fixed
circle. An example of this latter behaviour occurs when c – 1, for then the sequence
is 0, 1, 2, 5, 26, 677, 458330, . . . ; this also happens when c = –3, the sequence
being 0, – 3, 6, 33, 1086, . . . ; and also when c = i – 1, the sequence being O, i –
1, – i – 1, – 1 + 3i, – 9 – i5, 55 + i91, – 5257 + i10011, . . .
The Mandelbrot set, that is to say, the black region of our world of Tor’Bled-
Nam, is precisely that region of the Argand plane consisting of points c for which
the sequence remains bounded. The white region consists of those points c for
which the sequence is unbounded. The detailed pictures that we saw earlier were
all drawn from the outputs of computers. The computer would systematically run
through possible choices of the complex number c, where for each choice of c it
would work out the sequence 0, c, c2 + c,. . . and decide, according to some
appropriate criterion, whether the sequence is remaining bounded or not. If it is
bounded, then the computer would arrange that a black spot appear on the screen
at the point corresponding to c. If it is unbounded, then the computer would
arrange for a white spot. Eventually, for every pixel in the range under
consideration, the decision would be made by the computer as to whether the
point would be coloured white or black.
The complexity of the Mandelbrot set is very remarkable, particularly in view
of the fact that the definition of this set is, as mathematical definitions go, a
strikingly simple one. It is also the case that the general structure of this set is not
very sensitive to the precise algebraic form of the mapping z → z2 + c that we
have chosen. Many other iterated complex mappings (e.g. z → z3 + iz2 + c) will
give extraordinarily similar structures (provided that we choose an appropriate
number to start with – perhaps not 0, but a number whose value is characterized
by a clear mathematical rule for each appropriate choice of mapping). There is,
indeed, a kind of universal or absolute character to these ‘Mandelbrot’ structures,
with regard to iterated complex maps. The study of such structures is a subject on
its own, within mathematics, which is referred to as complex dynamical systems.
How ‘real’ are the objects of the mathematician’s world? From one point of view it
seems that there can be nothing real about them at all. Mathematical objects are
just concepts; they are the mental idealizations that mathematicians make, often
stimulated by the appearance and seeming order of aspects of the world about us,
but mental idealizations nevertheless. Can they be other than mere arbitrary
constructions of the human mind? At the same time there often does appear to be
some profound reality about these mathematical concepts, going quite beyond the
mental deliberations of any particular mathematician. It is as though human
thought is, instead, being guided towards some external truth – a truth which has a
reality of its own, and which is revealed only partially to any one of us.
The Mandelbrot set provides a striking example. Its wonderfully elaborate
structure was not the invention of any one person, nor was it the design of a team
of mathematicians. Benoit Mandelbrot himself, the Polish-American
mathematician (and protagonist of fractal theory) who first3 studied the set, had
no real prior conception of the fantastic elaboration inherent in it, although he
knew that he was on the track of something very interesting. Indeed, when his
first computer pictures began to emerge, he was under the impression that the
fuzzy structures that he was seeing were the result of a computer malfunction
(Mandelbrot 1986)! Only later did he become convinced that they were really
there in the set itself. Moreover, the complete details of the complication of the
structure of Mandelbrot’s set cannot really be fully comprehended by any one of
us, nor can it be fully revealed by any computer. It would seem that this structure
is not just part of our minds, but it has a reality of its own. Whichever
mathematician or computer buff chooses to examine the set, approximations to the
same fundamental mathematical structure will be found. It makes no real
difference which computer is used for performing calculations (provided that the
computer is in accurate working order), apart from the fact that differences in
computer speed and storage, and graphic display capabilities, may lead to
differences in the amount of fine detail that will be revealed and in the speed with
which that detail is produced. The computer is being used in essentially the same
way that the experimental physicist uses a piece of experimental apparatus to
explore the structure of the physical world. The Mandelbrot set is not an invention
of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is
just there!
Likewise, the very system of complex numbers has a profound and timeless
reality which goes quite beyond the mental constructions of any particular
mathematician. The beginnings of an appreciation of complex numbers came
about with the work of Gerolamo Cardano. He was an Italian, who lived from
1501 to 1576, a physician by trade, a gambler, and caster of horoscopes (once
casting a horoscope for Christ), and he wrote an important and influential treatise
on algebra ‘Ars Magna’ in 1545. In this he put forward the first complete
expression for the solution (in terms of surds, i.e. nth roots) of a general cubic
equation.* He had noticed, however, that in a certain class of cases – the ones
referred to as ‘irreducible’, where the equation has three real solutions – he was
forced to take, at a certain stage in his expression, the square root of a negative
number. Although this was puzzling to him, he realized that if he allowed himself
to take such square roots, and only if, then he could express the full answer (the
final answer being always real). Later, in 1572, Raphael Bombelli, in a work
entitled ‘I’Algebra’, extended Cardano’s work and began the study of the actual
algebra of complex numbers.
While at first it may seem that the introduction of such square roots of negative
numbers is just a device – a mathematical invention designed to achieve a specific
purpose – it later becomes clear that these objects are achieving far more than that
for which they were originally designed. As I mentioned above, although the
original purpose of introducing complex numbers was to enable square roots to be
taken with impunity, by introducing such numbers we find that we get, as a
bonus, the potentiality for taking any other kind of root or for solving any
algebraic equation whatever. Later we find many other magical properties that
these complex numbers possess, properties that we had no inkling about at first.
These properties are just there. They were not put there by Cardano, nor by
Bombelli, nor Wallis, nor Coates, nor Euler, nor Wessel, nor Gauss, despite the
undoubted farsightedness of these, and other, great mathematicians; such magic
was inherent in the very structure that they gradually uncovered. When Cardano
introduced his complex numbers, he could have had no inkling of the many
magical properties which were to follow – properties which go under various
names, such as the Cauchy integral formula, the Riemann mapping theorem, the
Lewy extension property. These, and many other remarkable facts, are properties
of the very numbers, with no additional modifications whatever, that Cardano had
first encountered in about 1539.
Is mathematics invention or discovery? When mathematicians come upon their
results are they just producing elaborate mental constructions which have no
actual reality, but whose power and elegance is sufficient simply to fool even their
inventors into believing that these mere mental constructions are ‘real’? Or are
mathematicians really uncovering truths which are, in fact, already ‘there’ – truths
whose existence is quite independent of the mathematicians’ activities? I think
that, by now, it must be quite clear to the reader that I am an adherent of the
second, rather than the first, view, at least with regard to such structures as
complex numbers and the Mandelbrot set.
Yet the matter is perhaps not quite so straightforward as this. As I have said,
there are things in mathematics for which the term ‘discovery’ is indeed much
more appropriate than ‘invention’, such as the examples just cited. These are the
cases where much more comes out of the structure than is put into it in the first
place. One may take the view that in such cases the mathematicians have
stumbled upon ‘works of God’. However, there are other cases where the
mathematical structure does not have such a compelling uniqueness, such as when,
in the midst of a proof of some result, the mathematician finds the need to
introduce some contrived and far from unique construction in order to achieve
some very specific end. In such cases no more is likely to come out of the
construction than was put into it in the first place, and the word ‘invention’ seems
more appropriate than ‘discovery’. These are indeed just ‘works of man’. On this
view, the true mathematical discoveries would, in a general way, be regarded as
greater achievements or aspirations than would the ‘mere’ inventions.
Such categorizations are not entirely dissimilar from those that one might use
in the arts or in engineering. Great works of art are indeed ‘closer to God’ than are
lesser ones. It is a feeling not uncommon amongst artists, that in their greatest
works they are revealing eternal truths which have some kind of prior etherial
existence,* while their lesser works might be more arbitrary, of the nature of mere
mortal constructions. Likewise, an engineering innovation with a beautiful
economy, where a great deal is achieved in the scope of the application of some
simple, unexpected idea, might appropriately be described as a discovery rather
than an invention.
Having made these points, however, I cannot help feeling that, with
mathematics, the case for believing in some kind of etherial, eternal existence, at
least for the more profound mathematical concepts, is a good deal stronger than
in those other cases. There is a compelling uniqueness and universality in such
mathematical ideas which seems to be of quite a different order from that which
one could expect in the arts or engineering. The view that mathematical concepts
could exist in such a timeless, etherial sense was put forward in ancient times (c.
360 BC) by the great Greek philosopher Plato. Consequently, this view is frequently
referred to as mathematical Platonism. It will have considerable importance for us
later.
In Chapter 1, I discussed at some length the point of view of strong AI,
according to which mental phenomena are supposed to find their existence within
the mathematical idea of an algorithm. In Chapter 2, I stressed the point that the
concept of an algorithm is indeed a profound and ‘God-given’ notion. In this
chapter I have been arguing that such ‘God-given’ mathematical ideas should have
some kind of timeless existence, independent of our earthly selves. Does not this
viewpoint lend some credence to the strong-AI point of view, by providing the
possibility of an etherial type of existence for mental phenomena? Just
conceivably so – and I shall even be speculating, later, in favour of a view not
altogether dissimilar from this; but if mental phenomena can indeed find a home
of this general kind, I do not believe that it can be with the concept of an
algorithm. What would be needed would be something very much more subtle. The
fact that algorithmic things constitute a very narrow and limited part of
mathematics will be an important aspect of the discussions to follow. We shall
begin to see something of the scope and subtlety of non-algorithmic mathematics
in the next chapter.
NOTES
1. See Mandelbrot (1986). The particular sequence of magnifications that I have
chosen has been adapted from those of Peitgen and Richter (1986), where
many remarkable coloured pictures of the Mandelbrot set are to be found. For
further striking illustrations, see Peitgen and Saupe (1988).
2. As far as I am aware, it is a consistent, though unconventional, point of view
to demand that there should always be some kind of rule determining what the
nth digit actually is, for an arbitrary real number, although such a rule may
not be effective nor even definable at all in a preassigned formal system (see
Chapter 4). I hope it is consistent, since it is the point of view that I should
most wish to adhere to myself!
3. There is actually some dispute concerning who it was that first came across
this set (see Brooks and Matelski 1981, Mandelbrot 1989); but the very fact
that there can be such a dispute lends further support for the view that the
finding of this set was more like a discovery than an invention.
___________________________
Penrose, Roger. The Emperors New Mind : Concerning Computers, Minds, and the Laws of Physics, Oxford University Press, Incorporated, 2002. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/utk/detail.action?docID=1107726.
Created from utk on 2021-02-16 17:10:29. Copyright © 2002. Oxford University Press, Incorporated. All rights reserved.