**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.

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