This is not a website offering a course in elementary mathematics, needful though such a course is (interested readers might try the Khan Academy); my concern is rather with the underlying rationale of mathematics, the theory of numbers and numbering. However, “If you want to be a blacksmith, go and work at the forge” as the saying goes and one cannot really understand numbers without actually using them and getting to know them well. Hate them and they will hate you. Once you have overcome a likely initial repulsion, you will find that numbers have a fascination all of their own. It is very easy to pick out patterns in batches of numbers that crop up and mankind is, above all, a pattern loving animal — indeed the search for pattern has probably done more for progress than the tool-making capacity. Fibonacci Numbers, Pascal’s Triangle, Primes and so on  retain their fascination nearly a thousand years since they were first discovered and people are still finding new things in them today.

As I have mentioned elsewhere, it was pattern that made me start studying mathematics in the first place : at school I was in the second set and just scraped through what was then Maths ‘O’ level. From then on I made haste to abandon such rubbish for good although I did at one time think I ought maybe to study mathematics simply in order to better attack it. But one day I came across in a book that had nothing to do with mathematics the statement that if you added together three successive odd numbers starting at unity, you get a square number. This worked for 1 + 3 + 5 = 9 = 32  and, seemingly, for longer strings of odd numbers. Even better, if you select three consecutive odd numbers at random you end up, if not with a square number the next best thing, the difference between two squares. Thus, 11 + 13 + 15 = 39 = 64 25 = 82   52That there was such an unexpected association between odd numbers and squares struck me as being little short of miraculous and from then on I became increasingly hooked on mathematics,  number theory in particular. The very first article I published in  M500, the magazine of the Mathematics Department of the Open University, was entitled precisely Sums of Odd Numbers.
Nearly three thousand years ago early Greek mathematicians messing around with pebbles and counters discovered the so-called figurate numbers — which is why we still to this day talk of ‘squares’, ‘triangular numbers’ and ‘odd’ and ‘even’ numbers. Before being marks on paper, numbers were objects and before becoming abstract entities that ‘obey the Axioms of Fields’, numbers were, well, what most of us think of as numbers. Arithmetic goes back to the Ancient Babylonians and Egyptians and scribes five thousand years ago were passing around mathematical puzzles just like people today passing round Sudokus. Contemporary pure  mathematicians look down on the integers, considering them small beer compared to ‘irrationals’, complex numbers and the like; it is even said that they make mistakes with the change though this is probably affectation. Mathematicians behave like this because they  despise everyday physical reality and since the ‘natural’ numbers are about the closest one ever gets in mathematics to the real world, they rate the lowest. The positive integers are the proles, disdained precisely because the modern mathematical elite can’t do without them but don’t want to admit it. As it happens,  the great classical mathematicians were usually excellent calculators: Gauss, sometimes called the ‘Prince of Mathematicians’, was a calculating prodigy and Euler spent his last years doing abstruse computations in his head since he became blind. More recently, the brilliant Indian mathematician, Ramanujan, spent his best years trawling through seas of numbers in the days when the computer was not even a pipe-dream, working with a slate board and chalk (because he found paper and ink too expensive) on his parents’ verandah near Madras far from the madding crowd of Oxford and Cambridge.

The best initial advice I can give to anyone who wants to understand and enjoy numbers is to use them on a more or less daily basis and do stints of mental arithmetic. Several excellent little books on mental arithmetic exist and I will mention one in due course but at the beginning you can just take whatever comes to hand in the nearest charity shop. Even when using a calculator it is well worthwhile making an estimate mentally first  because you can then spot at once if the answer the calculator comes up with is plausible or not. It is also well worth experimenting with ‘primitive’ methods such as finger counting or Russian Peasant Multiplication (about which I have already written). If professional mathematicians had given more thought to the diverse ways of writing numbers, they would have grasped much more quickly what seems painfully obvious today, namely that numbers can be written in any base, our chosen base, ten, being a consequence of our having five fingers and two hands. When one or two mathematicians at the end of the nineteenth century first put forward the notion of alternative bases, they were met with incredulity and scorn from within the profession itself.

After mental arithmetic comes written longhand computation. It pays to be proficient even in long division because the principle is very useful for dealing with polynomials.     When doing mathematics, it is essential to have plenty of space and one of the most useful practical tips I ever got from a tutor was to work with an A4 sheet in the landscape not portrait position, i.e. arrange the sheet so that it is wider than it is high. I had in fact already found something even better than this  (and which also reduced my expenditure), which was to  buy rolls of plain, so-called ‘lining paper’ (wallpaper without a pattern) and spreading this across a table, clamping down the ends at each side. You can now really spread yourself, use different coloured marker pens, have various calculations going on at the same time and so on. Results and formulae are then transferred to a Notebook or computer and you can tear off the sheet and throw it away. It is amazing how much paper a mathematician requires and few  mathematicians work directly on a computer. Recently, I  came across something to replace Ramanujan’s slate and chalk, namely the medium  size ‘whiteboards’ now available cheaply in hardware stores and which allow you to simply wipe off calculations with a Kleenex. This, along with the ‘biro’ pen that writes in real ink, are two recent inventions I would not like to be without.

One cannot get on these days without a pocket calculator but using one is not without its dangers. Apart from it being very easy to touch the wrong  button, everything is in decimal and this gives people get the impression that a perfectly straightforward fraction such as a third (1/3) is ‘really’ an unending expansion which goes on for ever 0.33333333….  In many cases, we actually need to see what is happening on the level of ordinary fractions (which are simply numbers with a temporary base other than ten), or transfer to another base such as 2.

As for the full-size computer, I personally never use it at all for doing mathematics, only to check results I’ve got out already. One of the present-day big mathematics programmes will most likely contain all the mathematics you are likely to need in your lifetime and everything you are likely to discover unless you are a second Euler, but you will learn nothing about mathematics by just clicking  the right buttons to get the ‘right’ answer. The entire joy of mathematics lies in discovery and you will never discover anything if you just look up the subject on Wikipedia. After spending one or two sometimes fruitless mornings on some problem, I do have recourse to Wikipedia but only when I’ve already got out some results for myself, or am completely flummoxed. It is gratifying to find that one has unwittingly been treading, as it were, in the footsteps of some great name from the distant past. Just last week investigating Unit Fractions, I was gratified to see that I eventually hit upon the same all-round formula for reducing any fraction to a series of unit fractions as Leonardo of Pisa (Fibonacci), the greatest mathematician of the Middle Ages, did working in 1203 with quill and parchment by the light of a candle. (He also gave several other artful methods and generalised results in a way that I never managed to.) A friend of mine, a retired civil engineer, Henry Jones, discovered for himself the so-called parametric equation of the ellipse — a result taught in all schools today and already known to Copernicus (though Jones was not aware of this). In its way Jones’s independent discovery of this formula is as great an achievement as Copernicus’s. A further boon of discovering something for yourself rather than reading it in a book, is that you are more likely to see a practical application : Jones proceeded to invent a compass that could draw ellipses, a substantial achievement at the time though subsequently rendered obsolete by computer aided design.

All in all, the trouble with professional mathematicians today is not that they know too little, but that they know too much. This website is in any case not for them but is aimed at the small band of people who practise mathematics simply for enjoyment. Out of all branches of mathematics the most accessible and the most satisfying for a beginner is Number Theory. It is the branch of mathematics where you can make little (or perhaps not so little) discoveries for yourself very rapidly, since, although the higher reaches of the subject still exercise the best brains, the fundamentals are easy to learn and easy to experiment with.

In this ongoing blog, eventually to be turned into a book, I intend to intersperse various fairly simple problems with theoretical and philosophical observations and give the answers in a subsequent post. They are not really so much problems as invitations to explore areas that are of interest or that I have found so.

All this does not rule out doing the necessary spadework including such things as learning off by heart times tables, lists of Fibonacci Numbers, primes and so on : this sort of drill is inevitable in every art or activity and is quite acceptable, even stimulating, if undertaken in small doses. A final word of advice to the aficionado : don’t go at a problem  hammer and tongs for hour after hour. It has been shown scientifically that it is best to take short breaks between stints of study, usually after about forty minutes at most. Hardy, a distinguished mathematician in his day, said that he could never work more than four hours at mathematics in a day (a whole morning) and he then did physical activity in the afternoon (cricket in his case). I can’t manage more than three hours at most before getting hopelessly muddled. Drop it, do something completely different and come back to it afresh.      S.H.  11/3/12

In an earlier post about Egyptian Fractions I discussed whether it was possible to present any (proper) fraction as a sum of unit fractions and posed the following questions :

(1) Can every proper fraction a/b (where a,b are positive integers)    be expressed as a sum of unit fractions?

(2) For any given a/b, how many ways are there of representing it as a sum of unit fractions, supposing it to be possible ?

(3) Is there a way to reduce the number of terms in the series of unit fractions to a minimum, supposing a/b can be so represented ?

These are my investigations (undertaken without looking up the relevant article in Wikipedia which I recommend you do.)
Take two positive whole numbers a and b and assume that a/b is a ‘proper’ fraction, i.e. a < b  as in  3/8 or  4/15
The challenge is to find a method to transform a/b into a sum of unit fractions, i.e. fractions which all have 1 as numerator such as 1/6, 1/47 and so on.
In the trivial case where a = 1 we already have a ‘series’ of unit fractions, so we shall assume a ≠ 1. Also, we will assume that a/b is in its lowest terms, i.e. that a and b do not have a common factor greater than unity (note 1).
Now
a       =   1  +  (na b)
b             n         nb

(since if you multiply out you obtain (b  +  na – b)   = na  =   a )
nb                   nb       b

For example,       7     =   1  +  (5.7 ‒ 24)   =  1   +  11

24          5          5.24             5      120

Note that if  na < b  the numerator  (na ‒ b) < 0  (is negative) and, since nb > 0 (i.e. is positive) this makes the whole second fraction negative and we end up with a subtraction, not an addition.
For example, if n = 2 we have  (na ‒ b) = 2.7 ‒ 24 = ‒10     and

7     =   1   ‒    10     =   1    ‒     5
24          2       2.24      2          24

With the values we have chosen for a and b, it is clear that we need to have n at least 4, n ≥ 4
Can we choose any value for n so long as it is greater or equal to 4 ?   Seemingly, yes. For example, if  n = 187

7     =   1     +  (187.7 ‒ 24)    =   1   +   1285
24        187         187.24           187      4488

However, we have still got the second fraction to reduce to a series of unit fractions and there may be more or less economical ways of doing this.
If we start with the lowest values of n that make an ‒ b positive, we find that, for a = 7, b = 24  we need to have n at least 4. In such a case

7     =   1     +  (4.7 ‒ 24)    =   1   +     4     =   1    +   1
24        4             4.24                4        4.24        4       24

So we have transformed 7/24  into a sum of just two unit fractions.
However, if we try out different values of a and b we will find that this was just a lucky fluke since, in general, the second fraction does not cancel down so readily. We can obtain the condition for this to happen, namely that (na b) = n   or   b = (a ‒ 1)n
This gives the useful rule :

If   b = (a ‒ 1)n     then     a     =   1  +  1   …………..(i)
b          n       b

We can check this by making up a/b in such a way that the condition is met. For instance, 65  =  5.13  and, according to  (i),

6     =   1     +  1       which is the case.
65         13       65

Similarly, since 51 = 3.17  we find that
4     =   1     +  1
51         17       51

This situation is, however, highly unusual.
In general, given an arbitrary proper fraction a/b, and arbitrary n, we will have a unit fraction and a second fraction which is not unitary, i.e.
a       =   1  +  c     where c ≠1
b            n      d

For example, with a = 7, b = 24, n = 5    we obtain

7     =   1 +   (35 24)   =   1  +    11
24        5            5.24          5       120

We can now express 11/120 as the sum of 1/n and c/d but this only pushes the problem further down the line. With n = 17

11   =   1     +  (17.9 ‒ 120)     =      1    +    33
120       17              17.120               17        2040

The second fraction is sufficiently small to be neglected in rough calculations but it is still there. And if we now express 33/240 as the sum of 1/n and e/f, we shall, unless we are very lucky, have a further non-unit fraction to deal with. Clearly, what we require is for the second fraction to disappear altogether, which will happen when (en — f) = 0, or f is an exact multiple of e, or when (en — f) = n in which case it will cancel with the n in the denominator (the case we have already considered). Can we, by a judicious choice of n, guarantee that this string of unit fractions will terminate either with zero or with a unit numerator?

Answer:  It is always possible to express a proper fraction a/b as a terminating series of unit fractions. To do this, we only have to look for the smallest possible choice of n where an b > 0 (positive). In the original case with a = 7, b = 24, it so happens that 4 is the smallest choice since 3 ×7 = 21 falls short of 24 whereas 5 × 7 = 35 is too far above.
In the case of 7/24 we achieved our goal immediately which will not usually be the case, so let us look at another case to see how this checks out.
7     =   1 +  (3.7 16)   =     1 +   5
16        3       3.16                  3      48

Here, 3 is the smallest choice giving a positive second fraction. We continue :

5     =   1 +  (10.5  48)        =    1 +   2
48       10       10.48                      10    480

nce again, we choose the smallest possible n. The second fraction reduces to  1/240 and if we hadn’t spotted that straight off we would obtain

2     =   1    +    (2.240 ‒ 480)   =   1 +    0
480    40             240.480             240

We have thus expressed 7/16 as the sum of unit fractions

7    =   1    +   1 +    1
16         3        10     240

Is something like this going to occur if we systematically choose the smallest possible number for n compatible with (an ‒ b) > 0 ?
In fact, yes. It all depends on the successive numerators of the non-unit fractions getting smaller each time. For example, 5 < 7 in

7    =   1 +          and  2 < 5 in   5       =   1 +    2
48       10       480                           48         10     480

If the numerators always decrease, and are always positive integers, we will eventually arrive at a numerator which is either zero or unity.  Can we show that this is necessarily the case ? Yes.
If a/b is a proper fraction, i.e. b > a, and b is not an exact multiple of a, then b/a will lie between (n ‒ 1) and n because we must have (an ‒ b) positive. Thus, b/a will be squeezed between successive integers or
(n ‒ 1)  <         b/a   <  n
The distance between b/a  and the smallest  integer n such that n > (b/a) will thus always be less than unity, for example the distance between 16/7 and 3  is less than unity since 16/7 lies between 2  and 3. Now, since

(n ‒ b)  < 1     means  (na ‒ b) <  a
a

However, (na ‒ b) is the numerator of the second fraction in

a       =   1  +  (na b) b
b
n         nb

This shows that the numerator of the second fraction in the expansion is always less than the numerator we started with. Expanding the second fraction, we have the same situation

(na b)   =   1  +  (m(na b) ‒ nb)
nb               m                mnb

Here, the numerator of the second fraction is less than the numerator we started with while the denominator is greater, since it has been multiplied by m. Thus, the last fraction in an expansion gets steadily smaller and, since the numerator is always a positive integer (or zero), we end up either with zero or a unit fraction.

More formally, is we define [b/a]  as the smallest integer equal to or greater than b/a, we have the relation

a       =    1        +     ([b/a]a b)
b           [b/a]               nb

and, if we have ([b/a]a b) > 0,  then ([b/a] ‒ b/a) < 1  so

([b/a]a  ‒ b) <  a

Supposing ([b/a]a  ‒ b) ≠ 1 we set  ([b/a]a  ‒ b) = c  whence

c      =       1           +     [nb/c] ‒ nb
nb          [nb/c]             [nb/c]nb

and we have  ([nb/c] ‒ nb) < c  < a   and so on.

Also, note that we have assumed a < b   which means that

na < nb  so that   (na – b) < (nb – b) or b(n – 1)

For example, in the case of we took n = 4

a      =   1 +  (na b)
b           n       nb

(na b)    =   1  m(na b) nb
nb               m             mnb

Let (na b) = c and let  m be the least multiple of c <  nb
Now, mc <  mb  <  a    so the numerator is always diminishing and since 0 £ numerator we must eventually get back to 1 or 0.

Try another example :   a = 8,  b = 57,   n = 8

8      =   1 +  (64 57)      =    1 +    7          c = 7  and 7 < 8
57          8          8.57                8      456

7    =   1   462 456   =   1   +       6            d = 6 < 7 < 8
456         66       66.456         66       30096

6    =    1 +      30096 30096   =    1 .   +  0
30096      5016      5016.30096          5016

Thus
8      =    1 +     7
57           8     456

=    1 +    1    +      6     .
8       66        30096

=    1 +    1    +      1     .
8       66        5016

We conclude that it will always be possible to express a fraction as a sum of unit fractions.     S.H.

“What a piece of work is mathematics! How noble in reason! How infinite in application! in form, how concise and elegant! In solving enigmas how admirable! in understanding how like a god! The beauty of the world! The paragon of the sciences!  And yet at the same time what is this  quintessence of argumentativeness? How dreadful in its applications! How cold as a cadaver to the fingers! How tortuous in its  reasonings! How pettifogging in its distinctions! How ridiculous in its assumptions! Mathematics delights and yet delights not me; no, nor logic neither.”
Hamlet, Act II, Sc. 2  (slightly adapted) ¹

One is sometimes tempted to consider mathematics as mankind’s greatest achievement : certainly without it our society’s unparalleled mastery of the natural world  would be inconceivable and, apart from that, once you have penetrated the formidable barriers of mathematics it offers much pleasure as well. However, everything has its day and mathematics is surely now in its declining phase, undone largely by its very success : it is an overvalued currency and, once enough people realise this, it will come tumbling down fast enough, indeed this is already happening.
The Greeks, Pythagoras and Plato at any rate and their followers, saw mathematics as the gateway to eternal truth, the first by way of  Numbers and the second by way of geometry. Supposedly, Pythagoras once passed a blacksmith’s shop and, hearing the various sounds coming from the interior, had the sudden intuition that sound was based on relationships between whole numbers and that the simpler relations like the octave, the fifth and the third were superior to more complicated and messy ratios. He, or one of his pupils experimenting with plucked strings, hit upon one of the very first scientific law : that the pitch of a musical string varies inversely with its length. Mathematics, in this case Number Theory, was thus from the very beginning intimately connected with scientific discovery.
By Plato’s time geometry and not arithmetic had become the leading branch of mathematics and Plato himself was responsible for making the first radical separation between mathematics and everyday physical reality. Geometry was the study of perfect forms whereas the natural world was, at best, only a poor imitation of the eternal. This was by no means such a foolish theory as it may sound since, as certain sophists pointed out, the theorems of geometry did not apply exactly to the physical world : a tangent to a circle was bound to touch the circumference at more than one point and, for that matter, a drawn straight line was not perfectly straight. And yet the theorems of geometry were ‘obviously’ true —  they could be proved ! The only way to resolve this dilemma was to posit a transcendent world, more beautiful and more real than this one, and it was this world to which the propositioons of geometry applied.  There lines really were perfectly straight, spheres perfectly round and a tangent to a circle touched it at a single point.
Plato’s conception of ‘eternal forms’ was given new life and became much more plausible when, at the Renaissance, the early scientists associated it with Christian theology. God was the supreme architect and mathematician and the truths of mathematics were literally ‘thoughts in the mind of God’ for people like Kepler and Galileo. “The truth which mathematical demonstrations give us [is] the same which the Divine Wisdom knoweth” as Galileo put it.  The mystery of why mathematics, a creation of the mind, could be successfully applied to the study of natural phenomena was now resolved : God had devised the natural laws that Nature was obliged to follow and these laws were, at bottom, strictly mathematical. There was thus no great separation between pure and applied mathematics which is why all the leading classical scientists were also leading mathematicians and vice-versa, Galileo, Huyghens, Newton and Leibnitz.
Thus to the nineteenth and twentieth centuries when mathematics expanded prodigiously but increasingly severed its connection with physical reality. Today it is fashionable, at any rate amongst pure mathematicians and they rule the roost, to view mathematics as a free creation of the human mind that neither has nor needs to have any connection with the actual world (we think) we live in. “The majority of writers on the subject seem to agree that most mathematicians, when doing mathematicvs, are convicned that they are dealing with an objective reality, but then of challenged to give a philosophical account of this rreality, find it easier to pretend that they do not believe in nit after all….  The typical mathematician is both a Platonist and a formalist — a secret Platonist.with a formalist mask that he puts on when the occasion calls for it”  (Davis and Hersh, The Mathematical Experience).            This is essentially having your cake and eating it, and it is incredible that mathematicians have been allowed to get away with such a sleight of hand. The Formalist approach, which reduces the whole of mathematics to the intellectual equivalent of embroidery, at once sabotages the teaching of mathematics in primary and secondary schools and is responsible more than anything else for the repugnance that mathematics inspires in a lot of people (ioncluding myself when I was at school). It is a great pity that no government would dare to impose on professional mathematicians the obligation to occasionally spend some time teaching eleven-year olds, as the author Lancelot Hogben perjhaps flippantly once suggested. I at one time had a pupil of about eleven who came to me after one session in great indignation, accusing me of having “said something that was not true”. I asked what this was. “You said that the sum of all the angles in a triangle is always 180 degrees, but I measured one and found that it wasn’t”. I asked her what result she got and she said, “One hundred amd seventy-nine and a half”.  I  was rather taken aback by this and mumbled something about the inaccuracy of measurements, finally getting her to agree that her triangle had ‘nearly’ 180 degrees. This girl, with such a strong empirical bent, has doubtless been completely put off studying mathematics and, for that matter, physics since the latter subject is now little but abstruse mathematics. Gauss was so bothered by the selfsame problem, i.e. “Is Euclidian geometry actually true?”, that he used surveying data to plot a giant triangle formed by three peaks in Hanover and test the well-known theorem. Fortunately for him, he found the result correct allowing for slight experimental error and breathed again.
This anecdote puts in relief the staggering change in attitudes to mathematics in little more than two centuries : on the one hand we have the greatest mathematician of his time and a man who toyed with the notion of non-Euclidian geometries bothered by a problem of ‘reality’, and on the other an era like the present where only fractious eleven year old children or fringe figures like myself worry about the ‘truth’ of mathematics.
Contemporary pure mathematicians do not simply have an indifference to ‘reality’ but a positive distaste for it : Mandelbrot, a brilliant thinker if ever there was one, scandalized orthodoxy by actually applying recondite mathematics to the ‘real world’ and, God forbid, even producing ‘pretty pictures’. But there is something more serious here : if mathematics is a ‘free creation of the human mind’, its successful applications in science and technology are utterly mysterious.
In reality (sic), present-day mathematics is a vast Art-Deco hotel with suites of empty rooms leading in opposite directions and some of them taking you straight into the icy waters of a non-existent lake : it is a confused medley of the realistic, the formal, the decorative and the useful. It would be nice if we could separate out those portions which are, or could be, applicable to the real world and those which are not. This is unfortunately not possible, but what we can do is to establish a sort of ‘reality’ grade for different branches of mathematics running from 0 to 1. Whole number theory is, if not 1, at least very close indeed to 1, Euclidian geometry somewhat less than 1, Banach Spaces a good deal less still with Cantor’s Theory of the Transfinite scoring very close to zero. In the past the major technical changes often came  directly or indirectly from mathematics but the Information Technology Revolution has been largely spawned by people outside or on the fringe of official mathematics — I was even told by a computer programmer that his firm actually regards trained mathematicians with some suspicion rather than respect. Similarly, Paulson, the head of the FED, os supposed to have said off the rcord that he would “rather meet with a trader than a mathematician”. The reason for all this is not far to seek : an excessive concern with logic and the formal aspects of mathematics makes one unfit for the hutly-burly of the real world wehere decisions have to be taken rapidly on the basis of inadequate and inaccurate data.

Notes :

¹ The original reads : “What a piece of work is a man! How noble in reason! How infinite in faculty! in form, how moving, how express and admirable! In action how like an angel! In apprehension how like a god! The beauty of the world! The paragon of the animals! And yet to me what is this  quintessence of dust? Man delights  not me; no, nor woman neither.”   from Hamlet, Act II, Sc. 2

In my poem, The Initiates, I have imagined a group of latter-day Greek aesthetes gathered in the house of one of their number:

“We met always by night: a household slave brought in
A tray of sand, giving each visitor a cane,
With joy we gathered round; the latest theorem
Imported from North Africa was scrutinised,
The argument abridged, occasional points of style
Touched up…Then silence fell, a sense of ultimate peace
Came over us; these lines and circles that we traced
Were clearly images of a superior world,
Indifferent to man, exempt from frailties,
War, death, disease, could not affect them and their truth
Did not depend on trial or experiment,
Each step self-evident, demonstrable and sure.”

It is often said that our current Hindu/Arab written numerals,  first introduced into Europe by Leonardo of Pisa (Fibonacci) in the thirteenth century, are the most convenient possible. They certainly make for a very concise way of representing quantity and have an indefinitely extendable range in both directions (to the very large and the very small by way of decimals) which other systems, such as the ancient Greek and Babylonian systems, did not possess. Most authors likewise believe, or rather assume, that our ways of handling the basic four operations of arithmetic, addition, multiplication, division and subtraction,  are also the most advantageous, in short that we have the best possible Number System imaginable.
However, “You don’t get owt for nowt” as the Yorkshire saying goes. A serious disadvantage of our way of doing things is that, in pre-computer  days, our arithmetic involved much tedious rote learning such as multiplication tables — schoolboys in the Edwardian era learned the thirteen times table as well. And ‘long’ division is so complicated that, even as late as the Restoration period, we find an important State official in one of the most powerful countries in the world, Samuel Pepys, getting up at five o-clock in the morning at the age of 30 to have lessons in it.
A more serious disadvantage from the conceptual point of view is that the artifice of cipherisation obscures the underlying rationale of all numbering which is the pairing off of sets of discrete objects against a selected ‘standard’ set. But in our written system a single sign such as ‘5’ is associated with a plurality of objects and there is no way of deciding a priori whether a sign such as ‘5´ represents a larger or smaller collection of objects than ’7’. If we represent a single item by a chosen object such as a bean or by a single mark such as a scratch, and repeat this an appropriate amount of times, we end up with collections of beans or scratches that we can compare visually. True, we will still need a ‘base’ because human beings are unable to cope perceptually with even quite small quantities of objects — try guessing the number of beads in a necklace or the number of seagulls on a lawn and you will be amazed how way out you are. But in our ciphered system, not only is there no very marked difference between one level and the next but there is no quantitative difference (more or less symbols) within a single level.
The system of bases and consequent scaling up (or down) relies on the artifice of a ‘moving unit’ which becomes progressively larger (or smaller) but always maintains the same ratio as we move from one ‘level’ to the next  — in our system a ratio of one to ten. Such scaling is immediately apparent in systems such as those used by ‘primitive’ tribes (and sometimes quite advanced societies like the Yoruba or the Incas) where a collection of discrete objects, twigs or shells, is bound together, piled up or threaded together, to make the next ‘one’, then these bundles are themselves tied together as we proceed to the next stage, and so on. Our way of representing bases by moving a  digit to the left or right makes for a very compact style of notation but it obscures the fundamental principle involved and is one of the reasons why many people find mathematics, even ‘ordinary’  arithmetic, so peculiar : the operational principles seem to have little connection with what goes on in the real world and are essentially just ‘rules’ that you are obliged to learn if you want to get through your exams.
The Ancient Egyptian number system, despite its ‘poor cousin’ status compared to the Hindu or Greek systems, is actually  by far the clearest, the most logical and the most ‘user-friendly’ of all number systems. It is, like ours, a base ten system and numerals are written, like ours, in  descending order of size (largest quantity first) though this is not immediately apparent since the Egyptians, like the Hebrews and the Arabs to this day, wrote everything from right to left instead of left to right like us (Note 1).
The Egyptian sysytem does not use positional notation as such but has a different sign for the unit, the ten, the hundred &c.  Numerals less than ten are repeated upright strokes, as many strokes as there are objects, i.e.   ⁄   ⁄   ⁄   is our ‘3’ and if we have ‘three hundreds’ the hundred   sign will be repeated. A simple glance shows the approximate size of the quantity being   represented, since a collection in the thousands will have several thousand marks which are readily distinguishable from hundreds or ten thousands. With our positional notation, one has to look closely to distinguish between say 10000 and 1000 especially since it has become the fashion to leave out the comma for the thousand. Admittedly, Egyptian notation does make it difficult to distinguish between two numbers less than our ten (or the equivalent within the tens or thousands). Some (though by no means all) Egyptian texts get round this by arranging the units or other repeated signs in two rows so that there is a maximum of five in the top row, e.g. writing ‘seven’ as

⁄  ⁄  ⁄  ⁄  ⁄    This makes instantaneous assessment much easier and in effect means having a sub-base of five.
⁄  ⁄
There are no symbols for quantities beyond 1,000,000 but this maximum would have been amply sufficient for normal purposes — since in effect this means that a scribe would be able to represent in writing any quantity less than a billion (million million). An ancient Egyptian child only needed to learn seven different pictorial signs  : an upright (stick) for the unit, a bent stick for ten, a coiled rope for a hundred, a lotus for a thousand, a snake for ten thousand, a tadpole or frog for a hundred thousand and a seated man holding up his hands in amazement for a million. There was no sign for zero and, since the system, though employing positioning, did not rely on positioning alone, none was needed.
The disadvantage, of course, is that, compared to our system, rather more signs are required which slows up calculation : a quantity we record as 1967 would require no less than twenty-three Egyptian characters (though our 20,000 would only require two Egyptian signs). Scribes, who were a respected and well-paid body of men, closely associated with but distinct from priests, did not in practice always paint careful ‘number pictures’ as seen on tomb murals : for everyday calculation they used a rapid freehand so-called ‘hieratic’ script which runs different signs together and which, to some extent, differs from one scribe to another. I have experimented with hieratic Egyptian numerals and, with some fairly natural simplifications (natural to me) I find that writing down numbers the Egyptian way is hardly more cumbersome than our present system, if at all.
Multiplication is incredibly easy using Egyptian methods since it depends wholly on doubling and then adding rather like Russian multiplication (see earlier article). An Egyptian child would thus only need to learn off  by heart the two times table and would soon be able to situate any given quantity between two successive powers of two, e.g see that, in our reckoning, 416 comes between 28 and 29. Every scribe would know his powers of two backwards (which most people do not today) and very likely be able to reduce at sight any number to combinations of powers of two, at any rate approximately. As Gillings writes in his excellent Mathematics in the time of the Pharaohs,
“If Egyptian multiplication was so clumsy and difficult, how did it come about that these same techniques were still used in Coptic times, in Greek times, and even up to the Byzantine period, a thousand or more years later? No nation, over a period of more than a millennium, was able to improve on the Egyptian notation and methods.”
Division for the Egyptians was simply multiplication in reverse. Instead of dividing our 134 by 7 the Egyptian scribe would lay out his powers of two on one side and 7 repeatedly doubled on the other.

1                                 7

2                               14

4                               28

8                             56

16                            112

He would stop here because he would see that the next doubling would take him well beyond his goal 134. He would then get as close as he could to 134 using the entries on the Right Hand Side, namely by adding together 112, 14 and 7 . He would then add the corresponding powers of two, namely 16, 2 and 1 giving as quotient 23 with remainder 1 since the nearest combination of 7s fell one short. This procedure is, once you have got the hang of it, no lengthier than our ‘long division’ which children at school (and beyond) often have a lot of difficulty with, indeed scarcely anyone bothers with any more. In the Egyptian system the powers of 2 form a framework within which every number can be situated and, with practice, one can juggle them around mentally to situate any number, at least to a fair degree of accuracy. The method works, of course, only because every number can be expressed as a combination of powers of 2 , in effect, as we have only recently rediscovered, can be written in  binary. The Egyptian scribes must have realized this though they do not say so specifically.

What of fractions ?  Here the Egyptians ran into difficulties because they did not have our stroke notation ¼, 5/6 &c. They got round the problem by using reciprocals of numbers, noting this by a bar placed over the top (which I cannot reproduce with the limited alphabet at my disposal here). In practice, this meant, in our terms, reducing every proper fraction to a series of unit fractions — with the important exception of 2/3 which has a special sign of its own. For some reason, a scribe doing a calculation would not write the sign for the reciprocal of 7 more than once (except in prepared tables). He would always express  our 3/7 as a series of unit fractions (i.e. all having 1 as numerator). It is not clear why the Egyptian scribes did this though one can guess certain reasons. One is that, if this reduction to unit fractions is done efficiently, the series tails off rapidly so that one can see at a glance what is significant and what is negligible. This is what we ourselves do since, although most people do not realize it, the decimal fraction notation is a sum of proper fractions with progressively decreasing denominators. For example, 0.2347 is a concise way of representing the sum
2/10   + 3/100 + 4/1,000 + 7/10,000

Conceptually, the Egyptian scribes were apparently unable to make the giant leap in thought involved in extending the base-10 system backwards to represent quantities smaller than 1. This is understandable since a fraction like 4/9 is actually a rather complex beast. It is not a single but a double number : it indicates the number of pieces we have (four) and the  number of (supposedly equal) pieces that make up the whole (nine).
One advantage of using reciprocals was that it was not necessary to invent any new signs for quantities less than the unit, and this seems to have been an important consideration. The Greeks themselves, who did not have a positional number system either, for some reason also baulked at extending  their ciphered base-10 numerals to small quantities and astronomers like Ptolemy used normal base-10 numerals for quantities greater than one, but sexagesimal (60-base) fractions inherited from the Babylonians for quantities smaller than the unit. The great advantage here was that it was rarely necessary to express a quantity to more than two places, since 1/(60)2 = 1/(3600) was already small and 1/(60)3 minute.
The Egyptians themselves needed fractions for eminently practical reasons : at an early stage in their development they, like most other  societies at the time, did not use currency for everyday payments, gold and silver being reserved for large scale State transactions. Temple personnel were paid mainly in bread and beer (with free lodging presumably thrown in). Since Ancient Egypt was a very hierarchical society, it became a matter of some importance to be able to divide up the bread and beer equitably.
“To divide 3 loaves equally among 5 men, each man would be given three separate portions, a 1/3, a 1/5 and a 1/15. One advantage of this division was that not only was justice done, but justice also appeared to have been done. In a modern distribution, three of the five men would get 3/5 of a loaf in one large piece, while the other two men would get two smaller pieces, 2/5 and 1/5 of a loaf, which division might be regarded as an injustice b y an ignorant workman.”
Gillings, Mathematics in the Time of the Pharaohs

The scribe would, seemingly, never write our 3/5 as 5*  5*  5*  (where I have used * instead of the Egyptian bar over the top of a number to indicate the reciprocal of 5). We have, then, the Egyptian ‘unit fraction equation’

3/5  =  1/3 + 1/5 + 1/15

This at once raises several interesting mathematical questions:

(1) Can every proper fraction a/b (where a,b are positive integers)    be expressed as a sum of unit fractions?
(2) For any given a/b, how many ways are there of representing it as a sum of unit fractions, supposing it to be possible ?
(3) Is there a way to reduce the number of terms in the series of unit fractions to a minimum, supposing a/b can be so represented ?

Sebastian Hayes

Note 1 In a version of this article published in M500 magazine, I stated erroneously that the Egyptians “like the Arabs today” write numerals in ascending rather than descending order. A correspondent, Rakph Hancock, pointed out that present-day Arabs speak and record their numerals much as we do, i.e. in descending order, but write them, like everything else, from right to left.
There are no less than three distinct issues here (1) writing anything sentences, names, numbers &c. from right to left or from left to right, (2) writing numerals in ascending or descending order, (3) the order in which we deal with numerals when performing addition/subtraction and so on.
My main source, Gillings, writes “Today most nations write from left to right, and our numbers are so written also; but the values of the digits in our ‘Hindu-Arabic’ decimal system increase in place value from right to left. So, if we have to perform an addition or subtraction, we begin with the units column on the right, and work toward the left through the tens, hundreds and so on.  Conversely, the Egyptians wrote their words and numbers from right to left. Of necessity, however, the Egyptian mathematicians, like ourselves, had to start adding in the opposite direction to that in which they were accustomed to write, so the place value of the Egyptians’ digits increases from left to right, and the Egyptian system therefore runs widdershins to ours.”  Gillings, Mathematics in the Time of the Pharaohs

The confusion about left to right and right to left has been compounded because renderings of ancient Egyptian texts into English naturally reverse the orientation with respect to sentences but often print the numerals in imitation hieroglyphs in the order in which they appear in the papyrus with a modern numeral written underneath each one (so that people can see the hieroglyphs). This gives the erroneous impression that the Egyptians wrote their numerals in ascending order which, seemingly, they did not.
At first sight  it seems something of a mystery why most, if not all, societies write their numerals in descending order.   Presumably this is so because of the way in which large quantities were assessed by State officials. Confronted with, say, a confused mass of prisoners or pottery imports, an official would start by working out the thousands or hundreds, then pass to the tens and finally to the units. He would call out the amounts as he worked them out and the scribe would record the numerals in the order in which he heard them, i.e. largest amount first (but writing from right to left) . Moreover, in this way, a visiting Egyptian official could get a rough idea of the size of a batch of prisoners or pottery imports simply by glancing at the first hieroglyphic numeral (which, remember, is a different pictogram for each power of ten). But, when it came to actual operations with numbers, the scribes like everyone else had to proceed the other way (though some mental arithmetic experts say they add the higher columns first).
The confusion demonstrates a basic conflict between two very different functions of numbers : (1) as devices for the compact recording of data and (2) as a means of drawing original conclusions from given data. The first process is a movement from the unknown (or very roughly known) to the known, the second a movement from the known into the unknown which, if the reasoning is valid, eventually transforms it into part of the known.  Professional mathematicians tend to think that numbers were invented for the purpose of getting out precise solutions to mathematical equations, but the recording function of numbers was by far the more important for millennia and arguably still is.    Sebastian Hayes

Ogilvy and Andersen, in their excellent book Excursions in Number Theory , recount the true story of an Austrian  colonel who wanted to buy seven bulls in a remote part of Ethiopia some sixty or so years ago. Although the price of a single bull was set at 22 Maria Theresa dollars, no one present could work out the total cost of the seven bulls — and the peasants, being peasants, didn’t trust the would-be buyer to do the calculation himself. Eventually the priest of a neighbouring village and his helper were called in.
“The priest and his boy helper began to dig a series of holes in the ground, each about the size of a teacup. These holes were ranged in two parallel columns; my interpreter said they were called houses. The priest’s boy had a bag full of little pebbles. Into the first cup of the first column  he put seven stones (one for each bull), and twenty-two pebbles into the first cup of the second column. It was explained to me that the first column was used for doubling; that is, twice the number of pebbles in the first house are placed in the second, then twice that number in the third, and so on. The second column is for halving: half the number of pebbles in the first cup are placed in the second, and so on down until there is just one pebble in the last cup. If there is a pebble remaining when doing the halving it is thrown away.
The division column (the right one) is then examined for odd or even numbers of pebbles in the cups. All even houses are considered to be evil ones, all odd houses good. Whenever an evil house is discovered (marked in bold), the pebbles in it are thrown out and not counted, and the pebbles in the corresponding ‘doubling’ column are also thrown out. All pebbles left in the cups of the left, ‘doubling’ column are then counted, and the total is the answer.”
from Ogilvy & Andersen, Excursions in Number Theory

The working on paper would be as follows :

Doubling Column       Halving Column

7                                     22
14                                     11
28                                      5
56                                       2
112                                       1
154

The priest worked out the result using holes and pebbles in the way I have demonstrated though instead of using different coloured beans the helper simply removed the stones from right-hand holes opposite ones with an even number in them. The colonel duly paid up, astounded to note that the crazy system ‘gave the right answer’.
Let us go further back in time. We suppose that a ‘primitive’ society had grasped the principle of numerical symbolism at the most rudimentary level, namely that a chosen single object such as a shell or bean could be used to represent a single different object, such as a tree or a man and that clusters of men or trees could be represented by appropriate clusters of shells — the ‘appropriateness’ to be checked by the time-honoured method of ‘pairing off’. This society has not, however, necessarily attained the stage of realizing that a single ‘one-symbol’ will do for every singleton, let alone reaching the stage of evolving a base such as our base ten. Now suppose  the chief wants each of the villages in a certain area to provide  ‘nyaal’  or
□ □ □ □ □ □ □  young men for some public works or warlike purpose. We have nyata’  or □ □ □ □ □ □   villages from which to draw the task force.  The chief relies on two shamans to carry out numerical calculations, both of whom are adept in the practice of ‘pairing off’ but one has specialized in ‘doubling’ imaginary or actual quantities, the other in ‘halving’ imaginary or actual quantities. Although both shamans know that every quantity can be doubled, the ‘halving’ shaman knows that this procedure does not always work in reverse. He gets round this by simply throwing away the extra bean or shell — the equivalent of our ‘rounding off’ a quantity to a certain number of decimal places.
The ‘halving shaman’ works with a column of holes on the left hand side of a ‘numbering area’ (a flat piece of ground with holes in it) and he has a store of short sticks, shells or some other common object, which he places in the holes, or simply in a cluster on the ground. The doubling shaman works with a similar column of holes on the right but he has a store of beans or shells which are in two colours, light and dark. (The use of colour to distinguish two different types of quantities, or to distinguish between males and females, was the invention of a revered shaman who taught the two current shamans.)
The halving shaman sets out the sticks or shells representing the villages and tries, if possible, to have two matching rows. The doubling shaman watches carefully and, if the amount on the left can be arranged in two rows exactly, as in this case, he starts off with a set of dark coloured beans to represent the young men to be co-opted for the task at hand from each village.  We thus have

Villages                                            Young males

□ □ □                                                       ■ ■ ■ ■
□ □ □                                                       ■ ■ ■

Now the Halving Shaman selects half the quantity in the first hole, i.e. a single row of □ □ □, and arranges it as evenly as possible in two rows. In this case, there is a bean left over, and the Doubling Shaman, noticing this, doubles the original amount on the right but also changes the colour of the beans. We have

□ □                                                □ □ □ □ □ □ □
□                                                   □ □ □ □ □ □

The Halving Shaman discards the extra unit on the second line of my diagram and once again halves what is left. This leaves just a single bean and, since we are not allowed to split a bean or shell, this signals the end of the procedure as far as he is concerned. The Doubling Shaman doubles his quantity and since the quantity on the left is ‘odd’ — it cannot be arranged in two matching rows — he once again chooses light coloured beans.

□ □ □ □ □ □ □
□ □ □ □ □ □
□ □ □ □ □ □
□ □ □ □ □ □

The two shamans collaborate to combine all the light coloured beans (but not the dark coloured ones) on the right hand side, giving a total of

□ □ □ □ □ □
□ □ □ □ □ □

□ □ □ □ □ □
□ □ □ □ □ □
□ □ □ □ □ □
□ □ □ □ □ □

The chief is given this amount of beans and thus knows how many young men he can expect to get for the task at hand. From experience, the chief will have a pretty good idea of what this collection of beans represents in terms of men and, if it seems inadequate for the task, may decide to increase the quota of young men impressed from each village. When preparing for battle, the chief might use human beings as counters, pair them off against the beans, then have them form square formations to judge whether he has a large enough army or raiding force.
If asked by a time traveller why the dark-coloured beans — which are always opposite an even number — are rejected, the Doubling Shaman would probably say that even amounts are female (because of breasts) and the chief doesn’t want effeminate men or boys who are still living with their mothers.

The multiplicative system just demonstrated is very ancient indeed : it is probably the very earliest mathematical system worthy of the name and was doubtless invented, reinvented and forgotten innumerable times throughout human history. Since it does not require any form of writing and involves only three  operations, pairing off, halving and doubling, which are both easy to carry out and are not troublesome conceptually, the system remained extremely popular with peasants the world over and became known as Russian Multiplication because, until recently, Russia was the European country with by far the largest proportion of innumerate and illiterate peasants. It is actually such a good method that I have seriously considered using it myself , at any rate as a visual aid in doing mental arithmetic — it is one of the tools employed by traditional ‘lightning calculators’ and mathematical idiots savants.
Actually, one could say that the three mathematical procedures predate not only the earliest tribal societies but even the existence of animals ! Viruses, the lowest form of ‘life’ — if indeed they are to be considered alive at all which is still a matter of debate — are incapable of doubling, i.e. cannot reproduce, let alone halving and have to get the DNA of another cell to do the work for them. They may be considered capable of ‘pairing off’ however, since a virus seeks out the nucleus of a cell on the basis of one virus, one nucleus. Bacteria, a much more advanced  life form, reproduce by mitosis, essentially duplicating everything within the cell and splitting in two, the ‘daughter’ cell being an exact replica (clone) of the ‘mother’ cell. Each prokaryotic cell is diploid, i.e. has a double complement of chromosomes, and this (even) number cannot be changed — it is 2(23) = 46 in humans. Eukaryotes, however, though still capable of pairing off and reproducing by mitosis (doubling) are also able to halve this diploid number by producing special so-called haploid cells (gametes) which, in our human case, come in two kinds, spermatozoid and ovum. Fusion of the ‘egg’ and ‘sperm’ cells restores the diploid number and incidentally introduces a further mathematical operation, combination, which may be considered the distant ancestor of Set Theory. It is thus maybe not at all surprising that peasants the world over have felt at home with ‘Russian’ Multiplication, living much closer than we do to the generative processes of Nature, even if they did not know what was going on.
A good written notation is not at all essential for Russian Multiplication, but it does speed things up. Using our Hindu/Arabic notation, suppose you want to multiply 147 by 19. This is a somewhat tedious enterprise if you are not allowed a calculator and these days two students out of three would probably come up with the wrong answer. So here goes

19               147
9               294
4              588                     147
2            1176                    294
1            2352                  2352

2793

Now do it with a calculator. The result:  2793.

Why does the system work? You might like to think about this for a moment before reading on. (It personally took me a long time to cotton on though someone I mentioned it to saw it at once.)
Russian Peasant Multiplication works because any number can be represented as a sum of powers of two (counting the unit as the 0th power of any number). Algebraically we have

N  =         An xn + An-1 xn-1 + …….+ A1 x1 + A0

with x = 2. In practice there are only two choices of coefficient for the An , An-1 …….A0  namely 0 and 1 because once we get to a remainder of 2 we move to the next column.  When 0 is the coefficient this term is not reckoned in the final count — is discounted just like the pebbles in the hole opposite an even cluster. Since 1 × xn = xn, we can simply dispense with coefficients altogether — which is not true for any other base.
If we look back at the pattern of black and grey in the right hand column and write 0 for black and 1 for grey, we have the representation of the number on the left in binary notation (though it is in reverse order compared to our system). Take the multiplication of 19 and 147  a couple of pages back.

19             147
9             294
4             588                  147
2            1176                 294
1            2352                  2352
2793

The pattern in the right hand column is, from the bottom upwards,

Grey
Black
Black
Grey
Grey =   10011   =  24      23     22     2     unit
1       0      0       1       1

= 1910

A hole in the ground functions as a ‘House of Numbers’ and can only be in two states: either it is empty or it has something in it (i.e. is non-empty). The Abyssinian priest’s assistant who removed the stones from a house opposite one with an even number of stones in it was placing the House in the zero state. The right hand column Houses were in fact functioning in two different though related roles: on the one hand they were in binary (empty or non-empty) while on the other hand they gave the quantities to be added in base-one.
Did people using the system know what they were doing? In most cases probably not although, judging by their confidence in handling arithmetical operations, the Egyptian scribes, using a very similar method I shall perhaps write about in a subsequent article, almost certainly did : the peasants using the system just knew it worked. There is nothing surprising or shocking about this — how many people today who use decimal fractions without a moment’s thought realise that the system only works because we are dealing with an indefinitely extendable geometric series which converges to a limit because the common multiple is less than unity?

One might wonder whether it would be possible to extend the principle of Russian Multiplication to tripling, quadrupling and so on?

Take 19 ­ ×  23 using 3 as divisor and multiplier

19             23

6             69

2             207

?

We have already run into difficulties since we cannot get back to the unit. On the analogy with modulus 2 Russian Multiplication, we might decide we have to take into account the final entry on the right nonetheless,  plus all entries which are not opposite an exact multiple of 3.  This means the answer is 207 + 23 = 230  which is way off  since 10  ×  23 = 230. What has gone wrong?

A little thought reveals that, whereas in the case of modulus 2  we only had to neglect at most a unit on the left hand side, in the case of modulus 3 there are two possible remainders, namely 1 and 2. If we are opposite a number on the left which is 1 (mod 3) we include the number on the right in the final addition. However, if we are opposite a number which is 2 (mod 3) we must double the entry on the right since it is this much that has been neglected. In the above 19 = (6 ×  3) + 1 and so it is 1 (mod 3) but 2 at the bottom is (0 × 3) +2 and so is 2 (mod 3).  Applying the above we obtain   23 + (2  ×  207) = 23 + 414 = 437  which is correct.
To make the system work properly we would thus need not one but two ways of marking entries in the right hand column to show whether they just have to be added on or have to be doubled first. This is an annoying complication, and even apart from this it is not that easy to divide into three and to treble integers. And if we move onto higher moduli there are much greater complications still. The Russian way of doing things ceases to be simple and user-friendly.  Russian Peasant Multiplication is a good example of an invention excellent in itself but which does not lead on to further inventions and discoveries: it remains all on its own like an island in the middle of the Pacific Ocean. Once the crucial improvement of distinguishing the entries to be added from the others was made, there was nothing much that could be done in the way of improvements except possibly the introduction of colour coding, my distinction between dark and light coloured beans. To actually find a better multiplication system you have to make a giant leap in time to the ciphered Greek system of numerals or the full place value Indian system — and even so the advantages would not have been apparent to peasants. If you are only dealing with relatively small quantities, Russian Multiplication is quite adequate, is easier to comprehend and there are less opportunities for making mistakes. In such a case we see that there is indeed a ‘simplicity cut off point’ beyond which it is not worth extending existing techniques, since the disadvantages outweigh the advantages. However, there may also be a ‘second time round point’ when technology has become so sophisticated that it has become ‘simple’ (= ‘user-friendly’)  once more. Computers, being as yet relatively unintelligent creatures, have reverted to base 2 arithmetic though I believe 16 is also used. Wolfram’s Cellular Automata based on simple rules which specify whether a given ‘cell’ repeats or doesn’t can perform complicated operations like taking square roots of large numbers.

This cycle of invention, stasis, disappearance and re-invention happens all the time : it is more often than not impossible to improve on an early invention without making a giant leap, a leap requiring not only new ideas but large-scale social and economic changes which are usually felt to be undesirable because disruptive, or are quite simply out of the question given the available technology. Short of hiring expensive modern haulage equipment the best way to move large heavy objects across uneven ground is the time-honoured Egyptian system of wooden rollers which are repeatedly brought round to the front. (I have often had occasion to use this system myself in inaccessible places and it is surprising how well it works.) The long bow made of yew and animal gut more than held its own against the far more advanced crossbow : English bowmen won  Agincourt against axe-wielding French knights and Genoese cross-bowmen largely because the crossbow is slow to reload and its effectiveness is much reduced in wet weather (the English kept their cat-gut dry until the battle began). In point of fact the longbow, an extremely rudimentary weapon, was only superseded in speed, range and accuracy by the repeating rifle ¾ one of Wellington’s military advisors seriously suggested re-introducing the longbow against Napoleon’s Grande Armée. And the horse as a means of transport was only superseded by the railway : messages were not transmitted much faster across Europe (if at all) under Napoleon than under Augustus Caesar.     S.H.  26/1/12

Acknowledgment  :  This article appeared in M500 Issue 243 , “M500” being the magazine of the mathematical department of the Open University, editor Tony Forbes, for whom many thanks. .

Russell and Frege viewed number — and in what follows ‘number’ should be taken to mean cardinal number, the answer to How many? — as the Set of All Sets that is ‘similar’ to a particular set M. Thus ‘seven’ is a label that English speakers agree to attach to certain collections of objects that are ‘similar’. How do we know the collections are similar ? Because we can pair them off member for member. We can do this for the two sets  ∏ ∏ ∏ ∏ ∏ ∏ ∏  and  Δ Δ Δ Δ Δ Δ Δ because we can form the pairs  ∏ Δ   ∏ Δ    ∏ Δ   ∏ Δ   ∏ Δ   ∏ Δ    ∏ Δ
No member of either set has been left out so if ‘seven’ or sept’  or 7 or VII is to be attached as an identifying mark to the first set, to be consistent, we must also apply it to the seond. As a further step we can form sets with different objects or exchange nembers of one set with members of another. Provided the resulting sets can be paired off with an original ‘seven’ set, we can legitimately apply the same mark or label.
This is a very useful approach provided we do not follow Russell into the quagmires of infinite sets, the Axiom of Choice and so on  : we may at this stage of the game assume  that all sets of objects we wish to number can be exhibited or listed.  It is not expected that we actually exhibit or group together all the members of a particular set since the objects in question may be far away like stars or very numerous like pebbles on a beach,  but it is essential to  believe  that in principle  this could be done.
A rather different approach to number is taken by von Neumann who identifies the Cardinal Number of a Set with a standard set which is selected from all the various sets similar to it. As it happens, everyone in society barring certain disabled or deformed persons is provided with a small portable standard set of numbers, namely their thumbs and fingers. For larger quantities the feet and other parts of the body can be brought into play. For more complex early societies the same principle applied : stacks of cowrie shells were made the ‘standard set’  amongst the Yoruba in West Africa, knots in cords amongst  the Inca. This ‘standard set’ approach, supplemented by the introduction of bases, even applies to the earliest Egyptian written numerals, the so-called hieroglyphic numerals. Here all quantities less than ten were represented by so many upright strokes   | | |  or | | | | |   and the same applied to groups of tens .
The beauty of such a system is that the label or marker for the entire set of all ‘similar’ groups of objects is itself a member of the set and can be paired off with anything you like to mention. One can, using one’s fingers pair them off with a small group of beans, a distant but clearly recognizable clump of trees or even with completely inaccessible groups of stars.  The crippling disadvantage of the system is
that we soon run out of bodily parts and, if we use beads or notches as numbers, we find we require a lot of space to represent even quite modest collections. Also, in the case of recorded numerals, it soon becomes difficult to tell at a glance what quantity is being represented as | | | | | |  looks more or less the same as | | | | |   Some Egyptian scribes got round this by grouping the numerals in clumps with no more than five identical signs in one clump. This makes it a little easier on the eye, but it was not long before the scribes moved on to a
ciphered system so-called where the ‘one symbol’ is not repeated endlessly but different single marks represent quite diverse quantities.
On the other hand, what is infuriating about our ciphered system of numerals is that the written or spoken words cannot be paired off with the groups they represent. Seven is a single word, and even if we take it letter  by letter we cannot pair it off with  Δ Δ Δ Δ Δ Δ Δ
Worse still, each of the following numerical labels 7, 8, 5  or again 785, 462 can be paired off with each other. This being so, it is something of a miracle if children finally catch on to numbering at all. Number books for young children generally present several pictures of sets of well known objects, apples, trees, goldfish and so on with the word SEVEN or SIX in bold at the head of the page. The hope is that by dint of varying the exhibits the child will eventually associate the word with the numerically similar groups portrayed and with no others.
Actually, this conflict between the Russellian and von Neumann approaches to number existed from the very earliest days of humanity. One can, I think, assume that number words came a long time before recorded systems of numerals. Many societies made do with no more than a handful of number words, sometimes just three or two. Thus the Bacairi of Brazil use only the words tokale (one) and ahage (two). Three is ahage tokale (though ahewao also exists), ‘four’ is ahage ahage  and so on (Closs, 1986). This means that even quite small quantities require a lot of words. But inhabitants of such societies often supplement spoken numerals with signs : “They [the Waica] show
exact numbers higher than two by raising their fingers. I have seen them crossing the dwelling to see if a person is holding up three or four fingers in the semidarkness. I have asked for as many as 12 objects and received the exact quantity by showing them four fingers of my hand three consecutive times” (James Barker, 1953 quoted Closs).
Today, we no longer have a standard set of objects that are universally recpognized as numbers : our written numerals are ciphered and calculations are carried out by arrays of minute lights that we do not even see. Numbers have been handed over to the care of professionals, electrical engineers on the one hand, pure mathematicians on the other. But old habits die hard : we still use our fingers
to point to objects on the rare occasions we need to count them and umpires in cricket matches transfer a marble or other small object from one hand to another as each ball is bowled. Curiously, the species which invented mathematics has a very poor innate sense of number, and the more advanced the society becomes the poorer the individual’s perception of quantity. Whereas many missionaries and traders in remote parts of Africa and Latin America were astounded to find that illiterate and innumerate tribesmen could tell at a glance whether a single member of their large herds of livestock and dogs was missing, we are perpetually losing things and making ludicrously inaccurate guesses at the quantities all around us. (Note 2) .   S. H.

Note 1  :     Closs (Editor), Native American Mathematics, 1986  University of Texas Press

Note 2 :   “The long train of mounted women was surrounded in front, in the rear, and on  both sides by countless numbers of dogs. From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected again. I have often wondered since how they, without knowing how to count, would tell at once, in spite of the confused throng, that one dog was missing.”
This is taken from an account written by made by of a missionary amongst the Abipones, a tribe of South American Indians and is quoted in Menninger, Number Words and Number Symbols, Dover 1992, p. 10-11.

“He who examines things in their growth and first origins will obtain the clearest view of them”  Aristotle

TODAY the natural numbers are treated as entirely abstract entities and as such are evolved either from the Peano Axioms (Note 1) or, if you want to be really stylish, from the Axioms of Zermelo-Frankel Set Theory, especially ZF6, the so-called Axiom of Infinity. Few people  apart from professional pure mathematicians find such ‘derivations’ convincing and those who are unmathematical find this whole way of proceeding both mystifying and repellent. One does not in fact recognize what we think we know as numbers in these abstruse procedures.
I propose to approach the question in a completely different, pragmatic spirit. What do we need if we are going to make — yes, actually make in the sense of ‘put together’, ‘construct’ — a number system? Well, to start off with you need a lot of ‘ones’, more or less similarly sized distinct objects that are readily manageable, like shells, beans, twigs. Such a set of ‘concrete numbers’, once officially consecrated , serves as a standard set against which you can pair off actual or imaginary collections of ‘ones’. Ideally, you want to be able to make as many new ‘ones’ as you are likely to need : but even in a relatively advanced society you are not likely to need any more shells or beads that you can either find or make.
Secondly, you need to be able to play around with this standard set, dividing it up, bringing little groups together and so on. This is not so easy to achieve since small objects get lost or otherwise ‘disappear’. We thus require objects that can be neatly piled up into stable structures like cowrie shells, or we require hollow objects like beads that can be imprisoned on the threads or wires of an abacus frame and slid to and fro. The Romans used marbles moving in grooves on hand-held calculi, the distant ancestor of our electronic calculator. There are other methods of a rather different nature, such as making knots in cords or marks on the surfaces of larger objects like walls. The important thing is that it should be relatively easy to group and regroup the concrete numbers. Manouevres with groups of vasrying size constitute the three basic operations of arithmetic, division, addition and subtraction. (Multiplication is a rather more comkplex procedure and is not, to my mind, a fundamental arithemtic operation.)
Thirdly, you will find, or might find, that you need to define standard subgroups of increasing size. Human beings are surprisingly inapt at assessing at a glance (subitizing) even quite small collections of objects accurately : indeed, it is in part because we are so bad at doing this that mathematics evolved in the first place. Take a guess at how many shrubs there are in a garden in front of you, how many gulls there are sitting on the grass in a park or on the beach, even how many different objects there are on the table in front of you. Unless you are something of a prodigy, you will be amazed to find how far out you usually are. Most people can only subitize reliably collections of seven or eight objects : any larger collections we have to count  one by one. For accurate assessment, it soon becomes necessary to break up larger collections into easily recognizable smaller ones, and, fairly early on, societies realized that it would be extremely convenient to introduce this feature into the concrete number system. The idea of ‘bases’ was born : one of humanity’s most useful inventions.
It is not absolutely necessary that the various subgroups should have a single ‘base’ like our 1, 10, 100, 1000… The Babylonians had 60 as their principal base — which is why we have sixty seconds to the minute and three hunded and sixty degrees — but, since 60 is relatively large, they found it necessary to have a ‘sub-base’ of 12.  The Yoruba used a mixed five-twenty system : “In the northern parts of the region, the cowrie shells were counted out in groups of five, while along the coast they were pierced and threaded,m generally in strings of forty” (Zaslavsky, Africa Counts p. 225). The same author states that the ‘Lagos system’ went like this :
“40 = 1 string
2000 =  1 head = 50 strings
20,000 = 1 bag = 10 heads”

There is, however, such an obvious advantage in keeping to a single base that the vast majority of advanced societies have done so.  There is only one ‘transformation rule’ to learn and one can conceptualize the entire process as a matter of collectimg together so many units where the value of the ‘new unit’ changes by the same amount each time, one becomes ten becomes ten tens or a hundred and so on. As long ago as 2000 – 3000 BC  the Egyptians evolved an excellent written system which went up to one million and is very clear and easy to use — I have tried it out myself. Usually the base chosen has been what we call ten and this is certainly no accident. 10 is only slightly larger than the limit of our subitizing capacity (8) so there is not too much of a gap and, more important still, we have ten fingers and thumbs. The first computer was the human hand and had we been born with twelve fingers and thumbs, mathematics would have got off to an even better start since 12 is obviously superior for calculation because it has more factors (we cannot even get a proper third  or quarter in our decimal system).
Nature is not consciously mathematical, or even numerate : if certain specific numbers keep coming up — and few do systematically — there is generally some physical or biological reason for their appearance and reappearance.  In this sense it is perfectly true that numbers, or at any rate number systems, are human creations but they are firmly based on features of the natural world that really exist objectively. Even the most recondite properties of the natural numbers are, in effect, ‘brought into existence’ just so soon as we have a universe where there are so many ‘different bits’ as opposed to one single, unified ‘bit. One might say, to paraphrase Guy Debord, “Number has always existed but not always in its numerical form” (Note 2) .
The axiomatic method whereby all sorts of properties of a system are deduced from an initial small number of key assumptions has had a long and glorious history but there is no reason to suppose that this has anything to do with how the universe became what it is today. Certain numerical features can be singled out and thrown into a deductive mould, but all this is done for our convenience, that is all. Practically speaking, there are two only two abilities are necessary to develop and use with confidence a number system. They are  :
1. The ability to distinguish between what is singular and plural, i.e. recognize a ‘one’ when you see it;
2. The ability to carry out a one-one correspondence (pairing off).
All the mathematicians who have developed abstract number systems, for example  Zermelo and von Neumann, had these two
perceptual/cognitive abilities — otherwise they would have been denied access to higher education and would not even have been able to read a maths book. Animals, with one or two possible exceptions, seem to have 1.) but not 2.) which is perhaps the reason why they have not developed symbolic number systems (though a more important reason is that they did not feel the need to). Computers are capable of 1.) and 2.) but only because they have been programmed by human beings.
Since numbering has been such a successful activity, there must seeingly be features shared by our man-made numerical systems and the inanimate world : in other words, there must be an objective, physical basis to numbering and calculation. In a future post I will consider what natural ‘principles’ are involved, i.e. I will look for basic facts of existence which are the equivalent of the Formalists’ and Platonists’ Axioms.     S.H.

Note (1)  The Peano Axioms are

P1. 0 is a natural number.
P2. If n  is a natural number, then n’ is a uniquely determined natural number.
P3. For all natural numbers m and n, if m’ = n’, then m = n.
P4. For each natural number n, n’ ≠ 0.
P5. If M is a subset of N such that 0 Î M  and m’ Î M, then M = N.

Note (2) :  Guy Debord wrote, L’histoire a toujours existé, mais pas toujours sous sa forme historique” (Debord, La Société du Spectacle)

So where does it come from according to these authors?   The answer seems to be :
“…[from] concepts in our minds that are shaped by our bodies and brains and realized physically in our neural systems” (p. 346).  One might consider this a rather obvious, not to say bland, conclusion but it is not a theory the mathematical establishment is going to accept any day soon. Why not?  Because it knocks out the ‘transcendental origin’ theory, otherwise known as Platonism, to which practically all professional mathematicians subscribe either openly or covertly. The authors point out that there is no way such a claim could be tested, so it cannot really be considered a scientific hypothesis.
The authors demonstrate fairly convincingly that many of the sophisticated mathematical
procedures we employ can be traced back to primitive schemas, such as the ‘Container Schema’ which underlies Set Theory and Boolean Logic, schemas which are themselves abstractions from physical sensations made by — wait for it — infants in arms. “Mathematics….is grounded in the human body and brain, in human cognitive capacities, and in common human activities and concerns” (p. 358).
All this is not news to me since I have, off and on, been advancing some such theory of the origin of mathematics for the last thirty years, but it is nice to see some of the details of these familiar cognitive ‘grounding metaphors’ fleshed out. The dreadful fact is that mathematicians, pure just as
much as applied, cannot get on without metaphors culled from sensory experience, and, far from ‘transcending’ these metaphors by abstraction, all too often mathematicians remain pathetically tied to these conceptions, the ‘metaphor’ of the Number Line being the most grotesque example. For, whatever numbers ‘really’ are, they certainly are not points on a line and they are not at a specific distance from a mathematical ‘origin’.
From my point of view, the book does not go far enough, since it (just) stops short of developing a truly empirical theory of mathematics, largely because of the excessive importance the authors give to what they call the ‘Basic Metaphor of Infinity’ (BMI) — for if there is one mathematical concept that is not grounded in our sensory experience, it is infinity.

Also, the book is too long — nearly 500 large pages —  though anything shorter would have been
dismissed by the establishment as superficial. For all that, this is a very welcome book and a brave one too, since the authors remark at one point,  with commendable understatement, that “it is not unusual for people to get angry when told that their unconscious conceptual systems contradict their fondly held conscious beliefs” (p. 339). Out of context, you might think Lakoff & Núñez were referring to hardline Creationists from the Bible belt in America — but no, ‘people’ in this quote simply means professional mathematicians.    Sebastian Hayes

Note:  This review appeared in M500, the magazine of the Mathematics department of the UK Open University.  S.H.

Arithmetic and the Natural Numbers

In principle the whole of contemporary mathematics can de deduced from the six or seven basic axioms of Zermelo-Fraenkel Set Theory.  No one, of course, ever learned mathematics that way (including Zermelo and Fraenkel) and doubtless no one ever will.

As far as we can tell, mathematics did not evolve as the result of philosophic speculation or as a formal exercise in symbol manipulation. It was the large, centrally controlled societies of the Middle East, Sumeria, Assyria and Babylon in particular, who developed both writing and numbering (2). Why? Their reasons are pretty obvious: a hunter/gatherer, goatherd or small farmer who is in constant contact with his small store of worldly wealth does not need much of a number system, but a state official put in charge of a vast area with varied resources does (3). Arithmetic was invented and rapidly brought to quite an advanced level for mundane and very unromantic reasons : it was needed for stock-taking, censuses and above all taxation. Geometria, literally ‘land measurement’, was developed by the Egyptians for similar reasons : it was found necessary to assess accurately the surface area of very dissimilar plots of land bordering the Nile so that the peasants working these plots could be taxed more or less fairly. It was only much later that the Greeks turned geometry into a recondite and stylish branch of higher mathematics.

J.S. Mill, almost alone amongst ‘modern’ writers on logic and mathematics, took a pragmatic view of arithmetic. “’2 + 2 = 4’ is a physical fact”, Mill dared to write in his Logic ¾ for which he has endlessly been ridiculed since by the likes of Frege, Russell and countless others. Strictly speaking, Mill is wrong. ‘2 + 2 = 4’ is not the alleged fact but the symbolic representation of the alleged fact —  but this is splitting hairs. What Mill meant is undoubtedly correct, namely that ‘2 + 2 = 4’ is a faithful representation of what happens when you take //, or ‘2’ objects and bring them together with another // objects, making up a group of //// or ‘four’ objects. Does anyone seriously doubt that this is what happens?

‘1 + 1 = 2’ is untrue if we are dealing with entities which merge when they are brought into close proximity. For droplets of water ‘1 + 1 = 2’. Droplets of oil are a little more complicated since I have it from a physics textbook that, if you keep on adding oil, drop by drop, to a blob on a sheet of water, the original blob eventually separates into two blobs. There is thus an upper limit on n in oil-droplet arithmetic. For the limiting value N, when  n < N   ‘1 + n = 1’, but if n >  N,  ‘1 + n = 2’.

In cannot for the life of me see that ‘1 + 1 = 2’ is a ‘truth of logic’ as Russell and Whitehead consider it to be. If it were to be so considered, then we would have the undesirable situation where two incompatible statements were both ‘logical truths’ — since ‘1 + 1 = 1’ seems to me just as valid, merely less interesting. The fact of the matter is that each statement is true in the appropriate context, that is all there is to it.

However, this does not mean that our elementary mathematics is a ‘free creation’ or that the rules of arithmetic we have are completely arbitrary. They apply exactly to objects that can be combined without merging : if they did not so apply, we would disregard them and use other ones. This has nothing to do with whether or not our rules of arithmetic can be deduced from the Peano Axioms : Nature did not consult Peano in the matter.

As Mill correctly said, it is a matter of fact, and not of logic, that if you have, say, a collection of stones, say  ΟΟΟΟΟΟΟΟΟΟΟΟΟ   and you are told to put them into containers Δ that have room for  ΟΟΟΟ only, you will need  ΔΔΔ  containers, no more, no less. In our rather muddled terminology, ’12 divided by 4 gives 3’ (it would be better to say ’12 divided into 4 gives 3’).

Theorems of so-called elementary Number Theory are not only ‘provable’ in the pure-mathematic sense, but in the many instances actually testable, i.e. they pass the Popperian test for empirical disqualification. For example, if I read in a textbook that a pyramidal number with base 24  24 is also a square number I can check whether this is the case by building up a pyramid on this base and then flattening the whole lot and making them into a square (which turns out to have side 70). Obviously, I am not going to test such statements most of the time since I have confidence that the normal rules of arithmetic are soundly based, but at least I know I have this possibility. It will be objected that, when dealing with general statements which apply to an unlimited number of cases, I cannot test them all. This is indeed so but what I can do is examine a particular case and then convince myself that what makes the proposition true in this case is not something specific to the particular case, but which will extend to all other cases of this type. Such a procedure does not cover non-constructive proofs of theorems which provide for the ‘existence’ of such and such a number without giving any indication of how such a number can be produced. However, such proofs do not have the persuasive power of constructive proofs and have rightly been treated with suspicion by many mathematicians. The proofs given in Euclid Books VII, VIII and IX, which are devoted to Number Theory, on the other hand are strictly constructive.

Moreover, theorems about the so-called ‘natural numbers’ are, in general, not just ‘roughly true’, ‘true in the limiting case’, ‘statistically true’  and so on, but are either completely true or wrong. Such a situation can only make practitioners of other sciences gasp with envy. Aristotle’s physics, in its day no mean achievement, had to give way to Newton and classical mechanics has had to give way to Quantum Mechanics. But the substance of Greek Number Theory has, apart from a greatly improved notation, scarcely changed in twenty-three centuries. It is in this sense that we should interpret the oft-quoted statement of Gauss to the effect that “Mathematics is the Queen of the Sciences and Number Theory the Queen of Mathematics”.

And the reason for the much greater sureness of results in Number Theory is that numbers (whole numbers) are far more basic than everything else. The distribution of the prime numbers is a fait accompli which does not depend on a formula, even if one could be found, it is ‘what it is’ and  follows ineluctably as soon as we have something which is repeatedly divided up into little bits. Physicists have imagined all sorts of universes where not only the basic constants but many of the ‘laws’ themselves would be different, but it is impossible to imagine a physical world where, for example, Unique Prime Factorisation does not exist ¾ if you don’t agree try to imagine one. The divisibility properties of numbers are ‘given’ and no intelligence is necessarily involved : Nature does not know and does not need to ‘know’ what quantities can be divided up in such and such ways. Perhaps, the same goes for physical laws but this is harder to believe : even though scientists have long since dispensed with an intelligent Creator God, they still need to appeal to certain ‘physical laws’ which are conceived somehow to have been there before even the universe existed.

Calculus and Infinitesimals

It is distressing in the extreme that practically everyone assumes that because Calculus is more difficult than ordinary arithmetic, it is in some sense ‘truer’. The exact opposite is the case. Except in very simple examples where it is not needed, Calculus always involves ‘rounding off’ whilst elementary arithmetic doesn’t. If I amalgamate two flocks comprising 100 and 200 sheep respectively, the resulting flock will have 300 sheep, not approximately but exactly. In such cases the mathematical model is 100% accurate.

In the Differential Calculus, and representing the increments in the independent and dependent variables respectively, can always be arbitrarily decreased, at any rate in ‘continuous functions’. This means, amongst other things, that time cab be chopped up into ‘infinitely small’ segments ¾ can one really believe this? Even if one can, the assumptions on which Calculus is based are obviously wrong if we are dealing with phenomena that are known to be discrete. Also, in Calculus the roles of the dependent and independent variables, x and y, can be, and very frequently are, inverted at will : this means, in realistic terms that effects can cause causes which is fatuous.

Suppose we have a machine powered by steam or diesel and we set it to work. Can the input we give to it be arbitrarily decreased? Obviously not. Any energy input beyond a certain level will not be sufficient to overcome internal friction and so no work will ne done at all. (To think otherwise is to quarrel with the 2nd Law of Thermo-dynamics.)

Are the roles of energy in put and work done interchangeable? No, they are not : output depends on input but input does not depend on output except in sophisticated machines which have feedback devices, and even then only to a small degree. Also, Calculus is blithely used in molecular thermo-dynamics even though (dn) can, in reality, never be less than 1, i.e. a single molecule. The same goes for population studies.

So how on earth does it come about that such an inaccurate mathematical model somehow ‘gives the right result’? “By virtue of a twofold error, you arrive, though not at science, yet at the truth!” as Bishop Berkeley exclaimed in wonderment. The good Bishop’s objections were more philosophical than technical though for all that unanswerable at the time. During the nineteenth century when the Queen of the Sciences parted company from her husband Natural Philosophy, the mathematical inconsistencies were sorted out and the conceptual problems swept under the carpet where they have remained ever since.

How does an increment in the dependent variable, call it (dy), change with respect to a small increment in the independent variable (dx) : this is essentially the issue which gave rise to the Infinitesimal Calculus. Newton needed to solve it to determine orbits amongst other things. Now, if y is strictly proportionate to x, y = Ax + C, with A, C, constants, then the rate of change will be the same no matter how large or small we make (dx) and the graph of the ‘derivative’ (giving the rate of change) will be a flat straight line (or nothing at all if f(x) is a constant function). In such cases we do not need Calculus. In every other case ¾ and this will come as a shock to most readers — the so-called derivative can only be determined by discarding non-zero quantities and so does not give the exact rate of change of any actual physical process.

The eventual mathematical solution was to view the ‘derivative’ not as a ‘final ratio’ as Leibnitz and Newton did, but as a ‘limit’, where ‘limit’ has a very precise mathematical sense. The way in which a limit is defined in Analysis neatly sidesteps the issue of whether the limiting value is actually attained, or not. Roughly, the idea is that if (and only if) we can make the difference between a proposed value and the actual value of a function as it approaches a certain point less than any given quantity, then the function  ‘goes to the limit’. For example, in the above example, we are allowed to consider the limit of the derived function to be 2x since, given any assigned quantity, we are free to make (dx) even smaller.

So how did it come about that geniuses like Newton and Leibnitz were incapable of grasping what most sixth formers today absorb in a single lesson ? The reason is rather an odd one. Newton and Leibnitz were mathematical realists and not Formalists : they believed that mathematics should, and could, provide a model for what actually went on in the real world. The analytic assumption does the trick but it involves the wholly unrealistic assumption that (dx) and (dy) can be made arbitrarily small.

Leibnitz, in his version of Calculus, always dealt in definite ratios between definite quantities however small, and could not rid himself of the conviction that there must be a final ratio between two quantities in tandem where one changes continually with respect to the other. Against all the odds, he has been proved right. For, as stated earlier, in all working machines there is a lower limit to the energy input that can overcome internal friction and produce useful work. We also know now for a fact that all energy, light, heat, chemical bonding and so on, is quantized so there is always a lower limit to all energy transfers. The quantities involved are, of course, very small by our standards but there is no longer any need to call them ‘infinitesimal’, a vague and meaningless word : in many cases the lower limits can actually be given a number. It is the nineteenth century analytic mathematical model which has been shown to be unrealistic in its assumptions —  not that this bothers the pure mathematicians. All that remains is Space and Time which today are still usually considered to be ‘continuous’ —  though this has not been proved and probably never will be. Some physicists are already suggesting that Space/Time may be ‘grainy’ at a certain level and Wolfram in A New Kind of Science writes,

“The only thing that ultimately makes sense is to measure space and time taking each connection in the causal network to correspond to an identical elementary distance in space and elementary interval in time” (my italics).     (p. 520)

He even puts a number to these Space/Time infinitesimals, guessing that the “elementary distance is around 10-35 metres, and the elementary time interval around 10-43 seconds” (5). Whitrow before him launched the concept of the ‘chronon’, an ‘atom of time’, evaluating it as the diameter of the smallest elementary particle divided by c, the speed of light.

The Integral Calculus is perhaps a better model of real life conditions than the Differential — which is why it was developed first –Ο but, nonetheless, in its modern form it involves treating a whole host of quantities, speed, momentum, force &c. as continuous when in most cases we know perfectly well that they are not. And the Integral Calculus works for much the same reasons as the Differential does : if the quantities under consideration are small enough, the unreal mathematical assumptions don’t make much odds.

Calculus was developed to deal with a situation where, typically, we have two widely different scales of values, macroscopic and microscopic. The macroscopic values correspond to things we can actually observe as human beings but we usually assume, and in many cases know for a fact, that these quantities are built up, or broken down, bit  by bit in very small stages, too small to be recorded other than with high precision instruments. The growth of a bacterial disease can be modelled by Calculus, but it depends on the number of bacilli or viruses within the human body, and  this number, though large, is certainly not infinite. Likewise, we may suppose that the growth of a bacterial colony is ‘continuous’, is ‘going on all the time’, but this is not the case since even the fastest growing bacteria require about twenty minutes to reproduce themselves.

Practically speaking, the microscopic changes can usually be safely neglected beyond a certain point : this is why throwing away all these little dxs doesn’t make a lot of odds. But how do we know where to draw the line? This is a matter to be decided by the practising physicist or engineer: the business of the mathematician is to provide a coherent model which can be adapted to varying circumstances. We, as human beings, consider that if dn, a small increment, or reduction, in the human population variable, is a single person and it is someone we know, then this increment is not negligible. But if we are dealing in billions, as in world population studies, such a quantity really is negligible and Calculus is perfectly acceptable as a model even though we know that the quantity we are concerned with is always discrete. In molecular thermo-dynamics, dn cannot be smaller than a single molecule and is usually negligible : nonetheless, the more reputable texts warn the student about the dangers of using the Integral Calculus at the quantum level ¾ it can quite simply give the wrong answers.

The centuries old ‘mystery’ of Calculus turns out to be nothing more than a failure to carefully distinguish between the very different requirements of pure and applied mathematics. The pure mathematician seeks consistency, generality and elegance, the applied mathematician wants fidelity to the facts of the matter. In the pure mathematical model dis quite properly left as a free variable without a lower limit even though in most (all?) applications it will have a precise non-zero value. In pure mathematics the ‘rate of change’ of one quantity with respect to another ‘converges to a limit’ and it is immaterial to the pure mathematician whether it actually attains the limit (generally it doesn’t). But in the real world there is always a final ratio between causes and effects as Leibnitz stressed. Instead of being ‘God’s shorthand’, Calculus simply turns out to be an ingenious method of getting approximately true results when we do not know the exact values of certain small quantities. Today, in the computer age, the tendency is, increasingly, to dispense with Calculus and to slog it out numerically. Sic transit gloria mundi.

Definite and Indefinite Totalities

it would be wearisome to give here an empirical underpinning to all branches of classical mathematics, though I believe it can and should be done. Pending a through reform of the subject which is long overdue, mathematical sceptics such as myself are willy-nilly obliged to attend divine service though we do not have to take at face value everything the minister says. To judge by the frequent occurrence of the infinity sign, ∞, in mathematical textbooks, you would imagine that we are surrounded by double infinities at every moment of our lives, but such is not my impression. Expressions like ‘n  goes to ∞’  are basically directives, not formulae : they tell the reader to allow n to increase indefinitely and one could quite easily dispense with the infinity sign and just write n followed by an arrow when we are letting n increase without bound and have the arrow going the other way when n  is decreased indefinitely while remaining positive. This does away with the temptation to view ‘infinity’ as a specific quantity which it certainly is not.

Similarly, I don’t as a matter of fact believe in the reality of n spatial dimensions with n > 3 but I don’t have any special problem with ‘n-dimensional phase space’ so-called because the mathematical treatment has in reality nothing whatsoever to do with spatial dimensions in the normal sense. Anything that can be quantified (stick a number to), e.g. speed, mass, pressure &c. can be treated as a ‘dimension’ and, since there are certainly more than three types of quantifiable entities, it is up to a point legitimate to  talk of ‘Hilbert Space’, or of ‘n-dimensional space’ — though I still dislike the term because it thoroughly confuses the layman as it is intended to. (In string theory there are supposed to be 8 or 11 spatial dimensions but one is never sure how literally one is to take this and anyway there is to date no evidence at all that they exist.)

Infinity, if it is a meaningful term at all which I doubt, lies outside our normal sense experience and is hardly something that falls within the remit of the natural sciences. The Greeks very sensibly would have nothing to do with it : Euclid is careful not to prove that the set of prime numbers is ‘infinite’, only that there is no greatest prime number. As late as the early nineteenth century the greatest mathematician of his time, Gauss, stated categorically, “There is and can be no actual infinity”. But Cantor, the mentally unbalanced founder of Set Theory, really believed in actual infinity ¾ he specifically rejected the concept of ‘potential infinity’ (which is all in practice we need) as inadequate. And hot on his heels Russell, despite being an agnostic and a positivist, introduced into his and Whitehead’s Principia Mathematicae the ridiculous Axiom, “That infinities exist” (RW, *120.03).

The belief in ‘infinite sets’ only comes about  because of a failure to distinguish between ‘definite’ sets  and what I call ‘indefinitely extendable’ sets. A ‘definite’ set is fully constituted once and for all, and its members are, at least in theory, listable, one, two, three…… There is also a last member (though the actual choice of which member comes at the bottom of the list is usually arbitrary). An ‘indefinitely extendable set’ cannot be listed ‘once and for all’ because it is capable of being extended. The following anecdote will perhaps make clearer what I am driving at.

Two schoolboy philosopher meet up during the lunch break. Sceptic  exclaims, “You know, there’s not a single thing I’m sure about!” His companion rejoins, “Ah! but there is one thing at least you’re sure about, and that is that you aren’t sure about anything!” This rather floors Sceptic —  for the moment.

But in fact there is no contradiction. Sceptic’s first statement only referred to the fully definite set of all beliefs he had actually considered up to that moment, and a standpoint of all-round scepticism was  not one of them. It would be quite perverse to consider his first statement as referring to the collection of all possible beliefs he as a human individual might conceivably entertain. The belief  “I don’t believe in anything” was not, at the beginning of the discussion, a member of the Set of All Beliefs Sceptic Had Considered So Far (a definite set) but after the end of the conversation it was. His first statement was time and context dependent : it was not an intemporal assertion.

All this hardly seems worth dwelling on. So why the fuss ? Because, when it comes to mathematics, the situation is very, very different. Mathematical assertions are not generally considered to be time and context dependent, they are in some sense held to be ‘eternally true’, true even before human beings or the universe we live in existed. Once true, always true, when it comes to mathematics.

A ridiculous amount of printing ink has been devoted to such earth-shaking issues as whether the Set of All Sets contains itself or what exactly is the status of the Ordinal of the set of All Ordinal Numbers. In common speech we don’t tie ourselves up in mathematical knots because we usually restrict ourselves to statements which have a meaningful context, and above all we do not confuse finite definite Sets with what one might call ‘partly definite Sets that are capable of being extended’. Most of the meaningful categories such as ‘the human race’, ‘members of the EU’, ‘species of insects’, and so on, are collections which are particle definite in the sense that we can list bona fide members but which are continually being extended as mew human beings are born, Eastern European countries are allowed into the EU, and the procedures of natural selection produce new species. This does not make these Sets ‘infinite’, only indefinitely extendable and no one except mathematicians ever have a problem with this.

Viewed in this light, many of the ‘paradoxical’ features of so-called infinite sets vanish like morning dew. According to Dedekind and Cantor, the Set of All Positive Integers, itself an ‘infinite’ set, can be put in one-one correspondence (paired off) with the Set of All Even Numbers, since for any positive number you like to mention I can match this with its double. What of it? This is only astonishing and mystical and extraordinary if we consider ‘all’ the natural numbers laid out on the grass in some Platonic Never-never Land and ‘all’ the even numbers lying on top of them. Even though there are ‘more’ natural numbers than evens ¾ since 3 and 5 and 151 are not even ¾  the two sets can be paired off ! Amazing! How does this fellow Cantor do it?

If, however, N, the Set of All Natural Numbers, is viewed simply as an indefinitely extendable set starting 2, 4, 6, then the paired set {(1,2), (2,4), (3, 6)…..} is also an indefinitely extendable set. So what? But for any given definite set of integers, a subset comprising the evens within this set cannot be paired off with the whole set.

As a matter of fact most of the mathematical sets we are interested in turn out to be ‘partly definite indefinitely extendable sets’, i.e. we can list a few members but are free to extend the list ‘indefinitely’. What above all we must not do is, however, to confuse or compare an open-ended indefinitely extendable set with a fully constituted one.

In real life, not only do most sets get extended all the time, some members drop out (through death for example) and certain individuals oscillate endlessly between different sets. This sort of situation wouldn’t make for good mathematics, since mathematics needs things to be clearcut, but the real world is actually not that mathematical.

A typical example of ‘oscillating membership’  is provided by Russell’s Village Barber Paradox though Russell did not realize this. Russell invites us to consider a Village Barber who claims he shaves everyone in the village who does not shave himself and only such persons. The big question is : Does he shave himself? If he does shave himself, he shouldn’t be doing so — since, as a barber, he shouldn’t be shaving self-shavers. On the other hand, if he doesn’t shave himself, that is exactly what he ought to be doing.

The contradiction only arises because Russell, like practically all modern mathematicians, insists on viewing sets as being constituted once and for all in the usual Platonic manner. Let us see what would actually happen in real life.  It is first of all necessary to define what we mean by being a self-shaver : how many days do you have to shave yourself consecutively to qualify ? Ten? Four? One? It doesn’t really matter as long as everyone agrees on a fixed length of time, otherwise the question is completely meaningless. Secondly, it is important to realize that the Barber has not always been the Village Barber : there was a time when he was a boy or perhaps inhabited a different village. On some day d he took up his functions as Village Barber in the village in question. Suppose our man has been shaving himself for the last four days prior to taking on the job, so, if four days is the length of time needed to qualify as a self-shaver, he  classes himself on day d as a self-shaver. He does not get a shave that day since he belongs to the Self-shaving set and the Village Barber does not shave such invididuals.

The next day he reviews the situation and decides he is no longer in the Self-shaving category — he didn’t get a shave the previous day — so he shaves himself on Day 2. On Day 3 he carries on shaving himself — since he has not yet got a run of four successive self-shaving days behind him. This goes on until Day 6 when he doesn’t shave himself. The Barber spends his entire adult active life oscillating between the Self-shaving and the Non-self-shaving sets. There is nothing especially strange about this : most people except strict teetotallers and alcoholics oscillate between being members of the Set of Drinkers and Non-drinkers — depending of course on how much and how often you have to drink to be classed as a ‘drinker’.

This example was originally chosen by Russell to show that the self-referential issue has nothing necessarily to do with infinity. Nor does it, but it does depend on the question of whether sets or collections are time and context dependent.

Important Numbers and Formulae

Specific numbers, sequences and formulae can be important in three rather different senses :

1. 1.   they are significant in Nature and the physical world;
2. 2.   they are key items in the construction of a workable symbolic system;
3. 3.   they are important for aesthetic reasons.

Ideally, that is to say Platonically, the really significant numbers and formulae ought to be important in all three ways at once, and certainly this is how most of us would like them to be. But usually they are not. The constant of gravitation and the fine structure constant are extremely important in sense (1) but not at all in senses (2) and (3). (This is in itself a strong argument against Platonism.) To date all attempts to deduce the basic constants found in Nature, g, c and so on, from logical or philosophic principles have failed dismally. But if mathematics really were a window on a timeless world which lies within and behind this one, it should theoretically be possible to do precisely this ¾ as Eddington for one tried to do. Plato and Hegel thought it was possible to deduce astronomy from mathematics : Plato decided that all planetary orbits must be circular because the circle was a perfect shape, Hegel proved from first principles that there could be no more than seven planets in the solar system.

The great Platonic argument from the alleged ‘simplicity’ of Nature has always struck me as extremely weak. The motions of the planets round the Sun are so complicated that it took humanity thousands of years to painfully work out a decent calendar. Had a mathematician been put in charge of the solar system, he would undoubtedly have arranged things very differently, perhaps making the orbits fit into the shapes of the Platonic solids as Kepler at first supposed was the case. More recently, Solomon Golomb has invented a genetic code far more elegant than the messy UAGT  one used in our bodies since it is not only ‘comma-free’ (cannot be read in two or more incompatible ways) but “can correct any two simultaneous errors in translation, and detect a third error”. As someone put it, “Life would be more reliable if Golomb had been put in charge”. Most of the really  elegant pieces of pure mathematics are of little utility while the mathematics of Relativity and Quantum Theory, though undoubtedly ingenious, could not remotely be called either simple or beautiful. A pure mathematician is supposed to have said about partial differential equations (which are useful), “This is not mathematics but stamp collecting” and I am inclined to agree with him.

There are a few numbers and functions ¾ but not that many ¾ which are equally significant to the pure mathematician and the scientist, e being a case in point. There is nothing surprising about the omnipresence of e, or rather the importance of exponential and logarithmic functions in Nature since they are central to biological processes. The more bacteria or humans there are, the more there are going to be in the near future when they reproduce, so we would expect population growth to be ‘exponential’ (proportional to size) ¾ until the food supply gets used up, or other limiting factors kick in.

The numbers we use the most are those which serve as bases, notably 10, the Babylonian base 60 (still used for seconds and minutes), also 2 which has had a new lease of life because of computers. Apart from 2, none of these numbers are especially important in Nature which, though it has a certain tendency to favour bilateral symmetry (right-left) does not use numerical bases. 10 is a dull number and its quasi-universality as a numerical base is the result of a historical accident ¾ primates have ten fingers and thumbs.

Conversely, many of the theorems and numerical conjectures that have delighted and obsessed  mathematicians are of no importance in physics. I doubt if it makes any great odds scientifically speaking if the Riemann Conjecture, the leading unsolved problem of pure mathematics, is true or not, and the same applies to Euler’s surprising discovery that the inverse squares 1 + ½2 + 1/32 + ….  sum to p2/6. Primes have limited use since they are currently used for encryption but otherwise there is no particular significance in their distribution amongst the natural numbers. Indeed, the most prized mathematical theorems are like emeralds : they delight humans but are of no great significance to the physical world.

The Unholy Alliance of Platonism and Formalism

Mathematical Platonism, apart from being a hard sell in a godless world, is difficult to maintain across the board. In practice most pure mathematicians sign up to some sort of amalgam of Platonism and Formalism : the best bits partake of the eternal but the more mundane parts are human invention. As one eminent contemporary mathematician puts it

“There are things in mathematics for which the term ‘discovery’ is much more appropriate than ‘invention’.(…) One may take the view that in such cases the mathematicians have stumbled upon ‘works of God’ ” (Penrose, The Road to Reality).

This is all very well up to a point and I am myself mathematician enough to have felt this frisson when confronted with certain theorems which, as Hardy said when being confronted by certain theorems of  Ramanujan, “must be true because no one could possibly have conceivably have made them up”. One is reminded of Coleridge’s distinction between Fancy and Imagination. But the lack of any clear guidelines as to how to distinguish the dross from the gold, the invented part from the divine, is embarrassing to say the least especially since we are dealing with a discipline that prides itself on its rigour. The criteria are vague : ‘simplicity’, ‘generality’, ‘elegance’, ‘profundity’…… On top of that, these criteria, for what they are worth ¾ and they are worth something ¾ are rarely met at the same time. I wouldn’t call the formula for solving a quadratic ‘beautiful’ (x = (b ±  Ö(b2 4ac))/2a  but it certainly has generality. Leibnitz’s formula

p/4 = 1 1/3 + 1/5 1/7 ……

is delightful but more a curiosity than anything else and useless for calculation purposes (because you have to add  hundreds of terms to get p even to three or four places after the decimal point). Indeed, I would go so far as to say that the elegance of pure mathematical theorems is inversely proportional to their usefulness (even in  strictly pure mathematical terms). But if a theorem of formula really were ‘universal’, it would crop up all over the place, both in nature and in man-made systems.

The extravagance of the claims made for mathematics (by Platonists) is sometimes really alarming : certainly no practitioner of any other art or science would dare to speak in such terms without fear of ridicule. Anyone passionately involved in something tends to envisage the whole universe in terms of it : the ballet dancer sees life as a dance and countless people from Empedocles onwards have envisaged the whole of life as a kind of erotic play. For the musician, in the beginning was the sound, for the calligraphist, all started with a brushstroke. But Penrose typically discounts all such claims except those made for mathematics :

“I cannot help thinking that, with mathematics, the case for believing in some kind of ethereal, 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.”

Penrose, (1989)

But why on earth should we believe that complex analysis is more ‘eternally true’ than Bach’s fugues or Highland Dancing? Certainly, of all symbolic systems yet invented, mathematics is by far the most efficient as a prediction system for physical phenomena ¾ but this is not what is being claimed.
Wonderful though it is (or can be), in recent years mathematics has become rather too big for its boots and needs to be brought down a peg or two. Mathematics arises in the main from our human concern to understand the workings of Nature, so at any rate I would like to believe. But this does not mean that Nature is consciously mathematical ¾ it neither is nor needs to be. In mathematical terms, every time a young child, or a cat, reaches out to snatch a dangling object it is, as someone put it in a popular maths book, ‘solving a succession of differential equations’. But is it? Of course not. Snatching a moving object is just something you have to learn to do if you want to survive. And if biologists are to be believed ¾ and no one these days wants to take them on ¾ evolution is blind and devoid of intelligence as we understand the term : the rather disappointing message is that if you’ve got the time, trial and error suffices for most things, for that is basically all natural selection is.

A final point that is worth making is that the much vaunted ‘objectivity’ of mathematics is self-limiting : it means that mathematics, has precisely nothing to say about the human condition, nothing about the joyful/painful sense of the ephemerality of life, the perennial theme of most  art, especially painting and lyrical poetry. We may reasonably doubt that a Platonic world of eternal Forms and/or mathematical formulae exists, but it is not possible to doubt that the passing world of ephemeral sensations exists, the ‘Floating World’ of the Japanese printmakers who in turn gave rise to French Impressionism. If everything changeless and timeless turns out in the last resort to be an invention (= illusion), as Zen Buddhists believe, then this ‘Floating World’ of sensations is ‘more real’ than the Platonic ¾ and indeed certain Indian Buddhists such as Vasubandhu and certain Chinese Taoists sais precisely this. Our mathematics, manufactured to deal with the changeless, is a clumsy and inadequate instrument for approaching the ephemeral.

Conclusion

The belief that mathematics, in its entirety, is ‘a free invention of the human mind’ is completely unacceptable 1. because this is not, as far as we know, how arithmetic and geometry were first conceived and why they were developed, and 2. because it would make mathematics’ success as a predictive system and model of physical behaviour a pure fluke. Mathematics is indeed not the physical reality itself but a product of human effort. However, that part of it which is the most useful and the most important is a constrained invention, constrained by the way things are even though the ‘things’ do not know how and what they are.
So much for Formalism. The rather different belief that mathematics ¾ those bits of it that we find ‘profound’ or beautiful ¾ is equally hard to sustain : certainly such a belief is, in the Popperian sense, unscientific since there is apparently no way this side of the grave of disproving (or proving) such an extravagant claim. The argument from simplicity is weak because, although the universe may have looked reasonably simple to Archimedes and Newton, it certainly does not look simple today. The alleged ‘simplicity’ of mathematics is perhaps merely a human convenience. The argument from design is no longer accepted in biology ¾ so why should it pass water in mathematics? The trend in evolution is rather from simplicity to greater and greater complexity. As to beauty I am responsive enough to some areas of mathematics to be sympathetic to such a line of argument but it is really not much more than the ‘argument’ of a nineteenth century vicar who cites Beethoven’s Ninth Symphony as a ‘proof’ that there must be a God. I have myself witnessed one of Europe’s most eminent mathematicians show the audience the Mandelbrot Set on a screen as evidence of…well, evidence that mathematics could not be entirely a human invention because ‘who could have invented this’? Yes, the world is full of strange and wonderful things, the Mandelbrot Set perhaps included (though I don’t find it beautiful) ¾ but why shouldn’t it be? One could equally argue that there must be a Devil by exhibiting many of the horrible and monstrous things also to be found in Nature.

The commonsense view on mathematics is that it is a mental construction intended to represent those parts of the natural world in which we humans, or some of us humans, were most interested. This at any rate is how mathematics started. From the empirical/constructionist point of view which is mine the success of (basic) mathematics is no puzzle at all : what has been extracted from the physical world can be returned to it. I believe that science’s main concern should be with truth and that scientific assertions should be if humanly possible empirically testable or, if not, have ‘explanatory power’. Now on these counts elementary mathematics up to and including early calculus is a science, indeed ‘Elementary’ Number Theory is probably the  most successful science that will ever be invented since the properties of ‘numbers’, their divisibility or not in particular, are very deeply rooted in the natural world.

As to the more fanciful branches of mathematics, especially all those involving the transfinite for whose ‘existence’ (other than mathematical) there is not a scrap of evidence, they do not in general have much physical content at all though occasionally they contain information about the world in a veiled form, usually unbeknown to their inventors. All this area of mathematics is not a science at all but at best an abstract art. Certainly, it is sometimes possible for science fiction to become science, and one example is Riemann’s hyperbolic geometry against all odds turning out to be invaluable as a tool in astronomy, but this sort of thing happens far less than mathematicians would have you believe. The typical modern pure mathematician has a distaste for physical reality and during the last hundred years immense effort has been expended to move mathematics away from contamination with the real. In return, society as a whole would do well to treat the more outlandish claims of mathematicians concerning the importance of their pet subject with distrust and scepticism.

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