With one or two possible rare exceptions that will not be dealt with here, animals (including birds, insects &c.) do not use numbers though some primates and birds can be (with difficulty) taught by humans to use them. The reason is quite simple: all species including our own until quite recently (on an evolutionary scale) got along perfectly well without number systems as such. What most, if not all, ‘advanced’ species *do* have is the ability to make *‘Rough Quantitative Assessments’ (*RQA). Rivals for food or females have to decide rapidly whether it’s safer to fight or flee and herbivores of whatever gender have to decide whether one locality has more, or less, nutritious plants. Such assessments usually, implicitly or explicitly, distinguish a **threshold: **below the threshold it is considered advantageous to fight, above it not. Experience, the great teacher, aids ― or rather obliges ― the species or tribe to hone their rapid assessment abilities since survival and reproductive success in a competitive world may well depend on it. When faced, say, with a threatening group of hungry rivals after you have made a kill (as often occurs), you don’t need to know the exact number of your assailants or their age and state of health ― and trying to work all this out would waste valuable time. You only need to make a snap decision on the information you’ve got ― but this is easier said than done. Subtle though such abilities are, they must be distinguished from numerical procedures.

A very important extension of this ‘*RQA’* sense is the ability to notice *at once* when something important from a set is missing. Primitive peoples so-called regularly astounded explorers or missionaries by their ability to keep check on their packs of dogs, horses or herds of cattle with staggering accuracy (**Note 1**) even when the culture barely had a number system at all. We still have this ability up to a point but it has been allowed to atrophy because we don’t practise it enough. A schoolteacher generally recognizes at once if a pupil is absent, a collector surveying a roomful of curios at once spots a missing item in a show case and so on. This ability is impressive and doubtless once again of evolutionary importance but it is a ‘pre-numerate’ ability.

To be able to develop a number sense and be capable of manipulating numbers reliably, two ― and as far as I can see only two ― cognitive abilities are required.

The first is the ability to sharply distinguish between ‘one’ and ‘many’, singular and plural, ‘one’ and ‘more-than-one’ (**Note 2**). But doesn’t everyone have this ability all the time? This is debatable. Some psychologists and philosophers claim that the newborn baby, though perfectly conscious, exists in a completely unified world where no proper distinction is made between itself and its surroundings ― everything is a “buzzing, blooming confusion” (Piaget) but a unified and coherent kind of confusion. According to this view, the great attraction of mysticism is that the practitioner temporarily regains this blissful unitary consciousness ― “Everything is One”. Certainly, it would seem that there can be no awareness of the ‘I’ without awareness of the ‘non-I’. Interestingly, in at least one ancient language, the word for ‘one’ or ‘single’ is the same as the word for ‘alone’ (**Note 3**).

The second absolutely essential ability for number development is the ability to ‘pair off’ two collections of objects. This ability does not come naturally and primary schoolteachers often have great difficulty in getting tiny children to develop it ― apart from anything else, it seems a rather pointless thing to do. So what, if we can line up two groups of apples (or boys and girls) so that each apple, or boy or girl, from one group is paired off exactly with an apple, or boy or girl, from the other group? Modern mathematicians call this carrying out a ‘One-One Correspondence’ and it is only since the latter 19^{th} century that mathematicians have realized it is the key to understanding numbering.

It is important to note that the child may perfectly well be able to form correctly two sets of paired *similar *objects, for example pens taken from a pool of pens, but be unable, or refuse, to pair off apples and pens.

Not only children but whole cultures strongly resisted the idea that *any* set of ‘ones’ ― i.e. discrete objects that do not fuse when brought close together ― can be ‘paired off’ with *any* other set of ‘ones’ (provided, of course, that there are enough objects in the second set). Several societies, when they did eventually develop spoken and written numerals, had more than one set depending on what sort of ‘things’ were being compared: the Nootka of British Columbia, for example, went so far as to use different number words for rounded objects and long, thin objects(**Note 4**). Other cultures, understandably, considered it blasphemous to use the same number words or number signs for humans as for gods, which is one reason why the Mayans had three different sets of numerals. Even in our own books, until recently, the date of publication was always given in Roman numerals ― as if ‘years’ somehow required a different set of numerals to everything else. More generally, even today, there is the persistent feeling that there is something degrading and dehumanizing about humans being numbered in the same way as cattle or pieces of wood. Accepting that two sets of objects, *no matter what the objects are*, provided they can be exactly paired off, item for item, are ‘numerically equivalent’, ‘represent the same number’, involves making a giant conceptual leap that we still baulk at.

**Note 1 : **“It was related by a missionary to the Abipones, a tribe of South American Indians compelled by a shortage of food to migrate (in the 18^{th} century): “*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 together again.” *(…) Yet they [the Abipones] had only three number words and showed the strongest resistance to learning the number sequence from white men.

(…) We can understand such phenomena if we remember the far closer relationship of these people with the world around them: the keen observation that unhesitatingly notes the absence of a single animal and can say which one is missing, and the translation of a number that cannot be visualized into a clearly perceived spatial form.” Menninger, *Number Words and Number Symbols *p. 11

**Note 2 : **“A few other South American languages are almost equally destitute of pure numerical words. But even here, rudimentary as the number sense undoubtedly is, it is not wholly lacking; and some indirect expression, or some form of circumlocution, shows a conception of the difference between *one *and *two*, or at least, between *one *and *many*” (Courant, *The Number Concept Its Origin and Development *p. 5)

**Note 3 : **“The Tacanas of the same country [Bolivia] have no real numerals whatever, but expressed their idea for “one” by the word *etama*, meaning alone” (Courant, *The Number Concept Its Origin and Development *p.5)

**Note 4 : **See *The Calendrical and Numerical Systems of the Nootka *by William Folan in *Native American Mathematics *edited by Michael Closs.

** **

** **

## EUCLID AND UNIQUE PRIME FACTORISATION

It is often said that Euclid (who devoted Books VII – XI of his *Elements *to Number Theory) recognized the importance of Unique Factorization into Primes and established it by a theorem (Proposition 14 of Book IX). This is not quite correct. Modern authors usually present UPF in the following way

THEOREM *Any positive integer N can be written as a product of primes in one and only one way barring changes in order. *i.e. * ***N = p ^{a} q^{b} r^{c}…..**

** **But what Euclid establishes by proving **Book IX Proposition 14 **— Heath, whose translation I use throughout, calls ‘theorems’ ‘propositions’ — is rather less than this, viz.

*“If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.”*

Now, from this one can, with the help of one or two other theorems, deduce Unique Prime Factorization (UPF), but Euclid does not actually do this. For one thing, Euclid would need to show that every (natural) number can be presented as a product of primes if **Proposition 14** is to have a universal application. He goes some way to doing this in **Propositions 31** and **32 **of **Book VII** : *Any composite number is measured by some prime number” *and *“Any number either is prime or is measured by some prime number”*. But, for some reason, we lack the clinching Proposition, that all numbers can be written as a product of primes and that there is only one way of doing this barring changes in order.

Euclid’s presentation of Number Theory is so idiosyncratic, not to say perverse, that many readers, flipping through the *Elements*, * *do not even realize that he ever dealt with numbers at all. This is because Euclid insists on presenting (whole) numbers as line segments A ______________ B __________ and not, as one would expect, as collections of discrete elements, e.g. by such sequences as ● ● ● ● ● ● ● ● or □ □ □ □ It is true that, by presenting numbers as lines Euclid gains generality : we can see in the above that A > B but we are not limited to specific magnitudes. Also, unlike us, Euclid did not have the mathematical etcetera symbol, ….

However, I doubt if this was the real reason. By Euclid’s time geometry had almost entirely ousted arithmetic as the dominant branch of mathematics much in the way that algebra subsequently ousted geometry. Pride of place in the *Elements *is given to the theory of proportion developed by Eudoxus. In the books devoted to Number Theory Euclid only deals with whole numbers (always imaged by line segments) and *ratios* between whole numbers which imitate ratios between sides of triangles and other figures. He does not mention ‘fractions’ as such though Greek housewives and practical people must have been well acquainted with them. Why this emphasis on geometry even when it is inappropriate? Part of the blame, if blame it is, must be assigned to Plato who, though not himself a mathematician, was well versed in the higher mathematics of his time and remains one of the most important theorists in the whole history of mathematics. Plato’s view that the ‘truths of mathematics’ are in some sense independent of human experience, while nonetheless underlying it, is the view still held by most pure mathematicians today. Plato considered mere calculation with numbers to be a lowly activity, the ‘affair of craftsmen and tradesmen’, while geometry was a discipline that ennobled the practitioner by fixing his eye on the eternal. Hence the radical ‘geometrization’ of number that we find in Euclid.

In his Books on Number Theory, one suspects that Euclid was building on a much older arithmetic tradition which not only presented numbers as discrete entities but actually used objects such as pebbles or shells in calculations and formed them into shapes — which is why we still speak of ‘triangular numbers’, ‘square numbers’ and so forth (**Note 1**). The material of Book VII, the basic Book dealing with Number Theory, looks as if it goes back a very long way indeed and this is at once an advantage and a drawback.

It is an advantage because Euclid kicks off with an eminently practical *procedure* (rather than an abstract theorem), the so-called Euclidian Algorithm, and makes it the foundation of the entire edifice. Most of Euclid’s proofs are by contradiction and thus ‘non-constructive’ but the Euclidian Algorithm not only demonstrates that a ‘least common measure’ of two or more numbers always exists, but actually shows you how to obtain it. Remarkably, the Euclidian Algorithm works perfectly well in any base, or indeed without any base at all — and this alone suggests that it is a very ancient procedure. It was quite possibly discovered before written numbers even existed : in effect, it shows you how to group or bag up two different collections of similarly sized objects (such as beads or shells) without anything being left over while using the *largest possible bag size*. **Proposition 1** is a special case of this : when the largest bag size possible turns out to be the unit. Such an outcome must have seemed extraordinary to the people who first discovered it, and indeed mankind has ever since been fascinated by ‘prime numbers’ — they were originally called ‘line numbers’ because they could *only *be laid out in a line or column, never as a rectangle.

However, probably because they are based on an ancient source, Euclid’s presentation in the Books devoted to Number Theory is not so impeccably logical as in the other Books. Euclid does not introduce any new Axioms in Book VII, the first of the four books dealing with Number Theory, though he does give twenty-two *Definitions*. He presumably assumed that the general Axioms, given in Book I, suffice. In fact, they do not. Operations with or on numbers differ from operations on geometric figures since plane figures and solids do not have ‘factors’ in the way that numbers do. As Heath notes, Euclid does not state as an Axiom that factorisation is transitive (as we would put it), i.e. “If **a /** **B** &** B/**** C, **then **a/****C**”, nor does he prove it as a theorem though he assumes it throughout. The Euclidian Algorithm would not work without this feature and a large number of other Propositions would be defective. Indeed, as Heath specifies, we not only need the above but the **Sum **and **Difference Factorisation Theorems** which, in Euclid’s parlance, would be

*If A measures B, and also measures C, then A measures the sum of B and C, also the difference of B and C when they are unequal and B is greater than C. *

** **An even more serious admission, from our point of view, is that Euclid does not explicitly state the *Well-Ordering Principle*, namely that *Every non-increasing sequence of natural numbers has a least member* though he assumes it in various propositions. Given the strong anti-infinity bias of Greek thought, Euclid would doubtless have thought it unnecessary.

Euclid proves **Proposition 14** (*If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it*) in the following way :

“Let **N = pqrs… **where ** p, q, r…. **are primes. Suppose a prime **u** different from primes **p, q, r… **and which divides **N**. Then **N = u × b**.

But if any prime number divides **(m × n) **and does not divide **m**, it must divide **n** [VII. 30].

Now, **p **divides **N**** **and **p **does not divide **u **since **u, p **are primes and **u ≠p **Therefore, **p **divides **b**. And the same applies to **q, r….
**Therefore,

**pqr…**divides

**b**

But this is contrary to the hypothesis, since

**b < N**and

**N**is the

*smallest*number that can be divided by

**pqr….**

Therefore,

**N**has no prime factors apart from

**p, q, r…**“

It should be noted that this is a Proof by Contradiction and that it applies only to the case where **p, q, r… **are each of them distinct primes.

What Propositions does this proof rely on?

Firstly, on **VII**. **Proposition** **30** *“If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.” *

This is one of the most important theorems in the whole of Number Theory and I call it the **Prime Factor Theorem**. What applies here is the special case when one at least of the two original numbers is prime — and a different prime from the ‘dividing number’.

But Euclid also needs to prove, or to have proved, that **N **really is, in our terms, the Least Common Multiple of **p, q** **, r…. **This he does in **Book VII. Propositions 34** and **35** which detail the procedure for finding the Least Common Multiple, first of two numbers (**Prop. 35**), and secondly of three or more numbers (**Prop. 36**). As a special case, Euclid shows that the LCM of two numbers **a, b **that are prime to each other is **ab ** and that the procedure can be applied as many times as we wish so that the LCM of **a,b,c….** where **a, b, c **are all primes is **abc…** He is also scrupulous enough to show (**Proposition 29**) that a prime and any other ‘number it does not measure’ are prime to each other, which makes any two primes ‘prime to each other’.

Euclid does not generalize **Proposition 14 **to powers of these primes, i.e. to our **p ^{a }q^{b} r^{c}… **though this extension is in effect covered by the propositions about Least Common Multiples

**VII.**

**34, 35**and

**36**taken together with

**VII. 31**and

**32**.

The propositions concerning LCMs are very much what one would expect and are easily assented to. The same does not apply to the **Prime Factor Theorem **which is by no means ‘intuitively obvious’ nor especially easy to establish.

In modern terms Euclid’s proof of the Prime Side Theorem is as follows:

“Suppose **p **divides **N (= ab) **where** p **is prime, and **p **does not divide **a**.

Then **(p, a) = 1 **[VII. 29]

Let **ab = pm = N **where **m **is some number.

Then **p/****a = b/m ** [VII. 19]

But since **(p, a) = 1**, **p/a **is in its lowest terms. Therefore **m **must be a multiple of **a** and **b **a multiple of **p **[VII. 20, 21].

So, if **p **divides **ab **where **p **is prime, then either **p **divides **a **or **p **divides **b **(or both).”

** **** **The key proposition here is **VII. 19**, the **Cross Ratio Theorem**: *“If four numbers be proportional, the number produced from the first* *and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.” ** *

This cumbersome statement shows the importance of algebraic notation which the Greeks did not have. Remember that Euclid is speaking only of *ratios *between hypothetical line segments, not of ‘rational numbers’ as modern mathematicians understand them. However, bearing this in mind, Euclid’s proof may be presented thus :

“Let **ac/ad = c/d = a/b
** But

**a/b = ac/bc**

So

**ac/ad = ac/bc**

But this can only be true if

**ad = bc**

Conversely, let **ad = bc
** Then

**ac/ad = ac/bc**

However,

**ac/ad = c/d**

Also

**ac/bc = a/b**

Therefore

**a/b = c/d**”

The above itself depends on the legitimacy of ‘cancelling out’, likewise the legitimacy of multiplying and dividing numerator and denominator by the same factor. Euclid has already dealt with such issues and I will not trace the derivation any further back. He has, I think, made a proposition by no means obvious — the ‘Prime Factor Theorem’ — entirely acceptable and, if we accept the latter, then seemingly we must accept **Book IX Proposition 14**. Apart from some tidying up and expansion, Unique Prime Factorization in the Natural Numbers has been established.

** ***SH 30/05/17*

** ****Note 1 : **“It seems clear that the oldest Pythagoreans were acquainted with the formation of triangular and square numbers by means of pebbles or dots; and we judge from the account in Speusippus’s book *On the Pythagorean Numbers*, which was based on the works of Philolaus, that the latter dealt with linear numbers, polygonal numbers, and plane, and solid numbers of all sorts….” (Heath, *History of Greek Mathematics *p. 76)

## MULTIPLICATION

Multiplication is a nonsense interpreted literally. ‘Increase and multiply’ ― but if you multiply something once it stays the same! And one could very reasonably conclude that if you get no change when you carry out an operation once, you will get no change when you repeat it: so no matter how many times you ‘multiply’ something it does not get any bigger. Abraham would never have had as many descendants as there are stars in the sky going about things this way.

Why is it, then, that we have this strange rule that **N × 1 = N**? The number system we work with today is formalised in terms of the Axioms of Fields: we have two basic operations ‘+’ and ‘×’ and both have a so-called ‘identity element’ which keeps things as they are, **0** for addition and **1** for multiplication. Why not have zero as the null operation for multiplication as well? On the face of it this makes a lot more sense since abstaining from increasing (or decreasing) something means it stays the same. In such a system

**1** **× 0 = 1 **and more generally

**N × 0 = N **where **N **is a positive integer.

The most natural sense of ‘multiplication’ is doubling what you already have, so

**1 × 1 = 2 **and more generally

**N × 1 = 2N **

It is not clear how we should proceed from now on. What happens when you have **N × 2 **? We could interpret this as an instruction to double again in which case multiplication becomes the initial quantity ‘times’ the appropriate power of **2**. But this lacks generality. Better to fall back on defining multiplication as repeated addition and interpret the ‘**×**’ as an instruction to “add on another **N**” or

**N × 2 = 3N **and more generally **N × m = (m + 1)N **

** **This sort of multiplication does not, as far as I can see, lead to contradiction and I have even attempted to use it. But it is extremely inconvenient because we lose the so-called commutativity of ‘**×**’ as an operation: the result would not generally be the same if we invert the two numbers involved. **3 × 1 **in this maverick system gives **6 **but **1 × 3 **gives **4**. What we *do *get in lieu is the peculiar

**N × m = m × N **if and only if **N = (m ─ 1**) so that

(**4 × 5) = (5 × 4) = 20**

This type of multiplication would also cause problems when related to ‘normal’ division since **
(N × m) /m ≠ N **in general. It thus requires a re-definition of division. And so on.

It would seem unlikely that these formal issues were the original reason for the rule that a single multiplication leaves the quantity unchanged. Arithmetic has only been formalised during the last 150 years or so while people have been handling numbers for thousands of years. There are two plausible explanations for the ‘null multiplication rule’.

It is generally accepted today that the earliest type of arithmetic was done using the fingers, sometimes the toes as well. This is shown by the abundance of number words that are related to the fingers: ‘digit, for example, comes from the Latin *digitus* meaning finger. And the widespread use of base *10 *throughout the world rather than the much more convenient *12* is doubtless due to the anatomical accident whereby mammals have ten fingers and thumbs rather than twelve. Finger counting and, more generally, ‘finger arithmetic’ was once widespread since most of the world was illiterate and was allegedly still used within living memory by pearl traders in the Middle East. The Venerable Bede wrote a treatise on finger counting and “the reader will be surprised to find that underlying these finger gestures is a positional or place-value system” (Menninger. *Number Words*).

Now I have actually experimented with a simple finger arithmetical system. Numbers, abstract, gestural or concrete, were not originally invented for ‘doing equations’ but in order to assess, and perhaps subsequently record, quantities of objects by representing them in a standard symbolic form ― even today numbers are used primarily for the recording of data. If you are walking along and want to assess how many trees there are in a clump you cannot operate with the trees since they are fixed. What you *can *do is to match each sub-clump of trees with the fingers of one hand and then use the other hand to record the number of handfuls if there aren’t too many. The eventual quantity can then be memorised and, if required, be subsequently recorded in a more permanent manner by way of charcoal marks on a wall, scratches on a bone, knots in a rope and so on — a Roman would have had a household slave with him holding a portable marble abacus.

Now such a procedure involves both division and multiplication. The collection of real objects is first of all ‘divided up’, at least in imagination, into so many fives or tens and each batch is ‘multiplied’ by repeatedly showing two open hands and the remainder neglected or shown with a single hand. There is, however, a significant difference between the two operations: it is the collection itself (the trees or beans or warriors) that is first ‘divided up’ into so many tens, but it is the copy, the handful, that is ‘multiplied’. This seems to be the true meaning of ‘multiplication’, namely ‘replication’ or ‘identical copying’ (cloning) and this, of course, is how mRNA goes about its business when it copies part of a strand of DNA in the nucleus while the actual assembly of amino-acids to form proteins takes place later *outside* the nucleus.

In the context of finger assessment or DNA replication, the rules for multiplication and division make perfect sense. The first copying is, as it were, an *inert *operation while ‘copying the copy’ by repetition is creative. The basic point is that the original collection being represented, the clump of trees or the group of men or the bases making up a gene, is not part of the arithmetic operation proper: it functions as a sort of template. And when we pass on to abstract operations where there is not necessarily a real object or collection in view, we retain the same mental picture to guide our operations. There is an original numerical quantity which is ‘out there’: we represent it by some written or verbal symbol and from then on operate on that.

Primitive commercial practice would have reinforced this schema. Arithmetic only got going with the rise of the large Middle Eastern empires, Assyria, Babylon and the like, when trade was extensive and an extensive bureaucracy was in place. It has been suggested that the development of writing, in the form originally of some sort of ideogram or recognizable picture, came about because of trade. Merchandise, say combs or olives or pins, were apparently often transported in sealed containers which could be checked on arrival to see if they had been tampered with. But how to know what was inside without breaking the seal? A simple stratagem would be to have a clay model of the object attached to the outside of the container indicating the contents. Later, a picture of the object replaced the clay model and later still a stylised representation and eventually a ‘word’. This symbol is then ‘multiplied’ so many times to indicate the sum total of the contents. Again, we have the strict separation between the representation and the actual object or objects without which the system would not work.

*SH 8/04/15*

## Even and Odd

Ο ΟΟ

ΟΟΟ ΟΟΟΟ

ΟΟΟΟΟ ΟΟΟΟΟΟ

ΟΟΟΟΟΟΟ ΟΟΟΟΟΟΟΟ

Animals and so-called primitive peoples do not bother to make nice distinctions between entities on the basis of number and even today, when deprived of technological aids, we are not at all good at it (**Note 1**). What people do ‘naturally’ is to make distinctions of *type* not *number* and the favourite principle of division by type is the two-valued either/or principle. Plato thought that this principle, *dichotomy*, was so fundamental that all knowledge was based on it — the reason for this being because the brain works in this way, the nerve synapsis is either *‘on’ *or *‘off’. *Psychologically human beings have a very strong inclination to proceed by straight two-valued distinctions, *light/dark, this/that, on/off, sacred/profane, Greek/Barbarian, Jew/Gentile, good/evil *and so on — more complex gradations are only introduced later and usually with great reluctance Science has eventually recognized the complexity of nature and apart from gender there are not many true scientific *dichotomies *left though we still have the classification of animals into *vertebrates* and *invertebrates*.

Numbers themselves very early on got classified into *even *and *odd* , the most fundamental numerical distinction after the classification *one* and *many* which is even more basic.

The classification *even/odd* is radical: it provides what modern mathematicians call a partition of the whole set. That is, the classification principle is *exhaustive* : with the possible exception of the unit, *all * numbers fall into one or other of the two categories. Moreover, the two classes are mutually exclusive: no number appearing in the list of *evens* will appear in thelist of *odds*. This is by no means true of all classification principles for numbers as one might perhaps at first assume. Numbers can be classified, for example, as *triangular* and as *rectangular *according to whether they can be (literally) made into rectangles or equilateral triangles. But ΟΟΟΟΟΟ turns out to be both since it can be formed either into a triangle or a rectangle:

ΟΟΟ ΟΟΟ

ΟΟΟ ΟΟ

Ο

The Greeks, like practically all cultures in the ancient world, viewed the *odd *and *even *numbers as male and female respectively — presumably because a woman has ‘two’ breasts and a male only one penis. And, since *oddness*, though in Greek the term did not have the same associations as in English, was nonetheless defined with respect to *evenness *and not the reverse, this made an *odd* number a sort of female *manqué. *This must have posed a problem for their strongly patriarchal society but the Greek philosophers and mathematicians got round this by arguing that ‘one’ (and not ‘two’) was the basis of the number system while ‘one’ was the ‘father of all numbers’.

On the other hand a matriarchal society or a species where females were dominant would almost certainly, and with better reasoning, have made ‘one’ a *female* number, the primeval egg from which the whole numerical progeny emerged. Those who consider that mathematics is in some sense ‘eternally true’ should reflect on the question of how mathematics would have developed within a hermaphroditic species, or in a world where there were *three* and not *two* humanoid genders as in Ian Banks’s science-fiction novel *The Player of Games*.

*Evenness* is not easy to define — nor for that matter to recognize as I have just realized since, coming across an earlier version of this section, I found I was momentarily incapable of deciding which of the rows of balls pictured at the head of this chapter represented odd or even numbers. We have to appeal to some very basic feeling for ‘symmetry’ — what is on one side of a dividing line is exactly matched by what is on the other side of it. A definition could thus be

**If you can pair off a collection within itself and nothing remains over, then the collection is called even, if you cannot do this the collection is termed odd.**

This makes *oddness* anomalous and less basic than *evenness *which intuitively one feels to be right — we would not, I think, ever dream of defining *oddness *and then say *“If a collection is not odd, it is even”*. And although it is only in English and a few other languages that *‘odd’* also means *‘strange’*, the pejorative sense that the word *odd* has picked up suggests that we expect and desire things to match up, i.e. we expect, or at least desire, them to be *‘even’* — the figure of *Justice* holds a pair of evenly balanced scales.

The sense of *even* as ‘level’ may well be the original one. If we have two collections of objects which, individually, are more or less identical, then a pair of scales remains level if the collections are placed on each arm of the lever (at the same distance). One could define *even* and *odd* thus pragmatically:

**“If a collection of identical standard objects can be divided up in a way which keeps the arms of a balance level, then the collection is termed even. If this is not possible it is termed odd.” **

This definition avoids using the word *two *which is preferable since the sense of things being *‘even’ *is much more fundamental than a feeling for *‘twoness’* — for this reason the distinction *even/odd*, like the even more fundamental *‘one/many’ *, belongs to the stage of *pre*-numbering rather than that of numbering.

Early man would not have had a pair of scales, of course, but he would have been familiar with the procedure of ‘equal division’, and the simplest way of dividing up a collection of objects is to separate it into *two* equal parts. If there was an item left over it could simply be thrown away. Evenness is thus not only the simplest way of dividing up a set of objects but the principle of division which makes the remainder a minimum: any other method of division runs the risk of having more objects left over.

Euclid’s definition is that of equal division. He says “An **even** number is that which is divisible into two equal parts” (*Elements ***Definition 6. Book VII**) and “An **odd number** is that which is not divisible into two equal parts, or that which differs by a unit from an even number” (*Elements* **Definition 7. Book VII**). Incidentally, in Euclid ‘number’ not only always has the sense ‘positive integer but has a concrete sense — he defines ‘**number**‘ as a “multitude composed of units”.

Note that Euclid defines *odd* first privatively (by what it is not) and then as something deficient with reference to an *even *number. The second definition is still with us today: algebraically the formula for the odd numbers is **(2n-1) **where **n** is given the successive values **1, 2, 3….** or sometimes (in order to leave **1** out of it) by giving **n** the successive values **2, 3, 4…. **In concrete terms, we have the sequence

Ο ΟΟ ΟΟΟ …….. …..

Duplicating them gives us the ‘doubles’ or even numbers

Ο ΟΟ ΟΟΟ ..….

Ο ΟΟ ΟΟΟ ……

and removing a unit each time gives us the ‘deficient’ odd numbers.

The unit itself is something out on its own and was traditionally regarded as neither even nor odd. It is certainly not *even *according to the ‘equal division’ definition since it cannot be divided at all (within the context of whole number theory) and it cannot be put on the scales without disturbing equilibrium. In practice it is often convenient to treat the unit as if it were *odd*, just as it is to consider it a *square *number, *cube *number and so forth, otherwise many theorems would have to be stated twice over. Context usually makes it clear whether the term ‘number’ includes the unit or not.

Note that distinguishing between *even *and *odd* has nothing to do with counting or even with distinguishing between *greater* or *less* – knowing that a number is even tells you nothing about its size. And vice-versa, associating a number word or symbol with a collection of objects will not inform you as to whether the quantity is even or odd — there are no ‘even’ or ‘odd’ endings to the spoken word like those showing whether something is singular or plural, masculine or feminine.

It is significant that we do not have words for numbers which, for example, are multiples of four or which leave a remainder of one unit when divided into three. (The Greek mathematicians did, however, speak of ‘even-even’ numbers.) If our species had three genders instead of two, as in the world described in *The Player of Games*, we would maybe tend to divide things into *threes *and classify all numbers according to whether they could be divided into three parts exactly, were a counter short or a counter over. This, however, would have made things so much more complicated that such a species would most likely have taken even longer to develop numbering and arithmetic than in our own case.

The distinction *even/odd *is the first and simplest case of what is today called a congruence. The integers can be separated out into so-called equivalence classes according to the remainder left when they are divided by a given number termed the *modulus*. All numbers are in the same class (**modulus 1)** since when they are separated out into ones there is only one possible remainder : nothing at all. In Gauss’s notation the even numbers are the numbers which leave a remainder of zero when divided by **2**, or are **‘0 (mod 2)’ **where **mod** is short for **modulus**. And the odd numbers are all **1 (mod 2) **i.e. leave a unit when separated into twos. What is striking is that although the distinction between even and odd, i.e. distinction between numbers that are **0 **or **1 (mod 2) **is prehistoric, congruence arithmetic as such was invented by Gauss a mere couple of centuries ago.

In concrete terms we can set up equivalence classes relative to a given modulus by arranging collections of counters (in fact or in imagination) between parallel lines of set width starting with unit width, then a width which allows two counters only, then three and so on. This image enables us to see at once that the sum of any two or more *even* numbers is always *even*.

And since an odd number has an extra Ο this means a pair of odd numbers have each an extra unit and so, if we fit them together to make the units face each other we have an even result. Thus *Even plus even equals even” *and *“Odd plus odd equals even” *are not just jingles we have to learn at school but correspond to what actually happens if we try to arrange actual counters or squares so that they match up.

We end up with the following two tables which may well have been the earliest ones ever to have been drawn up by mathematicians.

** + odd even ** ×** odd even **

** odd even odd odd odd even**

** even odd even **** even even even**

All this may seem so obvious that it is hardly worth stating but simply by appealing to these tables many results can be deduced that are far from being self-evident. For example, we find by experience that certain concrete numbers can be arranged as rectangles and that, amongst these rectangular numbers, there are ones that can be separated into two smaller rectangles and those that cannot be. However if I am told that a certain collection can be arranged as a rectangle with one side just a unit greater than the other, then I can immediately deduce that it can be separated into two smaller rectangles. Why am I so sure of this? Because, referring to the tables above,

**1.) the ‘product’ of an even and an odd number is even;
**

**2.) an even number can by definition always be separated into two equal parts.**

** **I could deduce this even if I was a member of a society which had no written number system and no more than a handful of number words.

This is only the beginning: the banal distinction between even and odd and reference to the entries in the tables above crops up in a surprising amount of proofs in number theory. The famous proof that the square root of **2** is not a rational number — as we would put it — is based on the fact that no quantity made up of so many equal bits can be at once even and odd. *SH 5/03/15*

**Note 1 **This fact (that human beings are not naturally very good at assessing numerical quantity) is paradoxical since mankind is the numerical animal *par excellence. *Mathematics is the classic case of the weakling who makes himself into Arnold Schwarzenegger. It is because we are so bad at quantitative assessment that playing cards are obliged to show the number words in the corner of the card and why the dots on a dice are arranged in set patterns to avoid confusion.

## Number Conservation Principle

There can be no doubt that mathematics was developed historically for very pragmatic and unromantic reasons. It was the Middle Eastern hierarchic societies, especially Babylon and Egypt, who not only started arithmetic as we understand it but took it to a surprisingly high level of development : the former, for example, gave a good estimate for the square root of 2 while the Egyptian scribes were adept at handling fractions. Such societies could not exist without efficient centralized planning, standardized weights and measures, effective methods of taxation and ‘fair’ remuneration of officials : indeed, they are not so very different from the EEC today ! This is not to say that the scribes and State officials did not have a ‘pure mathematical’ interest as well : like Civil Servants doing Sudoku in the morning break, their distant predecessors seemed to have enjoyed mathematical puzzles, as the Rhine papyrus shows.

The Greeks cast this mass of sporadic data, methods and formulae into a rigorous axiomatic mould with the results we know. However, in his *Elements of Geometry*, Euclid still has his eye on figures that *can actually be drawn *and many of the so-called ‘theorems’ (which Heath, Euclid’s best English translator, calls ‘propositions’) are better described as ‘Procedures’. For example, the very first ‘Proposition’ of Book I is *“[how ] On a given straight line to construct an equilateral triangle”*.

The success of Euclid’s *Elements* and the axiomatic method in general meant that many post-Renaissance early scientists attempted to cast their subject in the same mould and contemporary works on (classical) Mechanics and Thermo-dynamics are still written somewhat in this style. Newton in particular loosely imitated Euclid in his *Principia *but it is important not to see this as a retreat into some transcendental Platonic realm of pure mathematics. Notwithstanding the complicated mathematical formalism, Newton’s system of Mechanics was rooted in human sense experience, universal human experience, although it systematised, extended and idealised this experience in various ways. His audience, even those who were illiterate, could be expected to know what a ‘solid body’ was, what a ‘force’ was and even the somewhat metaphysical notion of ‘mass’ was not so very far fetched when defined in Newtonian terms as “the quantity of matter within a body”. Children know that if you push an object it usually moves in the direction of the push (‘applied force’), though they do not perhaps “know that they know this”. And although bowls on a bowling green and bar billiards are modern games, people have been playing around with balls from time immemorial and know from experience that if you strike something from the side rather than bang on from the back, it moves off at a slant. As far as we know, it was Leonardo da Vinci who first gave us the well-known diagram of the parallelogram of forces (in his *Notebooks*), and Newton who put the problem on a truly scientific footing by the notion of ‘resultant’ force — but this abstract treatment was entirely intelligible and by and large convincing to anyone who had mucked around with solid bodies.

The accepted procedure in such subjects, following Euclid, is to start off with certain ‘Axioms’, ‘Postulates’ and ‘Definitions’ and then proceed to derive conclusions, the ‘theorems’. For Euclid there was a difference between an ‘Axiom’ and a ‘Postulate’ : the former was an entirely general principle (Heath translates the Greek as ‘Common Notion’), while a ‘postulate’ had a more technical and *constructive* character. Thus, Euclid takes as one of his Axioms, *Things which are equal to the same thing are equal to each other* (Heath’s translation). In modern terms, Euclid is asserting the ‘transitivity’ of the ‘equality relation’ : something that is in a sense ‘obvious’ once it has been stated, but is well worth stating nonetheless. But the Postulates are introduced by *“Let the following be postulated” *and the first one is *: 1. To draw a straight line from any point to any point*. This doesn’t sound very good English but it is, I think, evident what Euclid has in mind. Practically, it may well be that I cannot ‘draw a straight line from point *A *to point *B*’ because the ‘points’ are too far apart, or one or both are inaccessible. But from a mathematical point of view, we need to assume that we can do this, and this assumption needs to be stated. In other words, any geometric conclusions we draw remain valid because the only reasons stopping us actually testing a particular claim are purely technical. This is all very sensible and, from a mathematical point of view, necessary. It does not mean that we can, or believe that anyone ever could, ‘draw a line from here to the Sun’, but that is not sufficient reason to stop us drawing certain conclusions which, hopefully, we *can* test in more mundane cases : it does not necessarily involve us in any supposed Platonic belief in a timeless world of Forms. Nonetheless, it was this gap between the observed and the imagined that, when it widened still further, started dissociating mathematics from the physical world, a process that has now gone the whole way in a manner that even Plato might not have approved.

Though this aspect is not so evident in Euclid, as the natural sciences developed in the West, it became necessary to make it very clear to what sort of entities the ‘principles’ and deductions therefrom applied. For example, Newtonian Mechanics only applied to ‘objects’, not (necessarily) to human beings in their entirety, and in modern times it became necessary to go even further and make it clear that Newtonian Mechanics only applied to relatively large massive bodies moving at modest speeds relative to each other (modest compared to the speed of light).

Most people would be surprised to learn that Euclid devoted four books of his *Elements* to Number Theory (Books VII – X). Though containing many important theorems, these Books are not quite so rigorous as the strictly geometric ones and they strike the modern reader as being quite perverse in their presentationof numbers as line segments instead of collections of discrete objects, blobs or squares say. Euclid was building on the earlier work of more ‘primitive’ Number theorists who actually worked with stones and pebbles, hence the interest in the visual appearance of numbers, in ‘square numbers’, i.e. collections of objects that can be made into a square, ‘triangular numbers’ and so forth. Also, there are, as Heath remarks, certain important ‘Common Notions’ (Axioms) that are not expressed such as the ‘transitivity’ of divisibility, as modern mathematicians would put it, i.e. if a number ‘goes into’ another exactly (‘measures it’), then it also ‘goes into’ any multiple of that number. Again, this is something we take for granted but which, for all that, is worth mentioning.

Euclid’s treatment of Numbers is, thus, already abstract and geometrical compared to what we surmise was the earlier approach. Today, ‘numbers’ are defined in a completely abstract way, so abstract that they are unrecognizable as such to the ordinary person. The only person I know of who in fairly recent times dared to treat mathematics, or at least arithmetic, in an empirical manner was John Stuart Mill, with the result that he has been pilloried ever since by Frege, Russell and more or less everyone else who has written on the foundations of mathematics. The philosopher Mackie once asked disingenuously, “Why cannot we have an *empirical* mathematics?” but, as far as I know, made no attempts to create one.

I believe that arithmetic and the theory of numbers can, and should, be presented as a *science*. So, what does this science depend on? Today, a large amount of classical physics is made to depend on conservation laws, themselves extensions of Newton’s Laws, thus we have the Conservation of Momentum, the Conservation of Angular Momentum, and even in an era where very little can ne taken for granted in physics, the Principle of the Conservation of Mass/Energy is still just about standing up. Now any completely generalised ‘principle’, though it can be shown to be ‘wrong’ or at least inappropriate in certain circumstances, can obviously never be justified completely : the validity of such principles is, firstly that they fit a good deal of the data we already have, simplify and make more intelligible the world around us and permit prediction which can in special cases be tested. Sometimes, the ‘principles’ are simply necessities, *sine qua non*s,m without which we just could not get started at all. For example, in physics, we usually have to assume that, given equivalent conditions, we will get equivalent results in a particular experiment, even though this is by no means self-evident and, if Quantum Mechanics is believed, is not strictly true!

Is there a key principle on which everything about numbers relies? Yes, I believe there is. It is what I call the **Number Conservation** **Principle** and it is made up of two sub-Principles, the **Principle of Replacement **and the **Disordering Principle :**

**Principle of Replacement**

The numerical status of a collection of objects is not changed if each individual object is replaced by a different individual object.

**Disordering Principle **

The numerical status of a collection of objects is not changed by rearrangement so long as no object is created or destroyed.

I think most people would agree, if they can accept the somewhat portentous language, that this is how things are, that the Principles are true. You think there are ‘*seven’ *objects on the table. I tell you to close your eyes and if, when you open them again, every previous object has been replaced by a different one, you will nonetheless (I hope) still say there are *‘seven’ *objects on the table. Similarly, if a completely change the arrangement, scattering the objects around (while taking care that none falls off the table), there will still be the same ‘amount’ of objects. Also, I can do these two operations in an y order and as many times as I like, and still ‘something’, what I call the ‘numerical status’ of the collection has not changed. * *

The formulation is open to the objection that both sub-Principles are stated in the negative: *“is not changed”*. However, it is perhaps impossible to avoid this since ‘number’, whatever it is, results from an “ignoring of differential qualities” as Piaget and Imfeld put it so well. ‘Number’ is what ‘is left’ when you have thrown away all distinctions of size, colour, race, weight, attractiveness, gender and so forth, and still have something left that is worth having or stating.

It should be emphasized that it is only when children in the Primary School have ‘understood’ the Number Conservation Principle that they are considered to have begun to be numerate : if they do not accept it, they will be classed as children with special needs. Neither the children, nor most likely the Primary Schoolteacher herself, have heard of the Peano Axioms, or the Axioms of Zermelo-Fraenkel Set Theory, but that does not stop them having made a beginning in ‘understanding number’. Zermelo and Fraenkel themselves had to go through this particular mill.

** **** **In other cultures different bases were used depending on the different objects being counted. Flat objects like cloths were counted by the Aztecs in twenties, while round objects like oranges were counted in tens. ** **The use of classifiers obviously marks an intermediary stage between the era when numbers were completely tied to objects and the era when they became contextless as now. We still retain words like ‘twin’ and ‘duet’ to emphasize special cases of ‘twoness’, note also ‘sextet’, ‘octet’ &c. The complete dissociation of verbal and written numerals from shape and substance is today universally seen as ‘a good thing’ especially by mathematicians. But classifiers were doubtless once extremely useful because they emphasized what people at the time felt to be important about certain everyday objects and activities, and they remain both a picturesque reminder of the origins of mathematics in the world of objects and our sense-perceptions. The removal of all such features from mathematics proper seems to be a necessary evil but at least let us recognize that it is in part an evil : the banning of contextual meaning from mathematics, the language of science and administration, is typical of the depoeticization of the modern world. In particular, there was strong cultural resistance to using the same set of words or sounds for divinities as for people, or for dead and alive people, for grains of corn and beetles. The Mayas found it necessary to have three sets of numerals where the first two, dealing principally with periods of time, were used exclusively by priests and only the third set was used by ‘ordinary people’. Likewise, a Vth century B.C. Athenian tribute list uses different symbols for the number 2 when the amount is respectively “2 talents”, “2 staters” or “2 obols” showing that even at this late date numbers were still at least partially tied to particular sets of objects (Menninger, *Number Words and Number Symbols *p. 268-9).

**Note 1 **Thus, for example, Japanese has the classifier *hon* for all cylindrical objects and Chinese *t’iao *for all elongated ones. The class word is placed between the numeral and its application in much the same way as we say (or used to say) “ten *head* of cattle”. Turkish has two classifiers ‘human’ and ‘non-human’.

** **Tribal languages used many classifiers or other methods of distinguishing between objects being counted. The Ojibway, a tribe from Northern Ontario, “classify objects according to their hardness. flexibility and dimensionality….while there are also classifiers for counting the two most important artefacts made within the traditional economy, the house and the boat. (…) All of these numeral classifiers for concrete objects ensure that, when counting, expression is given to essential aspects of the object counted, especially those that affect the handling of the object” (Denny, *Cultural Ecology of Mathematics *in Closs, pp. 148-9). The Nootkans used different terms for counting or speaking of “a.) people, men, women, children, salmon, tobacco; b.) anything round in shape such as the moon, clothing (except trousers), birds, vessels &c. c.) an object containing many things such as a block of matches, a herd of cattle, a bale of blankets &c., and several other classes of things.” (Folan, *Calendrical and Numerical Systems of the Nootka*, in Closs, Editor, p. 106)

** **

## Euclid’s Method of proving Unique Prime Factorisatioon

It is often said that Euclid (who devoted Books VII – XI of his *Elements *to Number Theory) recognized the importance of Unique Factorization into Primes and established it by a theorem (Proposition 14 of Book IX). This is not quite correct. Modern authors usually present UPF in the following way

THEOREM *Any positive integer N can be written as a product of primes in one and only one way barring changes in order. *i.e. * ***N = p ^{a} q^{b} r^{c}…..**

** **But what Euclid establishes by **Book IX Proposition 14 **— Heath, whose translation I use throughout calls ‘theorems’ ‘propositions’ — is

*“If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.”*

* * Now, from this one can, with the help of one or two other theorems, deduce Unique Prime Factorization (UPF), but Euclid does not actually do so. For one thing, Euclid would need to show that every (natural) number can be presented as a product of primes if Proposition 14 is to have a universal application. He goes some way to doing this in **Propositions 31** and **32 **of **Book VII** : *Any composite number is measured by some prime number” *and *“Any number either is prime or is measured by some prime number”*. But, for some reason, we lack the clinching Proposition, that all numbers can be written as a product of primes and that there is only one way of doing this barring changes in order.

Euclid’s presentation of Number Theory is so idiosyncratic, not to say perverse, that many readers, flipping through the *Elements*, * *do not even realize that he ever dealt with numbers at all. This is because Euclid insists on presenting (whole) numbers as line segments

A ______________ B __________ and not, as one would expect, as collections of discrete elements, e.g. by such sequences as ● ● ● ● ● ● ● ● or □ □ □ □ It is true that, by presenting numbers as lines Euclid gains generality : we can see in the above that A > B but we are not limited to specific magnitudes. Also, Euclid did not have the … facility which we have.

However, I doubt if this was the real reason. By Euclid’s time geometry had almost entirely ousted arithmetic as the dominant branch of mathematics much in the way that algebra subsequently ousted geometry. Pride of place in the *Elements *is given to the theory of proportion developed by Eudoxus. In the books devoted to Number Theory Euclid only deals with whole numbers (presented as line segments) and *ratios* between whole numbers which mimic ratios between sides of triangles and other figures. He does not mention ‘fractions’ as such though Greek housewives and practical people must have been well acquainted with them. Why is this? Partly no doubt because of the influence of Plato who, though not himself a mathematician, was well versed in the higher mathematics of his time and remains one of the most important theorists in the history of mathematics. Plato’s view that the ‘truths of mathematics’ are in some sense independent of human experience, while nonetheless underlying it, is the view held by many pure mathematicians today. Plato considered mere calculation with numbers to be a lowly activity, the affair of craftsmen and merchants, while geometry was a discipline that ennobled the practitioner by fixing his eye on the eternal. This explains the radical ‘geometrization’ of number that we find in Euclid.

In his Books on Number Theory, it would seem that Euclid was building on a much older arithmetic tradition which not only presented numbers as discrete entities but actually used objects such as pebbles or shells in calculations and formed them into shapes — which is why we still speak of ‘triangular numbers’, ‘square numbers’ and so forth (**Note 1**). The material of Book VII, the basic Book dealing with Number Theory, looks as if it goes back a very long way indeed and this is at once an advantage and a drawback.

It is an advantage because Euclid kicks off with an eminently practical *procedure* (rather than an abstract theorem in our sense), the so-called Euclidian Algorithm, and makes it the foundation of the entire edifice. Most of Euclid’s proofs are by contradiction and thus ‘non-constructive’ but the Euclidian Algorithm not only demonstrates that a ‘least common measure’ of two or more numbers always exists, but actually shows you how to obtain it. Remarkably, the Euclidian Algorithm works perfectly well in any base, or indeed without any base at all — and this suggests that it is a very ancient procedure. It was quite possibly discovered before written numbers even existed : in effect, it shows you how to group or bag up two different collections of similarly sized objects without anything being left over, using the *largest possible bag size*. Proposition 1 is a special case of this : when the largest bag size possible turns out to be the unit. Such an outcome situation must have seemed extraordinary to the people who first discovered it, and indeed mankind has ever since been fascinated by ‘prime numbers’ — they were originally called ‘line numbers’ because they could *only *be laid out in a line or column, never as a rectangle.

However, probably because they are based on an ancient source, Euclid’s presentation in the Books devoted to Number Theory is not so impeccably logical as in the other Books. Euclid does not introduce any new Axioms in Book VII, the first of the four books dealing with Number Theory, though he does give twenty-two *Definitions*. He presumably assumed that the general Axioms, given in Book I, sufficed. In fact, they do not. Operations with or on numbers differ from operations on geometric figures since plane figures and solids do not have ‘factors’ in the way that numbers do. As Heath notes, Euclid does not state as an Axiom that factorisation is transitive (as we would put it), i.e. “If **a ∣ ****B **&** B ∣**** C, **then **a ∣ ****C**”, nor does he prove it as a theorem though he assumes it throughout. The Euclidian Algorithm would not work without this feature and a large number of other Propositions would be defective. Indeed, as Heath specifies, we not only need the above but the **Sum **and **Difference Factorisation Theorems** which, in Euclid’s parlance, would be

*If A measures B, and also measures C, then A measures the sum of B and C, also the difference of B and C when they are unequal and B is greater than C. *

i.e. **a ∣ ****B & a ∣ ****C, **then **a ∣ ****(B + C)**, also **a ∣ ****(B ‒ C) **when ** B >** **C**** **

** **An even more serious admission, from our point of view, is that Euclid does not explicitly state the *Well-Ordering Principle*, namely that *Every non-increasing sequence of natural numbers has a least member* though he assumes it in various propositions. Given the strong anti-infinity bias of Greek thought, Euclid would doubtless have thought it unnecessary.

Euclid proves **Proposition 14** (*If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it*) in the following way :

“Let **N = pqrs… **where ** p, q, r…. **are primes. Suppose a prime **u** different from primes **p, q, r… **and which** **divides **N**. Then **N = u × b**.

But if any prime number divides **(m × n) **and does not divide **m**, it must divide **n** [VII. 30].

Now, **p****ô****N **and **p **does not divide **u **since **u, p **are primes and **u ≠p **Therefore, **p ∣ ****b**. And the same applies to **q, r….
** Therefore,

**pqr… ∣**

**b**

But this is contrary to the hypothesis, since **b < N **and **N **is the *smallest *number that can be divided by **pqr….
** Therefore,

**N**has no prime factors apart from

**p, q, r…**“

It should be noted that this is a Proof by Contradiction and that it applies only to the case where **p, q, r… **are each of them distinct primes.

What Propositions does this proof rely on?

Firstly, on **VII**. **Proposition** **30** *“If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.” *

This is one of the most important theorems in the whole of Number Theory and I call it the **Prime Factor Theorem**. What applies here is the special case when one at least of the two original numbers is prime — and a different prime from the ‘dividing number’.

But Euclid also needs to prove, or to have proved, that, **N **really is, in our terms, the Least Common Multiple of **p, q** **, r…. **This he has done in **Book VII. Propositions 34** and **35** which detail the procedure for finding the Least Common Multiple, first of two numbers (**Prop. 35**), and secondly of three or more numbers (**Prop. 36**). As a special case, Euclid shows that the LCM of two numbers **a, b **that are prime to each other is **ab ** and that the procedure can be applied as many times as we wish so that the LCM of *a,b,c….** *where **a, b, c **are primes is **abc…** (He is also scrupulous enough to show (**Proposition 29**) that a prime and any other ‘number it does not measure’ are prime to each other, which makes any two primes ‘prime to each other’.)

Euclid does not generalize **Proposition 14 **to powers of these primes, i.e. to our **p ^{a }q^{b} r^{c}… **though this extension is in effect covered by the propositions about Least Common Multiples

**VII.**

**34, 35**and

**36**taken together with

**VII. 31**and

**32**.

The propositions concerning LCMs are very much what one would expect and are easily assented to. The same does not apply to the

**Prime Factor Theorem**which is by no means ‘intuitively obvious’ nor that easy to establish.

In modern terms Euclid’s proof of the Prime Side Theorem is as follows:

“Suppose **p ∣ ****N (= ab) **where** p **is prime, and **p **does not divide **a**.

Then **(p, a) = 1 **[VII. 29]

Let **ab = pm = N **where **m **is some number.

Then **p ∣ ****a = b ∣ ****m **[VII. 19]

But since **(p, a) = 1**, **p/a **is in its lowest terms. Therefore **m **must be a multiple of **a** and **b **a multiple of **p **[VII. 20, 21].

So, if **p ∣ ****ab **where **p **is prime, then either **p ∣ ****a **or **p ∣ ****b **(or both).”

** **** **The key proposition here is **VII. 19**, the **Cross Ratio Theorem**: *“If four numbers be proportional, the number produced from the first*** ***and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.” ** *

This cumbersome statement shows the importance of algebraic notation which the Greeks did not have. Remember that Euclid is speaking only of *ratios *between hypothetical line segments, not of ‘rational numbers’ as modern mathematicians understand them. However, bearing this in mind, Euclid’s proof may be presented thus :

“Let **ac/ad = c/d = a/b
** But

**a/b = ac/bc**

So

**ac/ad = ac/bc**

This can only be true if

**ad = bc**

Conversely, let **ad = bc
**Then

**ac/ad = ac/bc**

However,

**ac/ad = c/d**

Also

**ac/bc = a/b**

Therefore

**a/b = c/d**“

The above itself depends on the legitimacy of ‘cancelling out’, likewise the legitimacy of multiplying and dividing numerator and denominator by the same factor. Euclid has already dealt with such issues and I will not trace the derivation any further back. He has, I think, made a proposition by no means obvious — the ‘Prime Factor Theorem’ — entirely acceptable and, if we accept the latter, we must accept **Book IX Proposition 14**. Apart from some tidying up and expansion, Unique Prime Factorization in the Natural Numbers has been established.

There was in fact another way to prove the **Prime Factor Theorem** which is more in keeping with the general plan of Book VII since the above theorem can be made to depend on the Euclidian Algorithm. The latter, for those who are not familiar with it or need a reminder, is a foolproof method for finding the lowest common factor of two numbers. Algebraically, take two different whole numbers **M **and **N **where **M > N **

** ****M = ***a ***N + R _{1
}**

**N =**

*b*

**P + R**

_{2 }**P =**

*c*

**Q + R**…….

_{3 }** **** **This sequence will eventually come to an end since the divisors are diminishing i.e. **Q < P < N < M
**

**(This is an example of Euclid appealing to the**

*Well-Ordering Principle*without stating it as such.)

** **Suppose, the very next line leaves no remainder.

**Q = ***d ***T**

For **N = ***b ***P + R _{2 }= **

*b*(

*c*

**Q + R**

_{3}) + R_{2}

**N =**

*b*

**P + R**

_{2 }=*b*(

*c*

**Q + R**

_{3}) + R_{2}

** ****Note 1 : **“It seems clear that the oldest Pythagoreans were acquainted with the formation of triangular and square numbers by means of pebbles or dots; and we judge from the account in Speusippus’s book *On the Pythagorean Numbers*, which was based on the works of Philolaus, that the latter dealt with linear numbers, polygonal numbers, and plane, and solid numbers of all sorts….” (Heath, *History of Greek Mathematics *p. 76) * *

* *

## ADDITION

**ADDITION **, the basic arithmetic operation, is not quite so straightforward and unambiguous as one might suppose. When we ‘add’ one thing to another, or to a collection, the originally separate items remain separate after combination : they do not fuse or merge.

Addition is a strictly numerical operation which tells us nothing about the sizes of the objects that are brought together, nor their colour, mass and so forth. The only assumption made is that the items with which arithmetic deals are discrete and remain so. Even when dealing with liquids which *do *combine together imperceptibly to form a whole, we often tend to still think in terms of discrete items : when we talk of ‘adding’ a bucketful of water to a pond, the implication is that the water in the pond is made up of ‘so many bucketfuls’, i.e. could be broken down into units. If we want to take merging into account we end up with the formula **1 + 1 = 1 **which, though it is a perfectly correct representation of what goes on when, say, we combine two droplets of rainwater, looks extremely peculiar. It would be quite possible, though probably not worthwhile, to develop ‘Rainwater Arithmetic’ where no matter how many items you add together the net result is **1**. Many liquids have an upper limit to merging : if you carry on adding droplets of oil to an initial droplet lying on water, the sheet of oil eventually splits in two. In such a case **1 + 100,000,000 = 2 **or something of that order of magnitude. Our arithmetic, then, concerns entities with an upper limit of **1** , i.e. substances which never merge at all or, if they do , immediately break apart.

There are at least three different senses to addition which we might call ‘adding’, ‘adding on’ and ‘adding up’. In the first case we join together two groups or collections of comparable size. In the second case we tack on a smaller collection to a larger, and in the third case we do not so much join together as rearrange. In abstract mathematics there is no difference in the operations involved but in concrete terms there is all the difference in the world.

The most ancient of the three types of addition is undoubtedly ‘adding on’. If we go back to the time when objects or events were recorded by notches on a bone or knots in a cord (which I shall call the tally system) there is nothing to ‘add up’ because there is no numerical base : what is on the stick or bone *is* the ‘total’ and that’s all there is to it. A new item, the birth of a child or a caribou kill, will be recorded at the end of the list which will, if there are very few items most likely be arranged in a row, or in several rows more or less underneath each other as on the Ishango bone, perhaps the earliest example of written numerals. In Wild West days, Billy the Kid “had so many notches on his gun” — one notch, one dead person. The amount grows by accretion just like an organism : the longer the list the more items, children born, midsummers, caribou kills and so on.

But when you add ciphered numbers the final amount does *not* grow noticeably : **2 + 3 = 5 **where **5** is no larger or longer than **2** or **3**. This is not in the least a trivial observation since it is the root cause of the current very widespread misunderstanding of what numbers essentially are, namely the symbolic representation of real or imaginary collections of certain objects, collections which increase or diminish in size.

And when you add up a column of figures all you are doing is getting the separate amounts into a tidier form. What was separate (distinct) before addition remains separate after addition : the difference is that the whole lot is now grouped in a systematic fashion according to the powers of the base, usually *ten*. But what is there at the end was there at the beginning.

However, when you ‘add’ fresh recruits to a regiment, new employees to a firm’s workforce or money to someone’s bank account you are not just rearranging what is already there but bringing in someone or something new from outside. It is at first sight somewhat surprising that this makes no difference arithmetically speaking, the reason being that by the time the new objects are actually joined to the existing group they are no longer completely ‘new’ since they are already in existence, even if, as in the case of an ‘addition’ to a family they have only just been born.

Despite the difference between adding up and adding on, the operation of addition remains, like division, an ‘inert’ one : strictly speaking nothing is created or destroyed. A primary schoolteacher teaching addition to a child has to have the extra building blocks required concealed to one side : if there is nothing there to add, addition cannot take place. This is quite distinct from a truly ‘creative’ operation where something new is produced from within as when a mother gives birth or when unicellular organisms like bacteria reproduce by splitting in two (mitosis). I did as a matter of fact start to construct an arithmetical system where the basic operations are *splitting* and *merging* instead of adding and subtracting. *SH 14/9/13*

## Bases

## Bases

The first thing to realize about bases is that Nature does not bother with them. Nature does not group objects into *tens*, *hundreds, thousands *and so on, not even into *twos, fours, eights*. Bases in the mathematical sense are *entirely* a matter of human convenience — Nature only uses base one. We are so used to thinking decimally and writing numbers in columns that we tend to consider that an amount, expressed in our modern Hindu-Arabic positional numerals, is somehow *truer *than if it were expressed in, say, Chinese Stick Numbers. Even, there is a tendency to think that the modern representation of a quantity is somehow numerically *truer *or *more real * than the *actual quantity * — an example of the delusionary thinking that doing mathematics all too readily gives rise to. Asked how many stones there are in a certain pile, an unhelpful but perfectly correct answer would be to transfer all the stones into a wheelbarrow and empty them out at the enquirer’s feet.

** **So why do we bother with bases? Partly for reasons of space : an amount expressed in base one requires a lot of paper or wood or whatever material we are employing and this was and is an important consideration. But the main reason is that our perception of number is so defective that we find it very hard indeed to distinguish between amounts of similar objects or marks beyond a certain very small quantity (*seven* at most). So what do we do? What we need is a second fixed quantity, a second ‘unit’, in terms of which we can assess larger quantities. What do we choose?

‘Ones’ are given us by Nature and the sort of objects we shall want to assess either are collections of ‘ones’, like trees or sheep, or can be treated as if they were, e.g. mountains, villages and so on. We speak of ‘whole’ numbers thus implying that they are in some sense ‘entire’, ‘indivisible’. If our standard ‘one-object’ is a pebble it really is indivisible in a pre-industrial society and a cowrie shell, though it can be broken in two, ceases to be a proper shell if this is done. There is, in concrete number systems, no question of dividing a ‘one’ and ‘fractions’ (‘broken numbers’), if defined at all, are represented by smaller objects or marks, not by splitting up the ‘one-object’. Whole number arithmetic was by Greek times firmly associated with the physical theory of atomism as put forward by Democritus who himself wrote works on arithmetic now lost.

A mass of ‘ones’ is perceptually unmanageable unless related to certain standard amounts we are familiar with. But Nature is extremely unhelpful in providing us with exact standard amounts. Litters of kittens and puppies are by no means standard, and apart from a slight prejudice in favour of the quantity *five*, the amount of petals in a flower or branches on a tree varies amazingly.

Familiar fixed amounts such as the ‘number’ of hills on the skyline could only be of strictly local relevance and at the end of the day about the only available standard amounts given to man by Nature are the fingers which, counting the thumbs, come in fives. It is thus no surprise that number systems are dominated by the amounts *five, ten *and *twenty*.

Alternatively, of course, we could start from the other end and opt for a ‘man-made’ secondary unit whose size would depend on our perceptual needs and what exactly it is we want to assess or measure. These three criteria **1.) **availability of a standard amount; **2.) **human perceptual limitations and **3.) **appropriateness for assessment purposes, conflict and one of the main problems of early numbering was how to reach an acceptable compromise between them.

*Two* is the first possibility for a ‘secondary unit’ but, although it has come into its own in the computer era (because of the two states *On* and *Off*), it is clearly too small to be of much use for ordinary purposes. A language spoken in the Torres Straits had a word for our ‘one’, namely *urapan* and a word for our ‘two’ *okasa* and that was about it. Their numerals went

**1. ***urapan ***4.** *okasa okasa*

**2. ***okasa* **5. ***okasa okasa urapan*

**3. ***okasa urapan *

Understandably, since even a number as small as **11 **would require six words, the natives referred to anything above **6** as *ras *— ‘a lot’ (Conant, p. 105).

A few things, or rather events, are viewed in *threes* witness phrases like “*third time lucky” *and we group quite a lot of things in *fours *(seasons, points of the compass &c.) but the obvious first choice for a ‘secondary unit’ is *five*. Beyond *five *we really feel the need for a ‘secondary unit’ since collections like ** l l l l l l l l **and

*re practically indistinguishable. Also, as it happens we have the five fingers to be able to check (by pairing off) whether we are separating out the items into groups correctly.*

**l l l l l l**The Old Man of the Sea in Homer *‘fives’ *his seals but for most herdsmen *five* would still have been rather too small as a secondary unit. So where do we go next? If we remain guided by the fingers the next possibilities are *‘both hands’* and what many primitive languages referred to as *‘the whole man’* i.e. *ten *and *twenty*. *Twenty *is in some ways a better choice since, if we keep the option open of reverting to *five* for trifling amounts we can cope with very sizeable collections using batches of *twenty*. The Yoruba used *twenty *cowrie shells as their principal counting amount after the unit. Some modern European languages which have long since become decimal show traces of an earlier vigesimal (twenty-based) system which probably suited farmers better. Hence Biblical terms like *‘three score years and ten’* in English and the French *soixante dix-huit *(sixty-eighteen).

A secondary unit is, unlike the unit, not actually indivisible — since it is still made up of standard ones — so how do we keep it together if we are using objects as numbers? This depends on the choice of standard object and in practice is one of the motivations for the choice of object in the first place (or second place at least). Heaps of pebbles are heavy enough not to blow away but can all too easily be disturbed by people bumping into them, while piles of flat objects unless they are paper thin readily tip over and in any case really flat objects are hard to come by in nature. This is where shells are advantageous since if of the cowrie variety they stack up neatly and, even better, can be pierced and threaded on strings to make number rosaries. Beads make good numbers but since they are manufactured items they would not have been amongst the very earliest examples of object numbers.

The clay Number Ball I have already mentioned would not be suitable for secondary (or tertiary) standard amounts precisely because the bits tend to adhere together : its use would be for assessing limited quantities in the field which, if required, could be recorded back at the Number Hut using a different system, knotting or incising.

On my island I opt for a stick of standard length as my ‘one-object’ and I instruct the natives to tie sticks together into a bundle when we reach the fixed secondary amount which tentatively I fix at our *twelve*. Lacking a sign on this computer for a bundle of sticks I represent this amount by **∏**. At the back of the Number Hut I set up partitions to make alleyways for concrete numbers and I suspend from the roof an example of the bundle the alleyway is to contain and its decomposition into smaller bundles or individual sticks as a sort of Système Internationale prototype. For the moment I only propose to use the extreme right two alleyways, the first for individual sticks and the second for bundles of the specified size. Whenever we have * ⁄*** **sticks in the extreme right alley, they are tied together and transferred (literally ‘carried’) to the next alleyway. The system can be used for the temporary recording of data but it is best to restrict its use to calculation, simple additions and subtractions, while using a different system for recording purposes. I can, for example, paint three vertical lines on a piece of bark to make it into a Number Board and paint in particular configurations of the sticks and bundles. When painting I do not use short cuts, I just represent the sticks and bundles as well as I can turning stick numbers into stroke numbers. As yet I do not proceed any further : all quantities are to be represented by sticks and/or Number Bundles of a single fixed amount and, for the time being, I do not set an upper limit to the amount of bundles. Thus the components of our number system so far are only ** l ** and **∏** where

** ∏ = l l l l l l l l l l l l **

A feature of this still very rudimentary system is that at any moment a bundle **∏** can be reduced to so many sticks simply by untying the cord and transferring them into the appropriate alley. Even, it is possible to have second thoughts about the ‘secondary unit’ and change it for another, since all you have to do is untie the bundles and tie the sticks up again using a different set amount. We might, for example, want to revert to *five* if the quantity to be assessed turns out not to be so great, or, conversely, jump ahead to *twenty *for a really large herd of goats or clump of trees. With systems that depend on threading objects on a string or wire, changing the secondary unit is either impossible or time-consuming and so would tend not to be done.

If the first ‘greater unit’ is set at *ten* and we are dealing with sticks, the problem of distinguishing different numbers less than *ten * remains — we have met requirements **1.) **and **3.) **but not **2.)** . The earliest Egyptian written numbers, perhaps based on still earlier number sticks, got round this problem, or tried to, by arranging the sticks in set patterns. But the patterns are not very distinctive or memorable. Far more striking are the excellent domino or dice dot numbers. Domino patterned numbers stop at *six *and the arrangements for the playing-card *seven, eight *and *nine *are not so striking — I have sometimes I caught myself having to look at the corner to distinguish between a *seven* and a *nine*.

The stratagem of arranging near identical marks in a pattern is an attempt to enlist shape in the service of number : if you recognize the shape you don’t need to count the dots. Shape recognition is distinction by type which the principle of distinction by number must displace in the cultural development of the species. For all that , even today, we feel at ease with shape and respond to it ‘naturally’ (perhaps because of the sexual instinct and childhood memories) while number is at first unappealing, it appears cold and inhuman. The supposedly artificial distinction between the arts and the sciences is rooted in this primeval struggle between distinction by shape and distinction by number, a struggle which the latter is obviously winning. In the pre-industrial past it was quite the reverse : most ‘primitive’ tribes considered that distinctions of shape and thickness were so much more significant than numerical distinctions that they developed a large and sophisticiated vocabulary to deal with the former while contenting themselves with half a dozen number words. And even in the domain of number itself shape cast its shadow: many societies used different number words depending on the overall shape of the objects being counted. The Nootkans, for example, used special terms for counting round objects and traces of this practice persist in the ‘numerical classifiers’ of modern Japanese and Chinese^{1}.

A drawback of numerical distinction by patterning is that every new number requires its own special arrangement which must then be committed to memory. This certainly limits the range but the same patterns could be re-used with slight differences or could be combined in various ways. One would not have thought the effort involved was that great, not that much more than is involved in learning the alphabet. Also, the idea of familiarising people with numerals by way of card and board games is delightful (though presumably not done deliberately). For some reason this promising system never got extended beyond *six* (otherwise we would have in our heads patterns for higher numbers) and taken in itself constitutes something of a dead-end in the history of numbering.

Domino numbers are a curious and attractive relic of days long gone.* *

*Q1. If we use dominoes as numbers, what base are they in? And what is the largest amount that can be represented by a single set of dominoes ? *

(To be continued)

## Bases

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

The first thing to realize about bases is that Nature does not bother with them. Nature does not group objects into *tens*, *hundreds, thousands *and so on, not even into *twos, fours, eights*. Bases in the mathematical sense are *entirely* a matter of human convenience — Nature only uses base one. We are so used to thinking decimally and writing numbers in columns that we tend to consider that an amount, expressed in our modern Hindu-Arabic positional numerals, is somehow *truer *than if it were expressed in, say, Chinese Stick Numbers. Even, there is a tendency to think that the modern representation of a quantity is somehow numerically *truer *or *more real * than the *actual quantity * — an example of the delusionary thinking that doing mathematics all too readily gives rise to. Asked how many stones there are in a certain pile, an unhelpful but perfectly correct answer would be to transfer all the stones into a wheelbarrow and empty them out at the enquirer’s feet.

** **So why do we bother with bases? Partly for reasons of space : an amount expressed in base one requires a lot of paper or wood or whatever material we are employing and this was and is an important consideration. But the main reason is that our perception of number is so defective that we find it very hard indeed to distinguish between amounts of similar objects or marks beyond a certain very small quantity (*seven* at most). So what do we do? What we need is a second fixed quantity, a second ‘unit’, in terms of which we can assess larger quantities. What do we choose?

‘Ones’ are given us by Nature and more often than not this is all we are given. The sort of objects we shall want to assess either are collections of ‘ones’, like trees or sheep, or can be treated as if they were, e.g. mountains, villages and so on. We speak of ‘whole’ numbers thus implying that they are in some sense ‘entire’, ‘indivisible’. If our standard ‘one-object’ is a pebble it really is indivisible in a pre-industrial society and a cowrie shell, though it can be broken in two, ceases to be a proper shell if this is done. There is, in concrete number systems, no question of dividing a ‘one’ and ‘fractions’ (‘broken numbers’), if defined at all, are represented by smaller objects or marks, not by splitting up the ‘one-object’. Whole number arithmetic was by Greek times firmly associated with the physical theory of atomism as put forward by Democritus who himself wrote works on arithmetic now lost.

A mass of ‘ones’ is perceptually unmanageable unless related to certain standard amounts we are familiar with. But Nature is extremely unhelpful in providing us with exact standard amounts. Litters of kittens and puppies are by no means standard, and apart from a slight prejudice in favour of the quantity *five*, the amount of petals in a flower or branches on a tree varies amazingly. Familiar fixed amounts such as the ‘number’ of hills on the skyline could only be of strictly local relevance and at the end of the day about the only available standard amounts given to man by Nature are the fingers which, counting the thumbs, come in fives. It is thus no surprise that number systems are dominated by the amounts *five, ten *and *twenty*.

**The Secondary Unit **

** **What there is no doubt we *do* need and have done from very remote times is a secondary unit. This must be clearly distinguished from a **base **since the latter** **is an extendable sequence of ‘unit’ sizes, a ‘geometric series’ like *1, 10, 100, 1000 *and so on. Why didn’t ‘early’ societies (with some exceptions) go straight for the base system? The answer is that they didn’t need it and that it doesn’t come naturally, at any rate to practical people. For most purposes two ‘significant amounts’ or the first and second powers of the base are quite sufficient. Even one will do if it is of reasonable size because you don’t actually have to stop at the ‘square’ of the base as if it were a brick wall cutting off all access to a numerical beyond. *Fifteen hundred*, apart from being more succinct, sounds a good deal more natural than the pedantic *‘one thousand and five hundred’ *which is what we ought to say by rights (**Note 1**). Without even defining the *hundred * we could still cope perfectly well with quantities up to *999 *reckoning in so many *tens *e.g. by speaking of *810 *as *eighty-one tens and a five*. Generally we do not need to go anything like so far and the language is littered with sets of number words which, though they show base potential, peter out into nothingness without ever even making it to the ‘cube’ stage.

The ‘natural’ way of cultivating the wilderness is to clear an area and then when you’ve planted that, to clear another. Natural at any rate if you have to do a fair amount of the work yourself. If you are a conqueror you may, of course, have an eye on infinity and eternity from the start but this is *folie de grandeur*. The first man to introduce wholesale decimalisation was the Ch’in Emperor and he is thought to have hastened his death by imbibing elixirs of immortality.

Numbers were measures before they became numbers ¾ even ‘one’ itself, the ‘father (better mother) of all numbers’, is essentially a measure, one drop, one mouthful, one foot. Our standard weights and measures are really only numbers that remain tied to particular contexts and functions. The Imperial system of liquid measures is an application of base two with *four* left out since the numbers involved are **1, 2 **and **8**.

** 1 pint = 1 pint**

** 2 pints = 1 quart**

** 4 quarts = 1 gallon = 8 pints **

Verbally, the number system stops here : although there are obviously larger quantities than the gallon we have no special words for them, it is someone else’s job to work them out.

In our number words and pre-decimal measures we find a surface order with an underlying picturesque confusion where all sorts of sets of numbers leave their traces. In the Avoirdupois weights we seem to have two sets of numbers, one proceeding from the ‘small’ end and one from the ‘large’ end, most likely developed by persons performing different functions. It makes sense to have the pound finely divided for sales over the counter to individuals, thus the appearance of the large, but not too large, secondary unit *16 *in *sixteen ounces to a pound*. From the wholesaler’s point of view we want a large quantity defined straight off, the hundredweight (which is not a hundredweight). We now quarter the hundredweight as it is always useful to divide something into four equal parts and we nearly but not quite converge with the rising *16 *system. But there are not *32* pounds to a quarter but the anomalous *28*. Is a systematic base system preferable even supposing we had a more suitable base than *10*? Not necessarily for the people doing the work. Their principal concern is not logical consistency but the ready availability of convenient set amounts which the chosen number system or systems should favour and promote. Moreover, once they have what they want, various landmark fixed amounts, they leave the system to its own devices.

There is certainly, within the context of a pre-industrial economy, no need for a number sequence stretching out into infinity : on the contrary this very feature would have in many cultures provoked a certain malaise as indeed it still does to persons like myself. The idea of really large magnitudes is frightening like the idea of really large intervals of time. Although it is now unfashionable, even politically incorrect, to speak of cultures having specific traits, it is surely no accident that it was the Hindu mathematicians who gave us the first fully positional indefinitely extendable written number system. Indian thinkers, both Hindu and Buddhist, were obsessed with large numbers and vast spans of time : the *kalpa *for example is a period which lasts 4320 million years. Armed with the decimal base number system the Indians built ‘number towers’ reaching unimaginable heights and not only could they write down these quantities but they had names for many of them. In one legend the Buddha, challenged to list the numbers (read ‘powers’) beyond *10 ^{7}*, answers with the names for all the powers up to the colossal

*tallaksana*or

*10*

^{53}*¾*

*i.e. 1*followed by fifty-three zeroes (

**Note 2**) . The Indian approach to large numbers is quite different from that of Archimedes who is, surprisingly in some ways, much more in line with the earlier ‘clear an area’ approach. He wrote a treatise on large numbers but showed none of the delectation and religious awe that the Hindu and Buddhist mathematicians clearly felt. Archimedes was concerned to show that the finite Greek number system

*could*be extended upwards and outwards to deal with colossal quantities like the amount of grains of sand in the, for him finite, universe. But his aim was to tame the beyond not to lose himself in admiration of it. To the amazement of modern commentators, he did not quite hit upon the artifice of full positional notation.

The acceptance (or imposition) of a

*single*indefinitely extendable base system has taken a very long time and is of comparatively recent date. For centuries individual and local numbering systems co-existed with the State imposed one, Roman or Napoleonic, especially in country areas. Until very recently by far the greater part of the inhabitants of Europe were illiterate and many of them used their own numbering systems and ‘peasant numerals’ like the notched sticks of Swiss cowmen. Inns could scarcely have carried on at all without the one-base slate and chalk system where the reckoning was totted up at the end of the evening or month, and in Spain it was once common for the innkeeper to toss a pebble into the hood of a traveller’s cape for each drink consumed. Today rustic numbering systems like the ‘milk sticks’ of cowmen in the upper Alps or the use of knots or pebbles are things of the past : everyone has at long last agreed to at least

*write*numbers in the approved standard decimal fashion. But for all that we do not think or feel in the way we write. In effect we still use the Babylonian secondary unit, sixty, for the subdivision of the hour, although we express the quantity in a ten-base. We think ‘sixty minutes’ as a ‘chunk of time’ divisible into so many units, not as

*six tens*of time. We do indeed have the availability of intermediate amounts,

*five minutes, ten minutes, quarter of an hour*, but they are subordinate to the

*hour*and the

*minute*. A day is experienced as a unity which is in the first place decomposable into the unequal ‘halves’ of daytime and nighttime. We do not experience or think the day as

*two tens*of time plus

*four units*.

To the administrator, of course, the use of a consistent base-system is as necessary as the use of a ‘universal’ official language (Latin, English in the Commonwealth &c.). To him numbers have finally ceased to be tied to objects or activities, have become contextless, in much the same way as, at a further level of abstraction, functions, to the contemporary mathematician, have ceased to be tied to numbers.

Alternatively, of course, we could start from the other end and opt for a ‘man-made’ secondary unit whose size would depend on our perceptual needs and what exactly it is we want to assess or measure. These three criteria **1.) **availability of a standard amount; **2.) **human perceptual limitations and **3.) **appropriateness for assessment purposes, conflict and one of the main problems of early numbering was how to reach an acceptable compromise between them.

*Two* is the first possibility for a ‘secondary unit’ but, although it has come into its own in the computer era (because of the two states *On* and *Off*), it is clearly too small to be of much use for ordinary purposes. A language spoken in the Torres Straits had a word for our ‘one’, namely *urapan* and a word for our ‘two’ *okasa* and that was about it. Their numerals went

** 1. ***urapan ***4.** *okasa okasa
*

**2.**

*okasa*

**5.**

*okasa okasa urapan*

**3.**

*okasa urapan*

* *

Understandably, since even a number as small as **11 **would require six words, the natives referred to anything above **6** as *ras *— ‘a lot’ (Conant, p. 105).

A few things, or rather events, are viewed in *threes* witness phrases like “*third time lucky” *and we group quite a lot of things in *fours *(seasons, points of the compass &c.) but the obvious first choice for a ‘secondary unit’ is *five*. Beyond *five *we really feel the need for a ‘secondary unit’ since collections like ½½½½½½ and ½½½½½½½ are practically indistinguishable. Also, as it happens we have the five fingers to be able to check (by pairing off) whether we are separating out the items into groups correctly.

The Old Man of the Sea in Homer *‘fives’ *his seals but for most herdsmen *five* would still have been rather too small as a secondary unit. So where do we go next? If we remain guided by the fingers the next possibilities are *‘both hands’* and what many primitive languages referred to as *‘the whole man’*‘ i.e. *ten *and *twenty*. *Twenty *is in some ways a better choice since, if we keep the option open of reverting to *five* for trifling amounts we can cope with very sizeable collections using batches of *twenty*. The Yoruba used *twenty *cowrie shells as their principal counting amount after the unit. Some modern European languages which have long since become decimal show traces of an earlier vigesimal (twenty-based) system which probably suited farmers better. Hence Biblical terms like *‘three score years and ten’* in English and the French *soixante dix-huit *(sixty-eighteen).

A secondary unit is, unlike the unit, not actually indivisible — since it is still made up of standard ones — so how do we keep it together if we are using objects as numbers? This depends on the choice of standard object and in practice is one of the motivations for the choice of object in the first place (or second place at least). Heaps of pebbles are heavy enough not to blow away but can all too easily be disturbed by people bumping into them, while piles of flat objects unless they are paper thin readily tip over and in any case really flat objects are hard to come by in nature. This is where shells are advantageous since if of the cowrie variety they stack up neatly and, even better, can be pierced and threaded on strings to make number rosaries. Beads make good numbers but since they are manufactured items they would not have been amongst the very earliest examples of object numbers.

However, on reflection I decide that introducing such a system would be premature. Within the bounds of a self-sufficient fishing, hunting or agricultural economy there would neither be any need for an indefinitely extendable number system nor would it have any special appeal. In the first place an inhabitant of such a society would not anticipate needing to assess really big quantities. Although a peasant needs more numbers than a hunter, in the past he probably rarely if ever needed numbers extending beyond about *400 *, supposedly the upper numerical limit of a typical 19^{th} century Russian peasant. Large amounts of fruit, potatoes and so on would, of course, not be counted but be assessed by weight just like coins that we hand in to the bank. (Banknotes are still counted but in most banks the work is now done by a machine.)

The practical man, craftsmen, herdsman or farmer does not deal in ‘numbers’, he deals in fixed amounts that are significant in terms of his or her daily work and/or perceptual apparatus. And such quantities do not usually correspond to the transition points of an extendable base system like our own. To judge by the traces they have left on our language the two most popular ‘significant amounts’ beyond the unit in English speaking countries were, and to a certain extent still are, the *dozen *and the *score*. Although *twelve *as such is beyond our perceptual capacity, the image of *two* boxes each containing *half a dozen *eggs must by now have penetrated to the collective unconscious, at any rate the English speaking one. *Twelve*, like *ten*, seems about the right size for making bundles or piles, but is better than *ten* in many ways because it can be halved, quartered and chopped into three equal portions. For dealing with larger amounts, the *score *which was originally a ‘score’ or notch a farmer made on a piece of wood as twenty animals passed through a gate, is about all you need so long as you know at least twenty number words off by heart. The publisher of the red book travel guides to Europe is said to have counted the steps leading up to Milan Cathedral by transferring a pebble from one pocket to the other each time he mounted twenty steps.

A brief list of ‘significant quantities’ on a world-wide scale with reasons for their significance would perhaps be:

** 5 ** hand, fingers, right size perceptually

**6 ** half of dozen

**10** both hands, right size for base

**12 ** right size for base, many factors

** 20 ** hands and feet, multiple of *5* and *10*

**60** many factors, multiple of all previous

**100 ** square of 10, many factors

Some of the above numbers are significant perceptually, notably *5* since this is around the stage when we cease to be able to assess objects numerically without counting them individually. Thus *5* combines significance because of its use in one-one correspondences (by way of the fingers) with significance as a perceptual ‘unit’. But it has the serious drawback that it cannot be divided up at all (has no factors). It is thus significant but not convenient as a ‘secondary amount’.

Having many factors is really more a matter of convenience than significance as such but since previous significant amounts are amongst the factors of *60* and *100* these numbers acquire significance acquire significance by proxy. *60* is particularly rich in factors and has the remarkable property of being a multiple of *all* previous significant amounts.

*100* means nothing to us perceptually though it undoubtedly did to a Roman centurion who would have had in his mind’s eye the terrain covered by his infantry when lined up ready to give battle. *100* is around the ‘acquaintance’ mark, i.e. near the maximum number of persons one is able to relate to personally ¾ I believe I have read somewhere (Desmond Morris?) that *128* is about the limit and that this generally corresponds to the maximum number of persons one has in one’s address book.

But of course *60* and *100* are above all significant because of their divisibility ¾ the main use of *100* is in percentages though it still has the defect that one cannot divide it into three properly. It must be stressed that a number’s divisibility is not just a matter of interest to modern number theorists : wholesalers or state suppliers receive commodities in bulk which they must subsequently sell or distribute to individuals and it is important that standard quantities should be easy to divide up. This is the reason why so many of the old Imperial measures are built around *20* or the powers of *2 *¾ as it is one of the main reasons why there is such hostility to metric weights and measures.

** ** One of the troubles with the transition points of a base-system, the ‘powers’ of the base, is that they are significant and convenient not in a practical but purely mathematical sense. Technically speaking, the unit, the base and its powers are the successive terms of a geometric sequence *1, b, b ^{2}, b^{3}, b^{4} ……*. with common ratio

*b*. My choice of

*twelve*for the bundles that are to go into the second alleyway means setting

*b*at

*12*. We would thus have

*1, 12, 144, 1728…*(since

*144 = 12*). Now

^{2}, 1728 = 12^{3}*144*and

*1728*apart from being too large are not meaningful amounts in our day to day experience. The same goes for smaller choices of base.

*5*is perhaps the most ‘significant amount’ of all in real-life terms but

*5*is nothing special and

^{2}= 25*5*

^{3}= 5*´*

*5*

*´*

*5 = 125*even less.

*6*certainly has some valid claims on our attention as a significant quantity but

*36*has none.

One suspects that the success of a

*hundred*as a ‘significant amount’ is due to its being a multiple of the significant amounts

*5*and

*20*¾ it is actually

*5*

*´*

*20*¾ rather than it being the square of

*ten*. A

*thousand*is just a word meaning ‘large quantity’ and the ambiguous meaning of

*billion*(a million millions or a thousand millions?) shows how vague such large quantities are. A

*million*only has meaning with respect to wealth — and even this sense has been eroded by devaluation so that we find it necessary to replace the word

*millionaire*by

*multi-millionaire*which is even vaguer

*.*

* ***Non-base extendable systems **

Most people assume that once you have defined your ‘secondary unit’ you are somehow obliged to turn it into a true base (and I tended to think along these lines myself before writing this book). But of course you aren’t. The ‘tertiary unit’ or next standard amount we choose to define can be anything at all in principle. One of the few numerically advanced peoples that still used object numbers, the Yoruba, took a pile of *twenty *cowrie shells as their ‘secondary unit’. They then combined *five *such piles of cowrie shells to make *100 *in our reckoning, and combined two such piles to define their second most important amount after the unit, *200 *in modern numbers. If they had operated a true base system the next halting point would have been *20 ^{2} *or

*400*which presumably they considered too large. (Algebraically the Yoruba sequence goes

*1, b, 10b*and not

*1, b, b*). For a somewhat different, but nonetheless pragmatic reason, the Mayans, who also took

^{2 }*20*as their ‘secondary unit’, then moved on to

*360*(instead of

*400*)

*for the next transition point in order to get close to the number of days in a year — or so it has been conjectured.*

It does not in practice matter too much for addition and subtraction if the transition points are not in proper sequence (‘proper’ as we see it today) though it is desirable that they should be *multiples *of the first ‘significant amount’. Thus, anticipating a further fixed amount in my stick system I have already decided to opt for *60* since it is a multiple of *12* while *20* or *100 *are not. Odd though it sounds, there is much to be said for defining a large ‘secondary unit’ and then defining ‘units’ rather *smaller* instead of larger than it. This is in effect what the Babylonians did by taking *60* as principal amount after the unit which they noted as . Such an enormous secondary unit makes it absolutely essential to have one or more halting points, or sub-bases, in the intermediary territory which the Babylonians provided at the *five* and *ten* points. They defined the *five* transition by grouping the one-symbols and introduced a special mark for *ten * but otherwise they used only the ‘one-symbol’ right up to *60* itself. For quantities > *60* the Babylonians proceeded by using *60* as a true base, i.e. the next halt was at *60 ^{2} = 3,600*. In their case they seemed to have no misgivings about using such a huge amount as tertiary unit which seems to contradict what I said earlier. But the Babylonian scribes who developed and used the sexagesimal number system were not hunters or herdsmen but officials helping to run a vast empire. They needed large numbers and spent their lives dealing in them as did the Egyptian scribes.

Practically speaking we require very different fixed standard amounts depending on the context. To divide a pound weight into sixty ounces would appear slightly crazy but we find it most convenient to divide up a fairly short interval of time, the hour, into sixty minutes, while we divide up a somewhat larger interval, the day, much less finely. The numbers *60, 12 *and *24* are not imposed on us in the way the number of days in the year is : we could divide up the day into *60* or *72 * or any (even) number of ‘hours’ and divide each ‘hour’ into *12 *or for that matter *17 *‘minutes’. The unsystematic way in which we divide up the day seems *right *: there is, as far as I know, no SI project to decimalise time (though the ancient Egyptians did just this) and the very idea fills me with horror.

___________________________

**Note 1 **

“Yet it was the Indians who reckoned the age of the Earth as 4.3 billion years, when even in the 19th century many scientists were convinced it was at most 100,000 years old ( the current estimate being 4.6 billion).” The apparent source for this is: Pingree, David, *Astronomy in India, *in *Astronomy Before the Telescope,* p.123-42. Quoted *Chasing the Sun* by Richard Cohen, p. 132

^{1} I recently came across an interesting example of how restricting the idea of always keeping to a base is. I noticed, or had read somewhere, that the Binomial Coefficients were powers of *11* and this made sense since they can be defined by starting with *1* and getting the next term by shifting what you’ve got across one column and adding. Thus

** 1 1 = 11 ^{0}**

** 1 0 **

** 1 1 11 = 11 ^{1}**

** 1 1 0 **

** 1 2 1 121 = 11 ^{2}**

** 1 2 1 0**

** 1 3 3 1 1331 = 11 ^{3} **

** 1 3 3 1 0**

** 1 4 6 4 1 14641 = 11 ^{4}**

** **

** **However, what happens now? The next line of Pascal’s Triangle is supposed to be

**1 5 10 10 5 1 **

** **This isn’t a power of **11** surely. But who says you can’t overstep the base if you want to?

(**1 x**** 10 ^{5}) + **(

**5 x**

**10**(

^{4}) +**10 x**

**10**(

^{3}) +**10 x**

**10**

^{2}) + (5 x**10) + 1**

** **** = 100,000 + 50,000 + 10,000 + 1,000 + 51 = 161,051 = 11 ^{5} **

** **

**Classifiers** ** **In other cultures different bases were used depending on the different objects being counted. Flat objects like cloths were counted by the Aztecs in twenties, while round objects like oranges were counted in tens. The use of classifiers obviously marks an intermediary stage between the era when numbers were completely tied to objects and the era when they became contextless as now. We still retain words like ‘twin’ and ‘duet’ to emphasize special cases of ‘twoness’, note also ‘sextet’, ‘octet’ &c. ** ** The complete dissociation of verbal and written numerals from shape and substance is today universally seen as ‘a good thing’ especially by mathematicians. But classifiers were doubtless once extremely useful because they emphasized what people at the time felt to be important about certain everyday objects and activities, and they remain both a picturesque reminder of the origins of mathematics in the world of objects and our sense-perceptions. The removal of all such features from mathematics proper seems to be a necessary evil but at least let us recognize that it is in part an evil : the banning of contextual meaning from mathematics, the language of science and administration, is typical of the depoeticization of the modern world.

**Base sixty **

It is not known why the Babylonians chose *60* as their most significant amount after the unit. The fact that *60 ^{2} = 360 * is close to the number of days in the year may have something to do with it. Certainly,

*60*would not have been the original choice. It has been suggested that

*60*evolved as a compromise solution to the separate claims of

*5*and

*12*which were already well established as ‘secondary units’ within the territories conquered by the Babylonians. Since

*60*has as factors all the main bases and significant amounts smaller than it, including

*10*and

*20*, it had something for everyone : it was a numerical

*Pax Romana.*

** **

## Gnomon :The World’s First Scientific and Mathematical Instrument ?

A *gnomon* was originally a sort of set-square that could be stood on its edge and was used to measure the lengths of shadows — present-day sundials have a ‘gnomon’ on the top though the shape is more complicated (**Note 1**). Thales is supposed to have used a gnomon to estimate the height of the Great Pyramid by employing properties of similar triangles and it was data amassed by similar methods that enabled Eratosthenes to estimate the circumference of the Earth by comparing noonday shadows cast at different localities (**Note 2**). The gnomon thus provided a precious link between three different disciplines : geometry, astronomy and, as we shall see, arithmetic : it was perhaps the first precision instrument of physical science.

Sets of gnomons put together — or drawings of them — became a surprisingly useful early calculator and enabled the early Greek mathematicians to investigate spatial properties of numbers.

Each coloured inverted L shape border in the above represents an odd number with the unit in the top left hand corner. The early Greek mathematicians deduced the important property that *Any square number can be represented by successive odd numbers commencing with* *unity.* As we would put it: * n ^{2 }*=

*1 + 3 + 5 + ……(2n + 1)*For example,

*4*=

^{2 }*1 + 3 + 5 + 7*.

*And this can be extended to the observation that*

*Every difference between two squares can be represented by a sum of successive odd*

*numbers*. Thus, 5

*– 2*

^{2 }*=*

^{2 }*5 + 7 + 9*(It was in fact this relation which struck me as being quite astounding that instigated my interest in mathematics which up to then I had despised.)

But much more can be got out of the simple diagram above. The Egyptians were certainly aware of certain cases of the the property forever associated with Pythagoras, namely that

*The Square on the hypotenuse is equal to the sum of the squares on the other two sides of a right-angled triangle*since they used stretched ropes with lengths in the ratio 3, 4, 5 to lay out an accurate square corner. However, they may not have realized that this property applied to all right angled triangles. The question provided a fruitful contact between geometry, the science of shape, and arithmetic, the science of exact quantity and the gnomon most likely played an important role here. Greek mathematicians were interested in sets of numbers that were ‘Pythagorean triples’, i.e. numbers

*a, b, c*where

*a*

^{2}*= b*+

^{2 }*c*.

^{2}Now, adding on a gnomon “preserves the square form” and, more significantly for the present discussion, that the

*difference of two successive squares is an odd number.*

+ =

** **

Some sharp sighted mathematician, perhaps Pythagoras himself or one of his disciples, realized that *if the gnomon is itself a square we have a Pythagorean triple. *(This follows from the observation that adding on the relevant gnomon leads from one square to the next.) So, if we select an odd square number, we can make it the gnomon and thus give an example of a Pythagorean triple. The first odd square is

** ** ** **(our *9*) and to make it into a gnomon we stretch it out into three parts , two equal and the third a unit

** ** ** ** This provides the outer framework for the two squares :

** ** **** The inner square has side *4* and the outer side *5*. This gives the simplest

** ** **** Pythagorean triple *5*^{2}*= 4 ^{2 }* +

*3*.

^{2}** **

****

** ****** ****

However, any odd square will do and, since *49 = 7 ^{2 }*we can construct a Pythagorean triple involving it. The gnomon is

*24 + 1 + 24*giving

*24*for the side of the larger square and

*23*for the smaller one. This gives the triple

*24*

^{2}*= 23*+

^{2 }^{ }*7*. The series of Pythagorean triples using this procedure is endless : it suffices to find an odd square number.

^{2}This procedure can be generalised if we allow a

*gnomon*to be made up of more than one row + column. For example, we might allow the gnomon to have

*three*rows + three columns.

**……………… **

**……………… **

**……………… **

**…
…
… **

If *r* is the side of the inner square, the outer square is *(r + 3) *instead of *(r + 1)* and the little square in the bottom right hand corner will be *3 x 3 = 9 *instead of *1*. The gnomon is made up of two rectangles (*r x 3)* *+ *the little square giving (*6r + 9) = 3(r + 3)* We must thus find a square which is equal to the gnomon or solve * 3(2r + 3) = m ^{2} * for some

*m*. Since

*m*is divisible by

^{2}*3*this makes

*m*a multiple of

*3*as well. We must also have

*m*large enough so that

*r*is at least

*1*. The first possibility is

*m = 21 = 7 x 3*so that

*3(2r + 3) = 21*This makes

^{2}*r = 72*This will be the side of the inner square while the outer one will be

*72 + 3 = 75*. So, if this reasoning is correct, we should find that

*75*

^{2}*= 72*+ 21

^{2 }*which is indeed the case (check this). So a rather more spaced out but still unending set of Pythagorean triples can be manufactured where the difference between the sides of the squares is*

^{2 }*3*rather than

*1*. It is left to the interested reader to concoct other sets.

As a matter of fact we have reason to believe that the early Pythaogoreans knew of such sets of triples and it is plausible that they hit upon them using some such method which has its basis in the manipulation of sets of wooden gnomons and/or actual counters on actual boards. Interestingly enough, the Babylonians a thousand years earlier were aware of Pythagorean triples and seem to have had some method of concocting them (

**Note 3**)

**.**The basic formula for all Pythagorean Triples is given in Euclid — or rather can be deduced from the argument given in Euclid which is mainly verbal since the Greeks did not have our algebraic notation. I shall not give it here — you can get it from Wikipedia or some other site by the click of a key — as I am more interested in seeing how such formulae arose in the first place and indeed in (re)discovering them for myself, something that I encourage you to do as well. In the next post I will examine the slightly more complicated problem of an isosceles right-angled triangle, i.e. one where the two smaller sides are equal. This provoked a trmendous rumpus at the time because it raised the issue of so-called ‘incommensurables”. If the short side is set at unity, the square on the hypotenuse comes out at of

*1*+

^{2 }*1*= 2 so the side itself is the square root of

^{2}*2*. But was there such a number? In the ideal world of Platonic forms (not yet elaborated) certainly there was, but in the Pythagorean world of number where number meant ratio between two integers there was apparently no such quantity and thus no such length.

*SH 1/1/13*

**Note 1 **“The word *gnomon* ….literally means an “indicator”, or “one who knows”. Specifically, it was the name of the sundial first brought to Greece from Babylonia by Anaximander, who was probably one of Pythagoras’s teachers. The word also serves to indicate any vertical object like an obelisk which serves to indicate time by means of a shadow.” Valens, *The Number of Things** *

“*gnomon* : Stationary arm that projects a shadow on a sundial” (Collins)

**Note 2** Actually, it seems that Eratosthenes’ data did not depend on gnomons as such but it did rely on the measurement of noonday shadows. Reputedly, Eratosthenes based his remarkably good estimate of the circumference of the Earth on the information. presumably relayed by a traveller, that the sun at noon at midsummer’s day at Syene was directly overhead because it was reflected at the bottom of a deep well. Eratosthenes, as Librarian at Alexandria some 500 miles or so due north of Syene, knew that the shadow of a pillar cast by the sun at the same moment in time was a little more than 7 and a half degrees off the vertical . This enabled him to come up with an estimate of *4,000* miles in our reckoning for the Earth’s radius using geometrical techniques. Current estimates put the Earth’s mean radius at about *3960 *miles.

**Note 3 **“The Babylonian tablet called Plimpton 322 (dating from between 1900 and 1600 B.C.) shows that the Babylonians had studied this problem [of Pythagorean triples] much earlier. The tablet merely lists a series of Pythagorean triples but the order in which they are listed makes us believe that the Babylonians had a general and systematic solution for the problem of finding Pythagorean triples.”

Bunt, Jones & Benient, *The Historical Roots of Elementary Mathematics *