In this sense it is perfectly true that numbers, or at any rate number

So how do we develop a number system? What are the minimal requirements?

Two, and as far as I can see,

- The ability to distinguish between what is singular and plural, i.e. recognize a ‘one’ when you see it;
- The ability to carry out a one-one correspondence (pairing off).

All the mathematicians who have developed abstract number systems, for example Zermelo and von Neumann, had these two perceptual/cognitive abilities — otherwise they would have been denied access to higher education and would not even have been able to read a maths book. Animals seem to have (1.) but not (2.) which is perhaps the reason why they have not developed symbolic number systems (though a more important reason is that they did not feel the need to). Computers are capable of (1.) and (2.) but only because they have been programmed by human beings.

What is number? One could describe ‘number’ as the ‘property’ that results when we have done away with all other distinctions between sets such as colour, weight, position, shape and so on. This is not much of a definition but it does emphasize the curious fact that number is more of a negative rather than a positive property since it results, as Piaget says, “*from an ignoring of differential qualities”*.

But, notwithstanding the difficulty of saying what exactly number is, practically speaking there is a perfectly simple and universally applicable test which can decide whether two sets of discrete objects are numerically equivalent or not, i.e. can be validly allocated the same number label. If I can pair each of them off with the *same* standard set of objects or marks, the two sets are numerically equivalent, if I can’t they are not. Of course, today if I want to assess the ‘number’ of chairs in a room, say, I associate the collection with a number word, *seven* or *four *or *six *as the case may be, but underlying this is a pairing with a standard set. As a matter of fact I find that, though I use the number words *one, two, three….. *when counting objects, I still find it necessary to use my fingers, either by pointing my finger at the object or pressing it against my side, one press, one object. And the umpire in a cricket match still uses stones or pebbles : one ball bowled, one stone shifted from the right hand to the left. It is not that the finger or stone pairing off is valid because of our ciphered numerals but the reverse : our written or spoken numerals ‘work’ because underlying them is this pairing off of items with those of a standard set.

Now, one could actually derive the Cantor definition of cardinal number — “*that **which results from abstracting from a set the order of appearance of the elements and their specific character”* — from what happens when I apply my test. If I rearrange the objects I am supposed to be counting, does that make any difference to the ‘number’ representing the sum? No. Because if I could pair off the original collection with items from a standard set, such as so many pebbles or marks, I can do the same after rearrangement. Does the actual identity of the objects matter? Apparently not, since if I replace each original item by a completely different item, I can still pair off the resulting set with my standard set (or subset).

We thus arrive, either by reflection or simply by applying the test, at the two basic numerical principles, the **Disordering Principle **and the **Principle of Replacement
Disordering Principle
**

** Principle of Replacement
**

* * Together these two principles make up a sort of **Number Conservation Principle **since whatever ‘cardinal number’ is, this ‘something’ persists throughout all the drastic changes the set undergoes just as, allegedly, a given amount of mass/energy persists throughout the interactions between molecules within a closed system.

These two principles may either be viewed as *Definitions* i.e. they tell you what we mean by cardinal number, or as *Postulates * since they are the generalisation of actual experiments (pairing off sets with a chosen standard set). They are not ‘logical truths’ and not strictly speaking axioms.

The **Principle of Correspondence **has a somewhat different status and is more like a true Axiom, i.e. something which we have to take for granted to get started at all but which is not directly culled from experience.

**The Principle of Correspondence **

** ***Whatever is found to be numerically the case with respect to a particular set A, will also be numerically the case for any set B that can be put in one-one correspondence with it. *

By ‘numerical’ features I mean such things as divisibility which has nothing to do with colour, size and so forth. We certainly do assume the **Principle of Correspondence** all the time, since otherwise we would not gaily use the same rules of arithmetic when dealing with apples, baboons or stars : indeed, without it there would not be a proper science of arithmetic at all, merely ad hoc rules of thumb. But, though the Principle of Correspondence is justified by experience, I am not so sure that it originates there : it is such a basic and sweeping assertion than it is more appropriate to call it an *Axiom* than anything else. Note that physical science uses a similar principle which is today so familiar that we take it for granted though it is far from ‘obvious’ (and possibly not entirely true), namely that “what is found to be physically the case for a physical body in a particular place and time is the case for a similar body at a completely different place and time”. Newton’s law of gravitation is not just true here on Earth but is assumed to be true everywhere in the universe — a fantastic generalization that many scientists at the time thought unwarranted and arbitrary.

These principles do not by any means exhaust the assumptions we implicitly make when we use or apply a Number System : indeed, if we listed all of them we could probably fill a sizeable volume. For example, we continually assume that there is a physical reality ‘out there’ to number in the first place (which solipsists and some Buddhists deny), that there are such things as discrete objects (which philosophic monists and in some of his writings even Einstein seems to deny) and so on and so forth. But these ‘axioms’ are best left out of the picture : they underlie most of what we believe and are not specific to numbering and mathematics. ** **

**Notes **

^{1} This is (perhaps) not true of the basic constants such as the gravitational constant or the fine structure constant : they seem to be ‘hard-wired’ into the universe as it were and there seems to be no special reason why they should have the values they actually do have, unless one accepts the Strong Anthropic Principle. In theory it should be possible to deduce the values of basic constants from *a priori *principles but to date attempts to do this, such as Eddington’s derivation of the number N, the number of elementary particles in the universe, have not been very successful to say the least. One could argue from ‘logical’ considerations that there must be a limiting value to the transmission of electro-magnetic signals but there is no apparent reason why it should be *3 × 10 ^{8} m/sec *

^{2} The quotation I have in mind is, *“L’histoire a toujours existé mais pas toujours sous sa forme historique” *(‘History has always existed but not always in its historical form’) from *La Société du Spectacle *by Guy Debord. The phrase sounds wonderful but means very little.

*SH *4/03/2018

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Early arithmetic and numbering generally was concerned **(1)** with recording what was already known (at least approximately) and **(2)** finding out and recording what was not known — but which could, hopefully, be extracted from the relevant data. A census carried out in a series of villages would tell a regional official how densely populated the area was, and such a piece of data needed to be recorded in a form that other officials would be able to comprehend. This is **(1)**, recording what is already known — at any rate locally . If we want to work out the food supplies necessary to keep all these people alive in a time of famine, or how many young men the region is likely to be able to provide for the army, we have a primitive kind of equation. This is a case of **(2)**, finding out and then recording what is, prior to the census or other data collection, is not known locally. There is, however, no hard and fast line separating **(1) **and** (2) **since simply combining the separate data about each village does provide new information, i.e. the* total* number of inhabitants in the region which, doubtless, no single villager knew.

Now, a number system is used both for *assessing* and *recording* certain quantities and in practice this means that *two* systems, or two versions of the same system, are needed, a *temporary * system and a more *permanent *system. If quantities are small, we can *assess* a given quantity (how many pigs? how many coconut trees?) using our hands as the temporary recording system but, since we need our hands for other purposes, we also need a separate, much more durable, recording system which could be clusters of shells (Benin Empire in Nigeria), knots in a string (Inca Empire in Peru) or marks on some long-lasting material such as bone, bark or papyrus (Egypt). Even today, numbers are still primarily used simply for recording data — rather than for pure-mathematical purposes. Coping with numerical data has, in fact, been a perennial problem for advanced societies from ancient Egypt right down to the present day.

The early Egyptian ‘hieroglyphic’ number system is perhaps the clearest and simplest number system ever invented. A single item, a datum, was originally represented by a picture of a papyrus leaf which soon just became a stroke. The Egyptians, like most (but by no means all) societies used a base-ten system, i.e. once you have a given collection of strokes, you make it into a ‘first base’ (our *ten*), when you have the same quantity of ‘first bases’ you make it a second base (our *hundred*) and so on. In principle the different bases could be distinguished by size ― if unity is a stroke, ‘ten’ is a longer stroke, ‘hundred’ a longer stroke still &c. &c. The inconvenience of such a number system is that it requires a lot of space if you are dealing with large quantities, which the Egyptian officials often were (it is thought that some Egyptian cities at their height had nearly a million inhabitants). Considerations of space have in fact played a very large part in the development of number systems and recording technology generally. The Egyptians did not distinguish the ‘one-symbol’ from the symbol for first base, the symbol for the first base from the second and so on by comparative size: they had *separate pictograms *for ‘one’, ‘first base’, ‘second base’ and so on. Our *ten *was a bent leaf, our *hundred* a coiled rope, our *thousand *a lotus flower, our *ten thousand *a snake, our *hundred thousand *a tadpole or frog and our *million *a “seated scribe holding up his hands in astonishment”. In this system you only had to learn the meaning of seven hieroglyphs whichis not a very great task. But with these seven symbols repeated when necessary any quantity less than a ‘million million’ (original meaning of ‘billion’) could be represented. “They [the Egyptian officials] could record the number of captives available for slave labour and share them out for public works. They could estimate how much food and drink, how many blocks of stone of different shapes and sizes, how many slaves and overseers would be needed from day to day to build the pyramids” (McLeish, *Number*).

Note that in the Egyptian system, as opposed to the ‘increasing size’ system which hardly any society ever used, a new *single* symbol is needed for each larger base; any given symbol is never repeated more than a certain number of times (nine times in a ten-base system). Each new symbol is thus not just a bigger and better version of the basic ‘one-symbol’ but something quite different. Some of the new symbols seem somewhat arbitrary since one sees no obvious connection between a quantity we call a *hundred* and a coiled rope for example. On the other hand, the Egyptian symbol for our 100, 000, either a frog or a tadpole, may well have been chosen because frog spawn contains a vast number of eggs, as someone recently suggested to me. Since, even today, our brain finds it much easier to store images of real things rather than abstract signs, the Egyptian system was extremely easy to memorise.

This is not really what we mean by a ‘cyphered’ number system, however, since, in the Egyptian system all quantities less than our *ten *are still represented by the one-symbol repeated the appropriate number of times. The Greeks took the ‘different symbol’ principle much further by introducing *single* symbols for *all quantities *greater than one and less than first base, as we ourselves do. Thus our ‘*four*‘ is not represented by a plurality of one-symbols such as * l l l l *but by a

The Greeks possessed an earlier number system that was not unlike the Egyptian, called the Herodianic, but it got rapidly displaced by the alphabetic system. The later Greek system is, in point of fact, in many ways inferior to the Egyptian: you have to learn many more symbols, there is a perpetual risk of confusion between letters used as letters and letters used as numbers, and eventually you run out of symbols for large quantities. Against this, there is the sole advantage: Greek numerals take up far less space and, surprisingly, this property outweighed all other considerations. The French historian Tannery employed the Greek alphabetic system in the sort of calculations that an engineer like Archimedes would have carried out and, surprisingly, claimed that the alphabetic numerals had many unexpected advantages and that arithmetic operations hardly took up more time than when carried out with our numerals.

The Roman system is semi-cyphered. The Romans still repeated their one symbol, I, a certain number of times but introduced a new sign for *five, *namely V, and used a subtractive principle for the number immediately before a base, for example IX for *nine*, which gained a little space. Nonetheless, their numerals still appear to us very unwieldy. Until recently the dates of publication of English books were given in Roman numerals and such dates look extremely long-winded compared to ours. To perform multiplications and divisions using Roman numerals must have been tedious in the extreme (but it can be done) and Roman scribes almost certainly used books of tables for standard multiplications as, it is conjectured, the Egyptian scribes also did. The economy of the Greek and later Hindu-Arabic system was, in its day, as important as the miniaturisation of the components of contemporary computers that has revolutionised the world of communication technology: saving space for the recording of data remains one of the most important of all human concerns.

*SH 23/11/17*

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The Greek geometers never speak of the ‘golden ratio’ and the first recorded use of the term is as late as 1835 ─ when Ohm referred to it as the *goldener Schnitt*. Nor does any ancient Greek give a numerical value for what we now know as phi or Φ. What we *do *find in Euclid and other ancient writers is repeated mention of a certain manner of dividing a line segment in “extreme and mean ratio”. Euclid VI Proposition 30 shows you how to do this. In our terms, this method of division results in “the ratio of the larger to the smaller part of the line segment being equal to the ratio of the whole to the larger part” i.e. *a:b = (a + b) : a *where *a > b *.

← *(a + b) →
*

Why was this important to the ancient Greeks? Not apparently because of the supposed aesthetic properties of the associated ‘Golden Rectangle’ (formed by making the smaller portion into one of the sides). Although it is sometimes claimed that Phidias used the Golden Section in some of his Parthenon statues this is mere speculation; it was only Renaissance painters and architects who superimposed the proportions of the golden rectangle onto the human figure as in the famous Leonardo da Vinci drawing and claimed there was something especially beautiful about the ‘divine proportion’, as they called it.

Nonetheless, to judge by the number of theorems relating to it in Euclid and numerous references to it in other extant ancient manuscripts, the ‘section’, as Proclus calls it, was famous. So why *did* the ancient Greek mathematicians consider the division of a line in ‘extreme and mean ratio’ significant? Because it was a prerequisite for the ruler and compass construction of a regular pentagon (five-sided figure with all sides and angles equal) and thus for the construction of the pentacle (regular pentagon within a circle) and the starry pentagram (five-pointed star). The pentacle already had a certain history as a ‘magic symbol’, being originally associated with the ‘morning star’ (Venus), and this esoteric reputation has lasted right up to the present day ─ Dr. Faust uses it and so do some contemporary Wicca groups. In ancient Greek times the pentacle had a more respectable, but still somewhat offbeat, reputation since the Pythagoreans, originally a kind of scientific secret society, used it as a sign of recognition amongst the Fraternity ─ compare the Freemason handshake. They sometimes put letters at each point of the five pointed star and these letters spelled out the Greek word for health (*u**g**i**e**i**a*) ─ so it was a sort of “Good Health to you, fellow Pythagorean” message.

But the pentagon had a more serious meaning still for educated Hellenistic Greeks and Romans. Although he did not invent them, Plato was an ardent propagandist for the importance of the regular solids, still called Platonic solids in his honour. For Plato, shape was more fundamental than substance and the supreme shapes were the perfect forms of geometry such as the circle and the regular polyhedral. These ideal Forms were changeless and harmonious whereas everything on the terrestrial physical plane was erratic and unpredictable. The five Platonic solids, which Plato identified with the four elements, Earth, Air, Fire and Water (plus a subtle fifth element Ether), had much the same status as the elements of the Periodic Table have in our eyes today. Indeed, it would hardly be going too far to say that, for Plato, these ideal Forms were cosmic computer programmes while the entire physical world consisted of the fallible execution of such programmes, software compared to hardware, genotype to phenotype. In consequence, it was very important for Platonists to know how to construct these forms, if only in imagination. The five solids are:

- The
**Tetrahedron**(four triangular faces); - The
**Octahedron**(eight triangular faces); - The
**Cube**(six square faces); - The
**Dodecahedron**(twelve pentagonal faces); - The
**Icosahedron**(twenty triangular faces).

Euclid concludes his great work with Book XIII which is entirely devoted to the construction of the five Platonic solids. Although Euclid is generally regarded today as the originator, or at any rate greatest early expositor, of the axiomatic method, this gives the modern reader the wrong impression. Today, the axiomatic treatment of a mathematical topic implies complete disregard of practicalities and ‘realistic’ concerns, but Euclid always has his eye on the actual construction of figures inasmuch as this is feasible. The very first Proposition (Heath calls ‘theorems’ *Propositions*) of Book I is “*On a given finite straight line to construct an equilateral triangle”*. And the penultimate Proposition of his *Elements *(Book XIII. 17) tells you how to *“construct a dodecahedron and comprehend it in a sphere”*. To be sure, this construction is so complicated, likewise that of a icosahedron (20-sided regular polygon), that one is hard put to follow the steps in the argument, let alone produce an actual model in wood or metal. Nonetheless, the mathematical presentation is not abstract in the way that, say, a theorem about Baruch spaces in modern mathematics is.

Such an approach is absolutely in line with the Platonic philosophy. For Plato was not so much an Idealist as a Transcendental Realist: his Ideal Forms were *more*, not less, real than actual artifacts while not being absolutely divorced from material things either. As certain Sophists in Plato’s own time observed, the figures of geometry, when drawn, did *not* have all the properties accorded to them by geometers: points on an actual circumference were not always exactly equidistant from the supposed centre, tangents cut a circumference in more than one point &c. &c. “Yes,” Plato might have replied, “but the drawn circle is not the circle of geometry, only a tolerable imitation of it. The *true *circle and true tangent, of which our human imitations are derivatives, really do have all the properties we ascribe to them, such a tangent really does touch the circumference at a single point only.”

It is interesting to note that Book XIII concludes with the dodecahedron rather than the icosahedron (whose construction is even more complicated) ─ the final Proposition 18 deals with the relations between the entire five Platonic solids and proves as a sort of coda that they are the *only *possible regular solids. The reason for terminating with the dodecahedron is most likely because the dodecahedron was traditionally associated, not with the four *earthly * elements, but with starry matter which was considered to be different from, and superior to, earthly matter. (Tradition has it that the Pythagoreans were especially delighted with their discovery of the dodecahedron and sacrificed a hundred oxen to celebrate the occasion.) And, as stated earlier, the division of a line ‘in extreme and mean ratio’ is essential for the construction of the regular pentagon which is itself essential for the construction of the dodecahedron (since all the faces are regular pentagons).

This may go some way to explaining why the ancients had a particular veneration for the ‘section’. Moreover, Allman makes the interesting suggestion that what we call phi, the golden section, was the very first irrational (the Greeks would have said ‘incommensurable’) to be discovered, rather than √2 as is today usually assumed. This would explain the mystery and slightly sinister glamour attached to figures incorporating the golden section such as the pentacle; for the discovery of incommensurables was, as we know, extremely disturbing for Greek mathematicians and philosophers alike. The Pythagoreans seem to have shifted from an attitude of hostility towards irrationals/incommensurables to one of veneration, at least as far as Phi was concerned since they eventually adopted the pentacle as a sort of logo.

Did Euclid have what we might call a philosophical, almost a quasi-religious, aim in giving the ancient world such a detailed exposition of the *Elements *of geometry? This was certainly the view of Proclus who wrote a commentary on Euclid in which he claimed that Euclid was himself a faithful follower of Plato and that “it was for this reason he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures”. Heath rejects this out of hand, arguing that Proclus was a biased source since he was himself the leading Neo-Platonist philosopher of his time and keen to claim Euclid as one of his own. Nonetheless, there can be no doubt that philosophical Platonism was inextricably mixed up with late Greek higher mathematics and Heath himself admits that “it is most probable that Euclid received his mathematical training in Athens from the pupils of Plato”. Whether Euclid was himself a Platonist is unknown but he seems to have faithfully transmitted to posterity not only the discoveries of Platonist (or Pythagorean) mathematicians but their overall ‘view of the world’. We do not today consider Book XIII to be the most important part of the *Elements *and usually single out the ingenious treatment of the problem of incommensurables in earlier books because this treatment anticipates the 19^{th} century approach to irrational numbers as pioneered by Weierstrass and Cantor. But the *Elements *was not just an exercise in pure mathematics; at any rate for many later Greek mathematicians, it was a sort of technical preamble to Platonic cosmology as laid out in the *Timaeus*. Kepler, to whom the Alexandrian cultural ambiance of Euclid’s day would have been most congenial, made a persistent attempt to match the orbits of the planets to the outlines of the Platonic solids and, incidentally, singled out the ‘division in extreme and mean ratio’ as the ‘chief jewel of Greek geometry’, on a par with the Pythagorean theorem itself. Although for a long time it was fashionable in scientific circles to look down on interest in the Golden Section as the affair of aesthetes and mystics, it is now known that one version of it, the Golden Angle, does have some importance as a ‘close packing constant’ as Irving Adler relates in his latest book on Phyllotaxis, or Leaf Arrangement. *SH 25/09/17*

I would guess the original ‘symbol’ for zero was something like the double open handed gesture that hunters still use to indicate that they have caught nothing that day. This gesture, common amongst country people in the South of France, does not quite signify “nothing” in the absolute sense, but rather “Nothing where something was to be expected” — which is somewhat different.

“In those systems that build by addition to five, counting usually starts with the little finger of one hand and proceeds by the addition of the appropriate fingers in sequence until five is reached. This number is generally denoted by a closed fist. For six, the little finger of the other hand joins in the counting, and the fingers of the second hand are used in the same sequence as those of the first” (Zaslasky, *Africa Counts *p. 49).

That gestures directly gave rise to full-scale finger counting seems unlikely : the sophisticated finger counting systems such as the Venerable Bede describes in his 8th century treatise *De computo vel loquela digitorum *(“On calculating and Speaking with the Fingers”) must surely have developed *after* an advanced spoken number system. No one in their right senses would use finger counting *alone* to represent really large quantities : what generally happened is much more likely to have been a combination of various systems, gestures, spoken words, the use of object numbers alongside recorded numerals and so on. Zaslavsky says that the Arusha Masai of Northern Tanzania “rarely give numbers without the accompaniment of finger signs” (*op. cit.* p. 248). Different ethnic groups had different ‘cut-off points’, most ending with our **50** at most while in the Luo system “there are no gestures for numbers beyond **19**” (*op. cit.* p. 254).

The, at first rather surprising, fact that African languages are predominantly base-five (rather than base-ten) suggests that ‘gestural number systems’ predated written and even spoken ones. Tylor writes: “Word-language not only followed Gesture-language, but actually grew out of it” (Tylor, *Primitive Culture*)

*SH 08/08/17*

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.

** **

** **

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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,

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

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

Conversely, let **ad = bc
** Then

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)

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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*

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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;
**

** **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.

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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)

** **

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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,

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

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

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

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

Conversely, let **ad = bc
**Then

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
}**

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

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

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

For **N = ***b ***P + 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) * *

* *