Today, of course, algebra has itself taken the place of geometry and mathematics has become something entirely removed from the contagion of the physical world, “a game played according to fixed rules with meaningless pieces of paper” as Hilbert famously described it. Mathematicians today shun the contagion of the real and indeed remind one of the early Christians who found it shameful to be born with a body. This, of course, is why mathematics today only appeals to professionals while lay people generally regard it with fear and loathing. I myself do not think it is either healthy or fruitful to remove oneself entirely from the delights and constraints of the actual world which is why I have formed the ambitious plan of evolving a

Nonetheless, some attempt at rigour must be made. My aim is by and large to establish most of the important theorems of Euclid VII – IX without any appeal to geometry but nonetheless using the Euclidian framework of

Where to start? The best definition of (cardinal) number is that given by Cantor of all people :

**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 a set of discrete objects may undergo (provided no object is destroyed) ― just as, allegedly, a given amount of mass/energy persists throughout the various complicated interactions between molecules within a closed system.

It is not clear whether these two principles should be viewed as *Definitions* i.e. they tell you what we mean by the term ‘cardinal number’, or as *Postulates * since, after all, they are generalisations based on actual or imaginable experiments (the pairing off of random sets of objects with a chosen standard set). They are not ‘logical truths’ and not strictly speaking ‘common notions’ (Heath’s term for *axioms*).

We also need a third Principle, the **Principle of** **Correspondence **. It 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 but which is not directly culled from experience.

** Principle of Correspondence
**

I pass now to the basic

** DEFINITIONS**** **

**Object:**Continuous solid.**Discrete object:**object surrounded by unoccupied space.**Unit:**Chosen standard discrete object that cannot be split.**Collection**. Plurality of units in close proximity to one another but which do not merge or adhere to one another.**Number:**Collection in the above sense, or the unit itself.**Numerical equivalence:**If a collection*A*can be made to cover*B*completely unit for unit with nothing left over, it is said to be numerically equivalent to*B*, or we write*A =n= B*.**Smaller than:**If a collection*A*fails to cover*B*counter for counter it is said to be less than*B*, and we write*A < B*.**Greater than:**If a collection*A*covers*B*counter for counter, with at least one counter left over, it is said to be greater than*B*, and we write*A > B*.**Joining:**If a collection*A*is pushed up against*B*so that they form a single collection*C*, we say that*A*is joined to*B*producing*C*and write*(A + B)**ð**C*, or*C = (A + B)*.**Subtracting:**If, from a non-unitary collection*A*, a smaller collection*B*(which may be unitary or multi-unitary) is taken aside leaving a collection*C*, we say that*B*has been subtracted from*A*, and write*A – B**ð**C*and*C = (A – B)*.**Replication:**If a flat (two-dimensional) collection*A*is covered exactly unit for unit by a collection*B*, then*B*, once removed, is a replica of*A*. And if*A*is a three-dimensional collection with base*a*and number of layers*h*, then if*B*is built up on a numerically equivalent base with equivalent height,*B*is a replica of*A*.**Reduction:**If a collection*B*is separated out into so many*numerically equal*collections, possibly unitary ones, then any of these collections*A*is called a reduction of*B*. (*Note*. The same result would be obtained if we had a collection*B*laid out with base*A*and uniform height*h*, and then we ‘reduce’ it by compression to a single level. But since the units cannot, by definition, be squashed together, this operation cannot strictly speaking be performed.)**Rectangular Collection:**Collection that can be arranged as so many numerically equivalent rows or columns.**Linear (or prime) collection:**Collection that can*only*be arranged as a line, i.e. with the unit as side.**Common Side:**If collections*A*and*B*can be arranged as rectangles with a pair of numerically equivalent sides, they are said to have a common side. Alternatively, if*A*and*B*can both be fitted between two parallel lines width*a*units where*a > 1*, they have a common side.**Relatively linear or prime to each other:**If collections*A*and*B*have no possible side in common apart from the unit, they are said to be relatively prime and we write*(A, B) = 1*.**Common Multiple:**If every possible side of*A*and*B*is a possible side of*C*, then*C*is said to be a Common Multiple of*A*and*B*.**Complete Common Multiple or Product:**The complete common multiple of*A*and*B*is the rectangle*sid*e*A × side B*.**Least Common Multiple:**Smallest collection which is a Common Multiple of*A*and*B*.**Result Equivalence:**If, given some arrangement of units*B*, it is impossible to say which of two or more operations evolved it from an initial arrangement*A*, then the operations are result-equivalent. And if the final configuration is evolved from two different starting points using the same or different operations, then the two ‘starting-points + operations’ are result-equivalent.**Ratio Equivalence:**If a pair of collections*(A, B)*can be transformed into*(C, D)*by replication and/or reduction*alone*, at all times keeping the levels the same, then we say that the ratio of*A*to*B*is the same as the ratio of*C*to*D*and write*A:B = C:D*

In the next Post I will detail the basic *Postulates *required for Concrete Number Theory and establish the first theorems. The more fundamental theorems, which are those the most closely related to physical operations, will be singled out and named *Basic Assertions*. They are ‘proved’ by a combination of appeals to sense-impressions, the more direct the better, combined with some rational argument. But, whereas in ‘normal’ mathematics all arguments are strictly deductive, in Concrete Number Theory, inductive arguments are perfectly valid provided (1)a concrete example is given and (2) a possible generalization to cover all, or a multitude of cases, seems to have nothing going against it. Inductive ‘logic’ lacks the finality of deductive logic but it has the advantage that it can *never *degenerate into complete meaninglessness. One cannot have one’s cake and eat it: the inductive approach does mean that at least you taste a morsel even if you are not entirely sure that you will get hold of the entire cake. *SH *

** **

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To Archimedes then. First off, Archimedes does not turn the area of a circle into an abstract algebraic function: he compares the circle to something that we can see and draw, namely a right angled triangle with known area *½ base × height * where the height is the radius of a circle (any circle) and the base is the same circle’s circumference straightened out. Area = *r × C) = ½ r C*.

[Apologies but currently I have problems putting diagrams on this website, see later versions of this post.]

The proof is by double contradiction (as Professor Dunham puts it) and though these kind of proofs are often rather artificial, in this case the reasoning is both impeccable and even entertaining. Archimedes says in effect, “Let us suppose my claim is wrong, and that the Area of a circle is* not *equal to that of my triangle”. It might, for example, be *less *than the area of a circle. We now ‘exhaust’ the circle in the inimitable Greek manner by fitting regular polygons (many sided figures with all sides and matching angles equal) inside the circle. We can start with an equilateral triangle which fits exactly into the circle (how to do this is demonstrated in Euclid Book I), we then double it to produce a hexagon (which, amazingly has six outer sides equal to the radius), double that to make a *12-sider, 24-sider, 48-sider &c. &c. *All these doublings can actually be performed by following instructions given in Euclid Book I using only straight edge and compass.

Clearly ─ because we can actually *see it happening* ─ the gap between the total area of the inscribed polygon and the area of the circle gets less each time. Next, very reasonably, Archimedes asks us to accept that, in principle at least, by constructing polygons with an ever growing number of sides we can *make the difference between the area of the polygon and the area of the circle as small as we please. *(Difficulties with computer graphics stop me giving too many diagrams at this point, see updated versions of this post.)

Now the polygon is made up of identical triangular sections where the ‘base’ is the outer side of each section and the ‘height’ is the length of the perpendicular from the centre to this base, technically known as the *apothem*. Note that the two other sides of the triangular section are radii and, important point, the radius is always *greater *than the apothem since the apothem is the shortest route to the base. Now, every triangular section of the (regular) polygon is the same, so, if the polygon consists of *6 *triangles, the total area will be *6 × ( ½ base × height) *or, to put it another way, *6 bases × ½ h *. The length ‘*6 bases’ *is the length of the *perimeter *of the polygon, the distance you would have to go if you walked all round it. We now increase the number of sides to *12, 24, 48, 96…. *and so on. At each stage, what we end up computing to give the area of the polygon is the perimeter of the inscribed polygon multiplied by half the height, or *½h × P *where *P *is the * *Perimeter. [Apologies but currently I have problems putting diagrams on this website, see later versions of this post.]

What about the original right angled triangle? The area of Archimedes’ right angled triangle is*½ base × height = ½ C × r *─ where *C *is the circumference of the circle and *r *is the radius. Archimedes has suggested, for sake of argument, that the area of the right angled triangle is *less* than the area of the circle. Now, since the area of the inscribed polygon can be made as close to the area of the circle as we wish, it should be a better fit than the area of the right angled triangle. So, according to this supposition, we have the inequality

** Area Triangle < Area n-gon for large n < Area circle
**

We thus have a contradiction so we must reject the hypothesis that the area of the triangle < area of the circle.

Suppose, then, the area of the triangle is

Archimedes now delivers the

But what is the area of the right angled triangle in modern terms?

It is

The only point in Archimedes’ excellent piece of reasoning that is somewhat questionable is the following. By making the circumference the base of his triangle, Archimedes is in effect

** ****The Calculus derivation
**The modern derivation of the formula for the area of a circle goes like this. Armed with a co-ordinate system (which Archimedes did not possess) we draw a circle centred at the origin and assess the ‘area under the curve’ ─ or rather the area of the first quadrant which is all we need (because we eventually multiply it by

[

This expression is guaranteed to make the non-mathematical reader give up in disgust and go to the pub instead, and, although I have on occasion taught Calculus to school age kids, I dislike the inverse trigonometrical functions and had to look up the answer in a book and differentiate the result to see if I got back to √

But how do I know this Calculus result is right? Only by accepting various assumptions which underpin Calculus, assumptions which were, in the days of ‘infinitesimals’, extremely dubious as Bishop Berkeley pointed out at the time and have only been made rigorous by disconnecting Calculus completely from physical reality. The Greeks (correctly) argued that curved lines and straight lines are different species and that areas bounded by curves can be

The ancient

**(1) **that it is a (human) *creation* and,

**(2) **that it is not a *free *creation ─ it cannot be if its aim is to study ‘objects in the *world’*.

**(2)** in effect means that mathematics is, or rather was, subject to serious constraints. What sort of constraints? If we examine the two earliest branches of mathematics, arithmetic and geometry, we see that the principal constraints are *discreteness *and *distancing*. *Discreteness* because arithmetic sees the world as made up of lots of little bits that *can be counted* ─ if everything was mixed up with everything else like the ingredients in a cake arithmetic would not give sensible results (and would not be needed). Secondly, experience tells us that whatever is happening *here* is not simultaneously happening *over there*, i.e. that objects and events are separated by something, thus geometry, metric spaces, causality, anything that depends on spatial separation.

Viewed thus, the two earliest and (arguably) still the two most important branches of mathematics depend either on *separateness *or *separability *: in the first case, that of arithmetic, the focus is on the *objects *themselves, in the second, the focus is more on what lies *between *and *around *the objects. Practically all pre-Renaissance mathematics was in effect the study

**(1)** *of particular objects separated by empty space *or

**(2)** *of particular objects such as points and curves embedded in empty space.*

No hint of movement so far ─ as we know the Greeks, though they initiated both statics and hydrostatics baulked at the creation of dynamics, the study of objects in motion and, by implication, of the unseen forces that give rise to motion.

^{1} From Lakoff & Núñez, *Where does Mathematics come from?*

** DISTINCTION BY NUMBER CAN ONLY BE ACHIEVED BY ABOLISHING DISTINCTION BY TYPE. **

This explains the surprisingly rudimentary nature of number concepts amongst hunting/food-gathering peoples, and their stubborn resistance to the introduction of more advanced number systems and number concepts by missionaries. This resistance is not to be attributed to a lack of intelligence since the complex language structures, imaginative myths, pictorial sense and elaborate rituals of such peoples show that they were capable of first-rate cultural achievements. No, the reason for this resistance to number is deep and essentially well-founded since for such peoples it would have been a rash move to automatically prefer distinction by number to distinction by type.

For classification according to type is absolutely essential to the hunter/food-gatherer: he or she must make quick-fire radical distinctions between plants that are comestible and poisonous, animals that are harmless or dangerous, strangers that are hostile or friendly &c. &c. ─ and errors can easily lead to death of the individual and even extinction of the tribe. But counting objects is of little utility: what is the point of attributing a number sign to objects that are in front of you every day of your life? Do *you *know how many suits of clothes or dresses you own? How many rooms there are in your house or flat? Arithmetic only becomes significant when it is essential to know when to sow or reap, when trade is extensive and, above all, when a state official needs to assess a whole country’s resources. It was the Assyrians and the Babylonians who developed arithmetic just as it was the founder of the short-lived Ch’in Dynasty, Ch’in Shih Huang Ti (he of the terracotta warriors), who imposed the metric system on his citizens nearly two thousand years before Napoleon did and who likewise standardized weights and measures throughout his vast Empire. *SH 17/1/19*

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.

An efficient number system is necessary both for *assessing* and *recording* important quantities and in practice this means that *two* systems, or two versions of the same system, are required, 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 great advantage of the Greek alphabetic system was its conciseness. The economy of the Greek, and later the Hindu-Arabic number 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 has been and 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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