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November 23, 2017

Mathematics seems to be not only an exclusively human activity but also a very recent one (in evolutionary terms). Animals, with one or two possible exceptions (bees, whales), do not use symbolic systems for communicating information or as behavioural aids. This is not to imply that our way of doing things is necessarily preferable or even ‘more advanced’: it would be hard to better the extraordinary feats of migrating birds returning to the selfsame nesting spots year after year or the accuracy of predators pursuing rapidly moving prey. Quite how animals achieve such feats is still not completely understood, at any rate with respect to migratory birds, but, certainly, they do not consult ephemerides or solve differential equations. Much the same goes for early societies: they did not develop complex numerical systems because, most of the time,  they did not need them. Why did they not need them? For two reasons, firstly because the quantities involved, e.g. number of personal possessions or number of members of the tribe, were small and, secondly, because the sensory apparatus of early humans was extremely acute, more acute than ours. Being able to review objects in their precise locations often does away with the need to count them. Do you know how many chairs there are in your house or flat? How many rooms even? You don’t need to know the numbers involved because you can mentally review such a familiar landscape and even approach it from different sides, go into it, behind it and so on. Moreover, since in earlier times a good visual and auditory memory was necessary for survival, it was intensely cultivated alongside physical fitness. Predominantly oral societies routinely produced individuals whose memory capacities seem scarcely credible to us today: sacred books as long as the Rig-Veda or the Koran were learned off by heart and reputedly still are in some parts of India and Pakistan. Mathematics itself (as we understand the term) only took off with the advent of large, centralised, bureaucratic societies such as Assyria, Babylon and Egypt. In such cases, a good visual memory was inadequate since the scribe or official would not be personally familiar with what he was supposed to be assessing. It was the necessity to record and process data efficiently that gave rise to mathematics in these imperial societies and, conversely, an advanced numbering system only becomes essential when what you are dealing with exceeds the range of your personal experience. “The main condition under which arithmetical operations  become useful is economic action at a distance and such conditions do not arise for hunters or for the simpler forms of agricultural society” (Denny, Cultural Ecology of Mathematics) (Note 1). Unromantic though it sounds, arithmetic seems to have been developed mainly for the purpose of stocktaking, taxation and large-scale warfare while geometry (literally ‘land-measurement’) was, according to Herodotus, invented by the Egyptians in order to survey accurately (and subsequently tax) the irregular plots of peasants bordering the Nile.

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 single symbol, δ. We are so used to this principle that we do not realize what a significant departure it really was. The Greeks managed this by making their alphabetic letters double up as numerals and so β, the second letter, became ‘one-one’ or our ‘two’.The 27 letters of the Greek alphabet, which they took over from the Phoenicians,  were divided up into three sets of nine, the first set for quantities up to, and including, our 9, the second set for our 10, 20… 90 and the third for the hundreds. Various artifices such as having a bar above the letter-number enabled one to extend the system beyond 900 but the Greek alphabetic system could not be extended indefinitely in the way that ours can — because the Greeks did not hit on the idea of place value and positional notation. Archimedes himself, the greatest mathematician of the ancient world, felt obliged to write a treatise, The Sand Reckoner, to argue that, in principle at least, all the grains of sand in the world could be numbered ― but even he never hit upon the stratagem of place value.
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



Extreme and Mean

September 25, 2017

‘Extreme and Mean Ratio’

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)                                                        →
                 ←                     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 (ugieia) ─ 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:

  1. The Tetrahedron (four triangular faces);
  2. The Octahedron (eight triangular faces);
  3. The Cube (six square faces);
  4. The Dodecahedron (twelve pentagonal faces);
  5. 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 19th 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


August 8, 2017

Gestures have been used from time immemorial either as a rudimentary number system, or as a supplement to spoken or recorded systems. Claudia Zaslasky (Africa Counts, Lawrence Hill 1999), gives visual examples of a ‘gesture for “six” ’ from Rwanda and a Xhosa woman showing by gesture the number of her children. Zaslasky notes that “in some African societies finger gestures have equal status with the spoken numerals and constitute a proper system of numeration which may or may not agree with the spoken words” (op. cit. p. 37).
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

Minimal Conceptual Abilities for Development of a Number System

July 29, 2017


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 19th 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 18th 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.





May 30, 2017

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 = pa qb rc…..

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

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

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

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

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

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

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

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

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

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

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

“Let N = pqrs… where  p, q, r…. are primes. Suppose a prime u different from primes p, q, r… and which divides N. Then N = u × b.
But if any prime number divides (m × n) and does not divide m, it must divide n [VII. 30].
Now, p divides N and p does not divide u since u, p are primes and u ≠p Therefore, p divides b. And the same applies to q, r….
Therefore, pqr…  divides b
        But this is contrary to the hypothesis, since b < N and N is the smallest number that can be divided by pqr….
        Therefore, N has no prime factors apart from p, q, r…

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

What Propositions does this proof rely on?
Firstly, on VII. Proposition 30 “If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.”

This is one of the most important theorems in the whole of Number Theory and I call it the Prime Factor Theorem. What applies here is the special case when one at least of the two original numbers is prime — and a different prime from the ‘dividing number’.
But Euclid also needs to prove, or to have proved, that N really is, in our terms, the Least Common Multiple of p, q , r…. This he does in Book VII. Propositions 34 and 35 which detail the procedure for finding the Least Common Multiple, first of two numbers (Prop. 35), and secondly of three or more numbers (Prop. 36). As a special case, Euclid shows that the LCM of two numbers a, b that are prime to each other is ab and that the procedure can be applied as many times as we wish so that the LCM of a,b,c…. where a, b, c are all primes is abc… He is also scrupulous enough to show (Proposition 29) that a prime and any other ‘number it does not measure’ are prime to each other, which makes any two primes ‘prime to each other’.
Euclid does not generalize Proposition 14 to powers of these primes, i.e. to our pa qb rc…  though this extension is in effect covered by the propositions about Least Common Multiples VII. 34, 35 and 36 taken together with VII. 31 and 32.

The propositions concerning LCMs are very much what one would expect and are easily assented to. The same does not apply to the Prime Factor Theorem which is by no means ‘intuitively obvious’ nor especially easy to establish.
In modern terms Euclid’s proof of the Prime Side Theorem is as follows:
“Suppose p divides N (= ab) where p is prime, and p does not divide a.
Then (p, a) = 1 [VII. 29]
Let ab = pm = N where m is some number.
Then p/a = b/m  [VII. 19]
But since (p, a) = 1, p/a is in its lowest terms. Therefore m must be a multiple of a and b a multiple of p [VII. 20, 21].

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

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

This cumbersome statement shows the importance of algebraic notation which the Greeks did not have. Remember that Euclid is speaking only of ratios between hypothetical line segments, not of ‘rational numbers’ as modern mathematicians understand them. However, bearing this in mind, Euclid’s proof may be presented thus :
“Let ac/ad =  c/d = a/b
       But a/b = ac/bc
So ac/ad =  ac/bc
But this can only be true if ad = bc

Conversely, let ad = bc
     Then ac/ad = ac/bc
However, ac/ad = c/d
     Also ac/bc = a/b
Therefore a/b = c/d

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

                                                                                                                                                                                                                         SH       30/05/17

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



March 8, 2015

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

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

1 × 0 = 1 and more generally

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

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

1 × 1 = 2  and more generally

N × 1 = 2N  

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

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

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

N × m = m × N if and only if N = (m ─ 1)  so that
(4 × 5) = (5 × 4) = 20

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

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

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

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

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

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

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

SH   8/04/15


Even and Odd

March 5, 2015

Ο                                   ΟΟ
ΟΟΟ                            ΟΟΟΟ
ΟΟΟΟΟ                      ΟΟΟΟΟΟ
ΟΟΟΟΟΟΟ                ΟΟΟΟΟΟΟΟ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ο     ΟΟ     ΟΟΟ  ……..                 …..

Duplicating them gives us the ‘doubles’ or even numbers

Ο     ΟΟ     ΟΟΟ  ..….
Ο     ΟΟ     ΟΟΟ  ……

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

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

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

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

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

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

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

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

­­­­­­­­­­­­­­­­­­­­­­­          +       odd      even                        ×     odd    even  

       odd      even    odd                     odd    odd    even

       even    odd    even                     even  even  even


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

1.) the ‘product’ of an even and an odd number is even;
2.) an even number can by definition always be separated into two equal parts.

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

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


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