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Cosines and Sines

December 31, 2012

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

Our most basic ‘mathematical’ impressions are neither numerical nor geometrical. We situate an object (desired or feared) relative to certain familiar objects whose position we know or think we know. We position the new object by means of words such as ‘near’, ‘under’, ‘on top of’, ‘alongside’ and so forth. Babies and toddlers are not alone in working this way : even adults (including mathematicians and scientists) fall back on this method when confronted by a real life situation as opposed to a situation in a laboratory). If I ask where the nearest church or Post Office is, no one turns on their mobile phone complete with SATNAP (?) and tells me the latitude and longitude or even  the exact distance and orientation with respect to the present location. He or she would say, “Go right, then left, it’s the other side of the Church”, where the church is an easily recognizable landmark. If I ask where my pen or pencil is, I won’t be given the co-ordinates relative to the bottom left hand corner of the room; I will be told “It’s under the table”.
Now these relations are neither numerical or geometrical, even less algebraic : they are, as Bohm pertinently pointed out in an interview ‘topological’. Topology is that branch of geometry that deals with proximity and connectivity to the exclusion of metrical concepts. Perhaps we ought to make topology the first ‘science’ learned at school, though it probably does not need to be taught at this stage. Regrettably, topology is an exceedingly abstruse (and largely useless) branch of mathematics that only specialists study.
After ‘topological’ concepts, we have numerical ones, ‘How many of this?’ ‘How many of that?’  So-called primitive tribes get on perfectly well in their environment without numbers and arithmetic operations but the latter become necessary when we have a commercial and bureaucratic society such as developed in the Middle East in Assyria, Babylon and so on. Geometry was a somewhat later development, becoming necessary only when it was vital to assess accurately the areas of plots of land (for taxation purposes and also when State buildings became so large and complex that the coup d’oeil and rule of thumb of the craftsman/builder was no longer adequate. You can raise a pretty good straight wall without any calculation or mathematics but if you want to build a pyramid orientated in a particular way with regard to the stars, you require both a relatively advanced arithmetic and geometry.
After geometry came algebra (unknown to the Greeks) and algebra has so swamped mathematics that even Euclidian geometry, itself thoroughly idealized and remote from the natural world, seems homely and ‘concrete’ today since you can at least draw lines and, if you wish, even make spheres out of Blu-tack and play around with them.
A problem that was posed in the mathematical magazine M500 prompted some reflections on the gulf between geometry and Calculus and the even greater gulf between both of these and physical reality. The problem was about proving the basic limit     lim   (sin θ)/θ  = 1
θ → 0 . As a mathematical fundamentalist, I view sines and cosines as essentially ratios between line segments rather than infinite series.

lim sin θ)/θ  = 1  as  θ → 0

 Cosine M500 diagram

If  angle PÔD = θ  in radians, PE, the arc subtended by the angle θ, is and  > PD = r sin θ. Also, QE = r tan θ > rθ> PD

So   r sin θ  <  rθ  < r tan θ 

This inequality holds for any circle with r > 0 and all angles θ for which sin θ, cos θ and tan θ are defined. We take θ as positive (clockwise from the x axis) and, since we are only concerned with small angles,  0 < θ < π/2.

Dividing by r sin θ which is positive and non-zero we have

         1  <  θ/(sin θ)   <  1/cos θ
        1 has limit 1 since it is never anything else.
If we can show that the limit of 1/cos θ is 1 as θ → 0  the expression θ/(sin θ)   will be squeezed between two limits.
What can at once be deduced from the diagram is :

1.  r cos θ must be smaller than the radius and so, for unit radius,  0 < cos θ < 1

2. As θ decreases, cos θ increases, or “If f < θ, cos f > cos θ

1/cos θ is thus monotonic decreasing and has a lower limit of 1 which is sufficient to establish convergence. If one wants to apply the canonical test, we have to find a δ  such that, for any ε  > 0  whenever 0 < θ <  δ

1– 1/cos θ)  < e
With δ < cos–1 ((1/(1+ ε)) we should be home and, applying the ‘sandwich principle’ for limits, we have

lim        θ/(sin θ)  = 1
                 θ → 0

Turning this on its head, we finally obtain
lim   (sin θ)/θ  = 1
θ → 0

Note, however, that θ is the independent variable — sin θ depends on θ and not the reverse.  

        From here we can find the  derivative of sin θ  in a straightforward manner by using nothing more than the definition of the derivative and the ‘Double Angle’  formula sin (A + B) = sin A cos B + cos A sin B which can be easily proved geometrically for all angles A, B  where
0 < A < π/2 and 0 < B <π/2  (Note 1).

However, what’s all this got to do with the well-known power series?

sin θ = θ – θ3/3! + θ5/5!  – θ7/7! + …… 

Define a convergent power series f(x) with the convenient property that d2 f(x)/dx2 = – f(x)

        Setting A0 = 0, A1 = 1 and equating coefficients we eventually end up with 

     f(x) = x –x3/3! +  x5/5! – x7/7!  ……+ (–1)n x2n+1 /(2n–1)!…

But I’m none too happy about identifying the above series with sin x (and its derivative with cos x). For, if sin x and cos x are  geometric relations between line segments, when there is no triangle, there can be no sine or cosine. For me, geometric sin x is undefined at x = 0 (and likewise at x = π/2 &c.) although the limit of sin x as x → 0 is certainly 0. (It is distressing how often it seems necessary to point out, even to mathematicians, that the existence of a limit does not in any way guarantee that this limit is actually attained.)
All in all, I would feel a lot easier if the ‘sin x power series’ were derived (or defined) recursively term by term along with a demonstration that the difference between f(x) and sin x is always decreasing as we add more terms with limit zero. For sin x is, in  my eyes, itself the limit of a power series as n increases without bound, i.e. If  0 < x < π/2

lim  f(x) = x –x3/3! +  x5/5! …+ (–1)n x2n+1/(2n–1)!   =  sin x
n →                           

Realirty is Discrete

A more general point needs to be made. Practically all proofs in analysis and Calculus depend on the assumption that the independent variable (in this case the angle θ) can be made arbitrarily small. This is quite legitimate if we restrict ourselves to pure mathematics. But Calculus was invented by Newton and Leibnitz to elucidate problems in physics. Translated into physical terms, the basic assumption of Calculus, is equivalent to the presumption that space and time are ‘infinitely divisible’. But I do not believe they are for both logical and observational reasons. There is a growing (but still minority) view amongst theoretical physicists that Space/Time is ‘grainy’, i.e. that there are minimal distances and minimal intervals of time just as there are minimal transfers of energy (quanta). If this proves to be the case, Calculus and a lot else besides constitutes a very misleading model of a reality that is essentially discrete. The great majority of differential equations are, in any case, unsolvable analytically and increasingly the trend is to slog things out iteratively with high-speed computers taking things to the level of precision required by the conditions of the problem and then stopping. Dreadful to say so, but it looks like Calculus’s reign, like that of the dinosaurs, is drawing to a close and that the future will go to algorithmic methods, genetic or otherwise.

Sebastian Hayes

Note 1   Theorem  d(sin x)   =  cos x
                                          dx

Proof     By the definition of the derivative we have

d(sin x)   =  lim       sin (x + h) – sin x
            dx          h → 0                    h

=  lim       (sin x) (cos h) + (cos x)(sin h) – sin x
            h → 0                               h 

=  lim    (sin x) (cos h – 1)   +     lim   (cos x)(sin h)
            h → 0                  h                   h → 0       h

It is not obvious that (cos h – 1)/h  has a limit but (after consulting a Calculus textbook) I substitute
– 2 sin2 (h/2)  for  (cos h – 1)  and, employing the limit we have already found ((sin θ)/θ = 1) we have

lim    (sin x) (cos h – 1)   =  sin x  lim  {– (sin h/2)(sin h/2)}
        h → 0                 h                             h → 0                            h/2

                        =   sin x  lim  (– (sin h/2)) (1)    =  sin x × 0 = 0
                                      h → 0                        

        So all we have left is       lim   (cos x)(sin h)
                                                 h → 0                 h                

Employing once again the ubiquitous lim sin θ/θ = 1 as θ → 1   I end up with the desired
d(sin x)   =  cos x  lim   (sin h)/h    =   cos x
            dx                    h → 0                    

Our dreadful mathematical terminology

December 11, 2012

Open just about any book on numbers (in the English language) and you will come across the boastful claim that we have the best number system there ever has been, so good that, according to one author, it is inconceivable that it could be improved upon in any significant manner. Granted, this claim has some justification : we do indeed have a remarkably supple system of notation since we can cope with quantities as inconceivably large as the American deficit or quantities as inconceivably small as the diameter of a proton. However, “you don’t get owt for nowt” and the flexibility of this system of notation — which we Westerners did not invent but owe to the medieval Hindu and Arabic mathematicians — comes at a cost. As I pointed out in my article on Egyptian numerals, a child at the time of the Pharaohs could  see at a glance that the quantity we note as 100000 was larger than the quantity we record as 10000 since different picture signs were used for hundreds, thousands and so on. More serious still, no one stranger to our language and notation could possibly tell whether the quantity we call seven and record as 7 was larger or smaller than the quantity we call nine and record as 9. Indeed, a visitor from another world might deduce that seven was ‘larger’ than nine since it has more letters.
Indeed, when you examine the language of basic arithmetic as it is still taught in Britain and America, you wonder how anyone ever manages to become numerate at all! The thoughtful child or adult — not quite the same as the intelligent one — is immediately repulsed by the illogicality of our far-famed system (as I was). Apart from cyphered numerals such as 7 and 9 which are perhaps a necessary evil, there is the complicated and incoherent way we form our number words beyond ten.  Instead of ten-one’  we have eleven’ which has nothing to do with either ten or one. Naturally, it takes a non-mathematician to see this and point it out to the world :

“In English we say fourteen, sixteen, seventeen, eighteen, and ineteen, so one might expect that we wouild say oneteen, twoteen, threeteen, and fiveteen. But we don’t. (…) We have forty and sixty, which sound like the words they are related to (four and six). But we also say fifty and tirty and twenty, which sort of sound like five and three and two, but not really. And, for that matter, for numbers above twenty, we put the “decade” first and the unit number second (twenty-one, twenty-two), whereas for the teens, we do it the other way around (fourteen, seventeen, eighteen). The number system in English is highlym irregular.” (Malcolm Gladwell, Outliers p. 119).

“Ask an English-speaking seven year-old to add thirty-seven plus twenty-two in her head, and she has to convert the words to numbers (37 + 22). Only then can she do the math: 2 plus 7 is 9 and 30 plus 20 is 50, which makes 59. Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there. embedded in the sentence. No number translation is necessary: it’s five-tens-nine.”
“For fractions, we say three-fifths. The Chinese is literally ‘out of five parts, take three’. That’s telling you conceptually what a fraction is. It’s differeentiating the denominator and the numerator” (Karen Fuson, quoted Gladwell) (Note 1).

Division      Let us go further. What about this nonsense about “division by” in phrases like “ten divided by five ?  Who on earth is doing the dividing? ‘Five’? This is what we do when we carry out the operation but numbers can’t ‘divide’ other numbers. In reality we asre modelling a situation where we have       ▄ ▄ ▄ ▄ ▄     objects and we sort them into groups each containing  ▄ ▄ ▄ ▄ ▄   no more, no less.
▄ ▄ ▄ ▄ ▄

How many groups do we have?  □ □  using a different symbol. If you envisage division as the sorting out a mass of similar objects into bundles or bags, an activity that still consumes a lot of time and energy in the world today, division at once makes sense. We are a magpie species and seem to have an obsessive interest in collecting objects and storing them in containers : hence the importance of division in our arithmetic system, indeed I consider it more fundamental than adding.
The rule that you are not allowed to divide by zero, which is supposed to be so bizarre and/or profound, is imposed on us by the world we live in like all the rules of arithmetic. It is simply impossible to divide up a mass of objects into so many bundles that have strictly nothing in them. Division ‘by’ zero is not allowed, not because the mathematical establishment have decreed this to be so, but because it actually is the case that you can’t divide a quantity into bundles with strictly nothing in each bundle. What you can, of course, do is divide up a massive composite object into smaller and smaller equal groups (the ‘equality’ being tested by pairing off the groups member to member) and stopping when you get to a certain point. We might decide to call it a day when we reach, for example, the size of a bean, or, more likely in modern times, the size of a molecule.

Infinite Series   In a different website someone queried my claim that infinity is “everywhere present in mathematics and everywhere absent in the real world”. It is true that infinity is not directly involved in the construction of the natural numbers themselves 1, 2, 3…. but even here we are confronted with a series that can be ‘indefinitely extended’. And every time you carry out a division of 1 into 3 and, using a claculator, get 0.33333333333,,,,,,,,  you are in reality being confronted with a sum which goes on for ever, literally 3/10 + 3/100 + 3/1000 +  3/10000 +  …..and so on. We have a cake or anything you like that can be divided up and we make ‘three’ roughly equal portions or bundles. Nothing mysterious about that. Not only is it impossible for the most sophisticated machines to divide up an object into say a hundred billion bits, but this monster 0.33333333333,,,,,,,,  does not even ‘equal’ 1/3 exactly since the series never terminates whereas 1/3 does.  (The subject of ‘equality’ in mathematics will be dealt with in a suibsequent post.)
The child is quite right to reject the absurd adult rule that a wretched stream of figures that never ends represents the simple operation of ‘dividing an object into three roughly equal bits’. It is lamentable that in this technological era, most people actually believe that 0.33333333….. is somehow ‘truer’ than the banal and homely 1/3 because this is what you get when you feed in the numbers plus the division sign into a calculator. Entirely the reverse is true : a non-terminating decimal fraction like 0.33333333333……   does not correspond to any actual state of affairs or operation in the real world that ever has or ever will exist but division into three does correspond to actual operations with actual objects. We do not in our daily life use non-terminating decimal fractions and even quite rarely do we use proper decimals since 10 is such a wretched because it can only be divided into fives and twos (as opposed to 12 where we have quarters and thirds as well as halves).  In day to day activities we use an appropriate temporary base when the quantity to be divided is small, or use the very convenient base ‘hundred’ for this is what a percentage is, i.e. so many out of a hundred. As I said, the wonder is not that there are so few people who take to mathematics with enthusiasm in the West but that there are any at all given the linguistic and conceptual muddle of our number system and its operations.                                             SH 11 December 2012

_____________________________

Note 1      These excerpts are taken from the extremely interesting book Outliers written by a non-academic, Malcolm Gladwell, a book which I thoroughly recommend along with his other insightful books, Blink and The Tipping Point. I trust the author if he ever hears of this  site will, because of the nice things I say about his books, forgive me for not obtaining official permission to quote him which would be time-wasting if not impossible. SH  

Desert Island Numbers : The Number Ball, Number Marks and Number Bearers

November 15, 2012

(New readers may find it useful to read the preceding post first.)

The ‘Number Ball’

For my island paradise awaiting its Robinson Crusoe or Raffles I hit upon the idea of a clay ‘Number Ball’.  The  advantage of this device is that, apart from being portable, it allows one to get rid of a number once it is of no further interest and start again. A native might be sent, for example, by a chief to find out how many palm trees there were on a particular beach. Equipped with his Number Ball issued at the Central Data Hut he would arrive at the site and tear off as many little bits of clay as there were trees. He would report back to Central Office where the bits of clay would be recorded by an equivalent amount of scratches on a bone or knots in a cord, and he would then squash everything together to recover the original ball.
This system has an interesting feature : it is two-way  in the sense that you can use the same apparatus for recording data but can then ‘de-record’ (wipe out) the data to recover the original set-up and start again. This means, firstly, that there is no wastage. There is also something aesthetically satisfying about such a simple apparatus having an  ‘inverse’ procedure built into it : once you have completed your task, the Number Ball is returned to what it was in the beginning like the visible universe being absorbed back into the Tao from which it sprang.
Most recording systems do not have this feature : if you make a scratch on a bone you cannot ‘de-record’ without damaging the recording device, and crossing out something written with pen and ink is both messy and inefficient (in films a crossed out line often gets deciphered and leads to the conviction of a criminal). Destroying data has in fact become a considerable problem in modern society, hence the sale of shredders and civil servants’ perpetual fear of e-mails being picked up.
Clay Number Balls would be too messy for modern interior use but Blu-Tack is an alternative I have experimented with a little. There is, however, a certain risk of the little bits of clay or Blu-Tack sticking together and thus falsifying the reckoning.
The Number Ball is something of an anomaly mathematically and even philosophically. The object-numbers produced, i.e. the little bits of clay, do not strictly fulfil the requirement that number objects should not merge on being brought into close proximity — they can be made to merge or kept apart at will, so we have an interesting intermediate case somewhat comparable to that of semi-conductors.
Also, and this is more significant, the Number Ball is not, properly speaking, a number but rather a source of numbers, a number generator. In this respect it resembles an algebraic formula since the latter is not in itself a number (in any sense) but can be made to spew out numbers, as many numbers as you require. (For example the formula f(n) = (2n –1)  gives you the odd numbers (counting 1) if you turn the handle by fitting in 1, 2, 3….. for n e.g. (2 × 1) – 1 =  1; (2 × 2) – 1 = 3; (2 ×3) – 1 = 5 and so on.)
Yet a Number Ball is not a formula or an idea : it remains an object. Of course, one could also call a box of matches or a set of draughtsmen  ‘number generators’ but there is a difference here : the object-numbers are present in the box as distinct items (matches, counters) and are thus already numbers at least potentially, whilst bits of clay of Blu-Tack are not. A Clay Number Ball is actually a special type of generator since everything it produces comes from within itself and can be returned to itself. I have coined a term for this particular case : I call such an object an Aullunn. Although there are no complete Aullunn Generators in nature — not even, seemingly, the universe itself —   many natural phenomena approximate to this condition. The varied life in and around a pond to all intents and purposes emerges spontaneously ‘from inside’ and dies back into it each winter; though we know that without some interaction with the environment, especially with sunlight, no generation would be possible.

Surprisingly I have not come across any accounts of tribes using clay Number Balls.

Number Marks and Number Bearers

A very different method of producing a set of numbers is to have an object or substance which is not itself a number (or a number generator) but a ‘bearer of numbers’ : the numbers are marks on the surface of the number bearer or deformations of it. This system, which at first sight seems a lot closer to the written system we use today, is extremely ancient and possibly pre-dates the widespread use of distinct number objects. The markings on the Ishango Bone, which dates back to about 20 000 B.C., are thought by archaeologists to have numerical significance. Other bones have been found dating almost as far back with scratches on them that are thought by some to  indicate the number of kills to a hunter’s credit — one thinks at once of Billy the Kid, the “boy who had so many notches on his gun” (or was it Davy Crockett?).
The limitation of the notch system is that an incision is permanent which means that once the ‘number-bearer’ gets filled up it has to be stored somewhere or discarded like a diary. It thus tends to be used in rather special circumstances, either when one does not expect to be dealing with large quantities (rivals killed) or when one wants the information recorded to be permanent as, for example, in the case of inscriptions on State monuments.
Making charcoal marks on a wall, also an ancient practice, is ‘two-way’ in that one can rub out what one has written but the system would not be reliable for long-term recording of data because of effects of weather, flaking of surface &c. But numbers on a number bearer do not have to be marks : they can be reversible deformations, the prime example being knots in a cord. The great advantage of such systems is that, though very long lasting if the material is itself durable, the numerical data can easily be got rid of when no longer needed since knots can be untied. On the other hand because they take a lot of time to tie and untie, knots are unsuitable for rapid calculation and it would seem that the Inca State officials used quipus for storing data whilst they had some form of a counting-board system for calculations. Knots in a cord constitute a partial ‘two-way’ recording system — what is done can be undone — but they are at the same time quasi-permanent, indeed are in a sense the arithmetical equivalent of semi-conductors.

Knotted cords were in widespread use all over the world at one time and it is thought that mankind may even have gone though a ‘knotted cord’ era. Lao-Tse, the author of the Tao Te Ching (VIth century B.C.) who was a Luddite hostile to new-fangled inventions and to civilization generally speaks nostalgically of the days when mankind used knotted cords instead of written numbers.
In practice both systems are required, a ‘two-way’ number system which allows one to carry out calculations and then to efface them, plus a more permanent system which is used to record results if they are considered important enough. Thus the Incas (so it is thought) used quipus for permanent or semi-permanent records while they used stones and a counting board for calculation. The lack of a suitable ‘number-bearer’ to receive marks meant that inscribed number systems were a rarity until comparatively recently — baked clay tablets and papyrus were reserved for the bureaucratic elite and paper, a Chinese invention, only entered Europe in the latter Middle Ages and was expensive. Traders, even money-lenders and bankers, when they did  not use finger-reckoning of which more anon, used a two-way system, namely counters and counting boards, right into the Renaissance. The abacus, a two-way system, was never widespread in Europe for some reason except in Russia, but in the East has remained in use right through to modern times. The soroban or Japanese abacus is still used today and as late as the nineteen-fifties a Japanese clerk armed with a soroban competed successfully with an American naval rating using an early electronic  calculator. However, it must be pointed out that the Japanese achievement with the soroban depends on extensive practice in mental arithmetic rather than any particular merits of the device itself.
The drawback of a ‘two-way’ system such as an abacus where you erase as you go is that you cannot check for mistakes and even the result itself, once reached, has to be erased when we perform our next calculation i.e. there is no inbuilt recording element, no memory. But when there is no easy way of erasing we oscillate wildly between conservation and destruction : we tend to accumulate a vast amount of stuff, then periodically have a sort out and throw it all away, the pearls with the dross. Like most authors and mathematicians from time to time I have to tip out the entire contents of a large dustbin to search for a scrap of paper with some idea or formula written on it.
The principal drawback of a one-way semi-permanent system such as ink on paper is that it is incredibly wasteful and was until recently so expensive that the bulk of the world’s population, the peasantry, practically never used it and employed a pocket knife and a flat piece of wood to record data. Even in the computer era we still use the chalk and blackboard two way  system though the chalked notice-board in the hall of buildings or private residences — to mark who is in or out — which was once commonplace is now virtually a thing of the past. I myself buy rolls of lining paper (which I clip down over a table) partly because I like to have plenty of room for drawing and calculation but also partly for reasons of economy — you get a lot of paper in a roll compared to an exercise book. It is a sobering thought that no less than a hundred years ago Ramanujan, one of the greatest names in Number Theory, like so many other Indian mathematicians of the time worked with slate and chalk because he found paper too expensive. Although to my knowledge no one has suggested this, I would guess that this is one of the main reasons why his early mathematical writings are so hard to follow — he left no tracks because he generally just copied out his conclusions, then literally wiped the slate clean (Note 1). To many people the results seemed to come from nowhere and indeed he was often incapable of explaining how he got them.   (Ramanujan lived a century too early : today we have an improved ‘chalk and board’ system, the Whiteboard. At last marks can be easily erased without mess. I use large boards everyday and have somewhat moved on from lining paper to a more up to date recording system.)

To be continued

Note 1 A brief article on Ramanujan “Is there a Ramanujan problem?” reprinted from an edition of the magazine M500 can be found on my website www.sebastianhayes.co.uk

 

Desert Island Numbers : Introduction

November 4, 2012

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

What exactly are numbers? This is by no means an easy question to answer despite the fact that we use numbers every day of our lives in some form or other. Do we not, then, know what we are doing?
One way of answering this question is to launch into an investigation of ‘number’ as a concept or basic principle. This approach quickly leads on to some deep and perplexing issues that go back at least to Pythagoras and are still with us today (though mathematicians try to avoid tackling them if they possibly can).
At the opposite extreme, we have the Set-theoretic approach according to which ‘numbers’ (integers, rational numbers, irrationals &c. &c.) are just ‘things’ that emerge as just one application of the six or seven basic Axioms (preliminary assumptions) of Zermelo-Fraenkel Set Theory. This way of proceeding avoids metaphysical speculation altogether but at considerable  cost : the end products are not recognizable as numbers. Moreover, no one ever learned mathematics this way and most likely never will.
I favour a more practical approach. Let us ask ourselves why mankind ever bothered with numbers in the first place? Do we really need them, and if so what for? Essentially, we need them to ‘represent’ or ‘stand in for’ certain objects, i.e. numbers are ‘symbols’ in the straightforward sense that they ‘stand in’ for something else, something they are not. So why bother with symbols? Why not use the real thing? The answer is that the real thing may be far away, may be too heavy to carry about, too small to see without a microscope and so on. It is frequently not practicable to deal with  the real thing, especially if you want to manipulate it in various ways, make it larger, smaller, join it to something else and so forth. So we employ a substitute which represents it.

     Historically, numbers did not evolve as the result of philosophic speculation or as an intellectual pastime. Arithmetic was developed for mundane practical reasons : numbers and operations with numbers were required for trade, stock-taking, taxation, carrying out censuses, assessing military strength and a host of other unromantic administrative tasks. Innumerable tribes got along pretty well without much of a number system at all — sometimes nothing more than ‘one, two, three’. It was the large, centrally controlled empires of the Middle East like Assyria and Babylon who developed both writing and arithmetic. The reasons are pretty obvious : a hunter, goatherd or subsistence farmer in constant contact with his small store of worldly goods does not need records , but a state official in charge of a vast area with varied resources does.
After cogitating about number for some time and not getting very far, I set myself a mind-experiment. I imagined myself marooned like Gauguin in a pre-mathematical society and asked myself the question: What exactly do you need to make a workable set of numbers? What are the minimal requirements?  And the answer is : all you need is a set of more or less identical portable objects that do not merge or stick to each other when brought close together. Before being ink marks on paper or dots on a computer screen

mankind’s numbers were objects — pebbles, shells, twigs, knots in a cord, things you can touch and handle.

Parable of the Goatherd

  Consider an illiterate goatherd such as existed in many parts of the globe until a few decades ago, and     possibly still does in very remote parts. He brings in his goats each night and leads them out through a gate each morning. On the right hand side of the gate is a pile of stones and as he lets each goat through the gate he shifts a stone from the right hand pile to make a similar pile on the left. One goat, one stone.
At the end of the day he lets the goats back in and shifts a stone back to the right hand side of the gate (right hand from the inside but on the goatherd’s left side if he is coming back.) If there are any stones remaining when the last goat has been ushered in, he knows there are goats missing — or at least one goat missing.
Does our goatherd know how many goats he has in his herd? In our sense of ‘how many’, perhaps not. In the past the society he lived inmay not even have had enough spoken or written words to represent such a quantity, at any rate if he had a sizeable herd. Innumerable tribal languages had no words to express quantities beyond our ‘forty’ and in many cases, incredible though this seems, the vocabulary of number was limited to the equivalent of our ‘one, two, three’ where ‘three’ had the meaning ‘inconceivably large’, ‘not numberable’  —. Karl Menninger, in his wonderful book, Number Words and Number Symbols, cites the true story of a venerable South Sea Islander who, being asked how old he was, answered, “I am three”.
But in fact our goatherd is not innumerate, he does have a set of numbers, his pile of stones. The stones are his numbers. If asked how many goats he had  in his flock, he would probably indicate with his hand the pile of stones on the right side of the gate. And if asked at the end of the day how many goats were missing, supposing some were missing, once again he would indicate the stones left. It is as ridiculous to suppose that numbers must be marks on a piece of paper or pictels on a screen as it is to suppose that poetry has to be words on a page.
 So-called ‘primitive’ peoples used shells, beans or sticks as numbers for thousands of years and within living memory the  Wedda of Ceylon carried out transactions with bundles of ‘number sticks’. Although the development of a centralised imperial state apparatus generally gave rise to written  number systems, this was by no means invariably the case. The state officials of the Inca of Peru managed a vast empire without any form of written records : they used the quipu system where knots in coloured cords served as numbers. And the Yoruba officials of the equally extensive Benin empire in Nigeria performed quite complicated additions and multiplications using only heaps of cowrie shells.

Number Objects and Object Numbers

 I imagine myself, then, Robinson Crusoe-like, looking for a set of objects which are to be the basis of a workable number system which I can use myself and, possibly, introduce to the inhabitants of the island supposing there are some and that they are as yet innocent of numbers. What criteria are going to influence my choice?
Firstly, it is important that the objects chosen should be more or less identical since I have already decided that the basic principle of number is that individual differences between objects do not matter. My ‘one-object’, whatever it is, is going to be used to represent indifferently a tree, a fish, a man, a god, indeed anything at all provided the ‘thing’ I want to represent is singular, is a ‘one’. If I used different number-objects to represent different objects there would be no net gain — I would soon need as many object-symbols as there are objects.
Secondly, since there are a lot of objects in the world, I need a plentiful supply of numbers to represent them, so my chosen ‘one-object’ must be abundant. Alternatively, if I am going to make my own numbers I require the raw material to be abundant — wood for example — and the manufacturing process to be relatively rapid and easy so that I feel I can always make more numbers if I run out of them.
Thirdly, the chosen ‘one-object’ must be portable and to be portable must be fairly small and light. For, once again, if I have a stationary set of numbers there is little net gain: one of my main goals in developing numbers is so I can move around to assess numerically a distant clump of trees or a distant village. Although in special cases such as censuses and elections  the (human) objects do actually come to the numbers — come to a place where they are numbered — it is generally necessary to take the numbering apparatus to the objects and, in the very important case of spying, this is essential (in the days before long-range surveillance was feasible).
Fourthly, it is essential that the number-objects (or object-numbers, the terms are perfectly equivalent) do not merge or adhere to each other when brought into close proximity. It must be possible to make the numbers into a group while the objects remain distinguishable whilst in this group. Why is this important? Because this is the commonest set-up  we shall be modelling numerically. If we were regularly confronted with entities that flowed into each other, fused, only to separate a little later,  we would need  to introduce this feature into our mathematics But we spend most of our lives amassing objects, removing them from one group’s ownership to another’s (commerce), shifting them from house to house or port to port and so on. And the majority of these objects do not merge when brought together — even liquids are transported in containers and so function numerically as solid objects.
Other requirements are that the ‘one-object’ be durable, easy to see and can be easily held or placed on the palm of the hand. Also, we do not want the number-object to have the power of locomotion or it might move off before we have finished the counting! This more or less rules out living things or at any rate mammals as being suitable numbers though human beings in very special circumstances (soldiers, prisoners of war) have been used as numbers of a sort, for example to make a rough guess at the enemy’s strength on the basis of terrain occupied and suchlike cases.
All these requirements make the choice of a number object by no means so simple as it might seem at first sight. Grains of sand are abundant  but not easy to see, grains of salt stick to the hand. Beads are about the right size and are today cheap but they would have been luxury articles on a desert island in the past. Also they tend to roll around as do marbles which is why beads were threaded onto wires in the usual ordinary abacus while marbles  were confined to grooves in the case of the Roman abacus (see right). The most suitable objects, at any rate for a  rural society, turn out to be exactly the ones actually used by tribes : shells, beans and sticks. Shells are abundant, light, portable, and can be neatly stacked into piles. Sticks have the additional advantage that they can be split in two and so, if one runs out of numbers on a field trip, one can make more numbers on the spot.
               The question is not for me entirely an academic one as I have been looking for suitable object-numbers for some time now (in order to practise concrete arithmetic) and haven’t found an ideal choice yet.                        Draughtsmen  stay flat and stack up beautifully if they are of the old type with rings on top, thus allowing for the representation of ‘powers’. They also have the advantage of coming in two colours though it would be even better if they were a different colour underneath so that in order to ‘change sign’ you could just turn a draughtsman upside down. But  they are not abundant : one set of draughtsman is not enough and if you buy different makes  you find they don’t stack up properly.
Coins are feasible numbers, especially ones with holes in them like old Chinese ones since they can be threaded together. The counters used on counting boards in markets and even banks right through to the seventeenth century were virtually indistinguishable from coins. Roman calculi, of glass or stone, were unmarked but from the Renaissance onwards counters were elaborately decorated.
Matches are not at all bad, especially outsize ones like Brymay Long Matches and you can even represent positive and negative quantities if so inclined by using ones with brown and red heads — though the latter are becoming scarce now for health and safety reasons. Number sticks, painted red and black, were used in China centuries before ‘double-entry’ book-keeping became current in Europe though the meaning was the reverse, black for negative, red for positive.            To be continued

Note 1

Is Mathematics a free Creation of the human mind? : Arithmetic and the Natural Numbers

October 28, 2012

 In principle the whole of contemporary mathematics can de deduced from the six or seven basic axioms of Zermelo-Fraenkel Set Theory.  No one, of course, ever learned mathematics that way (including Zermelo and Fraenkel) and doubtless no one ever will.
As far as we can tell, mathematics did not evolve as the result of philosophic speculation or as a formal exercise in symbol manipulation. It was the large, centrally controlled societies of the Middle East, Sumeria, Assyria and Babylon in particular, who developed both writing and numbering (2). Why? Their reasons are pretty obvious: a hunter/gatherer, goatherd or small farmer who is in constant contact with his small store of worldly wealth does not need much of a number system, but a state official put in charge of a vast area with varied resources does (3). Arithmetic was invented and rapidly brought to quite an advanced level for mundane and very unromantic reasons : it was needed for stock-taking, censuses and above all taxation. Geometria, literally ‘land measurement’, was developed by the Egyptians for similar reasons : it was found necessary to assess accurately the surface area of very dissimilar plots of land bordering the Nile so that the peasants working these plots could be taxed more or less fairly. It was only much later that the Greeks turned geometry into a recondite and stylish branch of higher mathematics.
J.S. Mill, almost alone amongst ‘modern’ writers on logic and mathematics, took a pragmatic view of arithmetic. “’2 + 2 = 4’ is a physical fact”, Mill dared to write in his Logic ¾ for which he has endlessly been ridiculed since by the likes of Frege, Russell and countless others. Strictly speaking, Mill is wrong. ‘2 + 2 = 4’ is not the alleged fact but the symbolic representation of the alleged fact ¾ but this is splitting hairs. What Mill meant is undoubtedly correct, namely that ‘2 + 2 = 4’ is a faithful representation of what happens when you take //, or ‘2’ objects and bring them together with another // objects, making up a group of //// or ‘four’ objects. Does anyone seriously doubt that this is what happens?
‘1 + 1 = 2’ is untrue if we are dealing with entities which merge when they are brought into close proximity. For droplets of water ‘1 + 1 = 2’. Droplets of oil are a little more complicated since I have it from a physics textbook that, if you keep on adding oil, drop by drop, to a blob on a sheet of water, the original blob eventually separates into two blobs. There is thus an upper limit on n in oil-droplet arithmetic. For the limiting value N, when     n < N ‘1 + n = 1’, but if n ³ N, ‘1 + n = 2’.
In cannot for the life of me see that ‘1 + 1 = 2’ is a ‘truth of logic’ as Russell and Whitehead consider it to be. If it were to be so considered, then we would have the undesirable situation where two incompatible statements were both ‘logical truths’ ¾ since ‘1 + 1 = 1’ is just as valid, merely less interesting and fruitful.The fact of the matter is that each statement is true in the appropriate context, that is all there is to it.
However, this does not mean that our elementary mathematics is a ‘free creation’ or that the rules of arithmetic we have are completely arbitrary. They apply exactly to objects that can be combined without merging : if they did not so apply, we would disregard them and use other ones. This has nothing to do with whether or not our rules of arithmetic can be deduced from the Peano Axioms : Nature did not consult Peano in the matter.
As Mill correctly said, it is a matter of fact, and not of logic, that if you have, say, a collection of stones, say  ¢¢¢¢¢¢¢¢¢¢¢   and you are told to put them into containers 5 that have room for ¢¢¢¢ only, you will need 555 containers, no more, no less. In our rather muddled terminology, ’12 divided by 4 gives 3’ (it would be better to say ’12 divided into 4 gives 3’).
Theorems of so-called elementary Number Theory are not only ‘provable’ in the pure-mathematic sense, but in the many instances actually testable, i.e. they pass the Popperian test for empirical disqualification. For example, if I read in a textbook that a pyramidal number with base 24  is also a square number I can check whether this is the case by building up a pyramid on this base and then flattening the whole lot and making them into a square (which turns out to have side 70). Obviously, I am not going to test such statements most of the time since I have confidence that the normal rules of arithmetic are soundly based, but at least I know I have this possibility. It will be objected that, when dealing with general statements which apply to an unlimited number of cases, I cannot test them all. This is indeed so but what I can do is examine a particular case and then convince myself that what makes the proposition true in this case is not something specific to the particular case, but which will extend to all other cases of this type. Such a procedure does not cover non-constructive proofs of theorems which provide for the ‘existence’ of such and such a number without giving any indication of how such a number can be produced. However, such proofs do not have the persuasive power of constructive proofs and have rightly been treated with suspicion by many mathematicians. The proofs given in Euclid Books VII, VIII and IX, which are devoted to Number Theory, on the other hand are strictly constructive.
Moreover, theorems about the so-called ‘natural numbers’ are, in general, not just ‘roughly true’, ‘true in the limiting case’, ‘statistically true’  and so on, but are either completely true or wrong. Such a situation can only make practitioners of other sciences gasp with envy. Aristotle’s physics, in its day no mean achievement, had to give way to Newton and classical mechanics has had to give way to Quantum Mechanics. But the substance of Greek Number Theory has, apart from a greatly improved notation, scarcely changed in twenty-three centuries. It is in this sense that we should interpret the oft-quoted statement of Gauss to the effect that “Mathematics is the Queen of the Sciences and Number Theory the Queen of Mathematics”.
And the reason for the much greater sureness of results in Number Theory is that numbers (whole numbers) are far more basic than everything else. The distribution of the prime numbers is a fait accompli which does not depend on a formula, even if one could be found, it is ‘what it is’ and  follows ineluctably as soon as we have something which is repeatedly divided up into little bits. Physicists have imagined all sorts of universes where not only the basic constants but many of the ‘laws’ themselves would be different, but it is impossible to imagine a physical world where, for example, Unique Prime Factorisation does not exist ¾ if you don’t agree try to imagine one. The divisibility properties of numbers are ‘given’ and no intelligence is  involved : Nature does not know and does not need to know what quantities can be divided up in such and such ways. Perhaps, the same goes for so-called physical laws

To be continued      S.H.  28/10/12

 

Is Mathematics a Free Creation of the Human Mind?

October 25, 2012

In these posts I defend the commonsense view that mathematics originates in  our sense perceptions and is neither a free invention of the human mind nor a window on the eternal. Although a certain part of mathematics ─ essentially that based on the properties of the ‘natural numbers’ ─ is empirically based, this is by no means true of all branches of mathematics.  Modern society has been unwise in accepting at face value the exaggerated claims made by late nineteenth-cnetury and  twentieth-century mathematicians about the origin and nature of mathematics and which have now become unassailable dogma.

1.Introduction  

Prior to the nineteenth century, practically all mathematicians in the West thought mathematics dealt with the ‘real world’, was indeed the surest way of getting a handle on it. In consequence, no very great distinction was made between pure and applied mathematics and the greatest names, Newton, Gauss, Euler, worked indifferently in both spheres. The important theorems were ‘ideas in the mind of God’    : some of these ideas the great Geometer had employed in the natural world, others He had seemingly kept up his sleeve.

Conversely, what made little sense in physical terms was treated with scepticism. It was a long time before negative numbers were accepted, let alone the square root of (−1) and Newton seems to have had serious doubts about his own greatest invention, the Calculus, which is why he returned to more cumbersome geometrical methods in his Principia.

All this changed dramatically during the second half of the nineteenth century : more or less at the same time as Gautier in France and Oscar Wilde in England launched the ‘Art of Art’s sake’ movement. A handful of analysts, especially in Germany, decided that mathematics was a law unto itself and could be developed along ‘pure’ or abstract lines, with no reference to material reality whatsoever.

The so-called ‘Formalist’ approach remains orthodoxy today. Ask any pure  mathematician and he or she will tell you that “mathematics is a free creation of the human mind” (Dedekind)  : any attempt to tie mathematics down to physical reality is met with incredulity and insdignation. The weakness of this position is, of course, that it makes the predictive power of mathematics utterly mysterious, indeed incredible. As one author put it disingenuously, “Mathematics is an abstract construction of the human mind, and it is really quite miraculous that it should have an immediate and practical application to the real world” (Backus, The Acoustical Foundations of Music).

Actually, this is not the whole story. As Davis and Hersch shrewdly point out (Davis & Hersch, 1983), the typical contemporary mathematician is two things at once, for he or she is both a Formalist and a Platonist. In the quiet of the study, mesmerised by the dancing  symbols, he feels he is looking in on ultimate reality. This was indeed what Kepler and Leibnitz thought when they stumbled upon their greatest formulae and theorems, and, within the context of rational deism, it made quite a lot of sense. But today? In the present agnostic or fiercely anti-religious climate of the West, such feelings hardly pass muster. So the contemporary mathematician takes the easy way out : if challenged to give some sort of rationalisation of his or her transcendental view of mathematics, he puts on the Formalist mask which has at least the merit of keeping the Empiricist at bay.

I believe that both these positions, the Formalist and the Platonist, have very little to commend them ─ except to persons who are specialist mathematicians (and even then). If we take the commonsense view that mathematics is rooted in our experience of the real world there is nothing mysterious about its success as a symbolic model and prediction system, quite the reverse. As to the more fanciful inventions of modern mathematicians, they should be classed more as art than as  science, and judged accordingly, i.e. on aesthetic, not empirical, criteria. To be sure what starts as science fiction can sometimes turn out to be little short of the truth ─ but there is certainly no guarantee that this will come about.

Euclidian Geometry

Euclid’s Elements is not a Formalist work. The author always has his eye on the actual construction of shapes and figures : the very first Proposition of Book I is “[how] To construct an equilateral triangle on a given straight line. The famous joke proof that every triangle is isosceles is obviously fallacious if you actually try drawing the figure (as Greek geometers would have done) since the two lines cannot possibly meet inside the triangle as the ‘proof’ requires.

Should one, then, view Euclid as a compendium of verifiably correct statements about the physical world? This would be going too far. If you actually measure  the angles in a triangle you will almost certainly find that they do not add up to 180°. An eleven year old girl once came up to me after a lesson to tell me this with deep indignation, accusing me of having told an untruth. I asked her what she had got and she replied, “179 and a half”3. And, as a Sceptic philosopher objected even in Plato’s time, if you draw a tangent to a circle you will find that it certainly touches it at more than one ‘point’.

Absolutely straight lines do not exist in Nature and raindrops are far from being true spheres. Euclidian geometry is a grid that we impose on the real world : most of the time Nature does not bother with it. The twisted tangles of branches on a tree haven’t the slightest resemblance to the clearcut shapes of school geometry and even fractals are far too regular. All this is a little worrisome since the basic propositions of Euclid must be true ─ after all, we can prove them! It is here that Platonism provides a very influential and, at first sight, satisfying way of resolving the problem. Euclidian geometry deals with an ideal world which exists independently of the actual world and is ‘more real’ than it. In such a world straight lines really are straight and tangents touch the circumference of a circle at one point only. Everything down here is an imperfect copy of these timeless Forms, or ‘Ideas’ 4.

Personally, I consider Platonism to be a delusion, though admittedly a seductive and historically very important one : the ‘real’ is best defined as what actually occurs  not what is supposed to occur or exists in a mathematical Fairyland. So how do I view Euclidian geometry? As an ensemble of true, i.e. empirically verifiable propositions, but only in a statistical sense. Speaking rather pedantically, one could put it this way:

“If you take the angle sum of any triangle drawn on a flat surface, the mean will be 180° (better, half a full turn) approaching 180° in the limit as n, the number of trials, increases without bound and p, the resolving power of the measuring device likewise increases.”

So at any rate, I believe ─ I have not put the proposition to the test. Gauss, the foremost mathematician of his time, was sufficiently bothered by the question that he took the trouble to work out the sum of the three angles of a triangle formed by the peaks of three mountains in the State of Hanover (using surveying data he had collected himself). He did not obtain 180° but was relieved to find that the discrepancy was “within the limits of experimental error”. It is rather pathetic to consider that the same concern with ‘reality’ exhibited by the ‘Prince of Mathematicians’ would be considered ludicrous today and, indeed, the contemporary tutors of such a person would probably advise him or her to drop mathematics in favour of biology or mechanical engineering. (I very much doubt that my eleven-year old empirical philosopher actually went on to study mathematics, or even physics.)

Clearly, it would be insufferable to have to translate all the theorems of Euclid and, for that matter, Newton’s Principia, into the language of statistics : it is not only convenient, but,  practically speaking, mandatory to formulate geometry  and mechanics in ‘absolute’,  not relative, terms. This does not make beautiful ‘perfect spheres’ or absolutely frictionless pulleys real, however, any more than Shakespeare’s definitive study of a reluctant revenge hero makes his Hamlet a real person. If there is a ‘hierarchy of realism’, it works the other way round : by my book it is the actual people, events and imperfect shapes that rate higher in the scale of what is, and the actual gives rise, via human invention, to the ideal, certainly not the other way round. In chemistry, we need the eminently Platonic concept of an ‘ideal gas’ (one that obeys Boyle’s Law exactly), but there are no such gases, nor are there likely to be any.

On the other hand, Euclidian geometry is not a free creation of the human mind : if it were, it would hardly be much use in industry. The subject matter of geometry is certain spatial properties which actual objects, to a high degree of approximation, are found to possess and, subject to  certain conditions, seemingly must possess. There is thus nothing mysterious about the successful application of Euclidian geometry to technology, nor anything paradoxical about Euclidian geometry not being the only pebble on the beach. If there is only one Set of Ideal Forms and only one prophet, Euclid, then there is a problem for the true believer. But Euclidian geometry is only one — though by far the simplest and most useful — out of several ‘geometries’, each of which have their appropriate  spheres of application such as spherical geometry, hyperbolic geometry, fractal geometry and so on. The key proposition of hyperbolic geometry, that the angle sum of a ‘triangle in space’ is less than 180°, started off in Riemann’s imagination, but is currently taken perfectly seriously by the practising astronomer because it gives rise to slightly better predictions than Euclidian geometry. This is not a matter of matrhmatical fashion but of empirical fact :  I believe that laser beams have been used to demonstrate that the sum of the three angles of a triangle in space is somewhat less than 180° as Einstein’s Theory requires.

To be continued      S.H.  23/10/12

Note: The image is Vivacity by Jane Maitland

Subitizing : Straight Quantitative Assessments

April 16, 2012

The natives on my imaginary island are innocent of numbers, possessing at most a handful of number words. So, how on earth did they manage to get on all this time without them? Is my claim that you don’t need numbers in a non-commercial society an adequate explanation?
The answer is that, to judge by first-hand accounts of similar peoples, my hypothetical natives undoubtedly had an  awareness of quantity which was in certain respects far more refined than ours. But this awareness of comparative size and quantity was not numerical in the accepted  sense of the word because it was not linked to verbal or recorded symbols.1 Man is the symbol-making animal par excellence and symbol-making — or symbol-mongering — does seem to be one of the main features which distinguishes the human species  from other mammals. Animals do not of their own accord seem to use symbols and the fact that apes can be taught to use them by humans is really neither here nor there1. Also, practically all societies at the hunting/food-gathering stage did not develop numerical symbols to any great extent though they did develop highly complex languages, myths, rituals and the like and in many cases produced superb paintings and sculptures2. But this does not mean that  had no awareness of quantity nor any way of expressing differences between different quantities, indeed they would not have survived if they did not have such a sense.
It is well known that a teacher entering a classroom often ‘instinctively’ notices if a single pupil is missing. In this case the teacher obviously  does ‘know about numbers’ but he or she does not count the number of pupils present, an extremely laborious task, and would certainly be unable to assess correctly the number of pupils present in a classroom just by glancing at them2. The point is that, if we are familiar with a particular quantity, we notice ‘immediately’ very small deviations from it — one item missing, one or two extra items. This sense is extremely primitive and at the hunting stage was much more developed than it is now3. Holmberg, in his field study of the Siriona of Bolivia writes, “A man who has  a hundred ears of corn hanging on a pole….will note the lack of one ear immediately” (Closs, p. 17). A  missionary to the Abipones, another South American Indian tribe, witnessed a mass migration.  “The long train …..was surrounded on both sides by countless numbers of dogs. From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected again” (Menninger, p.11). Yet these Indian tribes had only a handful of number words — for the Siriona “everything beyond 3 was etubiana, ‘much’, or eata, ‘many’ ” (Closs).
Note that the missionary himself describes the pack as made up of ‘countless numbers of dogs’ . Even if the numerate white man had laboriously counted the pack the previous night at the camp site,  this numerical knowledge would have been worse than useless once the train was on the move — far more useful was the innumerate  but finely tuned sense-perception of the Indian. Although as far as I know no extensive studies have been done on this, it would seem plausible that this ability for ‘direct quantitative assessment’ interferes with ‘number sense’ and vice versa, i.e. those who are good at the one are not good at the other. Certainly, Gay and Cole note how much better the African Kpelle were at ‘quantitative assessments’ such as evaluating how many cups of rice would fill a certain bowl than educated American Peace Corps volunteers4. This would explain the extreme resistance of primitive tribes to acquiring the white man’s numerical skills : the Spanish found they had to impose European number systems by force (backed up by threats of damnation) and in the priests strictly forbade the Indians to use their own number systems if they had them.
Even today when everything is measured, everything has a number attached to it, we still use non-numerical assessment of quantity  referring unknown or little known quantities to some well-known amount which is, at least to us, standard. A farmer assesses the size of a farm he is visiting not in acres but by comparing it to the size of his own and might well assess the size of a meadow in terms of the number of cows it would feed, a form of measurement which allows him to take into account qualitative features such as the lushness of the grass, features which the numerical assessment ignores completely. Here land is assessed in terms of animals since that is what the farmer is above all concerned with. In a converse ‘cross-measurement’ the Abipones assessed the size of a herd of horses “by indicating how much space the horses occupied when standing next to each other” (Menninger). Distance to a certain place was not measured in abstract SI units that one cannot visualize or ‘feel’ with one’s body, but in terms of the journey time it took to reach them : such and such a place was “two days sleep away”. This is a far more practical and succinct evaluation than the modern equivalent which might well be, “It’s 50 kilometres away but you’d probably have to make a detour……”
In the world of a hunter/nomad variable quantities are related to a known standard quantity just as they are today with our SI units : the difference is that the standard quantity is, for us, purely objective, remote from our experience, whereas for the hunter/nomad it is a personally experienced quantity that he carries around with him all his life. Even in the very different world of the agriculturalist standard quantities were still connected up to daily life — an acre was originally supposed to be the amount of arable land a ploughman could plough in a day. However, in the strictly scientific metric system introduced (or rather imposed) by Napoleon   every trace of subjectivity is deliberately removed and the basic unit of distance, the metre, is no longer related to the human body (cubit, foot &c.) but is fixed as a fraction of the diameter of the Earth.
It is because it is so important psychologically (and hence practically also) to have a set of standard quantities well fixed in our minds, that there has been such entrenched resistance right across the board to the introduction of the metric system. Prior to the change in the law, people in this country had their own Bureau International de Mesures in their minds and ‘at their fingertips’.
People of the present society, so ultra-educated and urban, and especially mathematicians themselves who have a vested interest in their ignorance of other ways of living, fail to realize how utterly irrelevant and indeed counter-productive purely numerical knowledge often is. There is  nothing at all absurd about a native mother not knowing numerically speaking how many children she has got — and since most South American Indian women presumably gave birth to many more than three children this state of affairs may well have existed throughout the entire history of ‘innumerable’ tribes incredible though this may sound. For why should she know ‘how many’ children she has when she knows them all individually? Even if she were more numerate than the Abipone she may still not have known because she never saw fit to ask the question ‘how many’. In a famous lawsuit of the East Cree against developers, a lawyer representing the latter thought he had made a great point when one of the Indians was forced to admit he did not know how many rivers there were in the disputed territory. As Denny says, “The hunter knew every river in his territory individually and therefore had no need to know how many there were” (in Closs, p. 133). I am, as a matter of fact, not absolutely sure how many rooms there are in the (small) house where I am writing this chapter. Do you know how many chairs there are in the living room of the flat or house where you live?
As a rule we only bother to number and count things in quite special circumstances and when we know things or people very well we precisely abstain from counting because it is not only needless but may actually give offence — one does not, for example, count the number of paintings on the walls of a friend’s house. Of course, part of this movement from knowing things individually to numbering or counting them is a consequence of mass production. The standardized products of an industrial society cannot be known individually — they are depersonalised in much the same way as numbers are — so the only way to differentiate them is by giving them a label such as ’40-74-88’, or a bar code. This is basically the reason why people instinctively object to being given a number — rightly or wrongly they do not regard themselves as standardized social products.
‘Progress’ as it is generally conceived is a movement from the concrete to the abstract, from the qualitative to the quantitative, from the quantitative itself to the merely numerical  — is this an unqualified ‘good thing’? I hardly think so even within the bounds of the most abstract science itself, mathematics. To the contemporary pure mathematician any relation to the physical everyday world is regarded as a sort of pollution. Even numbers themselves have been replaced by letters and the reduction of everything to algebra even includes the visual and tactile science of geometry : in a modern geometry textbook there are scarcely any figures and we prove theorems by messing about with co-ordinates. Lagrange actually boasted of writing a textbook on mechanics (of all subjects) which “contained not a single diagram”. There is something not only misguided but positively insulting in such an approach.
However, this is to jump the gun by several millennia, I am currently concerned to note the transition from the unnumerical hunting/foodgathering society to the highly numerate State bureaucracies that developed in the Middle East during the second millennium BC.5 Viewed from a distance individual differences between entities tend to disappear: trees which seen  close up are clearly differentiable one from another, become so many dark shapes on the horizon and, as we move farther away still, are eventually reduced to a vague mass, the ‘many’ has become ‘one’.  The hunter/pastoralist, and to a lesser extent the farmer, is so close to the important items of his life, is so immersed in the concrete, the here-and-now, that numerical reckoning is both largely unnecessary and conceptually difficult. The chief of a tribe would   know every one of his male tribesmen personally whether by name or not, thus no need for counting. To the absolute despot who perhaps, like the later Chinese emperors never ventured outside the Forbidden City, his millions of subjects were not individualized but were simply a vague mass, the ‘people’. However, the State official who represented the Emperor was sufficiently distanced from the concrete to lose sight of individual distinctions but sufficiently aware of the need for accurate assessment of goods and services, to be in exactly the right position for a ‘numerical’ view of things. A centralized bureaucratic society at once gives rise to the need for numbering and arithmetic and makes the necessary awareness possible in the first place. As Denny puts it:

“The main condition under which arithmetical operations become useful is economic action at a distance. (…) The basic factors we have associated with the need for mathematics, increased alteration of the environment and increased dependence on others to perform specialized tasks, must have developed to the point where some people are specialized managers of the man-made agricultural system [and they are the ones] who direct the efforts of specialized workers.”                    (Closs, editor, p. 156)

It is worthwhile noting that commercial activity by itself, though it obviously does provide motivation for the development of arithmetic, is not in itself decisive. A good deal of trade until quite recently went on in terms of barter and even when ‘money’ was involved it was ‘concrete money’ not figures on a bank balance : medieval Japan used rice as a means of exchange and in Virginia during the seventeenth and early eighteenth centuries tobacco was employed. In Nigeriaquite extensive trade was conducted using bars of copper or iron and even bottles of gin as currency5.
When, however, we have a State currency and State controlled weights and measures imposed throughout a large area, we expect (and find) an accompanying development of sophisticated number systems and arithmetic operations as, for example, in Chinaunder the Han and later dynasties6. Arithmetic, along with writing itself, is the necessary and inevitable offspring of a centralized, bureaucratic society, the sort of society Wittfogel (who seems to have go  ne out of fashion) characterized as ‘hydraulic’, i.e. based on State supply of water and other essential commodities.6 This is so because such societies necessarily involve ‘government from a distance’ and ‘quantitative assessment from a distance’, the precise opposite of conditions operating in tribal society.
Apart from wordless quantitative assessments ‘primitive’ man was perfectly able to indicate quantity other than by using number words or number marks. It would be ludicrous to suppose that a tribe possessing only the number words “One, two, many” was incapable of distinguishing between different larger quantities. Asked by the explorer Oldfield how many persons were killed in a certain tribal battle, the tribesman “answered by holding up his hand three times” (Conant,  ). In such a case I doubt whether the tribesman was expressing our ‘15’ by finger counting : he was simply indicating a certain quantity ‘more or less’ equivalent to three times the fingers of one hand. In the majority of situations even today a rough assessment is adequate, indeed an exact numerical assessment often appears pedantic and thus objectionable. If asked how many people you were with in the pub last night you would perhaps say, “About half a dozen”. This is sufficient to distinguish the group from an intimate one consisting of two or three persons or an amorphous one consisting of twenty, and this is really all that one wants to know. One might call this ‘qualitative quantitative assessment’ and this is not a contradiction in terms. In the world of the hunter and foodgatherer distinction by type is so much more significant than distinction by number that even his or her quantitative assessments are largely qualitative. This is deeply shocking to the scientific mentality of today but the merit of such procedures is that they concentrate on what is important to the individual. They are in the current jargon ‘user-friendly’. I do not require to know the exact temperature it will be tomorrow — in fact I’d rather not know —  but it might be useful to have an idea whether it is likely to feel cold, or conversely whether it is likely to be muggy, because I can them dress up appropriately. The expected exact temperature does not tell you what you want to know though the qualitative additional comments of the human weather reporter may do.

Notes

 1 The only tolerably convincing case of ‘mathematics’ in the animal kingdom is the waggle dance of honey bees where a worker returning to the hive communicates information about the distance and direction of the food source by means of repetitive swaying movements. But see Animals as Mathematicians by Dr. Kalmus (Nature June 20, 1964  vol. 202). I would call most of the examples cited quasi-mathematical activity rather than ‘mathematical’ in the true sense but it depends how you define mathematics.

 2 No slur on these ‘primitive’ peoples (or for that matter animals) is implied. Animals and hunting societies don’t use numbers much because they don’t need them and can’t be bothered with them. Well-intentioned contemporary writers who feel uneasy about their own implied superiority fall over backwards to try to persuade us that these ancient peoples did perhaps ‘use numbers’ after all and that animal species did evolve language ‘after all’. All the evidence is to the contrary: ‘primitive’ societies used altogether different quantitative methods which it would be misleading to call numerical.
Like Jean-Jacques Rousseau, I am often tempted to think that civilization (and possibly life itself) is a mistake : certainly I do not judge individuals or societies according to their technological and mathematical development. One could make out a strong case for the claim that writing and numbering skills are intimately associated with militarism and exploitation : after all it was the expansionist societies in the Middle Eastthat developed both. Napoleon was the metrifier of Europe, founded the prestigious Ecole Polytechnique   and was no mean mathematician himself. Even today, America, though hardly a cultured country, is pre-eminent in mathematics. Amongst recent historians Innis is about the only one who sticks up for ‘oral’ societies as against tgose that employed writing. “[For Innis] writing, even before it was clearly mechanised, represented a mechanization of the spirit… Small was beautiful because it was built on a human scale of tongue and ear and living memory” (Godfrey, Foreword to Innis, Empire and Communication).

3  It has been shown experimentally that the maximum number of items a normal person can assess without counting them is about seven. Interestingly, the calculating prodigy Dase went up to twenty.

4 See Gay and Cole, The New Mathematics and an Old Culture, (New York, Holt, Rinehart & Winston 1967). Galton, a British nineteenth century explorer, remarks how the Damara people, a nomadic African tribe, were able to recall the faces of all the animals in a herd and thus detect not only how many animals were missing but the precise animal or animals (quoted in Zaslavsky, Africa Counts, p. 33).
Interestingly, the Kpelle were not so good at assessing numbers of people or huts in the village — probably because it was considered unlucky (or simply impolite?) to count people you actually knew. 

5 “Bottles of gin passed from hand to hand for years without having been opened, and might represent the whole wealth of a chief. Basden reports seeing huge collections of gin bottles — a record of past transactions.”  (Zaslavsky, Africa Counts  Lawrence Hill 1973).

6 We are in fact today far closer to such societies than we are to the medieval feudal society where the lord lived on the land or to the commercial, slave owning societies of Ancient Greece. AlthoughAthens at its zenith controlled a wide-flung maritime empire, it never developed a powerful administrative class such as existed inPersia andEgypt. The mathematics of Ancient Greece was primarily geometry and the Greek number system compares unfavourably with that used by the earlier Babylonians who were magnificent calculators. Although geometry is necessary for surveying (and surveying necessary for taxation of land), centralised bureaucratic societies tend to be ‘arithmetical’ rather than ‘geometrical’. The vision of the administrator is a numerical vision, everything and everyone must have its number. Distinctions of type which include shape and colour are altogether secondary. The first society to impose wholesale decimalisation wasChina which, from the T’ang onwards, was controlled by an official class to which persons of any social origin (except merchants) had access, at least in theory. And Chinese mathematics, though very advanced, was not very geometrical either. It is notable —  and alarming —  that whereas Greek higher mathematics translated numbers into shapes by evolving a sort of geometrical algebra, the last two hundred years have seen shapes  reduced to numbers in the arithmetic geometry of nineteenth century analysis and beyond.

S.H.   16 April 2012

 

 

Number : a ‘negative’ evolutionary step

March 27, 2012

It is an amazing fact that both mathematics and physics are distinguished by being ‘negative’ sciences. Out of all the information provided to us by our senses arithmetic rejects pretty well the whole lot, about 99%. Numbering is an informational minimum. We have something in front of us : living, dead, black, green, beautiful, manufactured, organic, it is all the same numerically speaking. There is not much further we can go except perhaps by making the most basic distinction of all, between ‘something’ and ‘nothing’.  As Piaget and Inhelder put it, “Number results primarily from an ignoring of differential qualities” (P & I, 105).¹
          To become numerical entities must first of all undergo two negative transformations: they must first of all be depersonalised, i.e. all distinguishing features such as shape, size, substance, origin &c. must be wiped out (in imagination if not in fact). Secondly, if we are dealing with a group, this group must be disordered, i.e. all relative positioning, left, right, on top, under, and so on must be eradicated as well. This is, incidentally, why Georges Cantor, the brilliant though misguided infinity lover, wrote two bars, symbolising a double negation, on top of the letter he chose to represent the cardinal number of a set, M.²
Those who adhere to a strictly Darwinist position and do not hesitate to apply it to human society, find themselves in something of a predicament when they attempt to explain the origin of numbers and arithmetic. For not only does a knowledge of numbering not confer any immediate evolutionary advantage  but actually goes in the opposite direction to what one would expect. Numerical status can only exist when type status is abolished, at any rate for the time being. But for animals the important distinction to be made in practically all circumstances is one of type not number. Friend or foe, comestible or poisonous, known or unknown, such distinctions are all-important for animal species and mistakes spell disaster and possibly extinction. As to numbering, provided one has a vague idea of quantity, enough to allow you to decide when it is, for example, better to flee rather than fight, there is little need to bother with it. What sort of thing is this? Is the question to be asked, not How many?  Imagine someone saying to you, “There’s three out there”. The natural reply is, “Three what?”
Much the same applies to mankind when living in a hunting/food-gathering society. Any effort made in the direction of increasing numeracy would not only confer no obvious advantage but might even be counter-productive by interfering with the mental procedures that were of proven use. Missionaries, explorers, colonists and the like found that not only did hunting peoples have a very rudimentary number system but strongly resisted the introduction of more complex number systems. This is usually dismissed as ‘cultural resistance’ to anything new but it may also have been motivated by fear of losing their own quite different skills. Whether this was the underlying reason or not, there can be no doubt that hunting peoples along with herdsmen and early agriculturalists had very little interest in numbers. Even such an advanced but still semi-nomadic people as the Jews whose wealth lay in their flocks had a grotesquely rudimentary mathematics when one considers their epoch making achievements in religious thought and literature, law, hygiene and so much else.
It is rather paradoxical that the species which has developed mathematics and mathematical physics so very far is not, naturally, very talented numerically speaking!  Even today, in our ultra-educated, ultra-urban society human beings perform very poorly numerically which is precisely why we rely so cravenly on the ubiquitous calculator. Guess how many daisies there are in the lawn or the number of gulls sitting on the beach and then go and count them if you can. You will not only find that your guess was out but was wildly out. Compare this with the absolutely phenomenal ability we have for recognising faces after decades or recognizing places we have only perhaps visited once in childhood. We retain this ability to make fine qualitative distinctions even though we no longer need it to the same extent as the nomad or goatherd of the distant past. It is ironically exactly those abilities that come most easily to us (pattern recognition) that we find it most difficult to ‘teach’ to computers.
Why did early man not develop much number sense? Clearly, because he did not need it and any expense of effort in that direction ran the risk of warping his judgment in more important matters. Most really effective memorising was, and to some degree still is, essentially visual. Most of the calculating prodigies themselves relied on visualization techniques, turning numbers back into ‘things’ or even into ‘people’ to make them more memorable. One method is to visualize a street until you know it off by heart and store bits of numerical data in each of the houses as if they were inhabitants. Advocates of high speed mental arithmetic, now hardly practised, recommend visually associating signs with others so that, when for example you are asked to add or take away you ‘see’ the result without performing any sort of calculation. Thus the juxtaposition of 7 and 5 will at once ‘call to mind’ ‘12’, or alternatively ‘2’. In effect such techniques are reducing arithmetic, which we are not naturally good at, to visual association and pattern recognition which we are good at. In a mathematical era which concentrates on  deduction — a strictly non-observational process — pattern recognition is not prized and is even distrusted in mathematics as likely to lead to false judgment and rash hypotheses. The ‘classic’ mathematicians, up to the mid nineteenth century and the advent of analysis, on the contrary prized observation and developed it, especially the incomparable Euler.
So how and why did numbering and arithmetic ever develop at all? The answer is that, compared to all sorts of other arts and crafts numbering and arithmetic didn’t develop for a very long time indeed. So-called ‘primitive man’ — and we are all primitives at heart —  found he or she had more important and more agreeable things to be doing than assessing quantity. This instinctive feeling that there is something annoyingly irrelevant about numbering and thus about mathematics as a whole is at the root of the sullen hostility of the ‘man or woman in the street’ towards invoices, forms, censuses, statistics and so on. What the hell does it matter where and when such and such a person was born? What his or her NI number is? The number of the street where he/she lives? Things that are really important to know because your survival may depend on them, such as whether so and so is trustworthy or likely to help you if you’re in trouble, is a question that is never asked on forms (and hopefully never will be). Of course from an administrative point of view the numerical data is not meaningless — but that is just it, numbering and much of what goes with it, appeals to the administrative mind and administrators remain wildly unpopular, out of all proportion to the harm they actually do. Humanity only quite recently got into administration — by recently I mean  about five or six thousand years ago — and it is hardly something that comes naturally to us whilst plenty of prehistoric activities do still seem to come naturally to us, like fishing, which a lot of people still practice at considerable expense and without any economic need to at all.
The mystery is not that animals did not develop numbers, nor that for practically all his life span as a species mankind didn’t. What is astonishing is how very rapidly and effectively mankind developed numbers and the sciences heavily dependent on numbers (astronomy and physics most notably), once they got started. It was in the interests of bureaucratic, imperialistic societies like Assyria andBabylon to develop numbers, and once again during the seventeenth centuries for the expanding maritime powers likeHolland andEngland to develop more advanced mathematical systems like trigonometry and calculus. This trend, however, goes against the grain, there can be no doubt about this. Instinctively we prefer the qualitative to the quantitative, prefer distinctions of type to numerical distinctions. And yet  the great leap forward in western civilization has been brought about by quantitative assessment, by the final victory of distinction by number over distinction by type, quantity over quality. This ‘progress’ has a downside : mankind’s perpetual (and increasing) sense of alienation, of being a ‘stranger in a strange land’ — the land of numbers.
S.H.  27/3/12

 

 

Desert Island Numbers

March 19, 2012

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

 What exactly are numbers? This is by no means an easy question to answer despite the fact that we use numbers every day of our lives in some form or other. Do we not, then, know what we are doing?
One way of answering this question is to launch into an investigation of ‘number’ as a concept or basic principle. This approach quickly leads on to some deep and perplexing issues that go back at least to Pythagoras and are still with us today (though mathematicians try to avoid tackling them if they possibly can).
At the opposite extreme, we have the Set-theoretic approach according to which ‘numbers’ (integers, rational numbers, irrationals &c. &c.) are just ‘things’ that emerge as just one application of the six or seven basic Axioms (preliminary assumptions) of Zermelo-Fraenkel Set Theory. This way of proceeding avoids metaphysical speculation altogether but at considerable  cost : the end products are not recognizable as numbers. Moreover, no one ever learned mathematics this way and most likely never will.
I favour a more practical approach.   After cogitating about number for some time , I set myself a mind-experiment. I imagined myself marooned like Robinson Crusoe on a desert island where the natives, though amiable and intelligent, did not seem to have a number system. After I had demonstrated what could be done with numbers like calculating when the next full moon would be (which they considered to be a kind of magic) they asked me what they had to do to make their own numbers. I laughed and said that numbers were not things that could be made like boats and mats. But then I stopped short and asked myself the question: What in fact do you need to construct a workable set of numbers? What are the minimal requirements?  And the answer that came to me was : all you need is a set of more or less identical portable objects that do not merge or stick to each other when brought close together. Before being ink marks on paper or dots on a computer screen mankind’s numbers were objects — pebbles, shells, twigs, knots in a cord, things you can touch and handle.

Parable of the Goatherd

Consider an illiterate goatherd such as existed in many parts of the globe until a few decades ago, and possibly still does in very remote parts. He brings in his goats each night and leads them out through a gate each morning. On the right hand side of the gate is a pile of stones and as he lets each goat through the gate he shifts a stone from the right hand pile to make a similar pile on the left. One goat, one stone.
At the end of the day he lets the goats back in and shifts a stone back to the right hand side of the gate (right hand from the inside but on the goatherd’s left side if he is coming back.) If there are any stones remaining when the last goat has been ushered in, he knows there are goats missing — or at least one goat missing.
Does our goatherd know how many goats he has in his herd? In our sense of ‘how many’, probably not. Most likely the society he lives in does not have enough spoken or written words to represent such a quantity, at any rate if he has a sizeable herd. Innumerable tribal languages had no words to express quantities beyond our ‘forty’ and in many cases, incredible though this seems, the vocabulary of number was limited to the equivalent of our ‘one, two, three’ where ‘three’ had the meaning ‘inconceivably great’, almost our ‘infinite’. Menninger cites the true story of the venerable South Sea Islander who, being asked how old he was, answered, “I am three”1.
But in fact our goatherd is not innumerate, he does have a set of numbers, his pile of stones. The stones are his numbers. If asked how many goats he had  in his flock, he would probably indicate with his hand the pile of stones on the right side of the gate. And if asked at the end of the day how many goats were missing, supposing some were missing, once again he would indicate the stones left. It is as ridiculous to suppose that numbers must be marks on a piece of paper or pictels on a screen as it is to suppose that poetry has to be words on a page.
So-called ‘primitive’ peoples used shells, beans or sticks as numbers for thousands of years and within living memory the Wedda of Ceylon carried out transactions with bundles of ‘number sticks’. Although the development of a centralised imperial state apparatus generally gave rise to written  number systems, this was by no means invariably the case. The state officials of the Inca of Peru managed a vast empire without any form of written records : they used the quipu system where knots in coloured cords served as numbers. And the Yoruba officials of the equally extensive Benin empire in Nigeria performed quite complicated additions and multiplications using only heaps of cowrie shells2.

Number Objects and Object Numbers

 I imagine myself, then, Robinson Crusoe-like, looking for a suitable set of objects which are to be the basis of a workable number system which I can use myself and, possibly, introduce to the inhabitants of the island supposing there are some and that they are as yet innocent of numbers. What criteria are going to influence my choice?
It is important that the objects chosen should be more or less identical since I have already decided in my solitary cogitations on numbering that the basic principle of number would seem to be that individual differences between objects do not matter numerically speaking. My ‘one-object’, whatever it is, is going to be used to represent indifferently a tree, a fish, a man, indeed anything at all provided the ‘thing’ I want to represent is singular, is a ‘one’. If I used different number-objects to represent different objects there would be no net gain — I would need as many object-symbols as there are objects.
Secondly, since there are a lot of objects in the world, I want a plentiful supply of numbers to represent them, so my chosen ‘one-object’ must be abundant. Alternatively, if I am going to make my own numbers I require the raw material to be abundant — wood for example — and the manufacturing process to be relatively rapid and easy so that I feel I can always make more numbers if I run out of them.
Thirdly, the chosen ‘one-object’ must be portable and to be portable must be fairly small and light. For, once again, if I have a stationary set of numbers there is little net gain: one of my main goals in developing numbers is so I can move around to assess numerically a distant clump of trees or a distant village. Although in special cases such as censuses and elections  the (human) objects do actually come to the numbers — come to a place where they are numbered — it is generally necessary to take the numbering apparatus to the objects and, in the very important case of spying, this is essential (in the days before long-range surveillance was feasible).
Fourthly, it is essential that the number-objects (or object-numbers, the terms are perfectly equivalent) do not merge or adhere to each other when brought into close proximity. It must be possible to make the numbers into a group while the objects remain distinguishable whilst in this group. Why is this important? Because this is the commonest set-up  we shall be modelling numerically. If we were regularly confronted with entities that flowed into each other, fused, only to separate a little later,  we would need  to introduce this feature into our mathematics But we spend most of our lives amassing objects, removing them from one group’s ownership to another’s (commerce), shifting them from house to house or port to port and so on. And the majority of these objects do not merge when brought together — even liquids are transported in containers and so function numerically as solid objects.
Other requirements are that the object be durable, easy to see, especially when held in the hand or placed on the palm of the hand. Also, we do not want the number-object to have the power of locomotion or it might move off before we have finished the counting! This more or less rules out living things or at any rate mammals as being suitable numbers though human beings in very special circumstances (prisoners of war, conscripts) have occasionally been used as numbers of a sort, for example to make a rough guess of the enemy’s strength on the basis of terrain occupied and suchlike cases.
All these requirements make the choice of a number object by no means so simple as it might seem at first sight. Grains of sand are abundant  but not easy to see, grains of salt stick to the hand. Beads are about the right size and are cheap today but they would have been luxury articles on a desert island in the past. Also they tend to roll around as do marbles which is why beads were threaded onto wires in the usual abacus while marbles  were confined to grooves in the case of the Roman abacus. The most suitable objects, at any rate for a  rural society, turn out to be exactly the ones actually used by tribes : shells, beans and sticks. Shells are abundant, light, portable, and can be neatly stacked into piles. Sticks have the additional advantage that they can be split in two and so, if one runs out of numbers on a field trip, one can make more numbers on the spot.
The question is not for me entirely an academic one as I have been looking for suitable object-numbers for some time now (in order to practise concrete arithmetic) and haven’t found an ideal choice yet. Draughtsmen  stay flat and stack up beautifully if they are of the old type with rings on top, thus allowing for the representation of ‘powers’. They also have the advantage of coming in two colours though it would be even better if they were a different colour underneath so that in order to ‘change sign’ you could just turn a draughtsman upside down. But  they are not abundant : one set of draughtsman is not enough and if you buy different makes  you find they don’t stack up properly.
Coins are feasible numbers, especially ones with holes in them like old Chinese ones since they can be threaded together. The counters used on counting boards in markets and even banks right through to the seventeenth century were virtually indistinguishable from coins. Roman calculi, of glass or stone, were unmarked but from the Renaissance onwards counters were elaborately decorated.
Matches are not at all bad, especially outsize ones like Brymay Long Matches and you can even represent positive and negative quantities if so inclined by using ones with brown and red heads — though the latter are becoming scarce now for health and safety reasons. Number sticks, painted red and black, were used inChinacenturies before ‘double-entry’ book-keeping became current inEuropethough the meaning was the reverse, black for negative, red for positive.

The Number Ball

On another occasion thinking of my island paradise awaiting its Archimedes, I hit upon the idea of a clay ‘Number Ball’. The advantage of this choice is that, apart from being portable, it allows one to get rid of a number once it is of no further interest and start again. A native might be sent, for example, to find out how many palm trees there were on a particular beach. Equipped with his Number Ball issued at the Central Data Hut he would arrive at the site and tear off as many little bits of clay as there were trees. He would report back to Central Office where the bits of clay would be recorded by an equivalent amount of scratches on a bone or knots in a cord, and would then squash everything together to recover the original ball.
This system has an interesting feature : it is two-way  in the sense that you can use the same apparatus for recording data but can then ‘de-record’ (wipe out) the data to recover the original set-up and start again. This means, firstly, that there is no wastage. There is also something aesthetically satisfying about such a simple apparatus having an  ‘inverse’ procedure built into it : once you have completed your task, the Number Ball is returned to what it was in the beginning like the Tao.
Most recording systems do not have this feature : if you make a scratch on a bone you cannot ‘de-record’ without damaging the recording device, and crossing out something written with pen and ink is both messy and inefficient (in films a crossed out line often gets deciphered and leads to the conviction of a criminal). Destroying data has in fact become a considerable problem in modern society, hence the sale of shredders and civil servants’ perpetual fear of e-mails being picked up.
Clay Number Balls would be too messy for modern interior use but Blu-Tack is an alternative I have experimented with a little. There is, however, a certain risk of the little bits of clay or Blu-Tack sticking together and thus falsifying the reckoning.
The Number Ball is something of an anomaly mathematically and even philosophically. The object-numbers produced, i.e. the little bits of clay, do not strictly fulfil the requirement that number objects should not merge on being brought into close proximity — they can be made to merge or kept apart at will, so we have an interesting intermediate case.
Also, and this is more significant, the Number Ball is not, properly speaking, a number but rather a source of numbers, a number generator. In this respect it resembles an algebraic formula since the latter is not in itself a number (in any sense) but can be made to spew out numbers, as many numbers as you require. (For example the formula f(n) = (2n –1)  gives you the odd numbers (counting 1) if you turn the handle by fitting in 1, 2, 3….. for n e.g. (2 ´ 1) – 1 =  1; (2 ´ 2) – 1 = 3; (2 ´ 3) – 1 = 5 and so on.)
Yet a Number Ball is not a formula or an idea : it remains an object. Of course, one could also call a box of matches or a set of draughtsmen  ‘number generators’ but there is a difference here : the object-numbers are present in the box as distinct items (matches) and are thus already numbers at least potentially, whilst bits of clay of Blu-Tack are not. A Clay Number Ball is actually a special type of generator since everything it produces comes from within itself and can be returned to itself. I have coined a term for this particular case : I call such an object an Aullunn. Although there are no complete Aullunn Generators in nature — not even, seemingly, the universe itself —   many natural phenomena approximate to this condition. The varied life in and around a pond to all intents and purposes emerges spontaneously ‘from inside’ and dies back into it; though we know that without some interaction with the environment, especially with sunlight, no generation would be possible.
Surprisingly I have not come across any accounts of tribes using clay Number Balls.

Number Marks and Number Bearers

A very different method of producing a set of numbers is to have an object or substance which is not itself a number (or a number generator) but a ‘bearer of numbers’ : the numbers are marks on the surface of the number bearer or deformations of it. This system, which at first sight seems a lot closer to the written system we use today, is extremely ancient and possibly pre-dates the widespread use of distinct number objects. The markings on the Ishango Bone, which dates back to about 20 000 B.C., are thought by archaeologists to have numerical significance. Other bones have been found dating almost as far back with scratches on them that are thought by some to  indicate the number of kills to a hunter’s credit — one thinks at once of Billy the Kid, the “boy who had so many notches on his gun” (or was it Davy Crockett?).
The limitation of the notch system is that an incision is permanent which means that once the ‘number-bearer’ gets filled up it has to be stored somewhere or discarded like a diary. It thus tends to be used in rather special circumstances, either when one does not expect to be dealing with large quantities (rivals killed) or when one wants the information recorded to be permanent as, for example, in the case of inscriptions on State monuments.
Making charcoal marks on a wall, also an ancient practice, is ‘two-way’ in that one can rub out what one has written but the system would not be reliable for long-term recording of data because of effects of weather, flaking of surface &c. But numbers on a number bearer do not have to be marks : they can be reversible deformations, the prime example being knots in a cord. The great advantage of such systems is that, though very long lasting if the material is itself durable, the numerical data can easily be got rid of when no longer needed since knots can be untied. On the other hand because they take a lot of time to tie and untie, knots are unsuitable for rapid calculation and it would seem that the Inca State officials used quipus for storing data whilst they had some form of a counting-board system for calculations. Knots in a cord constitute a partial ‘two-way’ recording system — what is done can be undone — but they are at the same time quasi-permanent and so they combine some of the features of both types of system.
Knotted cords were in widespread use all over the world at one time and it is thought that mankind may even have gone though a ‘knotted cord’ era. The author of the Tao Te Ching (VIth century B.C.) who is a Luddite hostile to new-fangled inventions and to civilization generally speaks nostalgically of the days when mankind used knotted cords instead of written numbers.
In practice both systems are required, a ‘two-way’ number system which allows one to carry out calculations and then to efface them, plus a more permanent system which is used to record results if they are considered important enough. It has been agreed that the Incas only used quipus for permanent or semi-permanent records while they used stones and a counting board for calculation. The lack of a suitable ‘number-bearer’ to receive marks meant that inscribed number systems were a rarity until comparatively recently — baked clay tablets and papyrus were reserved for the bureaucratic elite and paper, a Chinese invention, only entered Europe in the latter Middle Ages and was expensive. Traders, even money-lenders and bankers, when they did  not use finger-reckoning of which more anon, used a two-way system, namely counters and counting boards, right into the Renaissance. The abacus, a two-way system, was never widespread in Europe for some reason except in Russia, but in the East has remained in use right through to modern times. The soroban or Japanese abacus is still used today and as late as the nineteen-fifties a Japanese clerk armed with a soroban competed successfully with an American naval rating using an early electronic  calculator. However, it must be pointed out that the Japanese achievement with the soroban depends on extensive practice in mental arithmetic which is still prized over there rather than any particular merits of the device itself.
The drawback of a ‘two-way’ system such as an abacus where you erase as you go is that you cannot check for mistakes and even the result itself, once reached, has to be erased when we perform our next calculation i.e. there is no inbuilt recording element, no memory. But when there is no easy way of erasing we oscillate wildly between conservation and destruction : we tend to accumulate a vast amount of stuff, then periodically have a sort out and throw it all away, the pearls with the dross. Like most authors and mathematicians from time to time I have to tip out the entire contents of a large dustbin to search for a scrap of paper with some idea or formula written on it.
The principal drawback of a one-way semi-permanent system such as ink on paper is that it is incredibly wasteful and was until recently so expensive that the bulk of the world’s population, the peasantry, practically never used it and employed a pocket knife and a flat piece of wood to record data. Even in the computer era we still use the chalk and blackboard two way  system though the chalked notice-board in the hall of buildings or private residences — to mark who is in or out — which was once commonplace is now virtually a thing of the past. I myself buy rolls of lining paper (which I clip down over a table) partly because I like to have plenty of room for drawing and calculation but also partly for reasons of economy — you get a lot of paper in a roll compared to an exercise book. It is a sobering thought that no less than a hundred years ago Ramanujan, one of the greatest names in Number Theory, like so many other Indian mathematicians of the time worked with slate and chalk because he found paper too expensive. Although to my knowledge no one has suggested this, I would guess that this is one of the main reasons why his early mathematical writings are so hard to follow — he left no tracks because he generally just copied out his conclusions, then literally wiped the slate clean. To many people the results seemed to come from nowhere and indeed he was often incapable of explaining how he got them.   (Ramanujan lived a century too early : today we have a relatively cheap improved ‘chalk and board’ system, the Whiteboard. At last marks can be easily erased without mess. I use large boards everyday and have somewhat moved on from lining paper to a more up to date recording system.)

 A Standard Set of Ones

 Prior to my arrival on the desert island in question the inhabitants did not bother with numbers at all and had no number objects, not even the bundles of number sticks used by the Wedda of Ceylon. How exactly they got on without any form of mathematics I shall discuss in a moment (not as badly as you might think). Almost certainly they had a handful of number words and were able to point and perhaps make intelligible signs with the hands. I decide on my chosen set of objects which are henceforth to serve as numbers, cowrie shells for example, and instruct the natives in their use.
The point is that I am taking this as an all-round numerical watershed : prior to having a set of standard objects or marks set aside for numbering I call such a society innumerate, while if it has such a set it is numerate. The line must be drawn somewhere. Most people today consider that, with one or two rare exceptions (bees, for instance), animals do not have number sense as we understand it. They are excluded by my definition because none of them use sets of number objects or signs. (I can hardly stress too strongly that this is not ‘putting down animals’ — animals, like many early societies  do not use numbers because they do not feel the need for them.) At the other extreme it would, I think, be ludicrous to take as a definition of numeracy familiarity with the Axioms for Fields as laid out in a modern algebra book. Such a criterion would exclude Newton, not only because he did not know of the Axioms for Fields which are a twentieth century concoction, but because, even had they been formulated in his  time, he would almost certainly have disapproved of them.1 My simple criterion lets in the Incas and the Yoruba who administered huge areas without any form of written or incised records, and even the Wedda.
More specifically I declare a society to be numerate if it operates a set of standard objects or signs which represent any set of discrete objects that can be put in one-one correspondence with it, or a portion of it.2
          The decisive step, then, is when a society accepts that a particular ‘one’, cowrie shell, bean, vertical stroke ½ or symbol 1, can validly represent any individual object, any ‘one’.
The step taken is really momentous — so momentous that most societies that reached such a stage baulked at a complete generalisation and made at the very least one or two exceptions. In very many societies certain sets of marks were permitted to represent specific objects alone, or certain categories of objects. Until modern times, craftsmen, tradesmen and the like often used special sets of numerals (generally marks on tally-sticks) for specific merchandise such as milk, wheat and so on despite having some familiarity with ‘ordinary’ numerals. Even in a society as advanced as Athensin the Vth century BC there was as yet no completely universal symbol corresponding to our ‘2’. In state documents dealing with the city’s income and expediture we find that different symbols are used, depending on whether the official is referring to “2 talents”, “2 staters” or “2 obols” (Menninger, p. 269).3
The step involved is not only momentous but shocking and indeed has still not been accepted psychologically by the vast majority of the population. Why does one feel an instinctive repugnance about being treated ‘as a number’? Why is there such deep-rooted though inarticulate hostility towards administrators and bureaucrats, the users of numbers? Because numbering wipes out every feature of individuality — that is what numbering is about.

“Where is mankind? These ciphers do not speak
          Of what is gay or sad, wistful or intimate,
          All persons and all feelings are equivalent;”

The two developments which have probably done the most to change the conditions in which we live, mathematics and physics, are not only non-human sciences but in a very real sense actually anti-human. This is the reason for the extraordinarily persistent popular antipathy towards the mathematical sciences — after all we are humans and cannot be expected to welcome with open arms something which does not recognize the fact! Numerically no distinction is made between a gnat, a star, a loved one or a murderer — they are all in some sense single objects and so the same mark can  be and is used for all of them. Hume wrote trenchantly, “From the point of view of the universe, there is no difference between the death of a man and the death of an oyster”. He might just as well have said, “from the point of view of arithmetic, there is no difference between a man and an oyster”. And as for physics, the numerical science, the same applies :Newton’s Laws predict that in a vacuum your mother or child would fall to the earth at the same rate as a piece of cardboard or a slug. ~
Prior to the twentieth century practically all societies were religious. For many it was not so much the implied threat to human beings that made people wary of numbers as the implied threat to the gods. For a god or goddess was also a ‘one’ and the immortals could be (and were) numbered in the same way as cattle were numbered. Practically, this led to several societies having two or three completely different sets of numerals, one kept for religious matters, one that dealt with mundane matters and perhaps a third for matters of state. Thus the Mayans had three sets in use. The Chinese had a special set of ordinal numbers for noting the length of the reigns of Emperors.  Even today I would hazard the guess that we still use the inefficient and not very picturesque Roman numerals for dates (because we somehow feel the passing of the years is ‘different’, is  semi-sacred and so needs special symbolic treatment. Certainly, it was the astronomer/priests who gave us the calendar in the first place and the last widespread reform, the Gregorian calendar, was instigated by the Papacy. Only quite recently  has the measurement of time finally fallen into the hands of entirely secular officials at Sèvres, Paris. The location (Paris) is fitting since it was the French Revolution which did away with the myriad local measuring systems that existed during the ancient regime and which deliberately invented (and imposed on most of Europe) a new ‘scientific’ system of measurement, the metric system. Whereas previous basic measurements had generally been based on human dimensions (the cubit, for example) or related to human activities — an acre was supposedly the amount of land a good worker could plough in a day — the metre was a sub-multiple of the circumference of the Earth.

Notes : 

Assumptions required to develop a Number System

March 14, 2012

Nature is not deliberately mathematical or even numerate :   if certain numbers keep coming up — and few do systematically — there is generally some physical or biological reason for them 1.
In this sense it is perfectly true that numbers, or at any rate number systems, are human creations but they are firmly based on features of the natural world that really exist objectively. One might say, to paraphrase Guy Debord, “Number has always existed but not always in its  its numerical form2.
So how do we develop a number system? What are the minimal requirements?
Two, and as far as I can see, only two abilities are necessary to develop a number system :

1. The ability to distinguish between what is singular and plural, i.e. recognize a ‘one’ when you see it;

2. The ability to carry out a one-one correspondence (pairing off).

All the mathematicians who have developed abstract number systems, for example  Zermelo and von Neumann, had these two perceptual/cognitive abilities — otherwise they would have been denied access to higher education and would not even have been able to read a maths book. Animals 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.
Cardinal number — and unless stated otherwise I shall be referring to cardinal (how many?) number  — cardinal number may be legitimately viewed as some sort of an abstract quality pertaining to all finite sets. Or can it? The best one can do to define ‘number’ is to say it is what 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 though it does emphasize the curious fact that number is a negative rather than a positive thing (in the usual sense) 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. If I can pair 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 — roughly, “what 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? 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

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

Principle of Replacement

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

           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 exactly 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, I think, ‘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 features 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. It is certainly not a theorem since it is quite untestable and not really a definition either.
This does 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 ‘something 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. There seems to be no special reason why they should have the values they actually do have, unless one accepts the Strong Anthropic Principle which claims  that only such values (or very similar ones) would permit the development of human life in the universe. 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 × 108 m/sec However, proponents of loop quantum gravity do claim to be able to deduce many physical laws from the equations of General relativity and nothing else, though I am not sure if they actually derive the precise values of  constants.
I am personally coming round to the view that there are strictly no numbers as such built into this universe or any other. What we do seem to have is certain ‘constraints’ which drive physical processes towards certain limits : these constraints must seemingly be independent of any actual universe, otherwise they would be part of it and thus vary along with everything else. By ‘constraints’ I have in mind principles like that controlling pressure in fluids, making air or water move from a region of relatively high pressure to one of relatively low pressure (2nd Law of Thermodynamics if you like). Newton’s Third Law of the equivalence of action and reaction would seem also to be a basic ‘constraint’ though it is not necessary to believe that such forces are ever exactly equivalent : there is a perpetual oscillation of pressure/reaction to pressure which tends towards a limit. Just recently, Adrian Bejan has come up with what seems to be a new ‘basic law’, one of a more positive type, namely his Constructal Law (his term) which states that “If a finite-size flow system is to persist in time, its configuration must evolve in such a way that provides easier access to the currents that flow through it” (see Design in Nature  by Adrian Bejart & J. Peder Zane).  
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.
S.H.     14/3/12