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

February 1, 2013

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

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

        
   
  
   

Each coloured inverted L shape border in the above represents an odd number with the unit in the top left hand corner. The early Greek mathematicians deduced the important property that Any square number can be represented by successive odd numbers commencing with unity. As we would put it:  n2  =  1 +  3 + 5 + ……(2n + 1)   For example, 41 +  3 + 5 + 7   And this can be extended to the observation that Every difference between two squares can be represented by a sum of successive odd numbers. Thus, 5–  2=  5 + 7 + 9    (It was in fact this relation which struck me as being quite astounding that instigated my interest in mathematics which up to then I had despised.)
But much more can be got out of the simple diagram above. The Egyptians were certainly aware of certain cases of the the property forever associated with Pythagoras, namely that The Square on the hypotenuse is equal to the sum of the squares on the other two sides of a right-angled triangle since they used stretched ropes with lengths in the ratio 3, 4, 5 to lay out an accurate square corner. However, they may not have realized that this property applied to all right angled triangles. The question provided a fruitful contact between geometry, the science of shape, and arithmetic, the science of exact quantity and the gnomon most likely played an important role here. Greek mathematicians were interested in sets of numbers that were ‘Pythagorean triples’, i.e. numbers a, b, c where   a2 = b +  c2 .
Now,  adding on a gnomon “preserves the square form”  and, more significantly for the present discussion, that the difference of two successive squares is an odd  number.

        +               =           
                                       
                                       
                          

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

          (our 9) and to make it into a gnomon we stretch it out into three parts , two equal and the third a unit
                   This provides the outer framework for the two squares :

              The inner square has side 4 and the outer side 5. This gives the simplest
              Pythagorean triple  52 = 42 + 32 .
      
      

      

However, any odd square will do and, since 49 = 72 we can construct a Pythagorean triple involving it. The gnomon is 24 + 1 + 24  giving 24 for the side of the larger square and 23 for the smaller one. This gives the triple  242 = 23  + 72 .  The series of Pythagorean triples using this procedure is endless : it suffices to find an odd square number.
This procedure can be generalised if we allow a gnomon to be made up of more than one row + column. For example, we might allow the gnomon to have three rows + three columns.

………………        

………………       

………………       

…          
…          
…          

If r is the side of the inner square, the outer square is (r + 3) instead of (r + 1) and the little square in the bottom right hand corner will be 3 x 3 = 9 instead of 1. The gnomon is made up of two rectangles (r x 3)  the little square giving  (6r + 9) = 3(r + 3)   We must thus find a square which is equal to the gnomon or solve  3(2r + 3) =  m2   for some m.   Since  m2  is divisible by 3 this makes m a multiple of 3 as well. We must also have m large enough so that r is at least 1.  The first possibility is m = 21 = 7 x 3  so that  3(2r + 3) =  212    This makes r = 72  This will be the side of the inner square while the outer one will be 72 + 3 = 75.  So, if this reasoning is correct, we should find that 752 = 722 + 21 which is indeed the case (check this). So a rather more spaced out but still unending set of Pythagorean triples can be manufactured where the difference between the sides of the squares is 3 rather than 1.  It is left to the interested reader to concoct other sets.
As a matter of fact we have reason to believe that the early Pythaogoreans knew of such sets of triples and it is plausible that they hit upon them using some such method which has its basis in the manipulation of sets of wooden gnomons and/or actual counters on actual boards. Interestingly enough, the Babylonians a thousand years earlier were aware of Pythagorean triples and seem to have had some method of concocting them (Note 3)The basic formula for all Pythagorean Triples is given in Euclid — or rather can be deduced from the argument given in Euclid which is mainly verbal since the Greeks did not have our algebraic notation. I shall not give it here — you can get it from Wikipedia or some other site by the click of a key — as I am more interested in seeing how such formulae arose in the first place and indeed in (re)discovering them for myself, something that I encourage you to do as well. In the next post I will examine the slightly more complicated problem of an isosceles right-angled triangle, i.e. one where the two smaller sides are equal. This provoked a trmendous rumpus at the time because it raised the issue of so-called ‘incommensurables”. If the short side is set at unity, the square on the hypotenuse comes out at  of  12 + 12 = 2 so the side itself is the square root of 2. But was there such a number? In the ideal world of Platonic forms (not yet elaborated) certainly there was, but in the Pythagorean world of number where number meant ratio between two integers there was apparently no such quantity and thus no such length.     SH   1/1/13

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

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

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

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

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

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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