46 Lessons in Early Geometry, part 10/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch


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Lesson 44


Dialogue with Ahmes, scribe of the Rhind Mathematical Papyrus, on the nature of numbers, part 1 (mistakes and bad style are mainly owed to the insufficient bablefish component of my astrologer's crystal ball, which, however, allows me to conversate with Ahmes over several thousand miles and years. I guess you would gladly endure the worst English if you could have a chat with, say, Archimedes)



Me: Good morning, professor Ahmes. Today I would like to ask you a principal question. What are numbers? What is your idea of numbers? What is their nature? How do they work? What are they good for?


Ahmes: Well, you announced that question last week, and so I prepared myself to provide a short and nevertheless complete answer. Numbers are basically things of the mind, I believe, and the very instrument by which Osiris overcame Seth.


Me: I can imagine what you mean by the first part of your answer, while the second one leaves me rather perplexed I must confess.


Ahmes: What is our greatest achievement? What do you admire most?


Me: I love Egyptian painting and sculpture. The most beautiful, charming, lovely and sensual lips ever carved in stone are found in your works. But since you ask for the greatest achievement I should perhaps mention the pyramids.


Ahmes: And if I tell you that we should be admired for a greater achievement?


Me: I can't imagine anything greater and more admirable than the pyramids of Sneferu and the Giza pyramids.


Ahmes: They are most impressive, and yet what I have in mind is a hundred times more admirable.


Me: Tell me, professor.


Ahmes: We succeeded in turning a desert valley into a blooming garden. We spent millennia in achieving this, and we are spending further millennia in maintaining our paradise. Without our being here, high floods would sweep away the fields, whereas low ones would make most plants wither, and within a few years the chamsin would cover with ever deeper layers of sand all our houses and even most of the temples. You may know that we worship Osiris in the Nile, and Isis in the land. But it isn't simply the river we associate with the god: it is the regulated Nile including dams, dykes, canals, ponds and even artificial lakes. And it itsn't simply the land we associate with the goddess: it is the cultivated land. Our forebears invested immense labor, and their almost superhuman task was a challenge to the mind. It gave rise to simple but clever devices like for example the nilometer, originally a simple pole allowing to measure the height of the rising river; or a shaduf for lifting water up to the fields. Building dams and canals and measuring out fields required geometrical skills. A series of low floods meant hunger; in preventing famines our forebears developed ways of storing goods, and in order to keep an overview over their many and various stored goods they developed signs for each good, and signs for the amount of a stored good, hence numbers, like 5 bags of dates, or 7 jugs of oil. Geometry lead to abstract reasoning, which turned out to be so successful that we not only considered free forms but also free numbers, which are no longer bound to objects. As long as we say One fish, Two fish, Three fish - or loaves of bread, or whatever -, we can mainly count, but as soon as we free the numbers from objects we can proceed from counting to calculating, from storing goods to adding numbers, from actual trading to symbolical trading by adding and subtracting numbers, from actually distributing goods to handling unit fraction series. A huge task requires new and intelligent ways of working. Turning a desert valley into a paradise and maintaining it a paradise was the very challenge, task and labor that made us develop geometry and mathematics, and without these most powerful tools of the mind we would never have succeeded. Now consider that we associate the desert with Seth, enemy of Osiris, and you may understand what I mean with numbers being a tool of the mind helping Osiris in overcoming Seth. May I tell you a fancy of mine? Me and my wife love to go for a pleasure walk in the late evening. We look up to the twinkling stars, and see the moon rise over the palm trees. That shining disk so far away can't simply be a marble disk as many believe. It must be a huge heavenly body, with valleys and mountains I guess. We can see no blue of water and no green of palm trees, only a yellowish white similar to limestone, so I guess that the moon is mainly a desert. Now for my fantasy. What if, say, in a million years a new generation of human beings were able to fly up to the moon and turn a heavenly desert  into a new paradise? That would again be an immense labor and a challenge to the mind, giving rise to many more mathematical skills, and to calculating devices that will make look our poor calculating boards we are so proud of mere toys of children. A fancy of mine, I told you. And when I say things like that on our pleasure walk in a mild evening my wife looks at me, calls me a dreamer, and gives me a kiss ...


(To be continued)




Lesson 45


Professor Ahmes? Hello? Professor Ahmes?


No response.


Well then, let me reconsider what he told me last time.


Turning a desert valley into a blooming garden and maintaining it a paradise was the challenge that spurred geometrical and mathematical skills. Rushdi Said will probably agree on that. He says that before around 7,500 years ago nobody lived in the Egyptian part of the Nile Valley, which was a dangerous place to be, ruled by an 'angry river' (Rushdi Said, The River Nile, Pergamon Press Oxford 1993). Taming that river and turning a 600 km long desert valley into a place to be surely was a great challenge.


The Greeks faced a completely different but no smaller challenge. Connecting the many distant shores of their vast archipelago required excellent ships, daring men, highly developed navigating skills, and may well have given rise to Greek reasoning including their fabulous achievements in geometry ... Like Isis, Penelope may symbolize the land, and like Osiris, Odysseus may symbolize the water, while the name Homer may go back on Greek homaereo = I unite. It was Homer's wish to unite Greece, I believe, and to hold together the many distant shores of the Greek archipelago, and this challenge might have been the very task that spurred the wonderful Greek achievements.


America faces yet another challenge: establishing a prospering global civilization and exploring space. Terraforming Mars will be an even greater task than taming an angry Nile, turning a desert valley into a garden, and uniting the Greek archipelago. It will again spur all kinds of skills, including geometry and mathematics and computing. John F. Kennedy should be given a Fields medal post mortem and honoris causa for having started the Gemini and Apollo space program ...


Enough chatting for today.


Professor Ahmes? Hello? Professor Ahmes?


Still no response. I will try again another day, hoping to ask him what exactly he means by saying that numbers are basically things of the mind.




Lesson 46


Me: Good morning, professor Ahmes. You told me that numbers are basically things of the mind. Which sounded fairly good to me. However, the more I think about your words the less I understand what you mean.


Ahmes: By handling numbers we participate in the creative power of the gods.


Me: You always succeed in puzzling me.


Ahmes: Build a wall of bricks or blocks. It may be a fine wall  and stay firmly for years or millennia. But even the best built and strongest wall has its weak points and will sooner or later decay, crumble away, come tumbling down. Add numbers instead and you will obtain a perfect wall, so to speak:


  1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 makes 8


Regard the ones as blocks, and the sum as the resulting wall. The numbers fit perfectly well, and the sum will last forever.


Me: I can see a block, while I can't see a number.


Ahmes: Numbers have a double nature. They are ideal things in the realm of building and constructing, but as such invisible. If you wish to make them visible, they turn from ideal objects into patterns.


Me: Can you explain this?


Ahmes: You may know that we teach our aspiring scribes from the tender age of five years on. We let them play for example with beans. I give each pupil nine beans and ask them to lay out as many patterns as they can imagine. That is a good way of introducing children into the realm of numbers.


Me: Playing with beans?


Ahmes: Ready for a game? Take twelve beans and call them one heap, or one line, or simply 1. Six beans are half a heap or simply '2. Four beans are '3. Three beans are '4. Two beans are '6. One bean is a twelfth of a line or simply '12. Now consider the following posibilities of dividing a line:


  oooooooooooo  1


  oooooo  oooooo  '2 '2


  oooooo  oo  oooo  '2 '6 '3


  oooooo  oo  o  ooo  '2 '6 '12 '4


A few transformations and we gain a fascinating pattern:


  one   '1

  one   '1x2 '2

  one   '1x2 '2x3 '3

  one   '1x2 '2x3 '3x4 '4


  one   '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7x8 '8x9 ...


We easily recognize mathematically gifted pupils. They love to play with patterns that emerge at every level and are the true guide in mathematics.


Me: Thank you, professor Ahmes.






Numbers are regarded as kinds of atoms, however, they can’t really be separated from patterns. Consider the following stairway decimals:






























All above numbers equal 1, nevertheless they are associated with all other numbers.


You can even calculate with these funny numbers:


    Square root of  1.000...6  equals  1.000...3

      Cube root of  1.000...6  equals  1.000...2


  1.000...2  times  1.000...3  equals  1.000...5


  1.000...4  minus  1.000...2

  --------------------------- = 2

  0.999...2  minus  0.999...1


One might build ever longer numbers:






  1.000...1  times  1.000...1  equals   1.000...2.000...1

  1.000...1  times  2.000...1  equals   2.000...3.000...1

  1.000...1  times  3.000...1  equals   3.000...4.000...1


  1.1   divided by  1.1           equals  1

  1.01              1.001                 1.00899...

  1.001             1.00001               1.0009899...

  1.0001            1.0000001             1.0000999...

  1.00001           1.000000001           1.00000999...


  1.000...1         1.000...0.000...1     1.000...0.999...



As long as we carry out a finite number of steps, the infinitesimals have no influence on the value of the reals:


  1 squared = 1  squared = 1  squared = 1  squared = 1 ...


This changes when we carry out an infinite number of steps:


  1.000...1  squared  equals  1.000...2.000...

  1.000...2  squared  equals  1.000...4.000...

  1.000...4  squared  equals  1.000...8.000...



Go on like this forever and you obtain more than 1. Where in this process do we pass from 1 to the next real? May it be that the infinitesimals are hindering us from actually well-ordering the reals? Leibniz regarded the numbers as monads that contain all other numbers. Can it be that he played with the same stairway numbers?






a paper for the Brno Conference 2003, Chairman Alan Rogerson, The Decidable and Undecidable in Mathematics Education, a tribute to Kurt Goedel.



Mathematical logic


As a teenager I read the first sixty pages of a book on quantum mechanics without understaning anything, but I was fascinated by a footnote which said that the basic equation of mathematics a = a has not yet been thoroughly studied. Finally something I could understand. A couple of years later I found the following explanation. Mathematics is based on the formulas


     a = a        a  unequal to  b


which represent the logic of building and constructing. Consider these examples:


     b = b = b = b = b = ...


A wall can easily be built and will stand firmly if the bricks (b) are of the same material, size and consistency, and have in common all other properties; in short: if they are equal.


     b = b


The bricks should neither soak up moisture nor crumble in the rain nor crack in the summer heat; their properties must remain stable, they must stay as they are.


     2 + 1 + 3  =  6  =  2 + 1 + 3


A closed door (1) should become a part of the wall (6); one might then wish to open the door (2+ 1+3).


     0.999... = 1


A door (0.999...) must fit into its frame (1); otherwise it will be too tight (“klemmen” in German) or there will be a draft.


     9  =  2 + 3 + 4  =  9


In order to clean or repair a machine (9) one dismantles it into single components (2, 3, 4); then one reassembles the parts, in order to return the machine to its former functioning state (9).


Sooner or later, a mathematical discovery finds its way into technology. The imaginary number i (square root of minus one) was first regarded as a strange number, yet without that funny little number no radio, television set or computer would work today.



The logic of nature, life and art


The logic of nature, life and art is based on another formula:


     "All is equal, all unequal ..."


(Johann Wolfgang von Goethe, Wilhelm Meisters Wanderjahre, Aus Makariens Archiv)


     a = a


An apple is an apple; yet one fruit may be red and sweet, another green and sour, and another yellow and juicy ...


     A rose is a rose is a rose  (Gertrude Stein)


You may imagine a red rose, a white rose, a sweet-smelling yellow tea rose, a budding rose, a flowering rose, or you may think of a girl named Rose, Rosy, Rosemary.


     A snowflake is a snowflake


Every snowflake forms a hexagon, yet seen under a microscope each has its own unique pattern.


     A mouse is not an elephant


Mice and elephants belong to the animal kingdom, and are both mammals. They have a common mouse-like ancestor with a kind of proboscis, while the hyrax, the elephant's nearest living relative, resembles a large mouse.


     We are all equal and all different


A fair and reasonable human society is based on the fundamental equality of all humans, while leaving room for our individualities.


     p = p = p = p = p ...


Physicists search for elementary particles that fulfill the above equation. They explore and expand the realm of technology, with no end.



In his Diary of the Italian Journey, Goethe speaks of an ever turning key. This key might well have been the formula All is equal, all unequal, which Goethe successfully applied to the morphology of plants and animals, and also to works of art. Here is a wonderful quote from a later essay on the fine arts, in the original German:


"Alles, was uns daher als Zierde ansprechen soll, muss gegliedert sein, und zwar im höhern Sinne, dass es aus Teilen bestehe, die sich wechselsweise aufeinander beziehen. Hiezu wird erfordert, dass es eine Mitte habe, ein Oben un Unten, ein Hüben und Drüben, woraus zuerst Symmetrie entsteht, welche, wenn sie dem Verstande völlig fasslich bleibt, die Zierde auf der geringsten Stufe genannt werden kann. Je mannigfaltiger dann aber die Glieder werden, und je mehr jene anfängliche Symmetrie, verflochten, versteckt, in Gegensätzen abgewechselt, als ein offenbares Geheimnis vor unsern Augen steht, desto angenehmer wird die Zierde sein, und ganz vollkommen, wenn wir an jene ersten Grundlagen dabei nicht mehr denken, sondern als von einem Willkürlichen und Zufälligen überrascht werden."



A key episode from my school days


Let me recall a key episode from my school days. A teacher told us that we are forbidden to divide a number by zero. I replied that I could carry out such a calculation by choosing ever smaller divisors, until finally, I obtained an infinitely large number; and that when I would then multiply this number by zero I would obtain 1 and any other number. My teacher took hold of his iron key and knocked me on the head. When I repeated my argument in the next lesson he knocked me on the head again. Thus I came to learn that even mathematics, the kingdom of pure logic, has its forbidden zones, where logic fails and iron keys are required.


In the light of the above insights I shall put it like this: the division of a number by zero is a case where we leave the realm of mathematical logic in favor of the realm of art, life and nature, where all is equal and all unequal. Mathematical logic is a mental tool of building and constructing. If we wish to have this tool ready to hand we are not allowed to divide a number by zero, or to carry out similar operations; but we are always allowed to investigate the reverse side of the mirror and to explore the other realm of logic.





Once I was invited to give private lessons to a boy who had difficulties understanding the concept of equations. I told him: let us put away the schoolbooks and instead have a look at language. I asked him to give me an example of a sentence, and he did so. I then showed him that he had just formulated a verbal equation. As I don't recall his example, I shall invent a new one:


     The football game will take place tomorrow.


This sentence contains an equation:


     the football game - is - a tomorrow's event


We considered other sentences, and they always contained one or several equations.


     A car passes by   /   a car - is - something that passes by


We see a movement, we recognize a car, and the car and the act of moving belong together, allowing us to formulate an equation.


     The flowers are blooming


We see a happy sway of colors, we identify it with plants, and we formulate an equation.



Language is the means of winning the help and care of others on whom we depend in one way or another. By speaking we are building a world into which we invite our listener, and we shape and color it in such a way as to please him and to win his help and support. In order to do so we require a mathematical logic that is present in the basic structure of a sentence. All children, even those not good at mathematics, can establish this 'verbal equation' easily and naturally. It may well be  that many children who fear mathematics are capable of building the best-formulated, most elegant and colorful sentences.



A further solace


A further solace for those less proficient at mathematics. There are two realms of logic: a mathematical one; and another - a treasury of mathematical and scientific laws which have not yet been discovered. Those good at mathematics may be clever in a limited way, in the realm of known laws; while others may be clever in handling the many still unknown laws that nevertheless rule our lives. Dear teachers, you know well that half of your work involves motivating your pupils, and that the best way to this is to recognize and name the various abilities and talents of each individual child.







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