Archive pour mai 2008

21
mai

La topologie de Leibniz: analysis situs

Voilà que je veux vous présenter ici une nouvelle topologie que vous n’avez très certainement jamais apprise, et qui est surtout supérieure à celles qui existent aujourd’hui (à ma connaissance).

Je vais commencer par une citation de Leibniz qui sera l’avocate de la supériorité de sa topologie:

“Dieu a choisi [le monde] qui est le plus parfait, c’est à dire celui qui est en même temps le plus simple en hypotheses et le plus riche en phenomenes”

Cela veut dire un minimum de règles, de principes, de lois et un maximum de possibilités. C’est ce principe qu’il a appliqué au développement de ce qu’il a appelé l’analysis situs, qui est devenu la topologie, mais dont il est le véritable fondateur.

Leibniz, comme il a si bien réussi à me convaincre aussi, trouve que le monde géométrique de Descartes est loin à la fois de la réalité et aussi de la simplicité. Comment exprimer une forme apparemment simple comme un cercle ou une sphère. On doit déjà commencer, systématiquement, par définir un repère dit “cartésien”, et ensuite on doit déterminer une équation. Pour un cercle par exemple, après avoir défini le repère cartésien, on a x^2 + y^2 = r^2. Cela fait un total de 11 symboles (on compte les puissances comme un symbole à part entière, même si on peut l’écrire en exposant, car il s’agit de l’expression d’un principe différent). 11 symboles pour un cercle. Et cela reste relativement simple, mais à quel point se complique-t-on la vie pour exprimer des formes ne serait-ce qu’un peu plus complexe (cône, tore, ellipsoïde, etc.). C’est un système clairement handicapant et contre-intuitif. Mais il a un mérite, tout est immobile, il n’y a pas de surprise. C’est un monde froid, mais rassurant.

Leibniz n’aime pas ce monde, tout comme moi d’ailleurs. Il propose une autre manière de décrire des formes, en utilisant le minimum de symboles et une approche purement constructive et intuitive. Il l’appelle l’analysis situs, ce qui signifie l’analyse des situations:
“Les méthodes géométriques auxquelles je songe sont au nombre de deux: la première consiste à exprimer complètement une figure en n’utilisant que des caractères, sans l’aide d’explications verbales et sans y adjoindre de figure; la seconde consiste à le faire en n’utilisant que des mots, sans l’aide d’aucun autre caractère et sans l’aide d’aucune figure.”

Une autre citation prenante:
“L’algèbre n’est autre chose que la caractéristique des nombres indéterminés, ou des grandeurs. Mais elle n’exprime pas directement la situation, les angles, et le mouvement.”

Assez de suspense, voilà son invention (ou découverte?!?):
Soit le symbole “\/” signifiant “pareil que”. Soient les lettres du début de l’alphabet A, B, C, etc. des points fixes de l’espace, et les dernières lettres de l’alphabet Z, Y, X, etc. des ensembles de points de l’espace.

Exemple simple:

  • AB \/ AZ = l’ensemble des points Z tel qu’ils aient tous la même relation avec A que la relation de B avec A; autrement dit l’ensemble des points Z de sorte que la distance AB soit la même que la distance AZ; à vous de trouver la forme correspondante…
  • AZ \/ BZ = l’ensemble des points Z tel qu’ils aient tous la même relation avec A et B; autrement dit, l’ensemble des points Z tel qu’ils soient à la même distance de A que de B; à vous de trouver…
  • AZ \/ BZ \/ CZ = l’ensemble des points Z tel qu’ils soient à la même distance de A, B, et C…
  • ABC \/ ABZ = l’ensemble des points Z tel qu’ils aient la même relation avec A et B que la relation qu’entretient C avec A et B; autrement dit, l’ensemble des points Z tel que la distance AZ = AC, BZ = BC; un peu plus dur, mais faisable…

Toutes les citations viennent du livre “la caractéristique géométrique” par Leibniz édité chez VRIN. Je pourrai écrire aussi des citations très intéressantes sur les applications possibles d’une telle topologie…

2ème partie

Pour essayer de prolonger la discussion et surtout montrer mon niveau d’approfondissement de ce langage, je te propose deux nouvelles formes très particulières qui, je le pense, laissent entrevoir des possibilités intéressantes:

  • AB \/ AZ \/ ZX;
  • AB \/ AZ \/ ZX \/ XY.

Je voudrais terminer avec les propos de l’auteur du langage en question, notre très cher Gottfried Leibniz, tiré d’une lettre adressée à Christian Huyghens en 1679:

“Si elle était achevée de la manière que je la conçois, on pourrait faire en caractères qui ne seront que des lettres de l’alphabet la description d’une machine quelque composée qu’elle pourrait être, ce qui donnerait moyen à l’esprit de la connaître distinctement et facilement, avec toutes les pièces et même avec leur usage et mouvement, sans se servir de figures ni de modelles, et sans gêner l’imagination. [...] On pourrait faire aussi par ce moyen des descriptions exactes des choses naturelles, comme par exemple des plantes et de la structure des animaux.”

“Je crois qu’on pourrait manier par ce moyen la mécanique, presque comme la géométrie. Et qu’on pourrait même venir jusqu’à examiner les qualités des matériaux, parce que cela dépend ordinairement de certaines figures de leurs parties sensibles, comme si on décrivait la constitution, on en pourrait déduire leurs usages.”

11
mai

The end of illiteracy?

I want to present to you here a new revolutionary technique that I have developed in order to put an end for good to the problem of illiteracy in the world. It is a technique that doesn’t require any special training to understand on the part of the illiterate person, something which, on the one hand significantly reduces the costs related to such a project, and on the other hand, gives greater independence and self-sufficiency to the learner. I first start by using the case of Haiti as an illustration of the problem of illiteracy because it is one of the countries where this problem is more significant.

More than half of Haitians are illiterate

Illiteracy is a serious problem in Haiti. According to several sources, more than half of the Haitian population that lives on the island doesn’t know how to read, and thus doesn’t know how to write either. Imagine for an instant that you are illiterate. Your communication is done exclusively through auditory and visually means with the outside world. In order to communicate with someone, you absolutely need to talk to that person face to face or by telephone. You must always be “live” with someone to communicate with him or her. Your communication is strictly instantaneous and momentary. You could use a recording device, but that would certainly be extremely tedious.

Moreover, in a poor country like Haiti, illiterate people are bound to live in a world that begins and ends as far as their ears and eyes can hear and see, that is to say, not very far. Everything that comes from further away than this perimeter comes to you through someone that you hear speak in person, or through the radio. And if this person is illiterate like you (a probability of more than 50%), he will only tell you what he saw himself, or what he has been told by someone else. Just like the game that children play where a phrase is spoken in a chain and comes back to the beginning, the original message is always deformed. Imagine then the quality of information and ideas that an illiterate person receives. He is dependent on the honesty and the intelligence of the people with whom he communicates to understand the world outside of his sensory perceptions.

Now, how can a society hope to prosper when more than half of its population is maintained in this weak and fragile intellectual state? Even if the elite is literate and knows how to read and write and can install a government dedicated to the welfare and to the development of all, if the majority of the population remains illiterate, the victory will only be short lived. Convincing an illiterate person becomes as easy as organizing a political upheaval in the country. Every lie and political manipulation becomes child’s play, and a coup d’état easily succeeds. Illiteracy is perhaps one of the main elementary causes of the political instability of Haiti in the last decades.

How to solve this problem?

The situation is certainly very alarming for Haiti, like many other poor countries, as well as for all illiterate people on Earth. But this crisis can come to an end. Illiteracy can really become history.

The method that I am proposing is an original way and very quick way to teach to people how to read and write. It is an original method, albeit not a new one, because it dates back to the period when hieroglyphs were being used, as was the case in Ancient Egypt and in America at the time of the Olmecs, and other civilizations as well. We found on their pyramids, their buildings, their temples and artifacts writing systems based on hieroglyphs. And just like any writing system, it consisted of a one-to-one correspondence between symbols and sounds.

In the majority of the world’s languages today, we use letters, and each letter represents more or les a sound, and by organizing these sounds in our mind, we create words and we are able to read and understand what is written. This is why we are able to read words that we don’t know, in that we are able to pronounce the words simply by combining the different sounds represented by the letters (symbols).

Hieroglyphs work in the same way, but instead of using letters as symbols to represent sounds, it uses drawings. Every drawing is a sound. And by combining the sounds represented by the drawings (symbols), we are able to read. It is the same process with the writing system based on letters, the only difference being in the nature of the symbols being used (hieroglyphs versus letters).

How, then were the Ancient Egyptians and Olmecs able to read? Before answering that question, let us examine rapidly how we, who use letters, learn how to read. It is fairly simple to explain even though it may not be the quickest way to learn. We memorize in school the correspondence of the sounds and the 26 letters of the alphabet, and the combinations of letters such as “th” or “ch”. It is only a matter of learning them by heart, and through practice, we can be able to read without any problem. It is a little axiomatic, but it works.

Now, how did the Ancient Egyptians and Olmecs do it? First, they obviously knew how to speak before learning to read. When they saw a hieroglyph, they were already able to pronounce the word corresponding to the drawing. For example, if a hieroglyph was representing a jaguar, a young Olmec might have said to himself: “I know what that is. It is a jaguar.” And that was the case for all the hieroglyphs that represented objects from ordinary life, and that everyone could easily identify. Now, the originality and the ingenuity of such a system was this: each hieroglyph corresponded to the first sound of the word of the object that was drawn.

Again, with the example of the hieroglyph of the jaguar, this hieroglyph represented the sound “J” because it is the first sound of the word “jaguar”. In adopting such a system to the English language, the combination of a hieroglyph of a jaguar, an ostrich and a bear would correspond to the combination of the three sounds “j”, “o” and “b”. By putting them together, we get the sound “job”, and thus the very word “job”. Thus, in our new system of hieroglyphs, a job would be written with the hieroglyphs of a jaguar, an ostrich and a bear. (I will not bother you with my drawing skills!)

The development of a new writing system

What I am suggesting is in reality very simple. It is a project that could help Haiti directly, and all the people who still suffer from illiteracy. The project stems from the realization that all people and all individuals have a language and know how to speak it. From this vocabulary, a number of hieroglyphs must be developed representing concrete objects from everyday life and whose first sounds cover all the sounds used in the particular language. For Creole for instance, all the sounds can be found in the Latin alphabet by the use of one letter or two: a, j, k, and ch, in, etc. It is important, for very language, to use hieroglyphs that are easy to draw, that do not cause any vagueness in their interpretation and that represent universal things as much as possible, and also things that stimulate the senses and the imagination. For example, for the sound “C” or “K”, if we had to choose between the drawing of a car and a cat, we would choose the car because it is easier to recognize and because it is something closer to our human identity. For the sound “A”, we could choose an apple. And for the sound “T”, we could choose a table.

For this particular combination, the hieroglyphs of a car, an apple and a table yield the word “cat”. In this new alphabet, that’s how one would write the word cat.

Thus, this project of literacy will require the formation of a hieroglyphic writing system that covers all the sounds of a particular language. Then, one would only need to explain to others that each hieroglyphs corresponds to the first sound of the thing it represents. Once that has been understood combined with some good will and a little practice on the part of the learner, and he will easily be able to read.

The superiority of such a writing system is that it doesn’t require any training! Indeed, the illiterate person only needs to understand the principle of the “first-sound” of the drawing, and he will be able to read alone, by himself, independently of the presence of anyone. His reading will be slowly at the beginning, but with a little time, he will certainly be able to read as fast as any average reader. He will even be able to write immediately after having understood the “first-sound” principle. A table of all the hieroglyphs will have to be given to him, and he will then be able to draw the different hieroglyphs to form words, phrases and express ideas.

But, although he will be able to read in this hieroglyphic system in particular, he will not know how to read in the writing system with letters that the literate individuals of his people use. How should this transition be done?

A transition from hieroglyphs to letters

Again, the solution is simple. We would only need to add inside every hieroglyph the letter or letters corresponding to the particular sound. For example, inside the hieroglyph of … (Like I have done in the example above.) By getting used to reading the hieroglyphs and making the association between the symbols and the sounds, the individual could after a while, drop the drawings altogether and remember only the letters. The letter that he will see will remind him of the hieroglyph, and the hieroglyph will remind him of the sound, through the hieroglyph. After some time, his mental process will override the step of reminding the hieroglyph and he will be able to make the direct association between the letter and the sound: he will know how to read!!

A computerized translation

The first step is thus to establish a hieroglyphic alphabet for a given language. Once this alphabet has been done, it is necessary to produce texts that are to be read in this new system. The texts could be translations or new original ones. To do so, I think the most effective way would be to develop this writing system by computer. The simplest way would be to develop a program that would make an automatic translation between Latin letters and hieroglyphic symbols. Every hieroglyph would be recorded in the program and would be associated to a letter or a couple of letters (corresponding to one sound). The program could then translate an entire text by replacing every letter or couple of letters with the corresponding hieroglyph. The texts thus translated could be printed and distributed to the particular illiterate people , in all the regions of the world, according to their mother tongue.

The first universal text could be written to welcome them to the world of books, in the form of a history, or a fable and advise them of its possible dangers, as well as the pleasures one can get out of reading and writing.

A proof

To show you concretely that this method really works all the time, I encourage you to translate this sentence yourself:

Some possible problems

The only problem that I can foresee now is the orthography and the grammatical rules. Also the fact that there are different ways to represent the same sound. But the project that I am suggesting is not intended as a substitute for the study of languages of all the people of the world, but rather a simple tool to enable everyone to read and write in a system that is only a step between illiteracy and literacy.

Our future

We, as Human beings, are living in a very particular moment in our history. We have come to the point where we are all connected, as single Human species. Communication and transport means have totally  revolutionized our lives, and we can now consider ourselves truly as neighbors. But despite these increasing proximities and interdependences, we still suffer from racism, sexism, paranoia and war. A elementary and crucial step to remedy these problems is the literacy of all Human beings to give each and everyone of them the necessary tools to develop their freedom of thought and communication, thus eliminating one of the barriers that are still separating us.