The notion of number and its extraordinary generalizations are closely linked to the history of mankind. And life itself is steeped in mathematics: much of man's comparisons, as well as everyday gestures and attitudes, consciously or not allude to arithmetic judgments and geometric properties. Not forgetting that science, industry and commerce put us in permanent contact with the wide world of mathematics.
The language of numbers
In every age of human evolution, even in later times, the sense of number is found in man. This ability allows you to recognize that something changes in a small collection (for example, your children, or your sheep) when, without your direct knowledge, an object has been removed or added.
The meaning of number, in its primitive meaning and intuitive role, is not to be confused with the ability to count, which requires a more complicated mental phenomenon. If counting is an uniquely human attribute, some species of animals appear to have a rudimentary sense of number. Thus, at least, competent observers of animal customs give their opinion. Many birds have a sense of number. If a nest contains four eggs, one can be taken without anything, but the bird will probably leave the nest if two eggs are missing. In some inexplicable way, he can distinguish two from three.
The murdered crow
A feudal lord was determined to kill a crow that had nested in the tower of his castle. Time and again he tried to surprise the bird, but in vain: when the man approached, the crow flew from its nest, stood vigilantly on top of a nearby tree, and only returned to the tower when it was already empty. One day you resorted to a trick: two men entered the tower, one stayed inside and the other went out and left. The bird was not fooled and, to return, waited for the second man to leave. The ploy was repeated over the next few days with two, three, and four men, always unsuccessfully. Finally, five men entered the tower and then four, one after the other, while the fifth readied the trebuchet waiting for the crow. Then the bird lost count and life.
Zoos with a sense of number are very few (do not even include monos and other mammals). And the perception of numerical quantity in animals is so limited that one can scorn it. However, in man too this is true. In practice, when civilized man needs to distinguish a number he is not used to, he consciously uses or not - to help his sense of the number - devices such as comparison, grouping, or counting. The latter, especially, has become so integral to our mental structure that tests of our direct numerical perception have been disappointing. This evidence concludes that the meaning visual straight from the number possessed by civilized man rarely exceeds number four, and that the meaning tactile It is even more limited.
Limitations come from afar
Studies of primitive peoples provide remarkable evidence of these results. Savages who have not yet evolved sufficiently to rely on their fingers are almost completely devoid of any notion of number. The jungle inhabitants of South Africa have no numerical words other than one, two and Many, and yet these words are unlinked that it can be doubted that the natives give them a very clear meaning.
There is really no reason to believe that our remote ancestors were better equipped, since all European languages have traces of these ancient limitations: the English word thrice, just as the Latin word Tue, has two directions: "three times" and "a lot". There is a clear connection between the Latin words three (three) and trans (further). The same thing happens in French: three (three) and three (much).
How was the concept of number born? From experience? Or, on the contrary, did the experiment simply serve to make explicit what was already latent in the mind of primitive man? Here is a fascinating subject for philosophical discussion.
Judging from the development of our ancestors by the mental state of today's savage tribes, it is impossible to conclude that their mathematical initiation was extremely modest. A rudimentary sense of number, no larger than certain birds, was the core from which our conception of number was born. Reduced to the direct perception of the number, the man would have gone no further than the raven murdered by the feudal lord. However, through a series of circumstances, man has learned to supplement his limited perception of number with a device designed to exert extraordinary influence on his future life. This device is the operation of tell, and it is to him that we owe the progress of humanity.
The number without count
Nevertheless, even though it may seem strange, it is possible to come up with a clear and logical idea of number without counting. Entering a movie theater, we have before us two sets: the armchairs of the room and the spectators. Not to mention, we can assure whether or not these two sets have the same number of elements and, if not, which one is smaller. Indeed, if each seat is occupied and no one is standing, we know that the two sets have equal numbers. If all the chairs are occupied and there are people standing in the room, we know that there are more people than armchairs.
This knowledge is made possible by a procedure that dominates all mathematics, and named after two-way correspondence. This consists in assigning each object in one set an object in another, and continuing on until one or both sets are exhausted.
The technique of counting in many primitive peoples is reduced to precisely such associations of ideas. They record the number of their sheep or their soldiers through incisions made in a piece of wood or through stacked stones. We have proof of this procedure in the origin of the word "calculation"from the Latin word calculuswhich means stone.
The idea of matching
THE two-way correspondence it boils down to a "match" operation. It can be said that the counting is done by matching each object in the collection (set), a number that belongs to the natural succession: 1,2,3…
We point to an object and say: one; points to another and says: two; and so on until the objects of the collection are exhausted; If the last pronounced number is eight, we say that the collection has eight objects and is a finite set. But today's man, even with poor knowledge of mathematics, would begin numerical succession not by one but for zero, and would write 0,1,2,3,4…
Creating a symbol to represent "nothingness" is one of the boldest acts in the history of thought. This creation is relatively recent (perhaps in the early centuries of the Christian era) and was due to the demands of written numbering. O zero not only allows you to simply write numbers, but also perform operations. Imagine the reader - do a division or multiplication in Roman numbers! And yet, even before the Romans, Greek civilization had flourished, where some of the greatest mathematicians of all time lived; and our numbering is much later than all of them.
From relative to absolute
It would seem at first glance that the one-to-one correspondence process can only provide a means of comparing, by comparison, two distinct sets (such as the sheep of the flock and the stacked stones) and is unable to create the number in the absolute sense of the word. However, the transition from relative to absolute is not difficult.
Creating sets modelsTaken from the world around us, and each characterizing a possible grouping, the evaluation of a given set is reduced to the selection, among the model sets, of the one that can be matched one-on-one with the given set.
It began like this: the wings of a bird could symbolize the number two, the leaves of a clover the number three, the horse's legs the number four, the fingers of the hand the number five. Evidence that this could be the origin of numbers is found in many early languages.
Of course, once created and adopted, the number disengages from the object that originally represented it, the connection between the two is forgotten, and the number becomes a model or symbol. As man learned to use language more and more, the sound of words expressing the first few numbers replaced the images for which he was created. Thus the initial concrete models took the abstract form of the names of the numbers. It is impossible to know the age of this spoken numerical language, but it undoubtedly preceded the appearance of writing by several million years.
All traces of the initial meaning of the words designating the numbers have been lost, with the possible exception of five (which in several languages meant hand, or extended hand). The explanation for this is that while the names of numbers have remained invariable since the days of their creation, revealing remarkable stability and similarity in all linguistic groups, the names of the concrete objects that gave birth to them have undergone a complete metamorphosis.
Words representing numbers in some Indo-European languages
Source: Knowing Encyclopedic Dictionary - April CulturalNext: History of Geometry <