8. Mathematics a children game?
To my little children.
Teach children to count? You are joking! Any kid of two and a half years, when he says “sleep” or “am stram gram, peak and peak and gram, bourgeois and bourgeois and Ratata, etc.” Is higher in mathematics without even know the tables of multiplication.
You will say perhaps: “Well, it seems that the child speaks as Broca’s aphasia?”And you are absolutely right. Some may wonder what it is Broca’s aphasia. Without going into details here, I will answer that Broca’s aphasia is a guy who pushes the syntax to its climax (though often say the opposite,) which can be expressed as the horizontal axis “verbal units and their chains.” To see clearly, you must have in mind what I said (in simplified terms!) Mediations theory of the sign, when we asked the question: “What do you think?” And, in particular, what I explained about the biaxiality analysis process that defines the word rationality. I quickly remind you that we left the traditional analysis (which dates from the Middle Ages, anyway, I learned it in school,) namely that analyzing a sentence, it was to isolate it, of course, elements (words or propositions,) and define their nature and function. But what we could not tell me before Gagnepain was that nature and function was, in fact, indivisible, there was a reciprocal relationship between the vertical axis identities, differences, quality, etc. and the horizontal axis counts, units their reports. Well, say that Broca’s aphasia was lost following an accident, the two axes, and one identity, that is to say the vertical, called the axis of the “taxonomy.” But, not having lost the other (the horizontal axis or “phrase,”) he can beautifully calculate, proving that the calculation is indeed the verbalism. That is what we must understand. Nobody knows how to define Math: Well, it is just a language that is sometimes called the language of science (as we call us the “meta,”) but in no case are acts of a science (contrary to what some people believe.) Whether it is the most common words we have to say things or concepts mathematically more sophisticated, in both cases is the same verbalism it is, in other words, this is nothing more than what is called in grammar (Pascal, in particular, was well aware of this close kinship uniting grammar and math.)
But in the case of mathematics, we are dealing with an amputated grammar, this means a uniaxiale. You can understand that mathematical analysis has remained essentially the axis (horizontal) functions exactly as Broca’s aphasia; I mean the axis of counting and quantity. We owe to Descartes the fact that amputation of mathematics has become a kind of inexplicable grammar. Grammar is not simple, and, insofar as it is implied, it is very easily accessible, but still, it seems more familiar, because everyone is talking “normally” this means it uses its legs to walk. But in the case of thought-legged (mathematical thinking,) there is nothing to explain. Go and ask children who say “am stram gram…” to explain what they mean, they can only answer you: “There is nothing to understand. That’s it!” And that is exactly what I answered to my math teacher, gently and without laughing: “This cannot be explained. It is like that. These teachers were, somehow, right. In reality, whether calculation, arithmetic, algebra or geometry, there was nothing to explain. But it was very bad pedagogy. These teachers would have told me: “I will, introduce you to the grammar of a language that has no lexicon or morphology or variations…a language that does not say much other than itself. A language that is not spoke, but only written or one spell at a pinch, etc. That, at least, was honest, but there is the height: “This is not explained. The height of absurdity! It was of Ionesco. I have included the day I stumbled by chance on a radio broadcast of “The Bald Soprano” in the early fifties (I was, therefore, a little younger than you.) I choked with laughter: “The roses of my grandmother are as yellow as my grandfather who was Asian …. That’s it! Your grandfather is Asian? You know nothing, basically, and I either. But, after all, we will ridicule: “It is like that.” You see that the “crisis” in mathematics education does not start today. But finally, we suffered more or less obediently, because the dream of all parents of my time was to see their offspring entering to the Ecole Polytechnique. But today, it seems that there would be enough for kids to do math. But finally, if this judge does not teachers of math, because, as I have said in the beginning, when he says “sleep” a child is a first-class mathematician. All the kids are born with the Fields medal around his neck! I would even say that grammar teachers (but where are they gone, those there?) should teach math, and math teachers should teach grammar. And when I see the textbooks available to those children! It is really a shame! In my opinion, is proof that intelligence does not understand these books there. I think that already today, the same teacher should teach grammar and mathematics. Ah yes, probably teachers know all about math, so I suppose, speak of calculating derivatives, logarithmic or arithmetic progression, but just ask a math teacher to give you the correct definition of that over which he “operates.” They know very well “operate,” but be sure that they do not pose the question of defining the concepts on which they work, beginning with the number concept!
But all of this is not the fault of teachers. Thought-legged, in this third millennium, we would rejoice or lament that it is a damn! But we will always need bridges and causeways engineers. Do not make me say otherwise. But what about the rest? Physicists, chemists, biologists realize every day that basically it was not worth they do it yesterday, so hard to look like mathematicians a pretext to become a scientist, because science changes appearance. We can model anything in science called “nature” that requires the consideration of quality, without taking into account the interaxiality. Well, we are going to a different conception of mathematics, to anthrop mathematics. And how can we do that revolution! You are not going to teach your child to unlock as Broca’s aphasia. All in all, it is better to remove the teaching of mathematics and to quickly train teachers of grammar. I say many professors of grammar, in the most traditional sense, not “language,” since this grammar has been completely betrayed by the post-Saussurean linguistics. I am well placed to tell you. The language has been promoting the “little things” to talk like that Daudet was “pawn” in high school Ales. This was the Raft of the Medusa for “modern” literary who felt lost in the world that they were contemporary. Then, after 68, all these guys have found it: linguistics, semiotics, etc. With this, they said: “We will become the humanities popes, the popes of the modern world.” So if math were betrayed by Descartes and grammar by people like Chomsky, you understand that this famous crisis in mathematics education is in fact a crisis of knowledge in its totality.
To conclude: we have begun by the easier things. It is true that Descartes we got off the track by insisting on the concepts of quantity and measurement: that we may be allowed to go to the moon, but to the price of a considerable simplification of the conceptual thinking mathematicians (except that of Einstein, as you know,) since they no longer have taken into account the quantity (this quantity to be continuous or discontinuous,) laughing less about the quality, or attempting (unsuccessfully) to reduce this quality to the quantity. Well, it is finished. Mathematics of tomorrow, if they still exist, will considerably be more difficult! A such point that we can say, in a century may be: “really, mathematics in the past, it was a game of children!”