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I am going to tell you a tale of mathematics. It is based on a mathematical theory: all the mathematical facts are proven. Their meaning is put into perspective so as to relate to the reality of the world around each of you.
Mathematics is built on numbers. Numbers are familiar to all of you, are they not? Let us suppose you choose at random a number between all possible mathematical real numbers between 0 and 1, what is the probability that the first digit is a 1 or a 9 for example? We can all agree that all digits from 1 to 9 have equal probability 1/9.
But what if I choose a number at random in the list of the altitudes of the 10 highest mountains in each country of the world? Then it is another story: the astonishing fact is that the first digit is 3 times more likely to be 1 than 9! And this would be true for any set of numbers produced by Nature. This is the signature that the world is nonlinear. This means that a quantity a will appear in Nature's computing formula under a product form like (a * a) or (a * a * a). This means also that the so-called "real" numbers in mathematics have different properties than the natural numbers which are produced by Nature. These natural numbers should indeed be viewed as the real "real" numbers.
Now numbers are used in practice to compute. And for complex and long computations, one uses a computer. But computers do not compute exactly!! Try to compute 1/3 times 3 on you favorite hand-held calculator: you will not get 1! This is well-known to high school students. But does it matter? Not always. This is why computers are useful in everyday life.
It matters at "singularities". A singularity is a technical word to describe a discontinuity where something new happens. Look at this table. On the surface, you cannot distinguish between this point and that point. They all look alike, because the surface is regular. But you can distinguish between any such point and a point at the edge. The four edges of the table give it its shape. At first sight, one could easily dismiss singularities because they are rare, they are at the border. But they shape the world. You see me because my shape appears cut on the white back screen.
Singularities, of course, play a significant role in physics. For example, you go from the wave optics of Fresnel to the geometric optics of Descartes, by letting the wavelength tend to zero. And this limit is highly singular in mathematical jargon. In the physical world we live in, this singularity creates beauty. It creates the glittering of the surface of the sea under the sun. It also creates the fascinating ever changing patterns that you see on the bottom of a shallow swimming pool. It makes the stars twinkle at night. And after a storm, when the sun light is deflected through rain drops floating in the air, it designs a rainbow in the sky.
Recognizing the role of singularities means a complete thought revolution. A revolution may be historically as important as the Copernican revolution which put the sun at the center of our planetary system. The shift of paradigm occurred gradually during the 20th century, from 1910 to 1980, as scientists began to abandon the prevalent dogma of continuity and started to focus more on discontinuity. They realized that new things can emerge only at singularities, that is at boundaries.
This paradigm shift led to the well popularized theory of "chaos". Chaos appears with nonlinearity. The same nonlinearity which is reflected in natural numbers by Nature. Chaotic phenomena are highly unstable: the flapping of the wings of a butterfly in Hong Kong may result in a tornado in Florida three weeks later. More flatly: small causes may bring huge consequences.
Because they are so unstable, such phenomena cannot be computed exactly, they can only be computed on a computer. But this is good news: exact computations can be totally misleading when it comes to the emergence of a new phenomenon. Exact with reference to what? If we know the exact reference, then it belongs to the past; it cannot be NEW.
Why are unstable phenomena encountered more and more often in high tech? Because one wants to get a more global view of a phenomenon in relation with its environment. This is done by coupling. Two independent basic phenomena can be coupled through a feedback loop. This coupling creates one single phenomenon of higher complexity. And the resulting more complex phenomenon has new properties which can be captured by the computer, but cannot be captured by the exact arithmetic of mathematics.
You see, it does not always pay to compute right. Specially when one deals with the emergence of the new. And what is newer under the sun than a new life?
For conventional Mathematics:
1 + 1 = 2
is a true statement, but it expresses a static truth. In the third millennium, it will be understood that it should sometimes be replaced by the dynamic expression:
1 + 1 ==> 3
which is closer to the life experience of every one of you, provided that the two ones are not identical!
This realization may lead to more qualitative Mathematics, to the "Mathematics of Life" which will emerge, no doubt, during the third millennium.
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