Analogía fractal de un campo romano. Generado por Myriam B. Mahiques. Archivos personales. Fractal analogy of a Roman field. Generated by Myriam B. Mahiques. Personal archives.
Analogy in a fractal urban morphology. Generated by Myriam B. Mahiques. Personal archives
Mathematicians therefore proceed “by construction,” they “construct” more complicated combinations. When they analyse these combinations, these aggregates, so to speak, into their primitive elements, they see the relations of the elements and deduce the relations of the aggregates themselves. The process is purely analytical, but it is not a passing from the general to the particular, for the aggregates obviously cannot be regarded as more particular than their elements. Great importance has been rightly attached to this process of “construction,” and some claim to see in it the necessary and sufficient condition of the progress of the exact sciences. Necessary, no doubt, but not sufficient!
For a construction to be useful and not mere waste of mental effort, for it to serve as a stepping-stone to higher things, it must first of all possess a kind of unity enabling us to see something more than the juxtaposition of its elements. Or more accurately, there must be some advantage in considering the construction rather than the elements themselves. What can this advantage be? Why reason on a polygon, for instance, which is always decomposable into triangles, and not on elementary triangles?
It is because there are properties of polygons of any number of sides, and they can be immediately applied to any particular kind of polygon. In most cases it is only after
long efforts that those properties can be discovered, by directly studying the relations of elementary triangles. If the quadrilateral is anything more than the juxtaposition of two triangles, it is because it is of the polygon type.
A construction only becomes interesting when it can be placed side by side with other analogous constructions for forming species of the same genus. To do this we must necessarily go back from the particular to the general, ascending one or more steps. The analytical process “by construction” does not compel us to descend, but it leaves us at the same level. We can only ascend by mathematical induction, for from it alone can we learn something new. Without the aid of this induction, which in certain
respects differs from, but is as fruitful as, physical induction, construction would be powerless to create science.
Let me observe, in conclusion, that this induction is only possible if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science, for the different moves of the same piece are limited and do not resemble each other.
From Science and Hyphotesis. By Henri Poincaré. Edition of New York, 1905 (Bolt letters in this blog)
From Science and Hyphotesis. By Henri Poincaré. Edition of New York, 1905 (Bolt letters in this blog)
Analogy of a fractal and an urban shape. Generated by Myriam B. Mahiques. Personal archives.
Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9
Poincaré was an influential French philosopher of science and mathematics, as well as a distinguished scientist and mathematician. In the foundations of mathematics he argued for conventionalism, against formalism, against logicism, and against Cantor’s treating his new infinite sets as being independent of human thinking. Poincaré stressed the essential role of intuition in a proper constructive foundation for mathematics. He believed that logic was a system of analytic truths, whereas arithmetic was synthetic and a priori, in Kant‘s sense of these terms. Mathematicians can use the methods of logic to check a proof, but they must use intuition to create a proof, he believed.
He maintained that non-Euclidean geometries are just as legitimate as Euclidean geometry, because all geometries are conventions or “disguised” definitions. Although all geometries are about physical space, a choice of one geometry over others is a matter of economy and simplicity, not a matter of finding the true one among the false ones.
For Poincaré, the aim of science is prediction rather than, say, explanation. Although every scientific theory has its own language or syntax, which is chosen by convention, it is not a matter of convention whether scientific predictions agree with the facts. For example, it is a matter of convention whether to define gravitation as following Newton’s theory of gravitation, but it is not a matter of convention as to whether gravitation is a force that acts on celestial bodies, or is the only force that does so. So, Poincaré believed that scientific laws are conventions but not arbitrary conventions.
From Mauro Murzi´s article at Internet Encyclopedia of Philosophy
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