## lunes, 2 de octubre de 2006

### Relación entre Física y Matemáticas

Un amigo me contó que existía este artículo:
http://pauli.uni-muenster.de/~munsteg/arnold.html
Del que extraigo un par de "extractos":
"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."
algo así como: "Las matemáticas son una parte de la Física. La Física es una ciencia experimental, es decir, parte de la ciencia de la naturaleza. Las Matemáticas son la parte de la Física en la que hacer experimentos es barato"
Seguimos:
"In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences."
" A Mitad del siglo veinte se intentó dividir la física y la matemática. Las consecuencias fueron catastróficas. Generaciones de matemáticos crecieron conociendo la mitad de la ciencia y, por supesto, con total ignorancia del resto de ciencias."

¿Qué os parece? Bastante atrevido. Está claro que la Física se estudia al mismo tiempo que las matemáticas, y que la intuición física ayuda a comprender mejor esas matemáticas. intentad enseñarle a alguien ecuaciones diferenciales de manera abstracta y luego contadle que cierta ecuación es la que describe un muelle y por eso la solución es un seno "amortiguado" o que esta otra describe movimientos planetarios. es bastante más sencillo. De hecho, ¿tiene sentido estudiar ecuaciones diferenciales si no se aplicasen a lo que observamos en la naturaleza?

On teaching mathematics by V.I. Arnold

This is an extended text of the address at the discussion on teaching of mathematics in Palais de Découverte in Paris on 7 March 1997.

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.

In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).

Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians - both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.

The ugly building, built by undereducated mathematicians who were exhausted by their inferiority complex and who were unable to make themselves familiar with physics, reminds one of the rigorous axiomatic theory of odd numbers. Obviously, it is possible to create such a theory and make pupils admire the perfection and internal consistency of the resulting structure (in which, for example, the sum of an odd number of terms and the product of any number of factors are defined). From this sectarian point of view, even numbers could either be declared a heresy or, with passage of time, be introduced into the theory supplemented with a few "ideal" objects (in order to comply with the needs of physics and the real world).

Unfortunately, it was an ugly twisted construction of mathematics like the one above which predominated in the teaching of mathematics for decades. Having originated in France, this pervertedness quickly spread to teaching of foundations of mathematics, first to university students, then to school pupils of all lines (first in France, then in other countries, including Russia).

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.

For example, these students have never seen a paraboloid and a question on the form of the surface given by the equation xy = z^2 puts the mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like x = t^3 - 3t, y = t^4 - 2t^2) on a plane is a totally impossible problem for students (and, probably, even for most French professors of mathematics).

Beginning with l'Hospital's first textbook on calculus ("calculus for understanding of curved lines") and roughly until Goursat's textbook, the ability to solve such problems was considered to be (along with the knowledge of the times table) a necessary part of the craft of every mathematician.

[...]
V.I. Arnold