Para los interesados en los fundamentos de las matemáticas:
“Matemáticas. La pérdida de la certidumbre” de Morris Kline.
Clase del 21 de octubre (1 hora)
Hoy hemos resuelto el problema número 13 de la hoja de problemas del Tema 1. La solución es un poco liosa, pero nos ha servido para repasar algunos resultados y técnicas relacionados con órdenes de elementos.
Hemos comenzado también el Tema 2 (Homomorfismos de grupos). Hemos definido el concepto de homomorfismo de grupos (que es el análogo al de “aplicación lineal” en álgebra lineal) y hemos definido los diversos tipos de homomorfismos según sean inyectivos, suprayectivos o biyectivos (monomorfismo, epimorfismo o isomorfismo). Un homomorfismo de un grupo $G$ en sí mismo se dice que es un endomorfismo. Un automorfismo es un endomorfismo que es también isomorfismo. Son conceptos que os resultarán familiares (por el álgebra lineal).
En la próxima clase resolveremos algún problema más y continuaremos con la teoría del Tema 2.
Problema propuesto
Problema B-4: “Consideremos los siguientes subgrupos de $S_9$ :
$$
H:=\langle(1,2)(3,4)\rangle \quad \text { у } \quad N:=\langle(5,6,7,8,9)\rangle .
$$
(a) Demuestra que $H N$ es un subgrupo de $S_9$.
(b) Demuestra que $G=H N$ es producto directo interno de $H$ y $N$.”
De «Introduction to Mathematical Philosophy»(Bertrand Russell)
«Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics. But it should be understood that the distinction is one, not in the subject matter, but in the state of mind of the investigator. Early Greek geometers, passing from the empirical rules of Egyptian land-surveying to the general propositions by which those rules were found to be justifiable, and thence to Euclid’s axioms and postulates, were engaged in mathematical philosophy, according to the above definition; but when once the axioms and postulates had been reached, their deductive employment, as we find it in Euclid, belonged to mathematics in the ordinary sense. The distinction between mathematics and mathematical philosophy is one which depends upon the interest inspiring the research, and upon the stage which the research has reached; not upon the propositions with which the research is concerned.»
