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Una Fundamentaci´On De La Historia De Las Matem´Aticas

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Una Fundamentaci´On De La Historia De Las Matem´Aticas UNA FUNDAMENTACION´ DE LA HISTORIA DE LAS MATEMATICAS´ UNA FUNDAMENTACION´ DE LA HISTORIA DE LAS MATEMATICAS´ Jesus Hernando P´erezAlc´azar OSCAR´ ARMANDO IBARRA RUSSI Rector ALEJANDRO ALVAREZ´ GALLEGO Vicerrector Acad´emico MARIO BALLESTEROS MEJ´IA Vicerrector Administrativo y Financiero NOHORA PATRICIA MORENO GARC´IA Vicerrectora de Gesti´onUniversitaria c Universidad Pedag´ogicaNacional c Jes´us Hernando P´erez Alc´azar Profesor investigador Universidad Sergio Arboleda ISBN: Primera edici´on,2007 Preparaci´oneditorial Universidad Pedag´ogicaNacional Fondo Editorial LUIS EDUARDO VASQUEZ´ SALAMANCA Coordinador Impresi´on Bogot´a,Colombia, 2007 Dedicado a: La memoria de mi padre, el educador matem´atico Jos´e Ignacio P´erez, Mi madre Aura Mar´ıaAlc´azar. Contenido Agradecimientos IV Presentaci´on V Prefacio VIII 1. Principios orientadores para la fundamentaci´onde la historia de la matem´aticas 1 1.1. (P1) Principio de Durkheim o de la divisi´onsocial trabajo . 1 1.2. (P2) Principio ´eticoy legal o de las tensiones entre permitido versus prohibido y conveniente versus inconveniente . 2 1.3. (P3) Principio acad´emico o de la dial´ecticaimaginarios versus teor´ıas 4 1.4. (P4) Principio de historicidad . 8 1.5. (P5) Principio de Chomsky o de la tensi´onentre finito e infinito . 10 1.6. (P6) Principio anfibi´oticoo de la tensi´onentre ser y no ser . 14 2. Ejemplos iniciales de documentos historiogr´aficos 19 2.1. Un art´ıculode divulgaci´on. 19 2.2. Un trabajo hist´orico-filos´ofico . 24 2.3. Historia de la historia . 27 2.4. (P7) Principio de Struik o de la dignificaci´onde la especie humana 30 2.4. Homenaje a grandes matem´aticos. 31 3. Documentos tipo memorias 35 ii 3.1. Unas memorias de Andr´eWeil . 36 3.2. (P8) Principio de contextualizaci´onde Dehn . 40 3.2. Unas memorias de Gaisi Takeuti . 50 4. Documentos tipo Autobiograf´ıasy biograf´ıascortas 59 5. Documentos tipo biograf´ıas 72 5.1. (P9) Principio de Infeld o de la excelencia literaria . 80 6. Documentos tipo grandes historias 81 6.1. (P10) Principio de Alberto Campos o de la tensi´onentre historia y educaci´on................................ 90 7. Investigaci´onprimaria 94 8. Historia de las matem´aticaselementales 98 8.1. (P11) Principio de Mar´ıade Losada o de la tensi´onentre los niveles 105 Ep´ılogo 106 Referencias 108 iii Agradecimientos A la Universidad Nacional de Colombia, que me brind´oel invaluable privilegio de ser uno de sus docentes por cerca de 35 a˜nos. A la Universidad Sergio Arboleda, que me ha acogido tan amablemente en estos ´ultimosa˜nos. A la Universidad Pedag´ogicaNacional que me ha permitido expresar p´ublica- mente mis puntos de vista metamatem´aticos. A Carlos Luque Arias y a su equipo de colaboradores de la Universidad Pedag´ogi- ca Nacional y la Universidad Sergio Arboleda quienes han construido y fortalecido un maravilloso espacio de encuentro de la comunidad acad´emica en matem´aticas. A Sergio Carrillo y Carlos Hurtado, sin cuya dedicaci´oneste trabajo no hubiera sido publicado. A mis familiares y amigos. iv Presentaci´on Ninguna disciplina cient´ıfica perder´ıa m´asque la matem´atica si prescindiera de su historia An´onimo En matem´aticasexiste una enorme continuidad hist´orica,el teorema de Pi- t´agorassigue siendo v´alidoy conservando su elegancia, en general los conceptos en matem´aticasse caracterizan por su larga vida, esto hace que los matem´aticos sean conscientes de los or´ıgeneshist´oricosde sus conceptos generando un respeto y un marcado sentimiento de deuda para con sus predecesores, de hecho grandes y destacados matem´aticoshan insistido en la continuidad y la coherencia de la evoluci´onde las matem´aticas,combinando de un modo muy fruct´ıferola exposici´on de sus propios resultados con consideraciones hist´oricas.En este sentido, podemos decir que la historia de la matem´aticase remonta a los griegos. La historia de la matem´aticases actualmente una ciencia que a la par de la matem´aticasu crecimiento en los ´ultimosa˜nosha sido exponencial, ha desarro- llado sus propias metodolog´ıas y, como otras disciplinas, organiza sus congresos internacionales y cuenta con instituciones y publicaciones que la estudian con rig- or, la divulgan y la promueven; socialmente, la historia de la matem´aticaes una ciencia ´utily necesaria, ha sido determinante en decisiones de pol´ıticaseducativas en los pa´ıses que la apoyan. La historiograf´ıaadem´asde recopilar el pasado, pro- porciona elementos fundamentales para la construcci´ondel presente y del futuro de la matem´atica. ¿C´omonos encontramos en Colombia frente a este contexto internacional? ¿Estamos haciendo bien las cosas en esta direcci´on? Pues bien, el autor nos muestra, de manera categ´orica,elementos que nos dan directrices y nos colocan en un punto de hacer una reflexi´onprofunda y tomar una o varias direcciones que nos lleven por el camino correcto. Algunos aspectos relevantes del libro son: La presentaci´onde unos principios orientadores para hacer investigaci´onen v Una fundamentaci´onde la historia de las matem´aticas historia de la matem´atica,desarrollando aspectos biogr´aficos,sociol´ogicosy bibli- ogr´aficos,y contextualizando el desarrollo de las comunidades matem´aticasy sus formas de organizaci´on.Es de resaltar como el aporte m´assignificativo, un cap´ıtu- lo sobre la historia de la matem´aticaelemental moderna, de la cual la literatura es pr´acticamente inexistente; adem´asde lo hist´orico,el tema permite establecer v´ınculos con la educaci´onmatem´atica. Cada uno de estos aspectos los presenta acompa˜nados de ejemplos contun- dentes y recreados con un lenguaje sencillo y emotivo, donde el autor se confunde muchas veces con las historias, nos muestra adem´asvarios principios y ejemplos vivos de nuestra historia actual que nos cuestiona en el sentido de “por qu´eno aplicamos estos principios al desarrollo y funcionamiento de nuestras comunidades acad´emicas para que hagan parte de nuestra cultura y tradici´onacad´emica”. De hecho estos principios han tenido alg´unimpacto en la Universidad Na- cional de Colombia como lo se˜nalael autor, pero especialmente en la Escuela de Matem´aticasde la Universidad Sergio Arboleda; all´ı, Jes´usHernando, en su labor cotidiana como investigador, catedr´atico,y en especial en su trabajo con ni˜nos tal- entosos, es fiel a los principios que expone; podemos afirmar, sin que ´else lo haya propuesto, que es su experiencia y vida acad´emica un excelente ejemplo de estos principios que le ha mostrado la historia. Por los testimonios que personalmente he tenido la oportunidad de escuchar de muchos de sus disc´ıpulos,estudiantes, y colegas el profesor Alfonso Castro, entre otros, coinciden palabras m´aspalabras menos con lo que aparece en este trabajo en la autobiograf´ıade Saunders MacLane titulada A mathematical autobiography en el prefacio escrito por el ge´ometray matem´aticoDavid Eisenbud, uno de los disc´ıpulosm´assobresalientes de MacLane, escribe Eisenbud: Saunder MacLane ha sido mi maestro, mentor y modelo desde el inicio de mi vida como matem´atico. Ha sido una relaci´onmuy especial para m´ı. El´ ha sido para m´ı una figura de gran honestidad e integridad, que ha realizado un trabajo muy fuerte para avanzar la investigaci´on y servir a la comunidad matem´atica. Sus creencias sobre el bien, lo correcto y lo racional, su inter´esen la esencia de las ideas matem´aticas, su enorme entusiasmo, y su gran optimismo fueron y han sido un gran atractivo para m´ı. En lo personal su ejemplo y sus consejos me han dado orientaciones muy im- portantes para hacer de la Escuela de Matem´aticasun centro acad´emico con un respeto muy grande a los maestros, con la puesta en marcha de los principios que se plantean, donde el celo por el conocimiento y los valores acad´emicos est´anpor encima de otros tipos de intereses. Me llama particularmente la atenci´onuno de los principios orientadores que ex- pone Jes´us Hernando, el principio anfibi´otico,que significa moverse en dos ´ambitos cualitativamente diferentes; por ejemplo, para trabajar en educaci´onmatem´atica se debe ser educador, matem´aticoy, por supuesto educador-matem´atico;para ser vi Presentaci´on historiador de las matem´aticasse debe ser historiador, matem´aticoe historiador- matem´atico;la anfibiosis implica entre otras cosas el reconocimiento de la autori- dad acad´emica de quienes solo se mueven en uno de los dos ambientes y lo hacen magistralmente, este principio resolver´ıa muy bien lo que el autor plantea cuando dice: En Colombia existen avances significativos pero a su vez dudas sobre los trabajos orientados al estudio del funcionamiento del mundo acad´emico de las matem´aticas. El d´ıa que comencemos a despojarnos de la desconfianza y del desconocimiento del otro, a aceptar ser evaluados sin descalificar al evaluador, habremos dado un salto cualitativamente grande. Aplicar el principio anfibi´oticoy los otros principios que se sugieren en este libro ser´ıaun buen comienzo para seguir el camino correcto para construir una historia de la matem´aticaacorde con el contexto internacional. Reinaldo N´u˜nez Director de la Escuela de Matem´aticas Universidad Sergio Arboleda Bogot´a,junio de 2007. vii Prefacio El presente trabajo ha sido elaborado paralelamente al desarrollo de varios cur- sos relacionados con el tema, durante diferentes semestres, en la carrera de Filosof´ıa de la Universidad Nacional y m´asrecientemente, en la carrera de Matem´aticasde la Universidad Sergio Arboleda, y forma parte del proyecto de investigaci´onEl Semic´ırculo de la Universidad Sergio Arboleda, en el cual venimos trabajando desde hace ya cuatro a˜nos. En cada una de las versiones del curso el prop´osito fundamental ha sido mostrar a los estudiantes c´omose realiza el trabajo investiga- tivo, tomando como caso el de esta interesante disciplina.
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