Infinitesimals via Cauchy sequences: Refining the classical equivalence

A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy’s sentiment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals.

DEMOSTRACIÓN DE TEOREMAS DE NÚMEROS NATURALES EN EL SISTEMA AXIOMÁTICO DE GIUSEPPE PEANO

Después de los trabajos del griego Euclides, fueron muchos, entre matemáticos y filósofos, su contribución hacia la axiomatización de la matemática, en sus diversas disciplinas matemáticas, tales como Dedekind, Grassmann, Frege, Hilbert, Peirce, Peano, entre otros. Lo que pretendemos, es presentar algunas demostraciones, principalmente con fundamentación axiomática del conjunto de los números naturales, trabajadas por el matemático italiano Giuseppe Peano, considerando la construcción de éste sistema, a partir del número natural cero, como punto inicial.

Geometry on my Mind

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory. Motion cannot exist without space—trajectories are the tracks of points, mathematical or physical, through it.