Arquivo da tag: spivak

Spivak sobre definições e teoremas

ATENÇÃO: Este conteúdo foi publicado há 7 anos. Eu talvez nem concorde mais com ele. Se é um post sobre tecnologia, talvez não faça mais sentido. Mantenho neste blog o que escrevo desde os 14 anos por motivos históricos. Leia levando isso em conta.

“The reader probably suspects that the modern Stokes’ Theorem is at least as difficult as the classical theorems derived from it. On the contrary, it is a very simple consequence of yet another version of Stokes’ Theorem; this very abstract version is the final and main result of Chapter 4. It is entirely reasonable to suppose that the difficulties so far avoided must be hidden here. Yet the proof of this theorem is, in the mathematician’s sense, an utter triviality — a straight-forward computation. On the other hand, even the statement of this triviality cannot be understood without a horde of difficult definitions from Chapter 4. There are good reasons why the theorems should all be easy and the definitions hard. As the evolution of Stokes’ Theorem revealed, a single simple principle can masquerade as several difficult results; the proofs of many theorems involve merely stripping away the disguise. The definitions, on the other hand, serve a twofold purpose: they are rigorous replacements for vague notions, and machinery for elegant proofs.”

Michael Spivak, prefácio de “Calculus on Manifolds”.