tag:blogger.com,1999:blog-3722233.post926482972311551134..comments2018-03-19T03:09:46.700-04:00Comments on Computational Complexity: What Makes a Great DefinitionLance Fortnowhttps://plus.google.com/101693130490639305932noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3722233.post-69908870062260712632017-08-03T15:30:09.511-04:002017-08-03T15:30:09.511-04:00Here is a favorite and famous quote (by Spivak) ab...Here is a favorite and famous quote (by Spivak) about the mathematicians' approach to definitions. It departs, or at least adds nuance, to your claim that definitions should be simple.<br /><br />"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."Sashohttps://www.blogger.com/profile/09380390882603977159noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-91878518894206402102017-08-03T08:56:57.271-04:002017-08-03T08:56:57.271-04:00This is a great exposition on good "external&...This is a great exposition on good "external" definitions for theorems, but what about good "internal" definitions for lemmas? Sometimes what allows a proof to go forward is a correct definition, that "abstracts away" the parts that get in the way of a mathematical bashing. Good enough internal definitions sometimes even become external with time (and the lemma then becomes the theorem).Eldarnoreply@blogger.com