tag:blogger.com,1999:blog-3722233.post926482972311551134..comments2023-12-05T03:54:35.127-06:00Comments on Computational Complexity: What Makes a Great DefinitionLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3722233.post-69908870062260712632017-08-03T14:30:09.511-05:002017-08-03T14:30:09.511-05:00Here 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-91878518894206402102017-08-03T07:56:57.271-05:002017-08-03T07:56:57.271-05:00This 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