My answer to a question on Quora:<\/p>\n
\nDo mathematicians feel embarrassed when a conjecture they claim is disproved by counter-example?<\/span><\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n<\/div>\nI do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.<\/p>\n
Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine.\u00a0Refutations\u00a0are as important as proofs. There is a famous book about the role of refutations in mathematics,\u00a0Imre Lakatos<\/a>\u2019\u00a0Proofs and Refutations<\/a>.<\/p>\n
And the famous\u00a0Lewis Carroll<\/a>\u2019s lines in\u00a0Through the Looking-Glass\u00a0<\/a>capture the spirit:<\/p>\n
\n“I can’t believe that!” said Alice.
\n“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”
\nAlice laughed. “There’s no use trying,” she said: “one\u00a0can’t\u00a0believe impossible things.”
\n“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”<\/p>\n<\/blockquote>\nProofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.<\/p>\n
Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he\/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims \u2014 to prove a conjecture and refute it \u2014 alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.<\/p>\n
But, if you look back at that zig-zag prowl in search of a kill, you may say that\u00a0half of the time the mathematician believed impossible. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.<\/p>\n
There is one extreme case of the proof\/refutation balance: the original proof of the\u00a0Classification of finite simple groups<\/a>. I quote Wikipedia:<\/p>\n
\nThe proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.<\/p>\n<\/blockquote>\n
As a rule, almost each of these \u201cseveral hundred journal articles\u201d\u00a0contains a proof of a particular theorem, a special case or an intermediate step of the \u201cglobal\u201d statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of\u00a0non-existence of a minimal counterexample\u00a0to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora:\u00a0What are some aspects of mathematics that are nonsense?<\/a>\u00a0But these arguments about eventually non-existent minimal counterexamples are\u00a0not nonsense\u00a0\u2014 for example, they can be re-used in proving theorems in other branches of group theory.<\/p>\n","protected":false},"excerpt":{"rendered":"
My answer to a question on Quora: Do mathematicians feel embarrassed when a conjecture they claim is disproved by counter-example? I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved. Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-78","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/78","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/comments?post=78"}],"version-history":[{"count":2,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/78\/revisions"}],"predecessor-version":[{"id":80,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/78\/revisions\/80"}],"wp:attachment":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/media?parent=78"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/categories?post=78"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/tags?post=78"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}