{"id":654,"date":"2009-06-07T15:23:31","date_gmt":"2009-06-07T15:23:31","guid":{"rendered":"http:\/\/meeseeks:5080\/blog\/?p=654"},"modified":"2024-09-15T10:06:23","modified_gmt":"2024-09-15T10:06:23","slug":"achilles-and-the-tortoise","status":"publish","type":"post","link":"https:\/\/vukutu.com\/blog\/2009\/06\/achilles-and-the-tortoise\/","title":{"rendered":"Achilles and the Tortoise"},"content":{"rendered":"
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An\u00a0amusing\u00a0account<\/a> (at least to a mathematician) by Harvey Friedman of an encounter with\u00a0eccentric Russian mathematician and dissident Alexander Yessenin-Volpin<\/a>. Friedman supervised<\/a> the Stanford PhD of John E. Hutchinson, who taught me calculus.\u00a0 (Hat tip: AB<\/a>)<\/p>\n

\nLet me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of ‘points’ of that length. There is the obvious ‘draw the line’ objection, asking where in , . . . , 2^100 do we stop having ‘Platonistic reality’? Here this . . . is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist [mathematician Alexander] Yessenin Volpin during a lecture of his.\u00a0 He asked me to be more specific.\u00a0 I then proceeded to start with 2^1 and asked him whether this is ‘real’ or something to that effect.\u00a0 He virtually immediately said yes.\u00a0 Then I asked about 2^2, and he again said yes, but with perceptible delay. Then 2^3, and yes, but with more delay.\u00a0 This continued for a couple of more times, till it was obvious how he was handling this objection.\u00a0 Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 than he would to answering 2^1.\u00a0 There is no way that I could get very far with this. (pp. 4-5).<\/p><\/blockquote>\n

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Note:<\/em> Of course, Friedman is wrong about the . . . being replaced by 100 items. We would expect it to be replaced with just 96 items, since 4 items in the list of 100 are already listed explicitly.<\/p>\n

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Reference:<\/em><\/p>\n

<\/blockquote>\n

Harvey M. Friedman [2002]: Lecture Notes on Philosophical Problems in Logic. <\/em>Princeton University.<\/p>\n

<\/blockquot\n<\/p>\n","protected":false},"excerpt":{"rendered":"

An\u00a0amusing\u00a0account (at least to a mathematician) by Harvey Friedman of an encounter with\u00a0eccentric Russian mathematician and dissident Alexander Yessenin-Volpin. Friedman supervised the Stanford PhD of John E. Hutchinson, who taught me calculus.\u00a0 (Hat tip: AB) Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,50,51],"tags":[],"class_list":["post-654","post","type-post","status-publish","format-standard","hentry","category-argumentation","category-mathematics","category-matherati","p1","y2009","m06","d07","h15"],"_links":{"self":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/654","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/comments?post=654"}],"version-history":[{"count":3,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/654\/revisions"}],"predecessor-version":[{"id":13149,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/654\/revisions\/13149"}],"wp:attachment":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/media?parent=654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/categories?post=654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/tags?post=654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}