{"id":3598,"date":"2011-11-24T09:43:59","date_gmt":"2011-11-24T09:43:59","guid":{"rendered":"http:\/\/meeseeks:5080\/blog\/?p=3598"},"modified":"2011-11-24T09:43:59","modified_gmt":"2011-11-24T09:43:59","slug":"a-brief-history-of-mathematics","status":"publish","type":"post","link":"https:\/\/vukutu.com\/blog\/2011\/11\/a-brief-history-of-mathematics\/","title":{"rendered":"A brief history of mathematics"},"content":{"rendered":"<p>Australian category-theorist Ross Street has an elegant, one-page summary of the first 2,500 years of western mathematics, <a href=\"http:\/\/www.math.mq.edu.au\/~street\/ToposPhil.pdf\" target=\"_blank\">here<\/a>.\u00a0 This was apparently a handout given in a talk to the Macquarie University Philosophy Students Society in 1984.\u00a0 I found Street&#8217;s high-level view of what (some important) mathematicians have (mostly) been doing illuminating and thought-provoking, and so I reproduce it here.<br \/>\nA nice way to think about topoi, of course, is that due to Rob Goldblatt: \u00a0a topos is the most general object that has all the properties of the category of sets.<br \/>\n&nbsp;<\/p>\n<blockquote><p><strong>Space, Sets and Beyond<\/strong><br \/>\n<strong>First Cycle:\u00a0 General spaces advance the study of naive geometry<\/strong><br \/>\n1. Naive geometry:\u00a0 Zeno, Eudoxus.<br \/>\n2. Axiomatic geometry (unique model intended): Euclid, Apollonius (c. 300-200 BC).<br \/>\n3. Algebraic techniques (coordinate geometry): Descartes 1596-1650.<br \/>\n4. Non-Euclidean geometry (independence of the &#8220;parallels axiom&#8221;:\u00a0 models without parallels axiom constructed from a model with it):\u00a0 Gauss, Bolyai, Lobatchewski (early 19C).<br \/>\n5. Locally Euclidean spaces:\u00a0 Riemann 1826-1866, Lie.<br \/>\n6. Relationships between spaces (continuity, linearity): Cauchy, Cayley, Weierstrass, Dedekind (1880-present).<br \/>\n&nbsp;<br \/>\n<strong>Second Cycle:\u00a0 Toposes can be viewed as even more general spaces<\/strong><br \/>\n1. Naive set theory: Peano, Cantor (c. 1900).<br \/>\n2. Axiomatic set theory (unique model intended): Hilbert, Godel, Bernays, Zermelo, Zorn, Fraenkel.<br \/>\n3. Abstract algebra (mathematical logic): Boole, Poincare, Hilbert, Heyting, Brouwer, Noether, Church, Turing.<br \/>\n4. Non-standard set theories (independence of the &#8220;axiom of choice&#8221; and &#8220;continuum hypothesis&#8221;; Boolean-valued models; non-standard analysis): Godel, Cohen, Robinson (1920-1950).<br \/>\n5. Local set theory (sheaves): Leray, Serre, Grothendieck, Lawvere, Tierney (1945-1970).<br \/>\n6. Relationships between toposes (a &#8220;topos&#8221; is a generalized set theory): 1970-present.<br \/>\n<strong><br \/>\n<\/strong><\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Australian category-theorist Ross Street has an elegant, one-page summary of the first 2,500 years of western mathematics, here.\u00a0 This was apparently a handout given in a talk to the Macquarie University Philosophy Students Society in 1984.\u00a0 I found Street&#8217;s high-level view of what (some important) mathematicians have (mostly) been doing illuminating and thought-provoking, and so [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[50],"tags":[],"class_list":["post-3598","post","type-post","status-publish","format-standard","hentry","category-mathematics","p1","y2011","m11","d24","h09"],"_links":{"self":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/3598","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=3598"}],"version-history":[{"count":0,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/posts\/3598\/revisions"}],"wp:attachment":[{"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/media?parent=3598"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/categories?post=3598"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vukutu.com\/blog\/wp-json\/wp\/v2\/tags?post=3598"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}