{"id":6663,"date":"2015-08-31T21:36:10","date_gmt":"2015-09-01T04:36:10","guid":{"rendered":"http:\/\/www.rudyrucker.com\/blog\/?p=6663"},"modified":"2015-09-01T10:42:18","modified_gmt":"2015-09-01T17:42:18","slug":"john-conway-and-the-absolute-continuum","status":"publish","type":"post","link":"https:\/\/www.rudyrucker.com\/blog\/2015\/08\/31\/john-conway-and-the-absolute-continuum\/","title":{"rendered":"John Conway and the Absolute Continuum"},"content":{"rendered":"

I just read this wonderful biography of the mathematician and recreational-math maven par excellence, John Horton Conway, best known (to his mild annoyance) as the inventor of the cellular automaton rule known as the Game of Life. The book, Genius at Play <\/em>, is by Siobhan Roberts<\/a>.<\/p>\n

Some great quotes. John Conway’s daughter: “There goes somebody looking strange. Ergo, it must be a friend of Dad’s!”<\/p>\n

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\n[This photo has no literal connection to what I\u2019m talking about, although the musician is, in a way, like Conway. It was shot inside the San Jose California Theater during the Jazz Festival, it\u2019s a group called Bombay Jazz.]<\/em><\/p>\n

A lot of my favorite mathy people are in the book, many of whom I\u2019ve managed to meet over the years: Kurt G\u00f6del, <\/a>Stephen Wolfram, Bill Gosper, Martin Gardner, and more. Bill Gosper (discoverer of the Life glider gun): \u201cConway is approximately the smartest man in the world.”<\/p>\n

One of the things Conway is most proud of is his expansion of the familiar real number line so as to become the so-called surreal numbers. The surreal numbers are very densely packed\u2014technically speaking there\u2019s so many of them that they comprise what set-theorists call a \u201cproper class,\u201d\u009d which is a collection so vast that it is in some sense impossible to think of it as a finished thing. The less daunting collections are called sets: an example is the collection of all subsets of the set of finite natural numbers {0,1,2,3,…}. This “power set of the natural numbers” is the same size as the set of all real numbers.<\/p>\n

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\n[Original Tenniel drawing of Carroll’s Alice holding a baby who turns into a pig.]<\/em><\/p>\n

So how do the surreal numbers arise? There\u2019s a good popular book about them: Donald Knuth\u2019s Surreal Numbers<\/em>, available in paperback and also for free online. <\/a> The basic idea is fairly simple. Whenever you have a set L of numbers that are smaller than the numbers in a different set R, then there is going to be a surreal number in between the two sets…the number can be called {L | R}.<\/p>\n

In the regular real number system we use a principle like this to say that there\u2019s a number on the line just beyond 3, 3.1, 3.14, 3.141, 3.1415, 3.1459, etc. This number is our friend pi.<\/p>\n

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In Conway\u2019s system, we go further, and squeeze more and more stuff in. Like there\u2019s a number that\u2019s bigger than 0, but smaller than 1\/2, 1\/3, 1\/4, … , 1\/n, … for all finite n. This infinitesimal number can be called 1\/omega, where omega is a name often used for the simplest infinite number that lies beyond 1, 2, 3, …, n, …<\/p>\n

In Conway\u2019s rich system, however, omega isn\u2019t the very first infinite number, you\u2019ve got omega-1, omega \/ 2, and even the square root of omega.<\/p>\n

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Getting technical, the surreal numbers are what the early 20th C mathematician Felix Hausdorff would call an eta-On set, where On is the class of all finite and transfinite ordinal numbers\u2014an ordinal being a generalized counting number. The class On is a subset of the surreal numbers…like the spine of integers inside the real number line. On goes on and on, including infinite numbers like alef-null, alef-one, and so on. When we fill in the surreal numbers, we can have cool things like (((alef-one divided by (square root of alef-one)) minus (pi divided by alef-seventeen) divided by two) minus 48.) It’s all there!<\/p>\n

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The surreal numbers make up what we can call an “Absolute Continuum” where there’s always yet another number lurking between any successive pairs of sets of numbers. Physics would make more sense, suggests Knuth, if it was based on surreal numbers instead of the so-called real numbers, and I think he’s right. Assume we have endlessly transfinite infinitesimals in the small, and space gets really <\/em>smooth.<\/p>\n

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I\u2019m fascinated by the notion that our physical space is in fact an absolute continuum. This is what we might call \u201cinfinities in the small\u201d\u009d instead of \u201cinfinities in the large.\u201d\u009d I\u2019ve always felt that it\u2019s a mistake to simply stare out at the stars and yearn for the vastness out there. In my opinion, we have huge infinities right here. Underfoot.<\/p>\n

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In a slightly different vein, the founder of transfinite number theory, Georg Cantor wrote a couple of great passages.<\/p>\n

\u201cThe fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.\u201d\u009d<\/p><\/blockquote>\n

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\u201cThe actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo<\/em>, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto <\/em>as a mathematical magnitude, number or order type.\u201d\u009d<\/p><\/blockquote>\n

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I wrote about infinity in two of my pop math books, Infinity and the Mind <\/em>and Mind Tools<\/em>. But getting back to the idea of the absolute continuum, let\u2019s just marinate a little more in the idea that physical space is not *feh* <\/em>quantized, and it\u2019s not just some mere real number line\u2014it might have levels, sublevels, subsub………subsublevels. Inconceivably rich smoothness.<\/p>\n

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That quantum bump stuff is just a glitch, like a rumble strip by a toll booth, we trundle past that in our Shrink-O-Tron device, going ever downward in our hypertransfinite subdimensional bathyscaphe. In 2008, the ezine Tor.com published my SF story along these lines called “Jack and the Aktuals, or, Physical Application of Transfinite Set Theory<\/a>.”<\/p>\n

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If you have having absolute continua (plural of continuum) for your space and, indeed, the very substance of your body, then there\u2019s no reason to suppose that you will ever be repeated, as a pattern, in the transfinite universe. Those pawky bean-counting quantum Lego-block arguments don\u2019t work if we\u2019re transfinitely smooth. I was talking about that issue in a series of 3 posts called “Against Recurrence” <\/a>in May, 2015—disputing the common (and false) claim that, \u201cin an infinite universe everything repeats.\u201d\u009d No repetition if your gnarly down to the alef-seventh level and beyond, my friend.<\/p>\n

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I\u2019ll work the details out for you with my occult analog continuum computer, which merely appears <\/em>to be an electric-cord entangled with a semi-reflective mirror…<\/p>\n

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I\u2019ll leave you with this image of Davina and the Vagabonds at the San Jose Jazz Festival. Davina was great, she never stopped mugging and making faces. Funny rootsy old songs. Davina looks like a WWII poster of a working woman here. She is, very clearly, embedded in an Absolute Continuum.<\/p>\n

Having read to Siobhan Roberts’s charming bio, I get the impression J. H. Conway would enjoy meeting Davina and she, in turn, might well find him fascinating. A mathemagician among us.<\/p>\n","protected":false},"excerpt":{"rendered":"

I just read this wonderful biography of the mathematician and recreational-math maven par excellence, John Horton Conway, best known (to his mild annoyance) as the inventor of the cellular automaton rule known as the Game of Life. The book, Genius at Play , is by Siobhan Roberts. Some great quotes. John Conway’s daughter: “There goes […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-6663","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/6663","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/comments?post=6663"}],"version-history":[{"count":8,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/6663\/revisions"}],"predecessor-version":[{"id":6671,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/posts\/6663\/revisions\/6671"}],"wp:attachment":[{"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/media?parent=6663"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/categories?post=6663"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.rudyrucker.com\/blog\/wp-json\/wp\/v2\/tags?post=6663"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}