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<!DOCTYPE html> <html dir='ltr'> <head> <link href='https://www.blogger.com/static/v1/widgets/55013136-widget_css_bundle.css' rel='stylesheet' type='text/css'/> <meta content='text/html; charset=UTF-8' http-equiv='Content-Type'/> <meta content='blogger' name='generator'/> <link href='http://recursed.blogspot.com/favicon.ico' rel='icon' type='image/x-icon'/> <link href='http://recursed.blogspot.com/2018/08/' rel='canonical'/> <link rel="alternate" type="application/atom+xml" title="Recursivity - Atom" href="http://recursed.blogspot.com/feeds/posts/default" /> <link rel="alternate" type="application/rss+xml" title="Recursivity - RSS" href="http://recursed.blogspot.com/feeds/posts/default?alt=rss" /> <link rel="service.post" type="application/atom+xml" title="Recursivity - Atom" href="https://www.blogger.com/feeds/20067416/posts/default" />  <meta content='http://recursed.blogspot.com/2018/08/' property='og:url'/> <meta content='Recursivity' property='og:title'/> <meta content='Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about ....' property='og:description'/> <title>Recursivity: August 2018</title> <style id='page-skin-1' type='text/css'></style> <link href='https://www.blogger.com/dyn-css/authorization.css?targetBlogID=20067416&zx=6f2283e8-cf00-48dd-b17a-9498c2c4d2f9' media='none' onload='if(media!='all')media='all'' rel='stylesheet'/><noscript><link href='https://www.blogger.com/dyn-css/authorization.css?targetBlogID=20067416&zx=6f2283e8-cf00-48dd-b17a-9498c2c4d2f9' rel='stylesheet'/></noscript> <meta name='google-adsense-platform-account' content='ca-host-pub-1556223355139109'/> <meta name='google-adsense-platform-domain' content='blogspot.com'/> </head> <body> <div class='navbar section' id='navbar'><div class='widget Navbar' data-version='1' id='Navbar1'><script type="text/javascript"> function setAttributeOnload(object, attribute, val) { if(window.addEventListener) { window.addEventListener('load', function(){ object[attribute] = val; }, false); } else { window.attachEvent('onload', function(){ object[attribute] = val; }); } } </script> <div id="navbar-iframe-container"></div> <script type="text/javascript" src="https://apis.google.com/js/platform.js"></script> <script type="text/javascript"> gapi.load("gapi.iframes:gapi.iframes.style.bubble", function() { if (gapi.iframes && gapi.iframes.getContext) { gapi.iframes.getContext().openChild({ url: 'https://www.blogger.com/navbar/20067416?origin\x3dhttp://recursed.blogspot.com', where: document.getElementById("navbar-iframe-container"), id: "navbar-iframe" }); } }); </script><script type="text/javascript"> (function() { var script = document.createElement('script'); script.type = 'text/javascript'; script.src = '//pagead2.googlesyndication.com/pagead/js/google_top_exp.js'; var head = document.getElementsByTagName('head')[0]; if (head) { head.appendChild(script); }})(); </script> </div></div> <div id='outer-wrapper'><div id='wrap2'>  <a href='#main'>skip to main </a> | <a href='#sidebar'>skip to sidebar</a> <div id='header-wrapper'> <div class='header section' id='header'><div class='widget Header' data-version='1' id='Header1'> <div id='header-inner'> <div class='titlewrapper'> <h1 class='title'> <a href='http://recursed.blogspot.com/'> Recursivity </a> </h1> </div> <div class='descriptionwrapper'> Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about mathematics, science, politics, music, religion, and Recurrent thoughts about .... </div> </div> </div></div> </div> <div id='content-wrapper'> <div id='crosscol-wrapper' style='text-align:center'> <div class='crosscol no-items section' id='crosscol'></div> </div> <div id='main-wrapper'> <div class='main section' id='main'><div class='widget Blog' data-version='1' id='Blog1'> <div class='blog-posts hfeed'> <div class="date-outer"> <h2 class='date-header'>Sunday, August 26, 2018</h2> <div class="date-posts"> <div class='post-outer'> <div class='post hentry uncustomized-post-template' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='20067416' itemprop='blogId'/> <meta content='8971514728785550971' itemprop='postId'/> <a name='8971514728785550971'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://recursed.blogspot.com/2018/08/creationist-physicist-doesnt-understand.html'>Creationist Physicist Doesn't Understand Mathematics, Either</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-8971514728785550971' itemprop='description articleBody'> <div dir="ltr" style="text-align: left;" trbidi="on"> </div> If there's one consistent aspect of creationism, it's that people lacking understanding and training are put forth as experts. Here we have yet another example, from the creationist blog <a href="https://uncommondescent.com/intelligent-design/a-physicist-looks-at-biologys-problem-of-speciation-in-humans/">Uncommon Descent</a>. There physicist Rob Sheldon is quoted as saying THere [sic] can even be uncertainty in mathematics. For example, mathematicians in the 1700’s kept finding paradoxes in mathematics, which you would have thought was well-defined. For example, what is the answer to this infinite sum: 1+ (-1) + 1 + (-1) …? If we group them in pairs, then the first pair =>0, so the sum is: 0+0+0… = 0. But if we skip the first term and group it in pairs, we get 1 + 0+0+0… = 1. So which is it? Mathematicians call these “ill-posed” problems and argue that ambiguity in posing the question causes the ambiguity in the result. If we replace the numbers with variables, do some algebra on the sum, we find the answer. It’s not 0 and it’s not 1, it’s 1/2. By the 1800’s a whole field of convergence criteria for infinite sums was well-developed, and the field of “number theory” extended these results for non-integers etc. The point is that a topic we thought we had mastered in first grade–the number line–turned out to be full of subtleties and complications. Nearly every statement of Sheldon here is wrong. And not just wrong -- wildly wrong, as in "I have absolutely no idea of what I'm talking about" wrong. 1. Uncertainty in mathematics has nothing to do with the kinds of "infinite sums" Sheldon cites. "Uncertainty" can refer to, for example, the theory of fuzzy sets, or the theory of undecidability. Neither involves infinite sums like 1 + (-1) + 1 + (-1) ... . 2. Ill-posed problems have nothing to do with the kind of infinite series Sheldon cites. An ill-posed problem is one where the solution depends strongly on initial conditions. The problem with the infinite series is solely one of giving a rigorous interpretation of the symbol "...", which was achieved using the theory of limits. 3. The claim about replacing the numbers with "variables" and doing "algebra" is incorrect. For example if you replace 1 by "x" then the expression x + (-x) + x + (-x) + ... suffers from exactly the same sort of imprecision as the original. To get the 1/2 that Sheldon cites, one needs to replace the original sum with 1/x - 1/x^2 + 1/x^3 - ..., then sum the series (using the definition of limit from analysis, not algebra) to get x/(1+x) in a certain range of convergence that does not include x=1, and then make the substitution x = 1. 4. Number theory has virtually nothing to do with infinite sums of the kind Sheldon cites -- it is the study of properties of integers -- and has nothing to do with extending results on infinite series to "non-integers etc." It takes real talent to be this clueless. <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'> Posted by Jeffrey Shallit at <meta content='http://recursed.blogspot.com/2018/08/creationist-physicist-doesnt-understand.html' itemprop='url'/> <a class='timestamp-link' href='http://recursed.blogspot.com/2018/08/creationist-physicist-doesnt-understand.html' rel='bookmark' title='permanent link'><abbr class='published' itemprop='datePublished' title='2018-08-26T10:20:00-04:00'>10:20 AM</abbr></a> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/20067416/8971514728785550971' onclick=''> 11 comments: </a> <a href='https://www.blogger.com/email-post/20067416/8971514728785550971' title='Email Post'> <img alt='' class='icon-action' height='13' src='https://resources.blogblog.com/img/icon18_email.gif' width='18'/> </a> <a href='https://www.blogger.com/post-edit.g?blogID=20067416&postID=8971514728785550971&from=pencil' title='Edit Post'> <img alt='' class='icon-action' height='18' src='https://resources.blogblog.com/img/icon18_edit_allbkg.gif' width='18'/> </a> <div class='post-share-buttons goog-inline-block'> </div> </div> <div class='post-footer-line post-footer-line-2'> </div> <div class='post-footer-line post-footer-line-3'> </div> </div> </div> </div> </div></div> </div> <div class='blog-pager' id='blog-pager'> <a class='blog-pager-newer-link' href='http://recursed.blogspot.com/' id='Blog1_blog-pager-newer-link' title='Newer Posts'>Newer Posts</a> <a class='blog-pager-older-link' href='http://recursed.blogspot.com/search?updated-max=2018-08-26T10:20:00-04:00' id='Blog1_blog-pager-older-link' title='Older Posts'>Older Posts</a> <a class='home-link' href='http://recursed.blogspot.com/'>Home</a> </div> <div class='clear'></div> <div class='blog-feeds'> <div class='feed-links'> Subscribe to: <a class='feed-link' href='http://recursed.blogspot.com/feeds/posts/default' target='_blank' type='application/atom+xml'>Posts (Atom)</a> </div> </div> </div></div> </div> <div id='sidebar-wrapper'> <div class='sidebar section' id='sidebar'><div class='widget Image' data-version='1' id='Image1'> <h2>-</h2> <div class='widget-content'> <img alt='-' height='227' id='Image1_img' src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEio37WuIFScFiF-SVXAe3OdqTu1s2jfLoJAcrgz5szwlqKND83BLy_Re7IgcuvRoRboz_dZKeaFUMxuSRAFBN1KnBgLhyNHg20-o7BqtHfFNBflQbrjxGeATth7SACfnfP61inGBQ/s227/jeff-math.jpg' width='226'/> Jeffrey Shallit </div> <div class='clear'></div> </div><div class='widget Text' data-version='1' id='Text1'> <div class='widget-content'> American mathematician, professor of computer science at a major Canadian university, & skeptic. <div> </div><div>All comments are moderated to remove spam and duplication. 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class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/'> 2012 </a> (133) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/12/'> December </a> (13) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/11/'> November </a> (11) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/10/'> October </a> (20) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/09/'> September </a> (8) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/08/'> August </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/07/'> July </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/06/'> June </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2012/05/'> May </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' 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href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/12/'> December </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/11/'> November </a> (12) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/10/'> October </a> (1) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/09/'> September </a> (7) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/08/'> August </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/07/'> July </a> (8) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/06/'> June </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/05/'> May </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/04/'> April </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/03/'> March </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/02/'> February </a> (11) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2011/01/'> January </a> (9) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/'> 2010 </a> (133) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/12/'> December </a> (10) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' 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class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/06/'> June </a> (12) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/05/'> May </a> (12) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/04/'> April </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/03/'> March </a> (18) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/02/'> February </a> (18) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2010/01/'> January </a> (20) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/'> 2009 </a> (116) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/12/'> December </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/11/'> November </a> (25) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/10/'> October </a> (16) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/09/'> September </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/08/'> August </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/07/'> July </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/06/'> June </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/05/'> May </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/04/'> April </a> (2) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/03/'> March </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/02/'> February </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2009/01/'> January </a> (6) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/'> 2008 </a> (102) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/12/'> December </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/11/'> November </a> (7) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/10/'> October </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/09/'> September </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/08/'> August </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/07/'> July </a> (13) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/06/'> June </a> (10) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/05/'> May </a> (13) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/04/'> April </a> (13) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/03/'> March </a> (14) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/02/'> February </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2008/01/'> January </a> (8) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/'> 2007 </a> (78) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/12/'> December </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/11/'> November </a> (2) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/10/'> October </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/09/'> September </a> (22) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/08/'> August </a> (9) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/07/'> July </a> (10) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/06/'> June </a> (8) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/05/'> May </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/04/'> April </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/03/'> March </a> (2) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/02/'> February </a> (3) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2007/01/'> January </a> (3) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/'> 2006 </a> (65) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/12/'> December </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/11/'> November </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/09/'> September </a> (4) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/08/'> August </a> (3) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/07/'> July </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/06/'> June </a> (1) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/05/'> May </a> (5) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/04/'> April </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/03/'> March </a> (6) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/02/'> February </a> (13) </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2006/01/'> January </a> (12) </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2005/'> 2005 </a> (14) <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> ►  </a> <a class='post-count-link' href='http://recursed.blogspot.com/2005/12/'> December </a> (14) </li> </ul> </li> </ul> </div> </div> <div class='clear'></div> </div> </div><div class='widget LinkList' data-version='1' id='LinkList1'> <h2>Other Blogs to Visit</h2> <div class='widget-content'> <ul> <li><a href='http://scienceblogs.com/dispatches/'>Dispatches from the Culture Wars</a></li> <li><a href='http://www.scienceblogs.com/evolutionblog/'>Jason Rosenhouse's Evolutionblog</a></li> <li><a href='http://lippard.blogspot.com/'>Jim Lippard's Blog</a></li> <li><a href='http://www.scienceblogs.com/pharyngula'>Pharyngula</a></li> <li><a href='http://www.scottaaronson.com/blog/'>Shtetl-Optimized</a></li> <li><a href='http://www.talkreason.org/'>Talk Reason</a></li> <li><a href='http://www.pandasthumb.org/'>The Panda's Thumb</a></li> </ul> <div class='clear'></div> </div> </div> </div> </div>  <div class='clear'> </div> </div>  </div></div>  <script type="text/javascript" src="https://www.blogger.com/static/v1/widgets/2538535903-widgets.js"></script> <script type='text/javascript'> window['__wavt'] = 'AOuZoY6id-8KMS-aW2IkhEHhJ22lpWgz3w:1744357410980';_WidgetManager._Init('//www.blogger.com/rearrange?blogID\x3d20067416','//recursed.blogspot.com/2018/08/','20067416'); _WidgetManager._SetDataContext([{'name': 'blog', 'data': {'blogId': '20067416', 'title': 'Recursivity', 'url': 'http://recursed.blogspot.com/2018/08/', 'canonicalUrl': 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