In my Dembski rant, I used a metaphor involving the undescribable numbers. An interesting confusion came up in the comments about just what that meant. Instead of answering it with a comment, I decided that it justified a post of its own. It’s a fascinating topic which is incredibly counter-intuitive. To me, it’s one of the great examples of how utterly wrong our
\nintuitions can be.<\/p>\n
Numbers are, obviously, very important. And so, over the ages, we’ve invented lots of notations that allow us to write those numbers down: the familiar arabic notation, roman numerals, fractions, decimals, continued fractions, algebraic series, etc. I could easily spend months on this blog just writing about different notations that we use to write numbers, and the benefits and weaknesses of each notation.<\/p>\n
But the fact is, the vast, overwhelming majority of numbers cannot be written That statement seems bizarre at best. But it does actually make sense. But for it to <\/p>\n A describable<\/em> number is a number for which there is some finite representation. An As a computer science guy, I naturally come at this from a computational So – if you can write a finite program that will generate a representation An indescribable number is, therefore, a number for which there is no notation, So, take an arbitrary computing device, φ, where φ(x) denotes the result of Most numbers cannot<\/em> be described in a finite amount of space. We can’t compute with In my Dembski rant, I used a metaphor involving the undescribable numbers. An interesting confusion came up in the comments about just what that meant. Instead of answering it with a comment, I decided that it justified a post of its own. It’s a fascinating topic which is incredibly counter-intuitive. To me, it’s one of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[30,43],"tags":[],"class_list":["post-773","post","type-post","status-publish","format-standard","hentry","category-information-theory","category-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-ct","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/773","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=773"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/773\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=773"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=773"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\ndown in any form<\/em><\/b>.<\/p>\n
\nmake sense, we have to start at the very beginning: What does it mean for a number to be describable<\/em>?<\/p>\n
\nindescribable number is a number for which there is no<\/em> finite notation. To be clear,
\nthings like repeating decimals are not<\/em> indescribable: a repeating decimal has a finite
\nnotation. (It can be represented as a rational number; it can be represented in decimal notation
\nby adding extra symbols to the representation to denote repetition.) Irrational
\nnumbers like π, which can be computed by an algorithm, are not<\/em> indescribable. By
\nindescribable, I mean that they really<\/em> have no finite representation.<\/p>\n
\nperspective. One way of defining a describable number is to say that there is
\nsome<\/em> finite computer program which will generate the representation of
\nthe number in some form. In other words, a number is describable if you can
\ndescribe how to generate its representation using a finite description. It
\ndoesn’t matter<\/em> what notation the program generates it in, as long as the
\nend result is uniquely identifiable as that one specific number. So you could use
\nprograms that generate decimal expansions; you could use programs that generate either
\nfractions or decimal expansions, but in the latter case, you’d need the program to
\nidentify the notation that it was generating.<\/p>\n
\nof the number, it’s describable. It doesn’t matter whether that program ever finishes
\nor not – so if it takes it an infinite amount of time to compute the number,
\nthat’s fine – so long as the program<\/em> is finite. So π is describable: it’s
\nnotation in decimal form is infinite, but the program to generate that representation is finite.<\/p>\n
\nand no algorithm which can uniquely identify that number in a finite amount of space. In theory, any number can be represented by a summation series of rational numbers – the indescribable ones
\nare numbers for which not only is the length of that series of rational numbers
\ninfinite, but given the first K numbers in that series, there is no algorithm
\nthat can tell you the value of the K+1th rational.\n<\/p>\n
\nrunning φ on program x. The total number of describable numbers can be no larger than
\nthe size of the set of programs x that can be run using φ. The number of programs
\nfor any effective computing device is countably infinite – so there are, at most,
\na countably infinite number of describable numbers. But there are uncountably many
\nreal numbers – so the set of numbers that can’t be generated by any finite program
\nis uncountably large.<\/p>\n
\nthem, we can’t describe them, we can’t identify them. We know that they’re there; we can
\nprove<\/em> that they’re there. All sorts of things that we count on as properties of
\nreal numbers wouldn’t work if the indescribable numbers weren’t there. But they’re
\ntotally inaccessible.<\/p>\n","protected":false},"excerpt":{"rendered":"