Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!wuarchive!usc!rutgers!cmcl2!adm!smoke!gwyn From: gwyn@smoke.brl.mil (Doug Gwyn) Newsgroups: comp.compression Subject: Re: IP gnitaluclaC rof margorP Message-ID: <15822@smoke.brl.mil> Date: 15 Apr 91 20:47:16 GMT References: <1991Apr11.022122.26142@garfield.cs.mun.ca> <1991Apr12.104200.1691@nntp-server.caltech.edu> Organization: U.S. Army Ballistic Research Laboratory, APG, MD. Lines: 45 In article stephen@estragon.uchicago.edu (Stephen P Spackman) writes: >In article <1991Apr12.104200.1691@nntp-server.caltech.edu> madler@nntp-server.caltech.edu (Mark Adler) writes: > "The ratio of the circumference of any circle to its diameter". >... How do I know what language to interpret this string in? >The amount of context I have absorbed over my life to enable me >to read news in English and understand this little joke is immense. >But it's also finite, while the expansion of Pi is not. Obviously that was just an example. We could negotiate some acceptable common language for expressing mathematical predicates, then use more precise language entirely within the agreed-upon terms, perhaps: Pi =def Real_num s.t. for_all c(r) s.t. (c(r) is_curve in E2 & p is_pt_on_curve c(r) <=> p is_point in E2 & |p| = r & r is_Real_num), Pi = Circumf c(r) / Diam c(2) Well, this could be improved but you get the idea. (Perhaps Mathematica would be a suitable language for such expressions.) The information required to express the formula is not very large, on the order of a couple of hundred bits or less, given the base language (whose information content would in effect be amortized over all uses of the language, and thus be negligible per usage). >Which fact causes me to have greivous doubts about the existence of Pi >in any meaningful sense. ? That's just the standard question, what does it mean to say that a mathematical object "exists". Clearly it is not meaningless, and you can decide for yourself what the best meaning is after consulting some works on the philosophy of mathematics. >What in fact characterises the set of bitstrings that humans will ever >be interested in representing? This is an important question for data >modelling. An important but often overlooked point is that, analogously to the fact that there are no absolute probabilities but only conditional probabilities, so also there is no absolute information content but only information for discriminating in favor of some event over another in a given context of knowledge. If you send me the first million bits of Pi, you haven't sent me much information since I'll recognize the probable message from its first few bits and the remaining bits will then all be redundant. However, if you send a similar message with several of the bits altered, it would have a comparatively high information content due to the relatively lower level of redundancy (I could predict most of the trailing bits with high probability, but not all of them.)