From: | Alex Fink <a4pq1injbok_0@...> |
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Date: | Sunday, November 25, 2007, 5:34 |

On Fri, 23 Nov 2007 08:19:02 EST, MorphemeAddict@WMCONNECT.COM wrote:>In a message dated 11/22/2007 9:16:12 PM Central Standard Time, >mike@STARMAN.CO.UK writes: > >> That's a different thing entirely. How can you derive (for instance) >> graphing, topology or set theory from just 0 and 1? >> Mike > >I was wondering the same thing after my last post. >The objects of those disciplines could be treated as abstract meanings of >various numbers, then relevant rules applied to them. >Maybe.You could do that, but you'd be using the numbers for little more than their value as labels; I at least wouldn't find that very satisfying. Sure, Jot can be written in 0s and 1s, but that's merely a notational convention. You don't need to increment or add or multiply them to evaluate a Jot program, so they may as well have been white boxes and black ones. It's a program with a long and illustrious history to lay down the foundations of mathematics, and in particular derive everything from a small select set of initial axioms and objects and such. In foundations, rather than taking numbers as fundamental, a typical approach is to begin by developing first-order logic and then to move on to axiomatic set theory, in terms of which one does everything else. This is the way Russell and Whitehead went about it in _Principia Mathematica_ (1910--1913). Numbers, just like any other sort of object, are in this approach realised as particular sets. You can, of course, look at systems for arithmetic in which numbers are fundamental; this is the context in which Goedel set his incompleteness theorems. Goedel's proof of the first incompleteness theorem relies crucially on Goedel numbering, which is very much like your "abstract meanings of various numbers". So doing an encoding into the domain of numbers can certainly be useful in foundations; it's just less useful as a place to start, in that fewer other areas of math are "naturally downstream" from it. Pushing the minimalism as far as possible, you could also ask just how much these axiom systems can be slimmed down. This is a question fitting right in with Stephen Wolfram's New Kind of Science program -- _exactly_ how simple can something be and still yield universal ~ arbitrarily complicated behaviour? He addresses it to a degree, for instance at http://www.wolframscience.com/nksonline/page-773 So, in the usual setup, it takes seventeen axioms and not a few symbols to get arithmetic. You can shrink this a fair bit, in the case of basic logic from seven axioms and three operators (plus equals) to one and one, but the results are extremely unintuitive and far too much of a pain to do anything substantial with (343 steps with lemmas to reproduce the Sheffer system there; see page 810). For other domains the shortest axiom systems aren't known, and we might never know them without exhaustive search, and they'll likely be even less intuitive and user-friendly. So if you're building a _conlang_ on these principles, well, you'll probably fail for essentially Mungojelly's reasons, that semantics of the real world just don't reduce usefully like that. But if it _weren't_ for that, what you'd have to look forward to is a conlang where it takes a couple hundred uses of your ten-odd words to say "mouse" or "have", and nobody wants something that extreme. Alex