hamscher@HT.AI.MIT.EDU (Walter Hamscher) (10/27/86)
Date: 23 Oct 86 17:20:00 GMT From: hp-pcd!orstcs!tgd@hplabs.hp.com (tgd) Here is a rough try at defining the analog vs. digital distinction. [ * * * ] I don't read all the messages on AiList, so I may have missed something here: but isn't ``analog vs digital'' the same thing as ``continuous vs discrete''? Continuous vs discrete, in turn, can be defined in terms of infinite vs finite partitionability. It's a property of the measuring system, not a property of the thing being measured.
ladkin@KESTREL.ARPA (Peter Ladkin) (11/04/86)
(weinstein quoting goodman) > > A scheme is syntactically dense if it provides for infinitely many > > characters so ordered that between each two there is a third. (harnad) > I'm no mathematician, but it seems to me that this is not strong > enough for the continuity of the real number line. The rational > numbers are "syntactically dense" according to this definition. Correct. There is no first-order way of defining the real number line without introducing something like countably infinite sequences and limits as primitives. Moreover, if this is done in a countable language, you are guaranteed that there is a countable model (if the definition isn't contradictory). Since the real line isn't countable, the definition cannot ensure you get the REAL reals. Weinstein wants to identify *analog* with *syntactically dense* plus some other conditions. Harnad observes that the rationals fit the notion of syntactic density. The rationals are, up to isomorphism, the only countable, dense, linear order without endpoints. So any syntactically dense scheme fitting this description is (isomorphic to) the rationals, or a subinterval of the rationals (if left-closed, right-closed, or both-closed at the ends). One consequence is that one could define such an *analog* system from a *digital* one by the following method: Use the well-known way of defining the rationals from the integers - rationals are pairs (a,b) of integers, and (a,b) is *equivalent* to (c,d) iff a.d = b.c. The *equivalence* classes are just the rationals, and they are semantically dense under the ordering (a,b) < (c,d) iff there is (f,g) such that f,g have the same sign and (a,b) + (f,g) = (c,d) where (a,b) + (c,d) = (ad + bc, bd), and the + is factored through the equivalence. We may be committed to this kind of phenomenon, since every plausible suggested definition must have a countable model, unless we include principles about non-countable sets that are independent of set theory. And I conjecture that every suggestion with a countable model is going to be straightforwardly obtainable from the integers, as the above example was. Peter Ladkin ladkin@kestrel.arpa
ladkin@KESTREL.ARPA (Peter Ladkin) (11/04/86)
In article <1701@Diamond.BBN.COM>, aweinste@Diamond.BBN.COM (Anders Weinstein) writes: > The upshot of Goodman's requirement is that if a symbol system is to count as > "digital" (or as "notational"), there must be some finite sized "gaps", > however minute, between the distinct elements that need to be distinguished. I'm not sure you want this definition of the distinction. There are *finite-sized gaps, however minute* between rational numbers, and if we use the pairs-of-integers representation to represent the syntactically dense scheme, (which must be isomorphic to some subrange of the rationals if countable) we may use the integers and their gaps to distinguish the gaps in the syntactically dense scheme, in a quantifier-free manner. Thus syntactically dense schemes would count as *digital*, too. Peter Ladkin ladkin@kestrel.arpa