Peircers, Let me see if I can get back in saddle on this topic, the dormitive virtues of tryptophan and a few pounds heavier notwithstanding.
Recalling the context: ET:https://list.iupui.edu/sympa/arc/peirce-l/2016-11/msg00081.html JS:https://list.iupui.edu/sympa/arc/peirce-l/2016-11/msg00154.html JA:https://list.iupui.edu/sympa/arc/peirce-l/2016-11/msg00155.html JBD:https://list.iupui.edu/sympa/arc/peirce-l/2016-11/msg00159.html I was addressing Jeff's question: > On Nov 9, 2016, at 6:34 PM, Jeffrey Brian Downard wrote: >> >> John Sowa, Jon Awbrey, Edwina, List, >> >> I wanted to see if anyone have might suggestions for >> thinking about the analogy between (1) mathematical >> models of the differentiation of spaces starting with >> a vague continuum of undifferentiated dimensions and >> trending towards spaces having determinate dimensions >> to (2) models for logic involving similar sorts of >> dimensions? How might we understand processes of >> differentiation of dimensions in the case of logic? >> By way of review, here are my blog posts on the discussion so far: https://inquiryintoinquiry.com/2016/11/11/time-topology-differential-logic-%e2%80%a2-1/ https://inquiryintoinquiry.com/2016/11/13/time-topology-differential-logic-%e2%80%a2-2/ https://inquiryintoinquiry.com/2016/11/14/time-topology-differential-logic-%e2%80%a2-3/ https://inquiryintoinquiry.com/2016/11/15/time-topology-differential-logic-%e2%80%a2-4/ https://inquiryintoinquiry.com/2016/11/19/time-topology-differential-logic-%e2%80%a2-5/ We require a few elements from the subject of topology and that last blog post copies the first few definitions from J.L. Kelley. Correcting my earlier typos, here again is the plain text version: <QUOTE> Chapter 1. Topological Spaces 1.1. Topologies and Neighborhoods A topology is a family T of sets which satisfies the two conditions: the intersection of any two members of T is a member of T, and the union of the members of each subfamily of T is a member of T. The set X = ∪ { U : U ∈ T } is necessarily a member of T because T is a subfamily of itself, and every member of T is a subset of X. The set X is called the “space” of the topology T and T is a “topology for X”. The pair (X, T) is a “topological space”. When no confusion seems possible we may forget to mention the topology and write “X is a topological space.” We shall be explicit in cases where precision is necessary (for example if we are considering two different topologies for the same set X). The members of the topology T are called “open” relative to T, or T-open, or if only one topology is under consideration, simply open sets. The space X of the topology is always open, and the void set is always open because it is the union of the members of the void family. These may be the only open sets, for the family whose only members are X and the void set is a topology for X. This is not a very interesting topology, but it occurs frequently enough to deserve a name; it is called the “indiscrete” (or “trivial”) topology for X, and (X, T) is then an “indiscrete topological space”. At the other extreme is the family of all subsets of X, which is the “discrete topology” for X (then (X, T) is a “discrete topological space”). If T is the discrete topology, then every subset of the space is open. (Kelley, p. 37). </QUOTE> Reference ========= • Kelley, J.L. (1955), General Topology, Van Nostrand Reinhold, New York, NY. -- academia: http://independent.academia.edu/JonAwbrey my word press blog: http://inquiryintoinquiry.com/ inquiry list: http://stderr.org/pipermail/inquiry/ isw: http://intersci.ss.uci.edu/wiki/index.php/JLA oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey facebook page: https://www.facebook.com/JonnyCache
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