[Ham] "frankly you've lost me with "topography of value sets" for measurement. What, exactly, are you trying to measure, and what does it have to do with philosophy? "
[Ron] I think you use the term "value awareness". Can this be termed as the measurement of ultimate reality? if being aware is the result of the measurement of ultimate reality at every moment by our senses, "Value awarness" means the derivation of finite from the infinite. Which I believe the limit of the round represents in mathmatics, which also derives the finite from the infinite. If you may agree to this concept, a "value set" derives finite from the infinite consider that we are a "topography of value sets" from our Thoughts to what we are composed of which constantly measures and responds and depends on the "value sets" of it's environment of which we are a "value set". [Case] When there is talk of non-Euclidian geometry it is easily explained which of Euclid's axioms are modified and simple to see at least some of the consequences. [Ron] "Propositions about a system are representable by a Heyting algebra associated with the topos. A Heyting algebra is a distributive lattice that differs from a Boolean algebra only in so far as the law of excluded middle need not hold, A Boolean algebra is a Heyting algebra with strict equality." "The main difference between theorems proved using Heyting logic and those using Boolean logic is that proofs by contradiction cannot be used in the former. In particular, this means that one cannot prove that something exists by arguing that the assumption that it does not leads to contradiction; instead it is necessary to provide a constructive proof of the existence of the entity concerned. Arguably, this does not place any major restriction on building theories of physics." "One deep result in topos theory is that there is an internal language associated with each topos. In fact, not only does each topos generate an internal language, but, conversely, a language satisfying appropriate conditions generates a topos. Topoi constructed in this way are called 'linguistic topoi', and every topos can be regarded as a linguistic topos. In many respects, this is one of the profoundest ways of understanding what a topos really 'is'." "The main difference with classical logic is that the logic of the topos language does not satisfy the principle of excluded middle, and hence proofs by contradiction are not permitted. This has many intriguing consequences. For example, there are topoi with genuine infinitesimals that can be used to construct a rival to normal calculus. The possibility of such quantities stems from the fact that the normal proof that they do not exist is a proof by contradiction. Thus each topos carries its own world of mathematics: a world which, generally speaking, is not the same as that of classical mathematics. Consequently, by postulating that, for a given theory-type, each physical system carries its own topos, we are also saying that to each physical system plus theorytype there is associated a framework for mathematics itself! Thus classical physics uses classical mathematics; and quantum theory uses 'quantum mathematics'-the mathematics formulated in the topoi of quantum theory." moq_discuss mailing list Listinfo, Unsubscribing etc. http://lists.moqtalk.org/listinfo.cgi/moq_discuss-moqtalk.org Archives: http://lists.moqtalk.org/pipermail/moq_discuss-moqtalk.org/ http://moq.org.uk/pipermail/moq_discuss_archive/
