List: > On Feb 10, 2021, at 7:44 PM, Jon Alan Schmidt <jonalanschm...@gmail.com> > wrote: > > On Tue, Feb 9, 2021 at 11:25 PM John F. Sowa <s...@bestweb.net > <mailto:s...@bestweb.net>> wrote: > and Jerry LRC, > > JFS> In mathematics and logic, equivalence means freely interchangeable in > all contexts without any change in meaning. > > JLRC> Really? > > Yes, indeed. As Casey Stengel used to say, "Ya could look it up." > > John >
The question of interchangeability of meaning of mathematical symbols is remote from the simplistic view you present, John. In standard mathematics, that is, the usual mathematics of analysis and applied mathematics of physical analysis, including such graphic structures as categories, topological spaces, Hilbert spaces, etc, the term equivalence relations is used to distinguish one form of mathematical structure from another. The constraints used to separate the meanings of equivalence relations can be interpreted as a triad, firstness (or transitivity), secondness (or symmetry) and thirdness (or reflexivity). One very simple usage of the notion of “equivalence relations” is to structure the ordering of mathematical spaces such as (x,y,z), by inclusivity, that is, x is contained in y and y is contained in z and therefore x is contained in z. With these antecedent definitions, one can order the meanings of the structural graphs for various spaces. For example, a Hilbert space is logically contained in a metric space and a metric space is contained in a general topological space. These distinctions are critical for interpreting the relationships between CSP usage of “topology” / graph theory and the usage of Hilbert spaces in modern chemical graph theory. (That is, perplex number theory of electrical relations.) Note that the “handedness” of biomolecules are NOT symmetric as a consequence of the existential graphic patterns of internal electrical relations of chemical atoms in these compositions. If you are not aware of these advanced mathematical concepts, to quote Casey Stengel, you could look it up. But, I will save you the time and effort and suggest that the term “equivalence” as you used it in the sentence: > JFS> In mathematics and logic, equivalence means freely interchangeable in > all contexts without any change in meaning. > does not have a triadic meaning and does not imply an “equivalence relation”. Cheers Jerry Research Professor Krasnow Institute of Advanced Study Headwater House Little Falls, MN
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