> On 19 Nov 2019, at 16:48, Philip Thrift <[email protected]> wrote: > > > > On Tuesday, November 19, 2019 at 8:46:18 AM UTC-6, Bruno Marchal wrote: > >> On 18 Nov 2019, at 15:28, Philip Thrift <[email protected] <javascript:>> >> wrote: >> >> >> >> On Monday, November 18, 2019 at 8:01:12 AM UTC-6, Bruno Marchal wrote: >> >>> Then a huge technical problem is that the term “model” is used in opposite >>> sense by physicists and logicians, and the sense of “model” used by >>> logicians is technical and required some good understanding of what is a >>> theory as considered in logic (basically a finite machine, actually). >> Bruno >> >> >> >> I have thought about this almost 50 years, and have come to the conclusion >> that 'model' as used in physics to mean a mathematical formulation of a >> theory is correct, and that mathematical logicians should have never used >> that word for what they are using it for. It should be 'interpretation', >> 'semantics', or domain' instead. >> >> So Peano axioms is a model of arithmetic, and is ℕ a possible interpretation >> (or semantics, or domain). > > Usually the domain is the set from which the model is built. N is the domain, > But the Model is the whole structure set (N, 0, +, *). The interpretation is > the function going from the syntactic symbol to diverse object or > construction made on the domain. > > In some more vague context, we can use “interpretation”, “semantic” and > “model” as quasi synonym. The term “domain” has acquired a more technical > sense in the theory of domain by Scott, but very often is used to described > the set used in the model. > > Logicians use “model" like painters. The naked model is the reality, and the > painting is the syntax or theory pointing to that reality. Physicists use > model, like in Toy model, a simplification, or a theory, and is used most of > the time as both a theory or its interpretation (taken for granted most of > the time, although this has evolved a little bit, notably through the > difficulties to interpret QM). > > >> >> >> Mathematical logicians just goofed up, that's all. > > Logic is mainly the study of proof theory, model theory, and the relations > between both. “Model” has acquired a technical meaning. I think the term has > been introduced by Löwenheim, probably in his "cornerstone paper” on this > subject “Über Möglichkeiten im Relativkalkül” (“On Possibility In the > Relative Calculus” in German). > > A good interesting book on the birth of Model Theory is the book by Calixto > Badesa: “The Birth of Model Theory”, 2004, Princeton University Press > (translated from Spanish). > > Bruno > > > The transition from syntax to semantics is not not as clean as may be > thought, but there is mathematical logic and programming language theory and > theorem proving systems, each with some different perspective and > vocabularies. > > In my own formulation > > Program > Language > Translation > Object > Substrate > > I could identify Substrate with Model (in the mathematical logic sense).
With Mechanism, the substrate are described by the Models of the Material hypotases: the very object of the number’s dream. It is not an illusion, but it is something emerging from the arithmetical reality. Bruno > > @philipthrift > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/f6297390-9c6e-4a54-97a5-988b0f0b6805%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/f6297390-9c6e-4a54-97a5-988b0f0b6805%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/5AB3E588-EA63-449B-B477-EB94928A3B5B%40ulb.ac.be.
