On Monday, October 22, 2018 at 6:05:41 AM UTC-5, Bruno Marchal wrote: > > > On 21 Oct 2018, at 13:55, Philip Thrift <[email protected] <javascript:>> > wrote: > > > > It is generally not considered applying Rorty and or Derrida to > mathematical language, but mathematics is a language* too, like English. > (or programming languages for that matter). > > > Mathematics is no more a language that physics. They use a mathematical > language, but the mathematical language is independent of the choice of a > theory (written in that mathematical language). > > We should always keep in mind the distinction between > - a mathematical language (usually defined by some grammar which determine > the well formed formula) > - a mathematical theory. (A precise choice of some formula) > - a model of that mathematical theory (a structure satisfying the axioms > of a theory, with truth preserving inference rule). > - a relation judged plausible between a model of a mathematical theory and > a portion or an aspect of some “reality". > > Exemple: take arithmetic: > - the mathematical language is given by -> f, E, A, “(“, “)”, x, y, z … > (logical symbols) with “s”, “0”, “+”, “*” (arithmetical symbols) + the > usual formation rule (if X and Y are formula, then X -> Y is a formula, > etc.) > - an arithmetical theory: here the one by Robinson, with only 7 axioms > (chosen formula). > > 1) 0 ≠ s(x) > 2) x ≠ y -> s(x) ≠ s(y) > 3) x ≠ 0 -> Ey(x = s(y)) > 4) x+0 = x > 5) x+s(y) = s(x+y) > 6) x*0=0 > 7) x*s(y)=(x*y)+x > + the inference rule of modus ponens > > - a model is given by any structure verifying (satisfying) the axioms and > truth preserving rule. The standard model is the set N together with the > usual addition and multiplication (but there are many models, not all > isomorphic to the standard model). > > > > Bruno > > > Mathematics, both pure and applied (e.g. physics), is a collection of paradigm-specific and domain-specific languages (PSLs, DSLs), just like programming languages.
For example ,quantum field theory can be expressed in Hilbert-space or path-integral dialects. http://www.fuw.edu.pl/~kostecki/daniel_ranard_essay.pdf A "deconstruction" of first-order logic gives theories which replace infinite models with finite ones: https://www.jstor.org/stable/2273942 A theory of physics expressed with a mathematical language inherits the metaphysics of the language (e.g. space and time). - pt -- 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 post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
