Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Stephen Harris on 30 Jan 2005 11:20 "William Elliot" <marsh(a)privacy.net> wrote in message news:20050129205439.M3897(a)agora.rdrop.com... > On Sat, 29 Jan 2005 tchow(a)lsa.umich.edu wrote: > >> (*) Formal sentences (in PA or ZFC for example) adequately express >> their informal counterparts. >> > A formal sentence could have an unintuitive or even incomprehensible > informal counterpart I have a question. Quantum theory uses a formal mathematics in order to make predictions. All of the major interpretations use the same mathematics. So don't the interpretations (which often seem contradictory) of quantum theory stand as an "informal counterparts" to the formal mathematical basis? I'm thinking there would be several informal mathematical intuitions each one mapping to one formal quantum basis-- I think somebody else mentioned something like this without an example.
From: Helene.Boucher on 30 Jan 2005 11:22 Torkel Franzen wrote: > Helene.Boucher(a)wanadoo.fr writes: > > > Presumably you would mean by 'ordinary mathematics' > > something which includes the truth of the successor axiom, so your > > additional phrase answers the question "Why is it trivially true...?" > > in a trivial way (the answer being, "because it's true by the > > definition of 'ordinary' mathematics") or turns the question into one > > of causality instead of grounds ("why has ordinary mathematics come to > > include the successor axiom?"). > > It's not an answer at all to the question why it is trivially true. > It is merely the observation that since you put in question trivial > theorems of ordinary mathematics, your regarding Con(PA) as not being > a faithful translation of "PA is consistent" becomes a side issue. You appear to be replying to my remarks about your inclusion of "in ordinary mathematics" after "is trivially true." I therefore do not understand your comments, since they do not appear to have any relevance. > > > Except (again!) the faithfulness of the translation was the issue of > > the thread. And the intensional equivalence of two sentences should > > not turn on whether something else is true or not. > > Naturally it turns on whether we take other things to be true. I think you misunderstood what I meant, so I must have been unclear. What I meant was: if S1 and S2 are only equivalent supposing that some other proposition is true, then they cannot be intensionally equivalent.
From: Torkel Franzen on 30 Jan 2005 11:26 Helene.Boucher(a)wanadoo.fr writes: > I therefore do not > understand your comments, since they do not appear to have any > relevance. I was responding to your statement that "..your additional phrase answers the question 'Why is it trivially true...?'...". > What I meant was: if S1 and S2 are only equivalent supposing that some > other proposition is true, then they cannot be intensionally > equivalent. So what do you take "intensionally equivalent" to mean?
From: Torkel Franzen on 30 Jan 2005 12:16 Helene.Boucher(a)wanadoo.fr writes: > > So what do you take "intensionally equivalent" to mean? > That S1 and S2 mean the same thing. This is a traditional and very problematic concept in philosophy, but I think it's clear that it's too strong when we're talking about the intensional adequacy of formalizations. I at any rate would not claim about any formalization that it "means the same thing" as the informal mathematical statement it formalizes. > For every question a question in turn: do you believe that the > successor axiom is true? Sure, I am a fanatical believer in the truth of all the axioms of ZFC, PA, ACA, and so on.
From: Helene.Boucher on 30 Jan 2005 12:09
Torkel Franzen wrote: > Helene.Boucher(a)wanadoo.fr writes: > > > What I meant was: if S1 and S2 are only equivalent supposing that some > > other proposition is true, then they cannot be intensionally > > equivalent. > > So what do you take "intensionally equivalent" to mean? That S1 and S2 mean the same thing. For every question a question in turn: do you believe that the successor axiom is true? |