Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Helene.Boucher on 30 Jan 2005 12:26 Torkel Franzen wrote: > 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. Then what do you take it to mean? > > > 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. On what grounds?
From: Torkel Franzen on 30 Jan 2005 12:29 Helene.Boucher(a)wanadoo.fr writes: > Then what do you take it to mean? I have no answer. Repeating my earlier response to Tim: in practice, we recognize formalizations as adequate, but there are many different ways of formalizing informal mathematical statements, and it is far from clear how to characterize what counts as an adequate formalization. > On what grounds? How is this relevant?
From: Helene.Boucher on 30 Jan 2005 12:43 Torkel Franzen wrote: > Helene.Boucher(a)wanadoo.fr writes: > > > Then what do you take it to mean? > > I have no answer. Repeating my earlier response to Tim: in practice, > we recognize formalizations as adequate, but there are many different > ways of formalizing informal mathematical statements, and it is far > from clear how to characterize what counts as an adequate > formalization. OK. I would say that one can be sure that two terms do not mean the same thing if one can imagine a possible world in which their extensions are different. So "human being" and "rational biped" do not mean the same thing, because one can imagine a world in which there are non-human rational bipeds. Similarly, two sentences are not intensionally equivalent if there are worlds/environments in which one is true and the other not. That is not to contradict anything you have said, just to explain my position and what I was arguing for. > > > On what grounds? > > How is this relevant? Relevant to what? Anyway, even if it is not relevant to anything (although clearly it seems to touch on points of our discussion) I would like to know, On what grounds?
From: Torkel Franzen on 30 Jan 2005 12:55 Helene.Boucher(a)wanadoo.fr writes: > I would say that one can be sure that two terms do not mean the > same thing if one can imagine a possible world in which their > extensions are different. This is all highly problematic, and its relevance to the question of adequate formalization is moot. Who would argue that some standard formalization of the fundamental theorem of arithmetic in PA or in ZFC is correct because one cannot imagine any possible world in which the extensions of two terms (which terms?) are different? > Relevant to what? To the question posed by Tim Chow. To keep the discussion on track, questions regarding the grounds for accepting this or that theory are best pursued in other threads.
From: Helene.Boucher on 30 Jan 2005 13:10
Torkel Franzen wrote: > Helene.Boucher(a)wanadoo.fr writes: > > > I would say that one can be sure that two terms do not mean the > > same thing if one can imagine a possible world in which their > > extensions are different. > > This is all highly problematic, and its relevance to the question of > adequate formalization is moot. Who would argue that some standard > formalization of the fundamental theorem of arithmetic in PA or in ZFC > is correct because one cannot imagine any possible world in which the > extensions of two terms (which terms?) are different? I don't know, who? On the other hand if one could show that S1, which purported to be a correct formalization of the FTA, was not true under the same conditions as FTA, then that would be good grounds for rejecting that S1 is a correct formalization. > > > Relevant to what? > > To the question posed by Tim Chow. To keep the discussion on track, > questions regarding the grounds for accepting this or that theory are > best pursued in other threads. I find that cowardly and, while technically correct, not in the spirit of newsgroups. But fair enough, I have to sign off here anyway ! |