Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: LordBeotian on 31 Jan 2005 09:39 <Helene.Boucher(a)wanadoo.fr> ha scritto > > > 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. It's not clear what are the limits of this imagination, for example I think I am not able to imagine a world where 2^n does not exist for some n but you don't... Can we imagine a world where (p AND ~p) is true or where sqrt(2)=1? How can we decide this kind of questions?
From: tchow on 31 Jan 2005 09:35 In article <vcbwttuhxou.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: < If you mean the work in "Arithmetization of metamathematics in a general <setting", this dealt with more pressing problems. He showed among <other things that the incompleteness theorem holds in a general <formulation provided we require the axioms of a theory to be defined <by what we now call a Sigma-formula. However, a Sigma-formula which <extensionally defines the axioms of e.g. PA can still be intensionally <incorrect. That sounds like the paper I was thinking of, although I don't have my copy on me to check the title. Perhaps I expressed myself too strongly. I didn't mean to imply that Feferman was trying to capture exactly what "intensionally correct" means. That project, as I understand it, is still too vague to be a strictly mathematical investigation, and maybe it is doomed to remain that way permanently. But it still seems to me that a background motivation of Feferman's work was to separate off certain deviant (i.e., intensionally incorrect) predicates from "nice" ones. Having a clear grasp of what's going on in the transition from informal to formal is a prerequisite for formulating such a project and carrying it out. Now, someone of Feferman's caliber obviously doesn't need the "thesis" I'm proposing spelled out and given a catchy name. Mere mortals, though, might benefit. William Elliot mentioned the term "Hilbert's thesis." Googling on this term turns up stuff that's similar to what I'm groping for. I'll have to think about whether it's exactly what I'm after. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Torkel Franzen on 31 Jan 2005 09:44 tchow(a)lsa.umich.edu writes: > But it still seems to me that a background > motivation of Feferman's work was to separate off certain deviant > (i.e., intensionally incorrect) predicates from "nice" ones. Yes, but only in order to deal with more pressing problems, such as how to formulate the incompleteness theorem in a general way. For this, we only need to weed out some of the intensionally incorrect predicates, as Feferman showed.
From: tchow on 31 Jan 2005 09:59 In article <vcbmzupr84i.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: >tchow(a)lsa.umich.edu writes: >> However, even then you still need the assumption that the "direct" >> formalization is intension-preserving. > Yes, but this is not really an assumption, any more than when we >translate "Every apple is green" into "(x)(x is an apple -> x is >green)". I'm not convinced that there is no assumption. For something to be an appropriate object of mathematical study, it has to be described in a "sufficiently precise" manner. Syntactic strings we can deal with mathematically; English sentences in their raw form we can't, until we clarify which properties of the sentence matter and which don't. For a "direct" formalization of "PA is consistent" we would presumably need to describe precisely a formal language of syntax, like Quine's "protosyntax" or something similar. Then at some point we would need to agree that all the relevant properties of the ordinary mathematical assertion that PA is consistent are adequately captured by the formal counterpart. Surely that's an assumption; after all, some people reject it (or similar statements). Even in your apple example, it seems to me that there is some kind of assumption going on. "Every apple is green" has potentially many shades of meaning in ordinary English language. The translation process fixes the assumption that we care only about the strict logical form of the sentence. Maybe your argument is that the vague shades of meaning are already implicitly excluded when we say "Every apple is green" by virtue of (for example) its being posted in a logic newsgroup? Or that "PA is consistent" is already perfectly precise because it's a mathematical statement and mathematics is precise? Granted, "PA is consistent" is very precise, but I don't think it's *sufficiently* precise for us to prove theorems about it as it stands. A precise context (such as Quine's protosyntax) needs to be set up and "PA is consistent" needs to be translated into that context, and we have to assume that nothing important is lost in that translation. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Torkel Franzen on 31 Jan 2005 10:10
tchow(a)lsa.umich.edu writes: > For a "direct" formalization of "PA is consistent" we would presumably > need to describe precisely a formal language of syntax, like Quine's > "protosyntax" or something similar. Then at some point we would need > to agree that all the relevant properties of the ordinary mathematical > assertion that PA is consistent are adequately captured by the formal > counterpart. Surely that's an assumption; after all, some people reject > it (or similar statements). It is as you say an agreement, not an assumption. People can reject anything as meaningless or incomprehensible or not adequately expressing their thinking. But the formulation, in ordinary mathematical language, of for example the fundamental theorem of arithmetic is not in fact ambiguous or unclear in any way whatever in any ordinary mathematical context, and its direct formalization is similarly unproblematic. > Even in your apple example, it seems to me that there is some kind of > assumption going on. "Every apple is green" has potentially many shades > of meaning in ordinary English language. The translation process fixes > the assumption that we care only about the strict logical form of the > sentence. In accepting the formalization, we explicitly state that this is what we mean. We are not assuming anything about what people might mean in general by the statement. It would be absurd to make any claim about "1=1" always or necessarily meaning this or that to people. There will always be those who interpret "1=1" as an expression of evil Aristotelianism, for example. Agreement on meaning is always a social matter, whether the language is formalized or not. > Granted, "PA is consistent" > is very precise, but I don't think it's *sufficiently* precise for > us to prove theorems about it as it stands. I don't see what you have in mind here. What imprecision makes itself felt, and how and when? |