Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: tchow on 2 Feb 2005 13:05 In article <36cfohF4v935nU1(a)news.dfncis.de>, Mitch Harris <harrisq(a)tcs.inf.tu-dresden.de> wrote: >I think also I mentioned natural language as informal, and some strict >syntax/semantics language as formal, but now I want to critique that. >It just doesn't seem enough. Who's to say that I can't consider the >subset of natural language that you wrote your informal example above >to -be- the formal language (stipulate formal rules on it). How do you >know the language of PA (or ZFC or whatever) is formal enough? Where >(if anywhere) is the demarcation between informal and formal? > >And, then, what more is there if only part of the difference is precision? See Mike Oliver's comments. Initially, I also thought that the difference was a matter of precision, but upon further reflection, I don't think that that is the right word. The difference, as Mike Oliver said, is the difference between intelligibility (informal statements have meanings that we can understand) and manipulability (formal statements have a precisely defined syntactic structure that can be analyzed mathematically). Drawing a clean bright line between the two is probably a misguided project. However, we can still address your question, "How formal is formal enough?" The proper response is, "Formal enough for what? What is your goal?" If your goal is to prove independence results, for example, then you just need to be formal enough to allow the appropriate mathematical analysis to be carried out. Figuring out how formal is formal enough then becomes a special case of the more general problem of figuring out how to formulate problems so that they are amenable to mathematical analysis, which is a skill that comes with practice. -- 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: Mitch Harris on 2 Feb 2005 17:32 tchow(a)lsa.umich.edu wrote: >Mitch Harris <harrisq(a)tcs.inf.tu-dresden.de> wrote: >>I think also I mentioned natural language as informal, and some strict >>syntax/semantics language as formal, but now I want to critique that. >>It just doesn't seem enough. Who's to say that I can't consider the >>subset of natural language that you wrote your informal example above >>to -be- the formal language (stipulate formal rules on it). How do you >>know the language of PA (or ZFC or whatever) is formal enough? Where >>(if anywhere) is the demarcation between informal and formal? >> >>And, then, what more is there if only part of the difference is precision? > >See Mike Oliver's comments. Initially, I also thought that the difference >was a matter of precision, but upon further reflection, I don't think that >that is the right word. The difference, as Mike Oliver said, is the >difference between intelligibility (informal statements have meanings that >we can understand) and manipulability (formal statements have a precisely >defined syntactic structure that can be analyzed mathematically). hmmm... but what does intelligible mean? is correctness involved with that? Wouldn't you say that some informal statements are imprecise, and so must not be intelligible? That is, to the extent we can understand a statement correctly, that must be formal enough (rather than informal). (I'm not trying to be contrary for arguments sake; I'm trying to get a reasonable understanding of what informal to mean for the purposes of your proposed "thesis") >Drawing a clean bright line between the two is probably a misguided project. >However, we can still address your question, "How formal is formal enough?" >The proper response is, "Formal enough for what? What is your goal?" If >your goal is to prove independence results, for example, then you just need >to be formal enough to allow the appropriate mathematical analysis to be >carried out. Figuring out how formal is formal enough then becomes a >special case of the more general problem of figuring out how to formulate >problems so that they are amenable to mathematical analysis, which is a >skill that comes with practice. but this sense seems to be captured fully by "precision" or "removal of doubt" Mitch
From: Lee Rudolph on 2 Feb 2005 20:07 Torkel Franzen <torkel(a)sm.luth.se> writes: >me(a)privacy.net (Jamie Andrews; real address @ bottom of message) writes: > >> I think the Wikipedia entry might not be entirely accurate. >> Hilbert's specific hopes for logic, that there could be a sound >> and complete proof system for arithmetic, were dashed by Goedel, >> but I thought that the viewpoint of logicism was broader than >> that. > > I have argued at length over the years in various groups that the >idea - often expressed - that Godel's theorem somehow disproved the >claims of logicism has no justification. At this point in the history of my brain, I can no longer be sure whether I remember (and, if so, whether or not I remember correctly) that Kreisel has also argued that, or perhaps argued against that. Unless he said nothing on the subject. It would be kind of you to remind me, so I'll have something more recent to forget. Lee Rudolph
From: examachine on 2 Feb 2005 20:48 tchow(a)lsa.umich.edu wrote: > In article <1107357768.662174.224930(a)g14g2000cwa.googlegroups.com>, > <examachine(a)gmail.com> wrote: > >What do you think Godel's theorem shows? What does it mean that a > >finite set of axioms is insufficient to capture all mathematical truth, > >whatever it is? > > Logicists don't claim that a finite set of axioms suffices to capture "all > mathematical truth," whatever that is. Ok. So, what was Torkel's concern about logicism? Regards, -- Eray
From: examachine on 2 Feb 2005 21:00
Lee Rudolph wrote: > Torkel Franzen <torkel(a)sm.luth.se> writes: > > >me(a)privacy.net (Jamie Andrews; real address @ bottom of message) writes: > > > >> I think the Wikipedia entry might not be entirely accurate. > >> Hilbert's specific hopes for logic, that there could be a sound > >> and complete proof system for arithmetic, were dashed by Goedel, > >> but I thought that the viewpoint of logicism was broader than > >> that. > > > > I have argued at length over the years in various groups that the > >idea - often expressed - that Godel's theorem somehow disproved the > >claims of logicism has no justification. > > At this point in the history of my brain, I can no longer be sure > whether I remember (and, if so, whether or not I remember correctly) > that Kreisel has also argued that, or perhaps argued against that. > Unless he said nothing on the subject. It would be kind of you > to remind me, so I'll have something more recent to forget. I am not clear what these "claims of logicism" are at this point. -- Eray |