Name the thesis: "Formal sentences capture informal ones"
in [Theory]
|
From: Torkel Franzen on 2 Feb 2005 01:42 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.
From: examachine on 2 Feb 2005 10:21 Torkel Franzen wrote: > 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. And instead we are to believe in what? That somehow -- Eray
From: examachine on 2 Feb 2005 10:22 Torkel Franzen wrote: > 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. And instead we are to believe in what? 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? -- Eray
From: tchow on 2 Feb 2005 10:58 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. -- 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 12:08
Helene.Boucher(a)wanadoo.fr wrote: > Mitch Harris wrote: > >>tchow(a)lsa.umich.edu wrote: >> >>>I agree with this. However, I would describe the situation as follows. >>>There are two steps involved: first, we translate informal mathematical >>>statements into formal ones. Second, the formal mathematical statements >>>are reduced to purely logical ones. >> >>The separation is good. But I am still bothered by the term "informal >>mathematical statement" and "formal mathematical statement". What do >>you want those to mean? How are they different? Is the difference >>merely precision? > > I'd give an example: > Informal: Every number greater than or equal to 2 can be written as > the product of primes. > Formal: A normal writing of this in PA. > > The difference (as I take it anyway) is the informal is in natural > language, while the formal uses a language which can be defined using > precise (recursive?) rules. So I'm not sure if I want to say the only > difference is precision, but it is certainly one part of that > difference. 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? -- Mitch Harris (remove q to reply) |