Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Torkel Franzen on 29 Jan 2005 15:50 tchow(a)lsa.umich.edu writes: > But by saying that it's "problematic to make precise," are you *objecting* > to my project of formulating the thesis? No, only that the thesis is very unclear, much more so than the Church-Turing thesis.
From: tchow on 29 Jan 2005 20:26 In article <ctgt5c$34t$1(a)phys-news1.kolumbus.fi>, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: >The problem with the thesis under consideration is that, unlike the >Church-Turing thesis, it doesn't equate a mathematically defined concept >with an informal one, it equates two equally informal and vague >concepts. O.K., let me try another version. (*) Intension-preserving formalization of informal mathematical statements is always possible. Maybe this should be thought of not as a thesis but as a "thesis schema"? Instances of the schema would be things like: (+) Con("PA") is an intension-preserving formalization of "PA is consistent." Yeah, I know I'm abusing the term "schema" here, but I think you know what I mean. Con("PA") is formal; "PA is consistent" is informal, so like the Church-Turing thesis I'm equating---or at least making a tight correspondence between---something formal and something informal. Something like (+) is tacitly assumed by most people. Whenever we draw philosophical conclusions from Goedel's 2nd theorem such as "The consistency of PA cannot be proved using methods formalizable in PA" we are tacitly assuming (+) and a whole host of statements like it. This degree of acceptance seems to me to be very parallel to the widespread acceptance of the Church-Turing thesis. Failure to recognize that (+) is being assumed leads to all kinds of confusions and misunderstandings. It seems to me that this warrants the attempt to be more explicit about what is happening. -- 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: |-|erc on 29 Jan 2005 21:57 <tchow(a)lsa.umich.edu> wrote in > > (*) Formal sentences (in PA or ZFC for example) adequately express > their informal counterparts. > > Any candidates for a catchy name for (*)? (*) The Conjecture Conjecture. Good forward thinking, I tried to formalise something similar with UTM(sentence, char) mod 27 There exists some t, UTM(t, true_sentence_number) that gives all the true assertions (sentence UTM#) for some natural language. In theory, there are very high level TMs that can parse a subset of questions in English UTM(t100, "what is the capital of australia") = "canberra" Herc
From: William Elliot on 29 Jan 2005 23:59 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
From: William Elliot on 30 Jan 2005 00:02
On Sat, 29 Jan 2005, r.e.s. wrote: > <tchow(a)lsa.umich.edu> wrote ... >> >> (*) Formal sentences (in PA or ZFC for example) adequately express >> their informal counterparts. > > That reminds me of what Davis & Hersh say about > Hilbert's "formalist premise" ... > It's the converse of Hilbert's thesis, that every informal mathematical statement can be formalized. > "Hilbert's program rested on two unexamined premises; > first, the Kantian premise that _something_ in mathematics -- at least > the purely "finitary part" -- is a solid foundation, > is indubitable; and second, the formalist premise, that a > solidly founded theory about formal sentences could validate > the mathematical activity of real life [...]" > > --r.e.s. > |