Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: tchow on 1 Feb 2005 14:29 In article <41ffd715$0$580$b45e6eb0(a)senator-bedfellow.mit.edu>, I wrote: >(1) reducing informal mathematics to informal logic*; >(2) expressing informal logic* by formal logic*. > >Then the thesis I'm interested in is that (2) is possible. But the >crucial step for logicism* seems to be (1). I should also add that yes, the formulation I proposed that mentioned ZFC does seem to conflate (1) and (2) here. That's partly why I rejected that formulation. -- 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: Helene.Boucher on 1 Feb 2005 14:50 t...(a)lsa.umich.edu wrote: > In article <1107283504.859465.50390(a)f14g2000cwb.googlegroups.com>, > <Helene.Boucher(a)wanadoo.fr> wrote: > <Anyway, look at it this way. Define logic* to be logic union set > <theory, and consider the thesis that mathematics can be reduced to > <logic*. Call this the logicism* thesis. > < > <There is now the following two-step division: expressing informal > <mathematics by formal mathematics; expressing formal mathematics by > <formal logic*. Do *you* agree that it is useful to distinguish these > <two steps (given that logic* is not "just" logic.) > < > <The (*) that I understood does not do make this distinction > > O.K., if we're talking about logicism*, then I would analyze the situation > differently. The natural division into two steps seems to be: > > (1) reducing informal mathematics to informal logic*; > (2) expressing informal logic* by formal logic*. > > Then the thesis I'm interested in is that (2) is possible. But the > crucial step for logicism* seems to be (1). I guess you can analyze it that way! As I think I already said, logicism seems to be the whole kaboodle (1 + 2), since the paradigm logicists - Frege and the early Russell - were of that bent. But historical analysis like this is, of course, open to dispute. Not having thought about it more than the ten seconds of this reply, I would guesstiimate that (1) seems to do all the work, and (2) could turn out to be fairly trivial. But could be wrong. Anyway, I've finished my two cents.
From: tchow on 1 Feb 2005 15:21 In article <1107287450.529419.62350(a)z14g2000cwz.googlegroups.com>, <Helene.Boucher(a)wanadoo.fr> wrote: >> (1) reducing informal mathematics to informal logic*; >> (2) expressing informal logic* by formal logic*. [...] >Not having thought about it more than the ten seconds of this reply, I >would guesstiimate that (1) seems to do all the work, and (2) could >turn out to be fairly trivial. I'd agree that (2) is trivial compared to (1), but my point was not to discuss logicism (someone else brought that up) or to compare (2) to (1), but to point out that it is a common and unstated assumption that nothing is lost during step (2). However, it is conceptually a critical step, and I think a lot of confusion (among students/amateurs/beginners) is generated when this assumption remains tacit rather than explicit. -- 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: Helene.Boucher on 1 Feb 2005 15:30 Mitch Harris wrote: > tchow(a)lsa.umich.edu wrote: > > <Helene.Boucher(a)wanadoo.fr> wrote: > > > >>In any case, *if* that is the logicist thesis, then indeed it would > >>seem to depend on your (*). That is, informal mathematics cannot > >>reduce to formal logic unless informal mathematical assertions can be > >>captured by assertions in formal logic. > > > > 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. > > -- > Mitch Harris > (remove q to reply)
From: Jamie Andrews; real address @ bottom of message on 1 Feb 2005 16:00
>> In comp.theory tchow(a)lsa.umich.edu wrote: >>> (*) Formal sentences (in PA or ZFC for example) adequately express >>> their informal counterparts. >>> Any candidates for a catchy name for (*)? > "Jamie Andrews; real address @ bottom of message" <me(a)privacy.net> wrote in > message news:367nq8F4qcoivU1(a)individual.net... >> Is this not the central tenet of "logicism"? In comp.theory Stephen Harris <cyberguard1048-usenet(a)yahoo.com> wrote: > Logicism is the theory that mathematics is an extension of > logic and therefore all mathematics is reducible to logic. > Kurt Gýdel's incompleteness theorem ultimately undermined > the purpose of the project. The attempted resurrection of > this theory is styled neo-logicism. Please cite Wikipedia when you copy from it. 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. Torkel could probably clear this up if he decides to be a little less gnomic. OK, a lot less gnomic. >> See for example >> http://www.eecs.umich.edu/~rthomaso/documents/logicism/ --Jamie. (a Dover edition designed for years of use!) andrews .uwo } Merge these two lines to obtain my e-mail address. @csd .ca } (Unsolicited "bulk" e-mail costs everyone.) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ |