Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: tchow on 1 Feb 2005 12:00 In article <1107276841.434068.138780(a)c13g2000cwb.googlegroups.com>, <Helene.Boucher(a)wanadoo.fr> wrote: <Because you put ZFC in (*), I just presumed that you would have said <that (s union s) = d - where s is your favorite way of representing 1 <in ZFC and d is your favorite way of representing 2 - adequately <expresses its informal counterpart, which is 1 + 1 = 2. Am I wrong in <that? < <Otherwise how can the formal expression of "ZFC is consistent" <adequately express the informal assertion that ZFC is consistent?? < <In short, it seems to me - or at least this is where my confusion lies <- that (*) does not itself make the distinction between the two steps <(mathematical informalism to mathematical formalism, mathematical <formalism to logical formalism) but conflates them. First of all, although I did mention ZFC in one attempted formulation, I rejected that later because it put undue emphasis on one particular axiomatic system. But never mind that, it's not so important for the question at hand. I don't completely understand your question. Are you perhaps identifying logic with set theory? Expressing mathematical statements as statements about sets does not, in my mind, reduce them to *logic*, only to *set theory*. If it is controversial to think of numbers as purely logical entities, then it is doubly controversial to claim that sets are purely logical entities. -- 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 1 Feb 2005 12:14 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? -- Mitch Harris (remove q to reply)
From: Mike Oliver on 1 Feb 2005 12:58 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 don't think so. Informal statements can be quite precise, in my view. The difference is rather a recognition of the tension between the goals of comprehensibility and manipulability. The informal statements are the ones we can understand; the formal, the ones we can manipulate and prove metatheorems about. In case of a conflict, it's the informal versions that take precedence. It's not a sharp dichotomy, of course; it's more of a continuum.
From: Helene.Boucher on 1 Feb 2005 13:45 tchow(a)lsa.umich.edu wrote: > In article <1107276841.434068.138780(a)c13g2000cwb.googlegroups.com>, > <Helene.Boucher(a)wanadoo.fr> wrote: > <Because you put ZFC in (*), I just presumed that you would have said > <that (s union s) = d - where s is your favorite way of representing 1 > <in ZFC and d is your favorite way of representing 2 - adequately > <expresses its informal counterpart, which is 1 + 1 = 2. Am I wrong in > <that? > < > <Otherwise how can the formal expression of "ZFC is consistent" > <adequately express the informal assertion that ZFC is consistent?? > < > <In short, it seems to me - or at least this is where my confusion lies > <- that (*) does not itself make the distinction between the two steps > <(mathematical informalism to mathematical formalism, mathematical > <formalism to logical formalism) but conflates them. > > First of all, although I did mention ZFC in one attempted formulation, > I rejected that later because it put undue emphasis on one particular > axiomatic system. But never mind that, it's not so important for the > question at hand. Sorry I didn't read all the thread. In the post to which I was replying, your quote of Jamie quoting you (!) had ZFC in it. I didn't realize this was stale (even though, now that I reread your post, it even says that!). > > I don't completely understand your question. Are you perhaps identifying > logic with set theory? Never not me!! We were talking about logicism. Frege and the early Russell are the paradigm logicists - if they aren't logicists, then no one is. As I understand the historical record - but I could well be in error - both Frege and the early Russell would claim that their formal systems, which were akin to set theory, were logics. 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 - informal into formal mathematics, formal mathematics into formal logic*. But again, apparently I was arguing from a stale quote, for which I apologize.
From: tchow on 1 Feb 2005 14:23
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). -- 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 |