Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: tchow on 31 Jan 2005 21:45 In article <367nq8F4qcoivU1(a)individual.net>, Jamie Andrews; real address @ bottom of message <me(a)privacy.net> wrote: <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 (*)? < <Is this not the central tenet of "logicism"? Not really. Logicism is the doctrine that mathematics reduces to logic. Exactly what is involved in this "reduction" varies from one thinker to another; for example, in one version, mathematical assertions such as "1+1=2" that are prima facie about natural numbers are actually logical assertions of the form "`1+1=2' follows from the axioms for arithmetic." The sentence (*), or improved versions of it elsewhere in this thread, doesn't have much to do with reducing mathematics to logic. It just says that mathematical assertions can be mirrored in a formal language. The formal language might express non-logical propositions. -- 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: Russell Easterly on 1 Feb 2005 02:41 "Mitch Harris" <harrisq(a)tcs.inf.tu-dresden.de> wrote in message news:366enqF4sadtnU1(a)news.dfncis.de... > Torkel Franzen wrote: > > tchow(a)lsa.umich.edu writes: > > > >>The Church-Turing thesis is familiar to many people, largely because it > >>has been widely discussed both in textbooks and in popular science writing. > >>Having a name helps, too. > >> > >>There is an analogous thesis that is relevant to logic and the foundations > >>of mathematics: > >> > >> (*) Formal sentences (in PA or ZFC for example) adequately express > >> their informal counterparts. > > > > (*) is rather too imprecise to be given a catchy name. > > Yes, but I think that's the kind of discussion Tim is looking for, to > help make it more precise, if possible. I think the statement has > difficulties not with precision (or possible precision) but about > self-reference. If it is to be made more precise, then the concept > "informal sentence" must be made more precise. I haven't been able to follow all of this discussion, but I know I often run into "hidden" assumptions. I think a good place to start would be to admit that FOL and other systems work with symbols and that we "map" these symbols to "real world" things like numbers. I am interested in the complexity of boolean circuits. The complexity of a boolean circuit that emulates a mathematical operation like addition depends a lot on how we map numbers to symbols. A catchy name might be the "assignment problem". How are "real world" mathematical objects assigned to the symbols we use in formal theories? How much faith do we have in those mappings? Does a huge natural number really exist simply because we have a symbol defined for it? I have found it useful to assume there are two functions. The first is used to convert natural numbers into symbols. The second function maps symbols to natural numbers. These two functions don't have to be the inverse of each other. Perhaps this would allow an "informal" statement to be formalized. Russell - 2 many 2 count
From: Helene.Boucher on 1 Feb 2005 03:26 tchow(a)lsa.umich.edu wrote: > In article <367nq8F4qcoivU1(a)individual.net>, > Jamie Andrews; real address @ bottom of message <me(a)privacy.net> wrote: > <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 (*)? > < > <Is this not the central tenet of "logicism"? > > Not really. Logicism is the doctrine that mathematics reduces to logic. > Exactly what is involved in this "reduction" varies from one thinker to > another; for example, in one version, mathematical assertions such as > "1+1=2" that are prima facie about natural numbers are actually logical > assertions of the form "`1+1=2' follows from the axioms for arithmetic." > > The sentence (*), or improved versions of it elsewhere in this thread, > doesn't have much to do with reducing mathematics to logic. It just > says that mathematical assertions can be mirrored in a formal language. > The formal language might express non-logical propositions. I don't follow the importance that the formal language might express non-logical propositions. True logicists - Frege and the early Russell - didn't just believe that mathematics reduces to logic. They believed it reduces to formal logic. Since the mathematics they were talking about was informal, one could write the logicist thesis as: "Informal mathematics reduces to formal logic." Maybe you disagree with this formulation of logicism, but it strikes me as reasonably faithful. 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. Whether one wants to call it a "central tenet" would seem to be a matter of taste; but if it fell, then the logicist thesis would not hold.
From: tchow on 1 Feb 2005 10:27 In article <1107246412.251830.121830(a)z14g2000cwz.googlegroups.com>, <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 possibility of performing the first step is what I was focusing on. The second step is, I think, the heart of logicism. If someone were to propose a slightly different philosophical position from what you're calling logicism, namely that informal mathematics reduces to informal logic, I would still be inclined to call that a variant of logicism. On the other hand, someone who only accepts the first step but rejects the second doesn't sound at all like a logicist to me. So I wouldn't call the first step any kind of "logicist thesis." Something like "1+1=2" prima facie speaks of natural numbers. It is rather controversial whether natural numbers are purely *logical* entities. Simply formalizing the statement "1+1=2" without explicating how numbers reduce to logic might be the *first* step to demonstrating how logicism "works," but it is really the subsequent step (reduction of numbers to logic) that is crucial for the logicist. -- 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: george on 1 Feb 2005 11:02
> On Sat, 29 Jan 2005 tchow(a)lsa.umich.edu wrote: > > > (*) Formal sentences (in PA or ZFC for example) > > adequately express their informal counterparts. > > William Elliot wrote: > A formal sentence could have an unintuitive > or even incomprehensible informal counterpart Or no informal counterpart, or more than one informal counterpart. In real life what the formal sentence tends to have is NOT counterparts BUT RATHER informal APPROXIMATIONS, of various degrees of closeness/accuracy. As well as of various degrees of understandability or pedagogical efficacy. OBVIOUSLY, THE WHOLE GOAL is to find, out of the MANY possible informal counterparts, the ones that are most accurate AND effective. |