Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: george on 1 Feb 2005 11:08 > 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.
From: george on 1 Feb 2005 11:08 > 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.
From: george on 1 Feb 2005 11:14 tchow(a)lsa.umich.edu wrote: > But by saying that it's "problematic to make precise," > are you *objecting* to my project of formulating the thesis? Not in principle, just As Proposed. > The very nature of my proposed > thesis prevents it from being *mathematically* precise, No, really, it doesn't. Rather, the very nature of "informal counterparts" prevents THEM from being mathematically precise. You could still be precise about what you're trying to say about them. > just as the > Church-Turing thesis isn't *mathematically* precise. !@#$%. It IS SO, TOO, *dumbass* (*-* emphasizing NOT that you might be dumb, which of course you are NOT, but rather simply that you have just said something blatantly easy to contradict technically, and re-emphasizing my personal determination to use this epithet as I deem appropriate). The notion of a Turing Machine is mathematically precise. The only other part of the Church- Turing thesis is a Pure HOLE, an UNdefined term. We usually use the term "computable". The so-called Church-Turing Thesis ISN'T a Thesis: IT IS a PROPOSED DEFINITION. It is the PROPOSAL that we DEFINE "computable" to mean "Turing Computable". Reactions-to or dispositions-of this proposal can range from outright rejection to codification as The Official Definition of The Technical Term. For the moment, however, the community is reacting by reserving its right to broaden the definition of "computable" (or whatEVER term you want to use to name the definiendum/hole-on-the-left- side of Church's Thesis) beyond "Turing Computable", if some compelling reason should arise. Hope springs eternal, I guess. Personally I don't know what people are still hoping for. Quantum computing? It is sort of already known that you can compute more functions if you postulate more/infinitary capabilities for the machine (e.g. oracles). Originally it was surely some sort of new way of combining natural language concepts, some weird sort of combinatoric trick that human thinkers in natural language knew how to perform with words, terms, and diagrams, but that the TM paradigm didn't capture. It is entirely reasonable that that seemed a lot more humanly possible 70 years ago than it does today. > That doesn't mean > that it's too imprecise to formulate as a snappy thesis. This is sort of a self-solved problem; if you had POSED the problem well enough for us to get what you mean, it would ALREADY be "snappy".
From: george on 1 Feb 2005 11:32 Torkel Franzen wrote: > tchow(a)lsa.umich.edu writes: > > > O.K., let me try another version. > > > > (*) Intension-preserving formalization > > of informal mathematical > > statements is always possible. That is just Obviously False; Godel's theorem is the classic [counter]example. Though that doesn't say that "first-order truth in arithmetic over the natnums" can't be formalized AT ALL, but rather only that it can't be recursively formalized in FOL. Obviously, if you relax those restrictions, something closer is possible; there is also my prior claim that ANYthing you might want to say about first-order arithmetic is formal whether you like it or not; that the alleged informality of some presentations is just irrelevant superficial surface structure; that to understand those informal statements IS, internally, to correctly translate them into formal ones; that the informal statement is just a STYLE of communicating the inherently formal content. Therefore, to prevent this from lapsing into triviality, you have to PRESERVE some relevant constraints. In the case of first-order PA vs. first-order true arithmetic, I suppose you could formalize meta- theorems-[about-first-order-theories]-that-can't-be- formalized-in-first-order-logic, in HIGHER-order logic, but that trivializes the whole enterprise. Suppose you have some natural-language way of saying something. What sort of formal transformation do you have to apply to that natural langauge to get "formal langauge" out of it?? A VERY trivial one, surely. You can put formal sheep's CLOTHING upon ANY informal wolf, EASILY. ACTUALLY "formalizing" it surely goes a little deeper, and there is almost certainly MORE than one way to do that, and they are NOT likely to be isomorphic. The best you can HOPE for, truly, is multiple mutual approximations, i.e., for any given informal conjunction, there are VARIOUS formal counterparts with VARIOUS degrees of desirability, AND CONVERSELY. > > 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." But again, if by Con("PA") you mean what Godel meant by it, then this is just blatantly false: PA + ~Con("PA") is consistent. If Con("PA") were ACTUALLY intension- preserving then there couldn't be any models of PA in which it were false, since the falsity of the "informal"/intended version is precisely the NON-existence of ANY models of PA. I agree with the following: > (+) is not on the face of it an instance > of (*), since it states not only that an > intension-preserving formalization of "PA is consistent" > is possible, but that a particular arithmetical > formula is such a formalization. What is required of an > intension-preserving formalization? Especially when you are MIXING levels! The question of the consistency of a first-order theory is INHERENTLY second-order! The POSSIBLITY of an intension-preserving FIRST-order formalization is INDEED problematic! > In formalizing Con(PA) or the fundamental theorem of > arithmetic in PA, we need to represent finite sequences > of numbers as numbers. This can be done in many ways, > but are they intension-preserving? In the case of any first-order anything, preserving intension SPECIFICALLY around the concept of "finite" is what is hard. Is there any way to augment the concept of a first-order language, or even just add a new inference rule to FOL, such that it "gets" Finitude, without going all the way to 2nd-order logic?
From: Helene.Boucher on 1 Feb 2005 11:54
tchow(a)lsa.umich.edu wrote: > 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 I agree with your division. 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. |