Name the thesis: "Formal sentences capture informal ones"
in [Theory]
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From: Aatu Koskensilta on 29 Jan 2005 15:46 tchow(a)lsa.umich.edu wrote: > Or maybe, "Informal mathematical statements are adequately > expressed by formal sentences"? Help me out here. There are usually numerous different ways of formalizing a given informal mathematical statement which are not logically equivalent. So we must have a qualifying clause saying something like "relative to the relevant background theory". But what is the relevant background theory? Something like this springs to mind: T is the relevant background theory if all obvious formalizations of the informal mathematical statement are equivalent according to T But this is still a far cry from the Church-Turing thesis when it comes to usability or level of trust one should place on the thesis. For example, we should have some notion of what exactly are the obvious formalizations of a given mathematical statement before the above determines any theory at all. I for one have no idea how such a notion could be mathematically defined. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Torkel Franzen on 29 Jan 2005 15:38 tchow(a)lsa.umich.edu writes: > Then how about, "All informal statements of ordinary mathematics are > expressible by formal sentences in ZFC"? The Church-Turing thesis only requires that for every function computable by an algorithm, there is a Turing machine that computes the same function. It is not required that the Turing machine embodies the same algorithm. The above is a lot more problematic to make precise. What is it we require of a formal sentence in ZFC if it is to express a given informal statement of ordinary mathematics? In practice, we recognize formalizations as adequate, but there are many different ways of formalizing informal mathematical statements, and it is far from clear how to characterize what counts as an adequate formalization.
From: tchow on 29 Jan 2005 15:38 In article <41fbf294$0$566$b45e6eb0(a)senator-bedfellow.mit.edu>, I wrote: >Then how about, "All informal statements of ordinary mathematics are >expressible by formal sentences in ZFC"? Sorry, scratch that...that's not at all what I meant, because it sounds like a statement about ZFC in particular, not about expressibility in the language of set theory. Something like what Aatu Koskensilta said is better. Or maybe, "Informal mathematical statements are adequately expressed by formal sentences"? Help me out here. -- 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: Aatu Koskensilta on 29 Jan 2005 15:53 tchow(a)lsa.umich.edu wrote: > I agree that it's far from clear how to characterize what counts as > an adequate formalization. In fact, a large part of the point of > articulating the thesis is to draw attention to this difficulty. 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. Unless this can be remedied the level of usefullness must remain at that of a seemingly good heuristic principle, instead of a generally relied on thesis. Of course, I don't mean that one shouldn't try to come up with a clear formulation of the thesis! In fact, such a project seems to me to be very interesting, even if in the end we do not get a clear generally accepted thesis but only a series of interesting mathematical and conceptual results and analysis. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: tchow on 29 Jan 2005 15:47
In article <vcbr7k4j7it.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: >tchow(a)lsa.umich.edu writes: >> Then how about, "All informal statements of ordinary mathematics are >> expressible by formal sentences in ZFC"? [...] >The above is a lot more problematic to make >precise. What is it we require of a formal sentence in ZFC if it is to >express a given informal statement of ordinary mathematics? In >practice, we recognize formalizations as adequate, but there are many >different ways of formalizing informal mathematical statements, and it >is far from clear how to characterize what counts as an adequate >formalization. I agree that it's far from clear how to characterize what counts as an adequate formalization. In fact, a large part of the point of articulating the thesis is to draw attention to this difficulty. But by saying that it's "problematic to make precise," are you *objecting* to my project of formulating the thesis? The very nature of my proposed thesis prevents it from being *mathematically* precise, just as the Church-Turing thesis isn't *mathematically* precise. That doesn't mean that it's too imprecise to formulate as a snappy thesis. -- 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 |