Torkel Franzen on truth
in [Logic]
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From: george on 18 Dec 2007 14:08 On Dec 18, 3:00 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Well, I'm not so sure that there *is* a standard usage of "language" > among logicians. But one difference between me and some other people is that *I* would *not* PRESUME to generalize over anything as broad as "among logicians". I'm NOT TRYING TO TALK about that. I'm talking about a standard usage of "language" that remains standard within a domain of discourse extending across ONE TEXTBOOK AND *NO* further, and it had better be a textbook ABOUT formal languages or theory of computing. I have no quarrel with everybody continuting to use "language" to mean something richer outside those textbooks. The question is, if you VOLUNTARILY limit yourself to such an impoverished conception of "language", how much CAN YOU STILL achieve? The answer is, with some reasonable axioms, quite a bit. If the axioms are as simple as PA, however, there are a lot of first-order truths about the naturals that you cannot achieve. You can achieve more of them, however, with an even SIMPLER first-order language, using first-order ZFC. My point is, the richness of what you can achieve is sufficiently great that nobody is, really, especially not in THIS room, ENTITLED to COMPLAIN that this sense of "language" is unnaturally or unreasonably impoverished, EVEN THOUGH IT IS. Some students actually do grow up to overcome the poverty of their beginnings.
From: george on 18 Dec 2007 14:12 On Dec 18, 3:36 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > We start off working in everyday mathematics. You don't even know what that means. You are beginning by presuming the existence of an intended model. That is just not the way it's generally done any more. > Wanting maximal rigour > and absolute clarity, we reflect on our practices, and regiment our > informal mathematical language, and regiment chunks of our everyday > mathematics into nicely disciplined axiomatic systems. That is the BEGINNING, NOT THE END, of the process. > We construct, > for example, the regimented theory most of us call first-order PA. Almost nobody more than 10 years younger than you ever did any such thing. The rest of us STARTED with PA. It embodies the basic things we know, at first-order, about this realm. > This theory is as semantically contentful as the informal inchoate > theory we started off with, No, it isn't, and more to the point, PURELY BY VIRTUE OF BEING a first-order syntactic theory, IT IS NOT semantically contentful AT ALL. The completeness theorem is fundamentally a proof that first-order semantics SIMPLY DOESN'T EXIST.
From: george on 18 Dec 2007 14:14 On Dec 18, 3:36 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Of course, once that regimented theory is on the table, It completely moots the relevance of any alleged prior model that may have inspired it. In the case of PA, this is more relevant than usual for the simple reason that the alleged prior model WAS INFINITE and therefore (arguably) not well-understood enough TO BEGIN WITH for ANYbody to have been able to WIN the argument over the question of whether he did or didn't KNOW WHAT HE WAS TALKING ABOUT (i.e. know what infinite model he originally had in mind).
From: george on 18 Dec 2007 14:30 On Dec 18, 3:36 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Wanting maximal rigour > and absolute clarity, we reflect on our practices, and regiment our > informal mathematical language, and regiment chunks of our everyday > mathematics into nicely disciplined axiomatic systems. This is oversimplified and missing the point. A GREAT MANY DIFFERENT things could qualify as "nicely disciplined axiomatic systems". We are NOT talking about that huge total generality. We are only talking about ONE thing. WE are talking about classically-defined first-order languages over finitary signatures. > We construct, for example, the regimented theory > most of us call first-order PA. This is A COMPLETE LIE. *WE* do NOT do ANYthing. The first-order language of PA is a purely abstract formal thing with some simple known signature that ANYbody could've written down or stipulated WITHOUT CARING A FIG about what informal mathematical realm it might or might not have been "about". Mathematical practice IS PROVING stuff. In the first-order paradigm, proof is syntactic and INDEPENDENT of semantics. > This theory is as semantically contentful as the informal inchoate > theory we started off with, just better disciplined. It's simultaneously more AND less contentful. It's UNIVERSALLY contentful in that the theorems must be true of ALL models, both actual AND POSSIBLE, of the axioms. It is about a great many MORE semantic universes than may have motivated your initial investigation. My point is simply that the initial investigation is just irrelevant. The axioms are what matter. That is what is ACTUALLY being investigated. This no doubt seems to you like the inmates taking over the asylum; the axioms were CREATED *by* you to *serve* you in your quest for more TRUTH about the INTENDED model, in your opinion. In my version it winds up seeming like the axioms are the masters and you are doomed to slave away deriving their consequences. Yes, I suppose it is a lot more fun to imagine yourSELF and YOUR creativity in charge.
From: tchow on 18 Dec 2007 14:38
In article <0a4c3780-80c9-48ed-99c5-227a8105be7a(a)e6g2000prf.googlegroups.com>, george <greeneg(a)cs.unc.edu> wrote: >Almost nobody more than 10 years younger than you ever did any >such thing. The rest of us STARTED with PA. It embodies the basic >things we know, at first-order, about this realm. George, you've *got* to be kidding here. Do you not have any clue how idiosyncratic your viewpoint is? Name one person other than yourself who "STARTED with PA." I've seen lots of "new math" movements, but never have I heard of anyone teaching arithmetic to five-year-olds by starting with PA. -- 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 |