Give me ONE digit position of any real that isn't computable up
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From: Daryl McCullough on 9 Jun 2010 20:30 Transfer Principle says... >In short, I want to see a counterexample to Little's >statement that: > >"The vast majority of posts arguing against ZFC or its >theorems in sci.math are actually from incompetent >cranks." > >Thus, I want to see a post arguing against ZFC that's >from a competent poster. You're not going to see it, because there is no point in a competent person "arguing against" ZFC. What would such an argument even look like? It's conceivable that there could be an argument in favor or against the consistency of ZFC, or there could be an argument in favor or against the usefulness of ZFC for this or that purpose. But none of those would count as arguing "against ZFC". So I think it is a good bet that anyone arguing against ZFC will be a crank of some sort, because noncranks would have no reason to do such a thing. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 10 Jun 2010 12:45 On Jun 9, 7:30 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > there is no point in a > competent person "arguing against" ZFC. What would such > an argument even look like? It's conceivable that there > could be an argument in favor or against the consistency > of ZFC, or there could be an argument in favor or against > the usefulness of ZFC for this or that purpose. Or one could have philosophical grounds for objecting to ZFC and perhaps preferring some other theory on such grounds. MoeBlee
From: George Greene on 11 Jun 2010 17:53 On Jun 9, 8:30 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > So I think it is a good bet that anyone arguing against ZFC > will be a crank of some sort, because noncranks would have > no reason to do such a thing. Exactly. Noncranks understand that ZFC HAS AXIOMS. ZFC arguably IS a set of axioms AS OPPOSED to the theory that follows from them, the point being that as long as you insist on operating in THE(UNIQUE, up to orthography) first-order language with ONE binary predicate (you don't need any constants at all; the empty set is definable via the axiom- schema of replacement), you could accomplish the SAME theory from OTHER axioms, but their very otherness would make it true that you were doing something OTHER than ZFC. Once something has been conceded to be a set of axioms, it canNOT POSSIBLY be "argued against" -- this, again, is something that noncranks just know. If you don't like THOSE axioms then you are OF COURSE PERFECTLY welcome to just PICK SOME OTHERS and start deriving stuff from THEM. Nobody (or no non- crank, anyway) is going to allege that one set of axioms is "right" and the other is "wrong". The closest an axiom-set can come to "being argued against" is being argued to be inconsistent, and that is itself almost ALWAYS crankish BECAUSE, in this particular specific case, THIS IS A STANDARD CLASSICAL RECURSIVELY AXIOMATIZABLE FIRST-ORDER THEORY OVER A FINITE SIGNATURE, which means that IF it is inconsistent, then THERE IS A PROOF of that -- wherefore there is simply NO NEED to ARGUE: EVERYbody (or every non-crank anyway) stands OBLIGATED to PUT UP OR SHUT UP, i.e. to JUST PRESENT the proof.
From: Daryl McCullough on 16 Jun 2010 10:18 Leland McInnes says... >On Jun 15, 5:00=A0pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> On Jun 12, 8:24=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >> > Transfer Principle <lwal...(a)lausd.net> writes: >> > > The response I'd like to see is one which defends classical analysis >> > > against these smooth infinitesimals >> > This idea, that classical analysis needs defending against smooth >> > infinitesimals, is bizarre. >> >> But there has to be a reason why most mathematicians use >> classical analysis and not smooth infinitesimal analysis. > >I expect that relative newness of SIA is a big part. To make robust >foundations for SIA possible you need to ground things in topos theory >with its more flexible logics. That meant that SIA wasn't developed as >a theory until the 1980s. Compare that to classical calculus which has >more then a centurey of established history. I think that there is another reason that most mathematicians prefer standard analysis (or, for that matter, even nonstandard analysis): because they feel more comfortable using classical logic. It's usually assumed that the rules of logic are the same across all areas of mathematics, and what varies from area to area is the subject matter: what mathematical objects are being talked about, what functions on those objects, what relations. Some mathematicians take a different approach to logic. Rather than having a fixed logic that applies across different fields, they try to find a logic that is most appropriate for each field. So they use fuzzy logic for dealing with uncertainty, or they (it's actually not the same "they") use quantum logic for dealing with quantum mechanics, or they use constructive logic for dealing with computable functions, or they use intuitionistic logic for dealing with differential geometry or real analysis. Using domain-specific logics makes it a little harder for people to switch areas, so many people are hesitant to use specialized logics. Also, most people feel that they understand a logic better if they can give a classical model for what is going on, so many people use classical logic for the metatheory and model theory, even if they use a nonclassical logic for the object theory. But that might just reflect their comfort level with classical logic. Perhaps nonclassical metatheory and model theory works fine. Anyway, I do think that it is a mistake to view people's preferences for this framework or that framework in terms of ideology and personal "belief". For practical purposes, you don't have to any metaphysical commitment to use a logic or a mathematical theory. You just need to understand the criteria for rigorous mathematics in whatever field. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 16 Jun 2010 12:49
On Jun 16, 9:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >I do think that it is a mistake to view people's > preferences for this framework or that framework in terms > of ideology and personal "belief". For practical purposes, > you don't have to any metaphysical commitment to use a > logic or a mathematical theory. You just need to understand > the criteria for rigorous mathematics in whatever field. Also, I wonder whether the poster Transfer Principle has the impression that mathematicians in general even think very much about such foundational matters as alternative logics, alternative theories, etc. Perhaps a great many mathematicians more simply go into various branches of classical mathematics as such mathematics gets passed on from generation to generation and mathematicians of new generations still find an OVER-abundance of interesting questions to address within classical mathematics. One may argue (I don't opine myself here) that this tradition-bound acceptance deserves to be shaken up; that mathematics would be better if we "leveled the playing field" for a variety of alternative foundations. So that would be a call for changes in the OVERALL institutions of mathematics. Whatever one may say about that sentiment, though, I don't think it entails that most INDIVIDUAL mathematicians are intolerant of alternative mathematics or even that they work in classical mathematics for any particular philosophical reason, but rather that they work in classical mathematics because classical mathematics has plenty of interesting questions, statements to prove or disprove, and even new branches within classical mathematics to create. MoeBlee |