In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 31 Jul 2010 09:41 Marshall <marshall.spight(a)gmail.com> writes: > On Jul 31, 2:26�am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Why do you think Nam is insincere? > > He slings insults at MoeBlee, then supposedly offers advice, saying > "Hope that would help you somehow." The advice, while apparently > expressing virtues, is the sort Nam himself entirely avoids > taking. Thus I don't believe he is sincere about the value of the > virtues thus expressed. We can be very blind when it comes to our behaviour, actions, even deeply held beliefs. It's possible Nam sincerely believes he's doing his best to be virtuous, to take his own advice. After all everyone gets their knickers in a knot every now and then, over wrongs real or imaginary inflicted on them, and rash words are exchanged, but in the end it's the thought that counts and so on and so forth. Perhaps it could be called a form of insincerity; such extreme blindness is almost invariably to an extent willful. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 31 Jul 2010 10:37 Nam Nguyen says... >At the heart of the matter is I'd like to point out to them that they >(notably AK, CM, MoeBlee, etc...) would use the word "proof" of a 1st >order formula in 2 different senses: there's the syntactical sense via >the rules of inference for sure but they have been talking about proofs >through the intuitive knowledge of the natural numbers. Generally, if someone says S is provable in T, then they mean that it follows from the axioms of T. If someone just says S is provable (with no T mentioned) then it means that S follows from generally accepted premises, without stating exactly what those premises are. There is potentially an ambiguity in saying that, but in practice, most mathematicians share enough common mathematical background that it doesn't cause a problem. Similarly with "disprovable". >What I've tried >to say to them is that, among other consequences of having another >definition of proof other than the one using the rules, there's an >incompatibility (as mentioned in the suggested Principle of Compatibility) >between the 2 and this incompatibility would cast down or even render >our reasoning incorrect, logically speaking. I think it's always clear which one is meant. >By going back and forth about what's possible to prove in a syntactically >inconsistent theory, and what's impossible to prove in a syntactically >consistent one, I'd like to contrast the strength and weakness of both >kinds of proof-definition. The hope is that they'd understand more about >the Principles Compatibility and Symmetry, for the better of reasoning. If someone is talking about an *inconsistent* theory, then they are always meaning the formal notion of provable and disprovable. >So for the n-th time, anyone including I would know what an inconsistent >theory is and _NO ONE_ would care to use it, as a FOL system. But as >far as matters of foundation are concerned, they have ignored the >meaning of inconsistency too long to the point they couldn't recognize >what the foundational issues are, in the edifice of FOL reasoning. There is no meaning of inconsistency, beyond the observations: 1. In an inconsistent theory every formula is provable. 2. An inconsistent theory has no models. >> But it doesn't make any sense to me to make definitions so that >> they don't apply to inconsistent theories. What is the point of that? > >What does it mean when we say we "prove" a theorem? As I said, if you specify a *theory*, then it means that we establish that the theorem follows from the axioms and rules of inference of that theory. If we don't specify a theory, then it means that we establish that the theorem follows from generally accepted facts about naturals, reals, etc. >Now then, what does it mean to not-be-able-even-in-principle-to-prove >a formula doesn't have a proof? I don't know what that means. I know what it means to say that something is not provable in a particular *theory*. When would such a phrase ever come up? There really isn't a well-defined notion of what is provable in principle, if you don't specify which theory you are talking about. This shows an asymmetry between the words "provable" and "unprovable". People sometimes use "provable" without specifying a theory, to mean that it is possible to demonstrate that a formula follows from generally accepted mathematical facts. But I don't think anyone ever uses the word "unprovable" in an analogous informal sense. "Unprovable" is always relative to a specific theory. >If used as is, that's too long a verb, >right? But if you use "unprove" that's not quite correct because to >"un-prove" could mean to simply show that the proof is incorrect, but >doesn't necessarily mean the formula doesn't have a proof. The word you are searching seems not likely to ever come up in a mathematical discussion, although it might come up in a philosophical discussion. If there were a formal theory T that captured *exactly* the theorems (about arithmetic, say) that mathematicians can eventually come to realize are indisputably true, then it would follow that anything unprovable in T would be unprovable in the informal sense. >So, among that very long verb, the verb "un-prove", and the verb >"dis-prove", what would you pick as the opposite of the verb to "prove"? You're wanting a *verb*? I don't see any kind of verb here. I thought we were talking about properties of sentences, not things people do. "Provable", "disprovable", and "unprovable" are properties of sentences. There are corresponding verbs "prove" and "disprove" (prove means "find a proof" and disprove means "find a proof of the negation), but I have no idea what "unprove" would mean. >If you could see this difficulty of choice here, you'd understand >why I said "disprove" should be used only in the context of a >consistent theory. No, I don't understand that. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 31 Jul 2010 10:55 Frederick Williams wrote: > Nam Nguyen wrote: > >> Be honest, straight forward, to the points, logical, conforming to the 4 >> Principles (Consistency, Compatibility, Symmetry, and Humility). > > People aren't symmetrical, though you may have to know them rather > intimately to discover this. > The immediate antecedent sentence before that is: >> I'd venture to guess that, in the context of making mathematical >> arguments, HE'd advise something like: Please note my "in the context of making mathematical arguments"; and it's obvious that "conforming" would refer to the made "arguments". We all do snip, cut writings, quotes, what have we in responses, and that might not have been a perfect written piece of English, but you couldn't really understand a simple paragraph to point of rather severely distorting (your "intimately to discover this") what people have said? I'm really not sure anymore what some people might be thinking these days, when it comes to discussing mathematical reasoning! -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Aatu Koskensilta on 31 Jul 2010 10:57 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > I'm really not sure anymore what some people might be thinking these > days, when it comes to discussing mathematical reasoning! Sometimes people are just trying to be funny. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 31 Jul 2010 11:21
Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> I'm really not sure anymore what some people might be thinking these >> days, when it comes to discussing mathematical reasoning! > > Sometimes people are just trying to be funny. At the expense of somebody's else arguments, while they have contributed zero effort in the post(s) where the "fun" is in? I do appreciate laughings and humors here and there even in "dried" technical discussions. But this doesn't seem so, imho. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt ----------------------------------------------------------- |