The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 24 Apr 2010 11:43 William Elliot wrote: > On Sat, 24 Apr 2010, Nam Nguyen wrote: >>>>> >>>>>> It's widely believed our intuition of the natural numbers has led >>>>>> to foundational understandings of mathematical reasoning and not >>>>>> the least of which is the validity of GIT proof. In this post, >>>>>> however, we'll demonstrate that if there's an intuition of knowing >>>>>> the natural numbers, there's also an intuition about not knowing >>>>>> them that would invalidate GIT proof. >>>> >>>> GIT = Godel Incompleteness Theorem (The 1st Theorem) >>>> >>>>>> Let F and F' be 2 formulas, we'll define the logical operator xor >>>>>> as: F xor F' <-> (F \/ F') /\ ~(F /\ F') >>>>>> >>>>>> Let's now consider thew 2 formulas in L(PA): pGC ("pro GC") >>>>>> and cGC("counter GC"): >>>>>> >>>> L(PA) = L(0,S,+,*,<) >>> A first order language with equality, constant symbol 0, binary >>> function constants +,* and binary relation constant < with Peano's >>> axioms? >>> >>>> GC = Goldbach Conjecture >>> Every even number greater than two is the sum of two primes? >> >>>>>> pGC <-> "There are infinitely many examples of GC" >>>>>> cGC <-> "There are infinitely many counter examples of GC" >>>>>> >>>>>> Now, let's consider the following formula: >>>>>> >>>>>> (1) pGC xor cGC >>>>>> >>>>>> First observation: if we have any intuition about the naturals >>>>>> then we'd also have the intuition that we can't know the arithmetic >>>>>> truth or falsehood of (1). > >>> Currently we do not know, nor do we know if GC is independent of PA. >>> In addition there's no proof that we can't know. >> >>> Indeed, your fallacious argument could have been made a hundred years >>> ago about FCT or FLT which we couldn't know then, but do now. >> >> Perhaps you'd want to reflect a bit on what TF said below, before >> making the judgment above: >> >> "So for any formal system S which incorporates a bit of arithmetic >> - the basic rules needed to carry out computations - a Goldbach-like >> statement is disprovable in S if false. On the other hand, we can >> make no similar observation about how a Goldbach-like statement can >> be proved if it is true." >> > You have jumped from the observation that we don't know a proof or a > disproof of GC, to the claim we cannot ever know a proof or a disproof > of GC. You have yet to substantiate your claim. First of all, in arguing foundational issues we should be very precise in what we say (and I think you'd agree with this). It is (1) - NOT GC - that I've claimed here as the (first observation), and the 2 formulas aren't supposed to be equivalent. So, your "rebuttal" started in a wrong track already. Secondly, your wrongly having thought I had meant GC instead of (1) aside, if the claim you think I should have substantiated is my first observation (that you've singled out here) then that's not what I claimed in observation 1. And I shouldn't substantiate on something I've not intended to claim. Hopefully our arguing would be on the right track after these 2 caveats/ disclaims I've made. > In addition TF's statement by itself without the context of his system > of logic is vague. I don't see how it fits into the discussion or how > it fills the gap in your thinking. For example, what's a Goldback like > statement and is the quote other than a distraction? Per my caveats above, let's hold off discussion of the relevancy of TF's comment until we're in agreement what it is we'd like to argue. It suffices to mention though a) his definition of Goldbach-like statement: "A property P of numbers which can be checked by applying an algorithm will be called a computable property. (This notion will be explored further in Chapter 3.) What has been noted above is that Goldbach�s conjecture can be formulated as a statement of the form "Every natural number has the property P", where P is a computable property. This is a logically highly significant feature of Goldbach�s conjecture, and in the following any statement of this form will be called a Goldbach-like statement. (In logic, these are known by the more imposing designation PI(0,1)-statements.)" and b) he also said in the same book: "The property of an arithmetical statement of being Goldbach-like will play a role at several points in the discussion of incompleteness." *** In summary, if you _precisely_ let me know what I should substantiate then I'll try.
From: Nam Nguyen on 24 Apr 2010 13:25 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> First observation: if we have any intuition about the naturals >>> then we'd also have the intuition that we can't know the arithmetic >>> truth or falsehood of (1). >> >> This isn't an observation. It's a bald assertion that is, on the face of >> it, vague, implausible, and entirely arbitrary. >> > > In any rate, what are you clear, plausible, specific technical reasons > for your belief that the knowledge of the naturals aren't of intuitive > nature, or that you'd know the arithmetic truth or falsehood of (1)? > > Iow, what are your _technical, non-bias, objective_ grounds for attacking > my observation above? My observation is of the form: If H then C. What is it that you're objecting? If H it is, what's your reason of objection? If it's C, why? If it's somewhere in between, exactly what and why? While waiting for answers to these questions from you or anyone, let me repeat my position that our knowledge of the naturals numbers is of intuitive nature, which is not precise. To see why such knowledge is just an intuition, we can (in meta level) equate the truth of any mathematical formula F in L(PA) to a syntactical notion as: F is true <-> (PA isn't inconsistent) and (PA |- F) However, since rules of inference will NOT permit a syntactical proof of "PA isn't inconsistent", primarily because the rules will not permit any _disproof_ , the meta truth of "PA isn't inconsistent" must be of an intuition, hence so is "F is true". If we could argue about my claim here about H (the hypothesis of my first observation), and bring the argument to a satisfactory conclusion - whether you or I would be on the correct or incorrect side - then I'd proceed further with the demonstration that would lead to the conclusion (C). But first thing first, let's work on H.
From: William Elliot on 25 Apr 2010 04:28 On Sat, 24 Apr 2010, Nam Nguyen wrote: > William Elliot wrote: >> On Sat, 24 Apr 2010, Nam Nguyen wrote: >>>>>> >>>>>>> It's widely believed our intuition of the natural numbers has led to >>>>>>> foundational understandings of mathematical reasoning and not the >>>>>>> least of which is the validity of GIT proof. In this post, however, >>>>>>> we'll demonstrate that if there's an intuition of knowing the natural >>>>>>> numbers, there's also an intuition about not knowing them that would >>>>>>> invalidate GIT proof. >>>>> >>>>> GIT = Godel Incompleteness Theorem (The 1st Theorem) >>>>> >>>>>>> Let F and F' be 2 formulas, we'll define the logical operator xor as: >>>>>>> F xor F' <-> (F \/ F') /\ ~(F /\ F') >>>>>>> >>>>>>> Let's now consider thew 2 formulas in L(PA): pGC ("pro GC") >>>>>>> and cGC("counter GC"): >>>>>>> >>>>> L(PA) = L(0,S,+,*,<) >>>> A first order language with equality, constant symbol 0, binary function >>>> constants +,* and binary relation constant < with Peano's axioms? >>>> >>>>> GC = Goldbach Conjecture >>>> Every even number greater than two is the sum of two primes? >>> >>>>>>> pGC <-> "There are infinitely many examples of GC" >>>>>>> cGC <-> "There are infinitely many counter examples of GC" >>>>>>> >>>>>>> Now, let's consider the following formula: >>>>>>> >>>>>>> (1) pGC xor cGC >>>>>>> >>>>>>> First observation: if we have any intuition about the naturals >>>>>>> then we'd also have the intuition that we can't know the arithmetic >>>>>>> truth or falsehood of (1). > First of all, in arguing foundational issues we should be very precise > in what we say (and I think you'd agree with this). Indeed. What's a GC example? What's a GC counter example? > In summary, if you _precisely_ let me know > what I should substantiate then I'll try. I agree, we currently don't know the truth or falsehood of (1). From that known observation, show that we can't ever know the truth or falsehood of (1). -- In the year 1900, the truth or falsehood of FLT wasn't known. By using the method you use to substantiate that we'll never know the truth or falsehood of (1), people in 1900 would conclude that they can never know. Explain how your method and it's conclusion contradicts the now known proof of FLT. ----
From: Nam Nguyen on 26 Apr 2010 00:38 William Elliot wrote: > On Sat, 24 Apr 2010, Nam Nguyen wrote: >> William Elliot wrote: >>> On Sat, 24 Apr 2010, Nam Nguyen wrote: >>>>>>> >>>>>>>> It's widely believed our intuition of the natural numbers has >>>>>>>> led to foundational understandings of mathematical reasoning and >>>>>>>> not the least of which is the validity of GIT proof. In this >>>>>>>> post, however, we'll demonstrate that if there's an intuition of >>>>>>>> knowing the natural numbers, there's also an intuition about not >>>>>>>> knowing them that would invalidate GIT proof. >>>>>> >>>>>> GIT = Godel Incompleteness Theorem (The 1st Theorem) >>>>>> >>>>>>>> Let F and F' be 2 formulas, we'll define the logical operator >>>>>>>> xor as: F xor F' <-> (F \/ F') /\ ~(F /\ F') >>>>>>>> >>>>>>>> Let's now consider thew 2 formulas in L(PA): pGC ("pro GC") >>>>>>>> and cGC("counter GC"): >>>>>>>> >>>>>> L(PA) = L(0,S,+,*,<) >>>>> A first order language with equality, constant symbol 0, binary >>>>> function constants +,* and binary relation constant < with Peano's >>>>> axioms? >>>>> >>>>>> GC = Goldbach Conjecture >>>>> Every even number greater than two is the sum of two primes? >>>> >>>>>>>> pGC <-> "There are infinitely many examples of GC" >>>>>>>> cGC <-> "There are infinitely many counter examples of GC" >>>>>>>> >>>>>>>> Now, let's consider the following formula: >>>>>>>> >>>>>>>> (1) pGC xor cGC >>>>>>>> >>>>>>>> First observation: if we have any intuition about the naturals >>>>>>>> then we'd also have the intuition that we can't know the arithmetic >>>>>>>> truth or falsehood of (1). > >> First of all, in arguing foundational issues we should be very precise >> in what we say (and I think you'd agree with this). > > Indeed. What's a GC example? What's a GC counter example? I hope you don't mind me saying that there's a distinction between assuming the trivial knowledge such as formalizing an example (or a counter example) of GC and not being precise in mis-recognizing, mis- interpreting what people stated or claimed. (No big deal really, but just so you know). Any rate, here are some of the relevant definitions: GC(x) <-> (x is even >=4) /\ Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] cGC(x) <-> ~GC(x) and given a unary property P, the statement "There are infinitely many examples of P", denoted say by *P, is defined as: *(P) <-> AxEy[P(x) -> ((x<y) /\ P(y))] then we'd now have: pGC <-> *(GC) cGC <-> *(cGC) > >> In summary, if you _precisely_ let me know >> what I should substantiate then I'll try. > > I agree, we currently don't know the truth or falsehood of (1). I'd agree. > > From that known observation, show that we can't ever know > the truth or falsehood of (1). But again, William, you weren't precise: was that what I really had said? If you read my (first) observation carefully, you'd see what I said basically is: if we only intuit the natural numbers the way we currently do, then such intuition would never let us to know the truth value of (1) be. And that's not the same as saying something like "since we don't know something we can't ever know that something". I think what you meant to request is to demonstrate the impossibility of knowing the truth value of (1), given out current intuition of the naturals. It'd be great if you could confirm this before I begin the demonstration. Any rate I'll begin it in the next few days; I just need some time to organize my thought. > -- > In the year 1900, the truth or falsehood of FLT wasn't known. > By using the method you use to substantiate that we'll never > know the truth or falsehood of (1), people in 1900 would > conclude that they can never know. Explain how your method > and it's conclusion contradicts the now known proof of FLT. It's a fair observation. However there's reason why TF, e.g., coined the term "Goldbach-like" statement and not "FLT-like" statement. The point being certain class of formulas might have certain relevant foundational implication and certain other classes might not. For example, the class of theorems of the form F /\ ~F would signify inconsistency while the class of "normal" theorems like FLT would not mean that. But this is a reasonable concern and I agree if not-knowing is part of reasoning framework, care should be taken to make a distinction between the 2 different cases: that you don't know something but such something is possible to know, and that it's impossible to know something because intrinsically it's impossible so within the reasoning framework.
From: Phil Carmody on 26 Apr 2010 03:18
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > William Elliot wrote: >> Indeed. What's a GC example? What's a GC counter example? > > I hope you don't mind me saying that there's a distinction between > assuming the trivial knowledge such as formalizing an example (or a > counter example) of GC and not being precise in mis-recognizing, mis- > interpreting what people stated or claimed. (No big deal really, > but just so you know). Any rate, here are some of the relevant > definitions: > > GC(x) <-> (x is even >=4) /\ Ep1p2[(p1, p2 are primes) /\ (x=p1+p2)] > cGC(x) <-> ~GC(x) So clearly there are an infinitude of "counterexamples" to the Goldbach conjecture. 1 is one. 3 is one. 5 is one too... > and given a unary property P, the statement "There are infinitely many > examples of P", denoted say by *P, is defined as: > > *(P) <-> AxEy[P(x) -> ((x<y) /\ P(y))] Let P(x) <-> (x is a positive integer) /\ (x is minus pi) Clearly *P holds, as P(x) never holds, and your implication is always vacuously holds. I.e., using your logic, because there are no examples of P, there are infinitely many examples P. I always try skip past your long drawn-out threads, and have never spend enough time to ascertain whether you are a loon or not. I think finally I have the evidence that you really don't have a clue what you're talking about. Which means, with no guilt or fear of missing anything useful, I never have to see another post of yours again - *PLONK*. Phil -- I find the easiest thing to do is to k/f myself and just troll away -- David Melville on r.a.s.f1 |