The End of an Era - That of The Natural Numbers
in [Math]
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From: William Hughes on 2 May 2010 12:33 On May 2, 12:35 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > In summary, you and I each should give, as William Elliot suggested, > "insights" to _defend_ our different positions on this issue about > the knowledge of the naturals. And though "insight" is not a precise > word, that doesn't at all mean unreasonable insights such as there > are only finite number of the naturals could be used to further > the arguments. Ok. My "insight" is "The Goldbach Conjecture is true". Therefore, unless "any intuition" does not include "The Goldbach Conjecture is true". your First Observation is false. - William Hughes
From: Nam Nguyen on 2 May 2010 12:46 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> William Hughes wrote: >> >>> If we do not assume T is sound, then there is no connection >>> between provable and true. Now we have to phrase things: >>> (1) is true iff cGC is false. >> >>> However, "cGC is false" is a >>> perfectly reasonable intuition, so even if we don't assume >>> T is sound, observation 1 is false. >> I'm sorry: that simply doesn't cut it. Anyone else could equally >> say "cGC is true" is a perfectly reasonable intuition! So who >> would be correct: you or they? > > People differe on their intuitions; > surely you admit that? And diverging intuitions can both be reasonable. Of course that's reasonable - _in general_ . But appropriate contexts will cap the limit how much divergent opinions could be. For instance, if you have the intuition that ~(x=x) then that's fine and you'd have to make logical/mathematical deductions in a FOL different from FOL=. Or if you'd would prefer the intuitions of Genzen over Hilbert's on rules of inferences that's fine too. But there should be _no blank-check on intuitions_, if we're talking about mathematics and the reasoning about it. > >> And why would that _intuition_ >> be correct at the expense of others? > > Why is *your* claim about intuitions on arithemetic better than > other people's? Not sure why I had to repeat it but in my last response I already mentioned something that in mathematics and logic context "intuition" can't be a blank-check to claim anything whatsoever (wouldn't that make no difference between the crank and the real mathematicians?). At least I've never denied my obligation here to explain, share "insight" about my intuitions. The fact remains I don't just use the word "intuition" on the naturals or about (1) without some explanation - or some plan to do it. Perhaps you want review my latest conversations with WH (and WE [William Elliot] where I just began the explanation).
From: Nam Nguyen on 2 May 2010 13:34 Nam Nguyen wrote: > Alan Smaill wrote: >> People differe on their intuitions; >> surely you admit that? And diverging intuitions can both be reasonable. > > Of course that's reasonable - _in general_ . But appropriate contexts > will cap the limit how much divergent opinions could be. For instance, > if you have the intuition that ~(x=x) then that's fine and you'd have > to make logical/mathematical deductions in a FOL different from FOL=. > Or if you'd would prefer the intuitions of Genzen over Hilbert's on > rules of inferences that's fine too. > > But there should be _no blank-check on intuitions_, if we're talking > about mathematics and the reasoning about it. For example, given the context of FOL= and of the kind of intuition of the naturals that PA's non-induction-schema axioms be true, then one has to _appropriately in those contexts_ explain (or plan to explain) why, e.g., cGC is true/false, and not just give any-what-so -ever kind of explanation, as WH did in his recent post.
From: Alan Smaill on 2 May 2010 14:31 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Nam Nguyen wrote: >> Alan Smaill wrote: > >>> People differe on their intuitions; >>> surely you admit that? And diverging intuitions can both be reasonable. >> >> Of course that's reasonable - _in general_ . But appropriate contexts >> will cap the limit how much divergent opinions could be. For instance, >> if you have the intuition that ~(x=x) then that's fine and you'd have >> to make logical/mathematical deductions in a FOL different from FOL=. >> Or if you'd would prefer the intuitions of Genzen over Hilbert's on >> rules of inferences that's fine too. >> >> But there should be _no blank-check on intuitions_, if we're talking >> about mathematics and the reasoning about it. > > For example, given the context of FOL= and of the kind of intuition > of the naturals that PA's non-induction-schema axioms be true, well, of course you can find many accounts of the intuitive acceptability of the induction principle. See Poincar�, for example. > then > one has to _appropriately in those contexts_ explain (or plan to > explain) why, e.g., cGC is true/false, and not just give any-what-so > -ever kind of explanation, as WH did in his recent post. There is absolutely no reason to require any such explanation. I mean, no-one is claiming that Peano arithmetic, or heyting arithemtic, provides the answer to all the questions about arithmetic, are they? In the absence of such an explanation, do you want to rule this intuition inadmissable, then? -- Alan Smaill
From: Tim Golden BandTech.com on 2 May 2010 16:21
On May 1, 11:49 am, A <anonymous.rubbert...(a)yahoo.com> wrote: > On May 1, 7:04 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > The original point was that we can regard the natural numbers as the > > modulo-infinity numbers, and this yields primitives prior to the > > naturals. I say again, we do not need the number four in order to > > You know, if m divides n, you have a quotient map from Z/nZ to Z/mZ, > which is a ring morphism. You can take Z/nZ for all n, together with > all those quotient maps, and "let n go to infinity" by taking the > inverse limit in the category of commutative rings. But you don't get > the integers when you do this; instead, you get the profinite > integers, the Cartesian product of one copies of the p-adic integers > for every prime number p. (In some ways the profinite integers are > extremely natural and important themselves; e.g. the absolute Galois > group of any finite field k is isomorphic to the profinite integers. > But the profinite integers are still not the integers themselves.) > > You can try to go the other way, taking a direct limit, instead of an > inverse limit. Here's how that works out: if m divides n, you have an > inclusion from Z/mZ to Z/nZ (the "multiplication by n/m" map), which > is a morphism of abelian groups, but not a ring morphism, since (for > example) it doesn't send the unit element 1 to the unit element 1. But > okay, you can still take Z/nZ for every n, equipped with these > inclusion maps, and "let n go to infinity" by taking the direct limit > in the category of abelian groups (you can't do it in rings, since > these maps aren't ring morphisms). But again, you don't get the > integers. When you take the direct limit of Z/(p^n)Z for all n, where > p is a prime number, you get the abelian group Z with p inverted; if > you're not familiar with that construction, you can read about it in > any book that talks about localization of rings and modules, e.g. > Atiyah and Macdonald's excellent, short, and very readable > "Introduction to Commutative Algebra." So when you try to take a > direct limit of Z/nZ for all n, "letting n go to infinity" in that > way, you get...the integers, but with every prime number p > inverted...in other words, the rational numbers. > > So I think the notion of "modulo infinity" is not as straightforward > as it seems! There is no need to use all of this quotient language. It is not used in real analysis and I do not see it as a clean construction. My own impedance at this level prevents me from following your argument, but certainly I accept your conclusion. The idea of a wrapping count which never wraps would seem inherently conflicted, but because its lower forms are simple and clean and have no need of pushing for higher numbers these low forms do not deserve to be treated as constructions of an infinite bearing number system. I am saying simply that it makes no sense to define modulo numbers in terms of the natural numbers. It is instead cleaner to do the reverse. Structured information will build more complicated things from simpler things, both formally and informally. Informally, every natural number that you use beyond nine is already making use of modulo principles. > > > count to three. It so happens that these modulo forms are meaningful. > > Though you are used to relying upon the polynomial to gain their > > behavior, these discrete constructions can provide similar results > > from a primitive level. > > Beyond that abstract algebra is falsifiable. The very phrase > > 'polynomials with real coefficients' > > is incompatible with the ring definition. That meaningless X that only > > carries a sequential positional value... that is toxic. You already > > had what you needed back here in the modulo number, and it is sign > > where it makes its presence felt. The standard +/- sign of the real > > This is a strange thing to say. The ring of polynomials in one > variable with coefficients in any ring R is itself a ring, and this > isn't hard to prove; most undergraduate mathematics majors prove it as > an exercise in an algebra course. It's elementary and it's used every > day, in mathematical practice. Consider the care with which the product was only just treated in the ring definition. This new product of marrying values of different types is not congruent and so to claim that this branch of math is coherent one would be forced to discuss this second product form to the same level of formality as the first, rather than just slide it under the table. Is a polynomial coefficient a product? If so, then you tell me where it is handled formally. It certainly is not in the ring definition. The X in these polynomial products is not declared to belong to any set and so cannot fit the ring definition. My own conclusion on this detail is that when this coefficient business is handled formally that the product of a real value with a complex value will remain as such and not resolve to a complex value, so that a value a in the Reals in product with z in the Complex will yield an unresolvable a z and this then is some tidbit of new mathematics, likely more useful to physicists than to mathematicians. In the higher dimension associative algebras one sees larger spaces like C x C and it would be foolhardy to claim that one can assimilate a real or complex coefficient into this 4D space, for which of these two sets will you interperet them as? In general dimension the procedure does not hold up. > > I seem to recall there being a thread where many people weren't (after > much time and effort was spent) able to convince you that polynomial > rings are a valid construction; and I doubt I could succeed where they > failed, so I'd better not try. But really, polynomial rings do make > sense. No, they do not. The value X, or rather, the X, is completely undefined. It is familiarity with real valued X that allows acceptance of the undefined X. This usage is an important instance that goes unadressed in the mathematics texts that I have studied, which I admit is limited. The actual ability to take the product of a real value and this ill-defined X is nonexistent. What then is this notation? Shall you accept from me the construction of 2.34 Y where Y has no set that it is declared to belong to? No, this math has stretched into a terrain that deserves more scrutiny. You would not accept the line above from me, and the only reason that you accept it from them is sheer mimicry. This devolution is far away from the original intent, and you have provided no falsification of my argument. Instead you dodge; and the long way around. You've not even addressed multiplication as addition here. The level of dodge is absurd. Your own credibility is exposed. I take that vacuous space as loose proof. - Tim > > > number is modulo-two behaved. Will you require another image of the > > natural numbers be deployed in order to provide sign mechanics? I > > don't think so. > > > Really, this is so simple I find it difficult to believe that you > > would insist on inflating a compact (informal) set of values out to an > > infinite length set. You are going to insist on infinity now when it > > is not at all necessary. Shame on you. You are fine instances of what > > mathematics has become. Furthermore, there is no direct falsification > > of my statement. There is no conflict in looking at discrete > > multiplication as addition. I used to think you were fair TP, but now > > when I wipe I'll be remembering your acronym. > > > - Tim |