Does inductive reasoning lead to knowledge?
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From: Marshall on 22 Jan 2010 15:42 On Jan 22, 11:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 21, 11:27 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > On Jan 21, 5:33 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > On Jan 20, 10:24 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > [...] > > > > > You're on: > > > > > There does not exist a number that is a member of > > > > the set of real numbers that is not also a member of the > > > > set of complex numbers. > > > > You've overstated the case significantly here. > > > Not at all; it was a statement of simple fact. > > Actually, after I posted, I had to fault myself for > > stating it in such a cumbersome way; it could have > > been done with just 3 symbols. (Not in ascii, though) > > > R subset-of C > > > Your analysis was quite sophisticated, however it > > is an analysis of constructions of numbers, rather > > than numbers themselves, and so does not > > contradict my statement. > > > Marshall > > Have you withdrawn your claim that b is complex then? Of course not. What gave you that idea? Observe how I said he "does not contradict my statement?" > There are many real lines that can be drawn through the complex plane. > I am trying to do some analysis on this, and might even take interest > in the interpretation of curved paths as valid mappings of the reals > onto the complex plane. Such methods do require transformations, the > trivial transformation being the one that you are thinking of. > Anyway, if I work out the mapping in a nice simple way I'll provide it > here for feedback. > If this math with transformations can be admitted then I believe that > your subsetting argument takes on different meaning, and the trivial > map must be included in order to fully cleanse your subset statement. I have no idea what you are on about. The statement "R subset C" requires no cleansing. > It may be that products of differing types simply maintain themselves > as static constructions so that the product of one type A with another > type B simply remains as > A B That would be the term-rewriting take on the matter; yet another viewpoint that shows there is no problem here. > Marshall, I've asked for falsification of my statements and thus far > your only falsification has been falsified Not true. I have pointed out various errors in what you are claiming, the simplest and most recent being that you cannot claim a number is a member of set A and not of set B when A is a subset of B. > I do sincerely believe that the subject of abstract algebra is flawed > as it stands in modernity. A grandiose claim, and one which you have gotten nowhere with. Given the number of people disagreeing with you, most people would take some pause, but not everyone. Marshall
From: jbriggs444 on 22 Jan 2010 15:50 On Jan 22, 3:25 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 22, 11:45 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > > > > > On Jan 22, 12:03 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > Such expositions concern constructions, encodings, > > > representations of numbers. If we remember that they > > > are such, and that constructions of numbers are not > > > the numbers themselves, then we're fine. If we > > > fail to make that distinction, then we may end up > > > with results that are flat-out wrong. We might > > > conclude that 1/2 =/= .5, or that 1 =/= 1+0i, > > > which are flat-out wrong. We might conclude that > > > the cartesian x,y point (1, 1) is the same point > > > than the polar r, theta point (1, 1), when in fact > > > they are different points. > > > In my view there is no such thing as "<mumble> number" as a really > > truly physical, authentic, "this is the one and only set of <mumble> > > numbers", accept no imitations. What we have instead are > > axiomatizations and constructions or models that satisfy the axioms. > > In my view, the best we can ever say is that "yes, these are the > > <mumble> numbers, up to isomorphism". > > I have no argument with that. We cannot directly process > anything except reified abstractions; we cannot get any > closer to "a _is_ b" than "up to isomorphism." > > However if we extrapolate from that fact to the claim > that only physically representable things exist, then > we have to throw out a lot of useful stuff: almost > all real numbers, for example. That would seem an > overreaction, as tempting as it might be. In view of this, > I have no qualms about drawing a distinction between > constructions and the things they represent. Fine by me. Draw the distinction. Now explain to me why that distinction supports your claim that the real zero and the complex zero are, in fact, identical rather than the opposing claim that the real zero and the complex zero are, in fact, different. Bear in mind that you've already as much as admitted that there is no fact of the matter. > I don't consider the claim that R subset C at all > challenging. In fact, we might as well move to > a simpler example for clarity: N subset Z. Whichever makes the treatment go through more cleanly. If I respect your claim that N subset Z then I darned well expect you to respect my claim that N is not a subset of Z. > That is a simple instantiation of a familiar theorem: > > for all x, militant x sure are annoying. Good point.
From: Marshall on 22 Jan 2010 16:39 On Jan 22, 12:50Â pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > On Jan 22, 3:25Â pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > On Jan 22, 11:45Â am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > On Jan 22, 12:03Â pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > Such expositions concern constructions, encodings, > > > > representations of numbers. If we remember that they > > > > are such, and that constructions of numbers are not > > > > the numbers themselves, then we're fine. If we > > > > fail to make that distinction, then we may end up > > > > with results that are flat-out wrong. We might > > > > conclude that 1/2 =/= .5, or that 1 =/= 1+0i, > > > > which are flat-out wrong. We might conclude that > > > > the cartesian x,y point (1, 1) is the same point > > > > than the polar r, theta point (1, 1), when in fact > > > > they are different points. > > > > In my view there is no such thing as "<mumble> number" as a really > > > truly physical, authentic, "this is the one and only set of <mumble> > > > numbers", accept no imitations. Â What we have instead are > > > axiomatizations and constructions or models that satisfy the axioms. > > > In my view, the best we can ever say is that "yes, these are the > > > <mumble> numbers, up to isomorphism". > > > I have no argument with that. We cannot directly process > > anything except reified abstractions; we cannot get any > > closer to "a _is_ b" than "up to isomorphism." > > > However if we extrapolate from that fact to the claim > > that only physically representable things exist, then > > we have to throw out a lot of useful stuff: almost > > all real numbers, for example. That would seem an > > overreaction, as tempting as it might be. In view of this, > > I have no qualms about drawing a distinction between > > constructions and the things they represent. > > Fine by me. Â Draw the distinction. > > Now explain to me why that distinction supports your claim that the > real zero and the complex zero are, in fact, identical rather than the > opposing claim that the real zero and the complex zero are, in fact, > different. How does Leibnitz equality grab you? Or how about some prose from Leslie Lamport? ---------------- Mathematicians typically define objects by explicitly constructing them. For example, a standard way of defining N inductively is to let 0 be the empty set and n be the set {0, . . . , n â 1}, for n > 0. This makes the strange-looking formula 3 â 4 a theorem. Such definitions are often rejected in favor of more abstract ones. For example, de Bruijn [1995, Sect.3] writes "If we have a rational number and a set of points in the Euclidean plane, we cannot even imagine what it means to form the intersection. The idea that both might have been coded in ZF with a coding so crazy that the intersection is not empty seems to be ridiculous." In the abstract data type approach [Guttag and Horning 1978], one defines data structures in terms of their properties, without explicitly constructing them. The argument that abstract definitions are better than concrete ones is a philosophical one. It makes no practical difference how the natural numbers are defined. We can either define them abstractly in terms of Peanoâs axioms, or define them concretely and prove Peanoâs axioms. What matters is how we reason about them. ------------------------------- From: http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-types.pdf > Bear in mind that you've already as much as admitted that there is no > fact of the matter. Um, remind me where I admitted that again? > > I don't consider the claim that R subset C at all > > challenging. In fact, we might as well move to > > a simpler example for clarity: N subset Z. > > Whichever makes the treatment go through more cleanly. > > If I respect your claim that N subset Z then I darned well expect you > to respect my claim that N is not a subset of Z. Is it okay if I just declare that I respect the distinction between different constructions of N and Z? Marshall
From: John Stafford on 22 Jan 2010 16:59 In article <48724aa6-8dd8-4b24-aa17-21313e7f25d9(a)h2g2000yqj.googlegroups.com>, jbriggs444 <jbriggs444(a)gmail.com> wrote: > In my view there is no such thing as "<mumble> number" as a really > truly physical, authentic, "this is the one and only set of <mumble> > numbers", accept no imitations. What we have instead are > axiomatizations and constructions or models that satisfy the axioms. > In my view, the best we can ever say is that "yes, these are the > <mumble> numbers, up to isomorphism". Something unexpressed, some subtext seems to be at work here. Are y'all looking sideway at Brouwer's intuitionism without mentioning it?
From: Tim Golden BandTech.com on 23 Jan 2010 10:52
On Jan 22, 3:42 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 22, 11:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > > > On Jan 21, 11:27 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > On Jan 21, 5:33 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > > On Jan 20, 10:24 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > [...] > > > > > > You're on: > > > > > > There does not exist a number that is a member of > > > > > the set of real numbers that is not also a member of the > > > > > set of complex numbers. > > > > > You've overstated the case significantly here. > > > > Not at all; it was a statement of simple fact. > > > Actually, after I posted, I had to fault myself for > > > stating it in such a cumbersome way; it could have > > > been done with just 3 symbols. (Not in ascii, though) > > > > R subset-of C > > > > Your analysis was quite sophisticated, however it > > > is an analysis of constructions of numbers, rather > > > than numbers themselves, and so does not > > > contradict my statement. > > > > Marshall > > > Have you withdrawn your claim that b is complex then? > > Of course not. What gave you that idea? Observe how > I said he "does not contradict my statement?" > > > There are many real lines that can be drawn through the complex plane. > > I am trying to do some analysis on this, and might even take interest > > in the interpretation of curved paths as valid mappings of the reals > > onto the complex plane. Such methods do require transformations, the > > trivial transformation being the one that you are thinking of. > > Anyway, if I work out the mapping in a nice simple way I'll provide it > > here for feedback. > > If this math with transformations can be admitted then I believe that > > your subsetting argument takes on different meaning, and the trivial > > map must be included in order to fully cleanse your subset statement. > > I have no idea what you are on about. The statement "R subset C" > requires no cleansing. > > > It may be that products of differing types simply maintain themselves > > as static constructions so that the product of one type A with another > > type B simply remains as > > A B > > That would be the term-rewriting take on the matter; yet another > viewpoint that shows there is no problem here. > > > Marshall, I've asked for falsification of my statements and thus far > > your only falsification has been falsified > > Not true. I have pointed out various errors in what you are > claiming, the simplest and most recent being that you > cannot claim a number is a member of set A and not > of set B when A is a subset of B. > > > I do sincerely believe that the subject of abstract algebra is flawed > > as it stands in modernity. > > A grandiose claim, and one which you have gotten nowhere > with. Given the number of people disagreeing with you, most > people would take some pause, but not everyone. > > Marshall Alright Marshall. I do think you have a strong position and admit that it has taken me some time to come around to seeing how you are seeing. Still, the formality of the subsetting is not quite there. For instance, the value + 5.1 is in R, the reals. Is + 5.1 in C? Well, it is in there quite a few times. For instance we have + 5.1 - 1.2 i , + 5.1 + 0.1 i, + 5.1 + 0 i , ... . Now, I'm pretty sure it is the last one that you meant but how am I supposed to know that it wasn't the first one? We are bumping into that same specification of zero that was used earlier. Beyond these simplistic versions there are even more means of subsetting a real line into a complex plane, and I admit that I am thinking graphically of those possibilities. For instance, I was considering a real valued line E whose origin E0 is at -1+i and whose unity position E1 is at 0+2i. Is this real line E a subset of the complex plane? Well, I'm pretty sure we're going to now bump into the operators and I will be called a fool and so forth. Still, this is the level of simplicity that I am looking at and I've already been called a fool, so what have I to lose? I am not afraid to be wrong, especially if that helps me to understand some fundamental possibilities. Beneath lays a tuple or a cartesian product and the treatment of the real number as elemental and then the claim of superelementals via cartesian composition, at which point this scrutiny is applied. The ring definition maintains a pristine form and calls upon any that wish to use it to provide elements in a set. Yes, I see how your subset argument is working, but I also see that careful usage denies the ability of a one dimensional entity to inherently be a two dimensional entity. It can be embedded, but those embeddings are many. Without this there would be no rise in freedom as one raises dimension. Call this information theory if you like. I'm still standing by the understanding that the complex value a + b i is composed of two real values a and b and one nonreal i, whose quality I think is best described as a unit vector. These details help explain how this product and sum do not evaluate to a single element via the operators defined in ring theory. They are incompatible with the ring definition. How can a math which provides a product and sum and then admits the complex value into its system allow this discrepancy to go unaddressed? This is a weak point in the topic of abstract algebra, where such operators are being scrutinized. It is as if someone needed to get something published and his buddy let him through, and then the thing took off with so many others needing to get published too. Could this be the humanistic reality? Certainly some chain of human events has brought this math to its current position. I admit the pristine nature of the ring definition, but also admit that modern math does not fit it quite so tightly as is claimed. - Tim |