Does inductive reasoning lead to knowledge?
in [Logic]
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From: Marshall on 23 Jan 2010 14:34 On Jan 23, 10:29 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 22, 2:00 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: My goodness, the number of fundamental concept of abstract algebra that you misunderstand is impressive. Until you correct these misunderstandings you have no hope with this. > "+" denotes addition, but is not compatible with the ring definition's > form of addition. Therefore at the very minimum the language of > abstract algebra is incomplete. This addition operator does not > resolve itself as does the ring form. This is double-plus-wrong. The "+" in the notation "a+bi" isn't even *supposed* to be the ring's definition of addition. It's not even supposed to be the addition operator of the complex numbers as a ring. So your argument there is completely misdirected. Even if you were right about that particular plus not being a ring plus, it wouldn't matter, because that's not intended to be the operator that is assigned to the ring addition operator that shows the complex numbers to be a ring! But the irony of the whole thing is, that the addition operator above actually *happens to be* a restriction of the complex addition operator that we do use in the complex ring. Come to think of it, this irony may contribute to your confusion about what is supposed to be happening here. > The fact remains that there is no actual failure of the > complex number and so the failure is in abstract algebra as a > theoretical subject. The failure is in your understanding of what abstract algebra is trying to do. > I do see that a simple logic of products > whose types do not match as simply nonresolving products may be > effective, but this is a fairly fresh thought. Under this > interpretation products of any mixed types are freely available but do > not resolve. This is essentially the same thing that is the cartesian > product on like types, except that these types are unique. I suggest you run with that thought. > > Ring theory tells you that if you "multiply" one ring element by > > another that the result you obtain will be a ring element. > > > Ring theory doesn't tell you how to parse an ambiguous expression. > > Exactly, yet it goes ahead and uses these values anyway, and they > work. The notation "a+bi" is not part of ring theory. No subpart of it is part of ring theory. The "+" in "a+bi" is not supposed to be the ring addition operator. By coincidence, it happens to be a restriction of the usual complex ring-addition operator, but don't let that confuse you (henceforth.) There is no "+" in other representations of complex numbers, such as the tuple, or your polar form, and yet those representations still work, still have a ring addition operator, and are all isomorphic to the "a+bi" representation. > > If the context is "abstract algebra" then all that has been provided > > is a framework. The operators are INTENTIONALLY LEFT UNSPECIFIED. > > No. What do you mean "no?" What do you think abstract algebra even is? It is exactly the process of abstracting away the specific operators, falling back instead on specifying properties that are common to many operators. The operations themselves are intentionally left abstract! > How would one specify an unspecified operator? The smart-aleck answer, which I apologize for not being able to resist, is "with abstract algebra." The ring definition, for example, says only that rings have two operators, and that they are associative, etc. etc. It doesn't say what they are. In fact, there are cases where an algebra may apply more that once in different ways. The natural numbers form a monoid under the natural's addition operator, and also a different monoid under the natural's multiplication operator. > For instance does the set of real numbers come with operators? This question has nothing to do with abstract algebra, which does not have real numbers in it. Abstract algebra has something called a field; the reals are a field, but abstract algebra concerns itself only with the field aspect, which applies to anything that meets the field axioms. > On the one hand I accept a view of the real line of numbers without > operators. > On the other hand I know that these operators are already working and > have been tought to perform their operations since an early age. Then > after twenty years of accumulated schooling one is presented with > abstract algebra, which formalizes these operators. Clearly this is > far from leaving them unspecified. I can accept partially specified, > but not unspecified, particularly not in capital letters. > > > The exact definition of the operators has been "abstracted away". > > Instead of dealing with the "natural numbers" we're dealing with (for > > instance) a "ring". > > > That allows us to concentrate not on the operators themselves but on > > their properties. Such as commutativity. > > Here I would point out that commutativity itself is a more pure > abstraction of operators than is the ring. Commutativity specifies no > operator at all, and may or may not be compatible with any given > operator. Ring theory points to two specific operations, and so is > more specific. This is partly incorrect; the definition of commutativity requires an operator. > The only product and sum granted within abstract algebra are the ring > and group ones. > If other products and sums do exist within its boundaries then they > have been ignored. > Closure does not apply to the operators in > a + b i > where a and b are real, and i is not real. Therefor these operators go > ignored by abstract algebra, which leaves abstract algebra incomplete > since it makes use of such values. What do you mean by "make use of?" Nothing in ring theory says anything about real or complex numbers. Ring theory speaks only of rings. It says nothing about components of real numbers, for example. Neither does it say anything about botany. This makes it other than all-inclusive, but we don't usually call that kind of incompleteness a flaw. > I have a hard time understanding this difficulty of communication but > do understand that such problems are very serious. > I do come from an engineering background where the usage of the unit > vector tends to be a very clean treatment. I typically look at i as a > unit vector rather than as square root of the real valued negative > unity. Whereas the cartesian product RxR builds two-tuples the unit > vector symmetrically builds sums so that > a + b i <---> ( a, b ) > are a unit vector form on the left, and a tuple form on the right. The > tuple form is a more explicit expression of dimension since if we > subtract off bi from this expression we get > a <---> ( a, 0 ) > but typically we've already fixed the dimension verbally above here so > that we prescribe a space that we are working within. Otherwise we > could just at will build > a + b i + c j > without any conflict, and consider this value as compatible with > a + b i , > but I think so long as the space was prescribed then this is not > possible, such as in a study of the complex numbers. > I have no idea if this little bit helps any, but it does lead us a > step closer to the oncoming polynomial. If the language of abstract > algebra is accepted as ambiguous at this level then at the polynomial > level there are some pretty strong problems. None of what you talk about above is part of abstract algebra. Marshall
From: Marshall on 23 Jan 2010 15:47 On Jan 23, 7:52 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 22, 3:42 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > 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. You are confusing yourself with various irrelevant distractions. Is 2 a natural number or an integer? It's both of course. But wait, can't I come up with various mappings from N to Z? If you are talking about the natural number 2, how do I know that when you say it's also an integer, you don't mean the integer -100? There does exist a mapping where that makes sense, right? See the point? 2 is 2, which is natural, integer, rational, real, and complex. > 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. You are confusing yourself. From the ring perspective, every ring element is atomic. Likewise every ring product and sum produces an atomic ring element. The usual + and * of the complex numbers always produce complex numbers, (closure) and also meet the other requirements for being a ring. Thus the complex numbers form a ring, with closure and everything else perfectly met. The product and sum in "a+bi" are not the definition of the ring product and sum. However they *are* compatible with the ring product and sum, specifically they are are restriction of those operators. Understanding this compatibility requires that you understand that R subset C. That there exist mappings from individual complex numbers to things that could be described as "components" or "decompositions" of a complex number is irrelevant. This is exactly like the mapping from 2 to -100 above. > 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? There is no discrepancy; there is only your confusion. > This is a weak point in the topic of abstract algebra, > where such operators are being scrutinized. It is a weak point in your understanding only. Marshall
From: Tim Golden BandTech.com on 24 Jan 2010 09:47 On Jan 23, 3:47 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 23, 7:52 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > > > On Jan 22, 3:42 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > 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. > > You are confusing yourself with various irrelevant distractions. > > Is 2 a natural number or an integer? It's both of course. > But wait, can't I come up with various mappings from N > to Z? If you are talking about the natural number 2, how > do I know that when you say it's also an integer, you don't > mean the integer -100? There does exist a mapping where > that makes sense, right? > > See the point? 2 is 2, which is natural, integer, rational, > real, and complex. Marshall, I do see a flaw in this logic. Particularly it is the dimensional quality which has broken open in your progression from real to complex which does not exist in such an impressive form in the earlier number morphings of your series. For instance shifting from the real to the complex numbers is like shifting from the integers Z to a 2D lattice ZxZ . Should we attempt to put this 2D lattice into your subsetting argument? I don't think so. This is a very differently structured argument and so does not belong in the progression. Likewise the complex numbers do not belong in that progression. It is another progression that they are fitting into: R --> R x R --> R x R x R --> ... At some level the technicalities which we are discussing require the granting of the real number as an infinite dimensional entity, all components being zero except the first. This is a logical inversion of the cartesian product and is near to the heart of the matter. At least, this is one way of looking at the problem. There are likely other ways. Now in addition to identifying a conflict in abstract algebra we have located one in the more traditional mathematics. Well, it is all connected, isn't it? > > > 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. > > You are confusing yourself. > > From the ring perspective, every ring element is atomic. > Likewise every ring product and sum produces an > atomic ring element. Yes. Very good. When we speak of 'elemental' qualities this atomicity that you speak of here is in use. Whether the tuple form ( a, b ) can be granted this atomic nature is actually dubious, particularly when we define ( a, b ) as RxR with a in R and b in R. The concept of product and superposition is being blurred here in a meaningful way, for by rectifying the blur we may find a new math form. It will likely contradict the old form and so the act of contradiction itself is not a problem. It is a question of the new form's consistency that arises as its own test of itself, and then also the consequences of its construction. These are the proper criteria; not consistency with the old math. The old math is merely a cross study and an aid to further development. It may well be that in this new math for a value a in the real numbers and a value z in the complex numbers the product a z does not resolve itself. This then is a branch of mathematics which conflicts with the old math. Has it already been done? Perhaps. Who is to say these days with the accumulation that abounds. This new form az is awfully close to RxC, yet is an arithmetic product, not a cartesian product. How one gets off the ground floor of a1 a2 where a1 and a2 are in R is problematic since this product does resolve itself. Whatever this branch of mathematics would be I have a route based on generalization of sign that already recovers these dimensional entities without any cartesian product and provides for an emergent form of spacetime; a structured spacetime with unidirectional time that lays beneath the real number rather than being built out of the real number, thus disengaging the time reversal physics which is coming along thanks to Kaku et al. If you are interested please see http://bandtechnology.com/PolySigned Now we can have these new product types without the need for the old problems, and this is exactly the format of the emergent spacetime structure. I call it the tatrix, for triangular matrix, but here it is being developed nearby to the cartesian product concept, though still with arithmetic flare. The granularity of this step is exremely primitive and in this way it is a very small thing mentally and so to you I can understand that it may not be perceived. My mind is in such a primitive state that I can perceive this thing, and so I apologize, for miscommunication is nearly guaranteed. - Tim > > The usual + and * of the complex numbers always > produce complex numbers, (closure) and also meet > the other requirements for being a ring. Thus the complex > numbers form a ring, with closure and everything else > perfectly met. > > The product and sum in "a+bi" are not the definition of > the ring product and sum. However they *are* compatible > with the ring product and sum, specifically they are > are restriction of those operators. Understanding this > compatibility requires that you understand that > R subset C. > > That there exist mappings from individual complex numbers > to things that could be described as "components" or > "decompositions" of a complex number is irrelevant. This > is exactly like the mapping from 2 to -100 above. > > > 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? > > There is no discrepancy; there is only your confusion. > > > This is a weak point in the topic of abstract algebra, > > where such operators are being scrutinized. > > It is a weak point in your understanding only. > > Marshall
From: Marshall on 24 Jan 2010 11:44 On Jan 24, 6:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 23, 3:47 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > Is 2 a natural number or an integer? It's both of course. > > But wait, can't I come up with various mappings from N > > to Z? If you are talking about the natural number 2, how > > do I know that when you say it's also an integer, you don't > > mean the integer -100? There does exist a mapping where > > that makes sense, right? > > > See the point? 2 is 2, which is natural, integer, rational, > > real, and complex. > > Marshall, I do see a flaw in this logic. Particularly it is the > dimensional quality which has broken open in your progression from > real to complex which does not exist in such an impressive form in the > earlier number morphings of your series. > > For instance shifting from the real to the complex numbers is like > shifting from the integers Z to a 2D lattice ZxZ . Should we attempt > to put this 2D lattice into your subsetting argument? If you have a problem with ZxZ, then you have a problem with the rational numbers. Clearly in your view, the rational number 2/1 is different than the natural number 2, because rational numbers can be constructed with a cartesian product of ZxZ. Clearly, by your arguments, the rationals cannot form a division ring because closure is violated in the construction of 2/1, since the "/" in "2/1" is not the same division as in the ring; it only accepts integers. In fact, you must also have a problem with the integers, since there exists a construction of the integers as NxN. Somehow you have stopped talking about the perceived closure issue and have moved on to your own mathematical musings. Alas, I don't have the resources to follow you. Marshall
From: Tim Golden BandTech.com on 25 Jan 2010 08:50
On Jan 24, 11:44 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 24, 6:47 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > If you have a problem with ZxZ, then you have a problem with > the rational numbers. This has nothing to do with the topic that I am discussing, but then, the topic I am discussing has little to do with the topic of this thread, and then too the convenience with which content is deleted to suit ones own needs further clouds the issue, and on the other a cloud of undeleted material may equally cause an overcast state. To be as direct as possible seems to me the resolution. Your methods are of indirection. - Tim |