Does inductive reasoning lead to knowledge?
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From: Tim Golden BandTech.com on 23 Jan 2010 13:29 On Jan 22, 2:00 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > On Jan 22, 9:23 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > On Jan 21, 1:27 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > > On Jan 21, 11:07 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > > > wrote: > > > > > On Jan 21, 8:33 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. > > > > > > _IF_ we define the complex numbers as ordered pairs of reals under the > > > > > obvious cartesian coordinate method _THEN_ there is no real number > > > > > that _is_ also a complex number. [My background is real analysis. > > > > > This is the obvious construction] > > > > > > _IF_ we define the complex numbers as the closure of the real numbers > > > > > plus i under the obvious rules for how addition and multiplication > > > > > treat imaginary numbers _THEN_ there is no real number that _is not_ > > > > > also a complex number. [I've never been exposed to the foundations of > > > > > the complex numbers from an algebraic point of view, but I expect that > > > > > this is the kind of basis you might want to put under them] > > > > > > _IF_ we define the "foobar numbers" as ordered pairs of reals under > > > > > the obvious cartesian coordinate method and then define the "complex > > > > > numbers" as the isomorphic set produced by replacing each (x,0) pair > > > > > in the "foobar numbers" with the real number x _THEN there is no real > > > > > number that _is not_ also a complex number. [This is the obvious > > > > > foundation an analyst could put under the complex numbers if somebody > > > > > wants to get bitchy about subnet relations] > > > > > > To an analyst, all three statements are obviously true. (*) And the > > > > > distinction is irrelevant. Whether there is a sub-ring of the complex > > > > > numbers that _is_ the real numbers or whether the sub-ring is merely > > > > > isomorphic to the real numbers is of little consequence. > > > > > > I was trained as a Dedekind cut guy. But I feel no need to declare > > > > > Jihad against the infidel Cauchy sequence dudes. > > > > > Very nice Briggs. > > > > > I have to admit I have become a one man jihad on abstract algebra. > > > > I did once attempt to understand the quotient ring and found that I > > > > could not. > > > > At times like these, what is one to do? > > > > Accept that the flaw is in yourself and refrain from pontificating > > > until such time as you reach an understanding? > > > I still remain open to the possibility that I am wrong. However I am > > wrong then my language should be falsifiable. > > It's not a one-dimensional spectrum with "right" on the one end and > "wrong" at the other. There's also a dimension of "clear" versus > "ambiguous" and "obvious" versus "abstruse". Well, I appreciate your ability to do continuum thinking. I do accept that the qualities are continuous, at least for the human being, even while the constructions are symbolically discrete. Always we should leave a shadow of doubt. For instance even after the real number has seven definitions and is scrutinized by millions for many generations one is still free to ponder the continuum. > > It is difficult to find places where what you write has a clear and > meaningful interpretation. > > > > Should I simply accept the constructions which have been repeated many > > > > times by accomplished minds? > > > > To my way of thinking, the main point of a construction is to be > > > assured that a model exists. It gives you some grounds to be able to > > > talk about "the complex numbers" without worrying that the whole > > > notion is self-contradictory. > > > Well here you've steered to the complex number as the focus > > Actually not. You were using the complex numbers so I used the > complex numbers as an example. > > > whereas > > for me the focus is on abstract algebra, and I am saying that abstract > > algebra contains a self-contradiction. > > Oh my. So when you wrote about "a+bi" you are writing about abstract > algebra, not about the complex numbers. Silly of me not to realize. > > > > If you're satisfied that the complex numbers exist and form a ring > > > then it really doesn't matter (for most purposes) what foundations can > > > be put under them. > > > I am satisfied that the complex number in the z form is a clean match > > to the ring definition. > > But you've just finished announcing that abstract algebra contains a > self-contradiction. Is the ring definition somehow contradiction- > free? > > For that matter, what is the "z form" to which you refer? I would say that the ring definition is an attempt at a pristine form and calls for singular elements in a set and so if we instantiate elements z1,z2,z3,z4 in C (the complex numbers) then we have a clean match, even while we don't necessarily have a clean representation. For instance we can write z1 + z2 = z3 , z3 z2 = z4 . and the direct mapping to the ring definition is clear. > > > > Here lays a humanistic problem where the > > > > mathematical cannot be separated. > > > > I get uncomfortable talking about things that are this wishy washy. I > > > don't see a humanistic problem. I don't know what you're trying to > > > separate from what else. Why can't you just say what you mean for > > > goodness sake? With examples. > > > The complex value > > a + b i > > contains one product and one sum which are incompatible with the > > operators granted in the ring definition. > > Still no "humanistic problem" lurking there that I can see. Still no > indication of what you're trying to separate from what else. > > And still no indication that you have the cognitive ability to > recognize the various possible interpretations of a simple > expression. May we add parsers, parser generators and ambiguity to > the list of things you've never had to deal with? Below I make note of your own abuse with regard to software. > > Does "a" denote a real number, a complex number with a zero imaginary > part or an arbitrary complex number? a denotes a real number.. > > Does "+" denote complex addition or an extension to complex addition > to handle a real operand? "+" 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. > > Does "b" denote a real number, a complex number with a zero imaginary > part or an arbitrary complex number? b denotes a real number. This is the tradition. > > Does "i" to be interpreted as a complex literal or as an imaginary > number. or as a unit vector. It matters little for my argument which applies directly onto the ring requirement. This excludes operations such as a + b i from being ring compatible operations. Because these are not ring operations but goes on to use such values then the theory is incomplete, at a minimum. > > Does the juxtaposition of b and i denote complex multiplication or an > extension to real multiplication or an extension to complex > multiplication? What is an extensional multiplication in terms of abstract algebra? I can clearly state what this product is not, and point out the fact that the subject treats these operators very carefully, and so there is a void. 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. I do see that by introducint a product such as a unit vector product that the cartesian product is essentially built, as is the tuple form as well. 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. This may invalidate the cartesian product, or at least still leaves room for that invalidation by offering another route to constructive freedom. It seems to me that taking this route is like opening a can of worms of various types and I do not need to resolve this mess in order to make my statement without error. > > Ring theory does not address these issues. > > 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. Ring theory does not tell us that it is using ambiguous language. This extends into a more critical region of the polynomial construction, where I think you'll find that these same pricnciples are more consequential. > > > > > As you say, some level of > > > > flexibility is healthy, and at some level I am happy to cast this > > > > argument off as silliness. Yet, abstract algebra has gone to the > > > > trouble of providing operators formally. > > > > Personifying abstract algebra? Not much hope for a formal > > > understanding in that direction. > > > Here I smell insincerity, but I will operate as if you are sincere. > > People did construct abstract algebra. In this way all of human > > knowledge is personified. > > Yeah, right. And centrifugal force results from a body trying to go > in a straight line. > > Personification = obfuscation in my book. It can be useful as a > crutch. But it obscures correct understanding. I do not mean to obfuscate. I am arguing that abstract algebra contains details which have been obfuscated. Does abstract algebra do the obfuscation or do humans? The two are inseperable. The mathematician will never fully disavow himself from his human form, or at least this hasn't happened yet. Kurzweil works at it though: http://singularity.com . I have little doubt that if such an entity can be assembled that it will view our behaviors and abilities as quite askance to where we put them. It will be deeply frustrated to communicate its pure form back to an interpretation within a flawed language. > > > I should perhaps have written that the > > people who constructed abstract algebra have gone to the trouble of > > providing operators formally, and it is this content which you have > > cast off which I do believe is highly relevant. > > Rewording it doesn't change the central fact that "providing operators > formally" is wrong. > > If the context is "abstract algebra" then all that has been provided > is a framework. The operators are INTENTIONALLY LEFT UNSPECIFIED. No. How would one specify an unspecified operator? For instance does the set of real numbers come with operators? 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 formal granting > > of operators means that any identification of operators which do not > > fit should be scrutinized and addressed by this subject, particularly > > those operators in use within its range of applicability, such as the > > complex numbers > > a + b i . > > If you are trying to say that it would be nice if "a+bi" were > precisely and unambiguously specified so that we could nail down the > formal meanings of "a", "+", "b", " " and "i" before using the > construct in the formal definition of the complex numbers then... > > I AGREE! > > But going back 15 or 20 posts, I don't see any context that would make > me need to worry about the level of formality in "a+bi". Best guess > is that it's used in an exposition in some textbook or other. > > And my best guess is that it's used in the context of providing a > notation in which arbitrary complex values may be presented. That is > to say, it's not of any great formal relevance but is mostly important > for pedagogical purposes. > > If I can look at "1+2i" and understand what is meant, it is a WASTE OF > TIME trying to dot all the i's and cross all the t's and specify a > precise formal meaning for all the pieces parts. It is enough that it > is possible, in principle, to dot all those i's and cross all those > t's. And it _is_ possible to do so. > > > Is your own belief system personified? Certainly it is. It is the duty > > of the mathematician to identify false or incomplete beliefs. > > I do not even know what you mean by a belief system that is > personified. > How you jump from there to "duty" is a complete mystery. > > > Now you > > can harp on 'duty'. But please do not avoid the operator discussion. > > OK. > > > > > Here we see a unique product > > > > operator in common use which goes ignored. > > > > Here? Where? > > > b i . > > In the post to which I responded with "Here? Where?" there was no > appearance of either b or i or a juxtaposition of the two. > > Hence my request for clarity. Thank you for responding. > > Now. What datatype do you infer for b? What datatype do you infer > for i? What operator do you infer for the juxtaposition. Apparently > you see a conflict. In the past you've talked about "closure" as > being relevant. In the past I've stated that "closure" is irrelevant. > > Can you show why closure is meaningful? 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. > > > > > > Unique product operator? What's that? And how does it tie into a > > > This is actually quite a good question. The product > > b i > > does not resolve itself. > > What would it mean for a "product to resolve itself"? closure. > > > In this way we may actually have a new > > product model. > > What do you mean by "product model". Do you, for instance, mean that > the "product" operator here does not have the same profile as the > "product" operator in the ring of complex numbers? > > By "profile" I mean the combination of the data types of the result > and of the two operands. The cartesian product is nearby to this as is the tuple form. Here we see it as a type form inherently algebraic. This really blurs product and superposition so again is opening a large can of worms that I don't need to fully discuss, for my attack is on abstract algebra. > > > It is very much nearby to the cartesian product model, > > but within a subject which treats the product so carefully how can you > > go on denying what I have so directly under your nose so many times? > > I do not know what you mean by "nearby to the cartesian product > model". I have difficulty imagining how you read cartesian coordinates > into the notation. "bi" > > I do not know what you have said directly under my nose so many > times. Many people yell at you constantly for being unclear. Perhaps > you should be more explicit. 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. > > > > notation that is traditionally overloaded and disambiguated by custom, > > > context and "mathematical maturity". > > > > And, for that matter, what do you think "closure" means? Hint: it's > > > not about what operations outside the ring produce when used with > > > operands outside the ring. > > > Ahh. So maybe you do get it even while you deny getting it. These > > operators do seem to be beneath or within the ring of complex numbers. > > The notation ab (a juxtaposed with b) can denote integer > multiplication, rational multiplication, real multiplication, complex > multiplication or concatenation of strings. > > > The closure principle stated that an operator will function on two > > elements of the same set and return a value within that same set. The > > product > > b i > > does not fit the closure principle because b is real and i is not > > real. > > If we decide that the reals are a subset of the complex numbers then > both b and i are complex numbers, the operator is clearly complex > multiplication, the result is a complex number and closure of the > complex numbers under complex multiplication is upheld. > > If we decide that the reals are not a subset of the complex numbers > then you are correct. The left hand operand is not a complex number. > The operator is not complex multiplication. The closure principle > does not fit. So the closure principle has NOTHING WHATSOEVER TO SAY > ON THE MATTER! I'm getting pretty tired here and this tendril is obnoxiously long. You again provide a conditional statement and I believe that I can work around that conditional, or perhaps create a small opening by discussing what an element is. In that a product operates on two elements and returns a single element then the product b i is not ring behaved, for these two elements do not resolve to a single element. I think you might have a better time claiming that there is no product here at all, but I definitely see a product operation. Again, I have no need to rely on Mr. Clarke's criteria, for the information within the ring definition is sufficient to provide my clean argument. By breaching this problem we are indeed down at a very fundamental level. If a conflict exists here then it may be expressible in many terms for that is how conflicted constructions go. It happens that we have many terminologies for the cartesian product idea in use, and then the polynomial version gets added to this list. I say I smell a rat. You seem to say yeah, whatever. If it is a rat cooking then gross. It is a live rat then it smells lively. If it is dead rotting rat then gross. If it a rat in heat then there may be more rats about soon... > > > I know that I have said this ad nauseum > > The closure principle (for the complex numbers under complex > multiplication) says that: > > "If a and b are complex numbers and * denotes complex multiplication > then a*b is a complex number" > > It does not say that: > > "If a is not a complex number than a*b is syntactically invalid" > > At most one could say > > "The complex ring does not provide a defined meaning for bi where b is > a real number and i is the complex literal i" The ring definition only provides formal operators of the form foo RingProduct( foo z1, foo z2 ); foo RingSum( foo z1, foo z2 ); When b is real and i is not real then the product b i is not a ring product. That is all that my statement in its simplest form has ever been. When confronted with whether b is real or complex I say b is real. Marshall says b is complex. You say you could go either way, but refute Marshall and refute me. This is a fine level of logical complexity at which point we see that we are dealing with a flawed language. This is my ultimate claim and I have no need to resolve this conflict here. I have resolved it elsewhere in such a way that those who adhere to abstract algebra insist on going through its rat maze of inconsistent complexity to arrive at what I see as a fundamental construction devoid of these ambiguities. All that I seek here is the verification of those ambiguities, and still, with Smaill's wording I'm left confused, and with yours too. This again is the poor ability of the human to communicate things mathematical and is a strong indication that this very language in which we communicate is flawed. I apologize for my long word form, but do feel that it is not so mistakable as is say Smaill's short word form, nor your own form which lays loaded down with conditionals. I simply repeat myself ad nauseum here. The story has changed only slightly, but we have covered some options, such as assigning vast quantities of zeros, and your open argument on subsetting, but still the product b i will not resolve, which language I think you must get by now. - Tim > > > here but this is roughly > > where you've stepped into the conversation and this is exactly why > > Marshall insisted that b is complex, at which point you and I have > > both falsified his statement. > > Alan Smaill has correctly pointed out that the notation is ambiguous. > > My disagreement with Marshall is confined to the question of whether > is it a matter of formal fact that the reals are a subset of the > complex numbers. (That's my take on the disagreement. I suspect that > his take is different. As Aatu has pointed out, the question is of no > particular consequence and tends not to arise) > > I'm perfectly willing to consider the reals as a subset of the complex > numbers if it will make a particular treatment go through more > cleanly. > > If we grant that b is complex (with a zero imaginary part or with a > zero angle or however you want to express the fact that is in the > subset of the complex numbers that is isomorphic to the reals under > the respective ring operations and which maps 0 to 0 and 1 to 1) and > that i is complex then juxtaposition can denote complex > multiplication, the result is a complex number and so closure is > upheld. > > If you refuse to grant that b is complex then we can still type- > promote b to the unambiguously determined complex number b' according > to the uniquely determined isomorphism mentioned > above. > > If you don't like type-promoting operands then we can overload the > juxtaposition syntax to cover the case of multiplication of a real by > a complex. > > If we pretend to be cooperating adults then there is no real ambiguity > about the result that will be obtained, regardless of what the > formally ambiguous syntax is taken to mean. > > Surely you aren't trying to argue that 1+2i = 17-5i? > > > > > The same offense is in use > > > > further along but with much more density of information which further > > > > clouds the discussion. > > > > You haven't described the first "offense" yet. Or indicated why it's > > > an "offense" worthy of pejorative language. > > > Jeeze, I guess you didn't get it. > > If you would write clearly, I wouldn't have to guess at your meaning. > > > > > The a+bi complex representation is a very > > > > simple instance. > > > > I can understand this notation adequately without worrying about how > > > it is formally grounded, whether the a and b are is supposed to be > > > real or "complex with no imaginary part", whether i is supposed to be > > > complex or "imaginary", whether that discinction is even meaningful or > > > whether it's valid to multiply a real by an imaginary. > > > Ahhh. So you do get it. But as I read these words I see an incomplete > > statement. This is somewhat what I am claiming: the subject of > > abstract algebra may be incomplete. > > No. You don't get off the hook that easily. You claimed that > abstract algebra is self-contradictory. You're not allowed to > backpedal to some handwave that it "may be incomplete". > > > perhaps you are mustering up support for Marshall's argument. Please > > do clarify your statement for it does not read cleanly. > > "If presented with the expression "a+bi" and provided with real values > for a and b, I can correctly evaluate it and give you the complex- > valued result" > > Is that clear enough for you? > Would you like to negotiate further on the coding scheme that I will > use when conveying the result? > > [I had in mind relying heavily on the identity mapping and handing you > the ordered pair (a,b), but I'm flexible] > > > > You've probably never been exposed to Ada, (a computer language in > > > which an infix operator can be overloaded with multiple meanings > > > depending on operand type(s) and in which numeric literals are also > > > implicitly overloaded so that expression evaluation and type > > > determination is an interesting problem). > > > I haven't worked with Ada, but I do use C++ which allows limited > > operator generalization. Leaving these complicated languages if one > > were to create in software a function > > foo RingProduct( foo f1, foo f2 ) > > (foo should really be a templated type and have code for a real, > > complex, and whatever else you'd like to implement) one would receive > > a compiler error when passing a real and a unit vector, or whatever > > you want to call i. > > Not if you've _also_ provided: > > foo RingProduct ( real f1, foo f2 ) Come on, this is not the ring product. The ring product very strictly calls for elements in the same set as its arguments and returns an element in that same set. All types here must match. This I guess is the value of the template of C++, for it enforces that generalization. I also happen to never have actually needed to use a template until this moment so forgive me if I've generated a virtual compiler error with template <class RingSet> class Ring { RingSet operator +( RingSet z1, RingSet z2 ); Ringset operator *( RingSet z1, RingSet z2 ); }; I don't think we should take this code paradigm too far, but you've just plain abused it. > > Or, in Ada, barring syntax errors. > > foo "*" ( real f1, foo f2 ) return foo; > ... > foo "*" ( foo f1, foo f2 ) return foo; > ... > real b := 2; > foo bfoo := makefoo ( b,0 ); > const foo i := makefoo ( 0,1 ); > > c = b * i; -- Multiplication interpreted in terms of > first definition > c = bfoo * i; -- Multiplication interpreted in terms of > second definition > > Hopefully I don't have demonstrate overloading for "+" as well. > > > They are not of the same type and this is how > > Marshall comes to make the claim that b is complex, for if it were > > then the problem would be resolved. > > But you don't _want_ the problem to be resolved? I want the truth, and the best way to it that I know is via a skeptical approach. As I apply this skepticism to AA I see trouble. This is consequential. > > > > > As you point out the taking of a cartesian tuple > > > > ( a, b ) > > > > and then pulling out the first component and calling it real > > > > Calling it real? Why bother? Who cares what name it has? If we're > > > trying to be formal we're trying to get away from names. We can be > > > talking about bloo-blars and nightgowns. As long as they match the > > > behavior required by the axioms we're working to fit we don't give a > > > darn about what names they have. > > > I'm not talking about bloo-blars and nightgowns, nor horses in > > pajamas. I'm talking about well established mathematics, and have > > identified a mismatch and so to invert your statement you should give > > a darn. > > ROFL. You're funny when think you've caught someone out. > > > > I believe that it's irrelevant. And I wouldn't recognize your > > > argument if it bit me. > > > > > The value > > > > ( a, 0 ) > > > > is not a one dimensional entity. > > > > Who cares how many dimensions it has? If you multiply it by ( 0, 1 ) > > > you'll still get ( 0, a ). > > > Hmmm. Let's see, who cares how many dimensions a value's > > representation has? A mathematician perhaps? Are you a mathematician? > > Why would a mathematician care about representation? If he's an > algebraist, he's interested in properties relative to some operators. > The representation has been abstracted away. The operators may > themselves have been abstracted to some degree. Even if you're > paying lip service to foundations, the phrase "up to isomorphism" > covers things nicely. > > If you're trying to come up with a construction for the complex > numbers and make sure it works, representation is your bread and > butter. It has to work. It has to work right. If you are writing > "RingProduct", you had to choose a representation and use it. Hmmm. > Let's try explaining it to you that way... > > When you defined your function RingProduct, did you hide the > representation of "foo" from your users? You should have. You don't > want those jokers reaching into your data structure and trying to > retrieve (a,b) when you shifted your representation to (r,theta) six > months ago. > > If you designed it right, you provided interface routines "realpart" > and "imaginarypart" and maybe "makecomplex" to provide the relevant > user access. > > Or maybe you don't want to bother with makecomplex... > > You could provide an externally visible complex constant "i", overload > "+" and "*" and have your users write: > > real a := 1; > real b := 2; > complex x := a + b * i; > > With nary a compilation error to be seen. > > In any case, the property of dimensionality is not intrinsic to an > entity. It is extrinsic. It depends upon a containing set and a > topology on that set. [Topology is a weak point for me, but I'm > pretty sure that it is the standard way of formalizing > dimensionality. Vector spaces are also a weak point, but I suspect > that's where you'd go next. [Damn -- so much more fun I could have > had if I'd stayed for four years rather than graduating in three] > > I believe that it is a correct statement that _as a set_ the complex > numbers have no built-in dimensionality (a vector space that spans the > set need not respect the ring operators) but that _as a ring_, the > vector space would need to respect the ring operators and any vector > space that spans the ring would necessarily have two dimensions [i.e. > can be generated two generators but not by one]. > > [Corrections welcome from the peanut gallery -- I'm trying to feel my > way to the relevant definitions from first principles and not "cheat" > by consulting reference material]
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 |