Does inductive reasoning lead to knowledge?
in [Logic]
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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
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] |