Does inductive reasoning lead to knowledge?
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From: Marshall on 22 Jan 2010 12:03 On Jan 22, 8:29 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > > > If you had the least bit of sincerity you'd at least admit > > that your claim that "b" is not complex but rather real > > cannot be true since R is a subset of C. > > Here you are glossing over the ambiguities in the situation > that were admirably well out earlier in the thread. > > You will easily find expositions of the complex numbers as a set in which > the reals are embedded, but do not literally appear as a subset > (such as complexes as pairs of reals). > > Are such expositions flat-out wrong, as your claim suggests? 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. Marshall "First there is a mountain, then there is no mountain, then there is a mountain again."
From: jbriggs444 on 22 Jan 2010 14:00 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". 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? > > > 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? Does "a" denote a real number, a complex number with a zero imaginary part or an arbitrary complex number? Does "+" denote complex addition or an extension to complex addition to handle a real operand? Does "b" denote a real number, a complex number with a zero imaginary part or an arbitrary complex number? Does "i" to be interpreted as a complex literal or as an imaginary number. Does the juxtaposition of b and i denote complex multiplication or an extension to real multiplication or an extension to complex multiplication? 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. > > > 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 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. 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. > 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? > > > 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"? > 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. > 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. > > 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 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" > 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 ) 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? > > > 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: jbriggs444 on 22 Jan 2010 14:45 On Jan 22, 12:03 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 22, 8:29 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > > > > > > If you had the least bit of sincerity you'd at least admit > > > that your claim that "b" is not complex but rather real > > > cannot be true since R is a subset of C. > > > Here you are glossing over the ambiguities in the situation > > that were admirably well out earlier in the thread. > > > You will easily find expositions of the complex numbers as a set in which > > the reals are embedded, but do not literally appear as a subset > > (such as complexes as pairs of reals). > > > Are such expositions flat-out wrong, as your claim suggests? > > 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". Maybe we can't even say that. Maybe all we can say is that "yes, this model satisfies the axioms and god damn you to the hell of the infidel if you do not therefore accept that this model has as much claim to the title of <mumble> numbers as your version... Wait you didn't even _have_ a model, you just took it for granted that one existed? Well I guess I'm not a Platonist. And I have no problem with Platonists. *glances around and inches away from the guy with the thick glasses and Greek accent in the next seat*. As long as they don't try to pretend that what they're selling is truth with a capital T and that my point of view is wrong with a capital W. I didn't declare jihad on the Cauchy boys. But these militant platonists sure are annoying.
From: Tim Golden BandTech.com on 22 Jan 2010 14:47 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? 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. 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 just as the cartesian product performs such maintenance, and the unit vector too. Thus the product of a one-tuple (a1) with a two-tuple (a2,b2) as (a1)(a2,b2) has no resolution as the closure law provides for similar types. This then would be a new construct which is actually quite compatible with some math that I work on. I would hesitate to call this product a cartesian form, for it is the uniqueness of the types which prevents their resolution. Marshall, I've asked for falsification of my statements and thus far your only falsification has been falsified, though it may be that Uncle Briggs will change his mind. Obviously it is beyond my ability to change your mind. That is something for you to do yourself. It is important that we play out variations in order to fully explore things. Certainly most of these variations will be problematic, but nonetheless they expose where there may be some spare room, and just a sliver of room can be deeply meaningful. I do sincerely believe that the subject of abstract algebra is flawed as it stands in modernity. Opportunities have been overlooked. Fundamental constructs are still possible. I do have a few of these but they should not be relied upon here for this conversation. The admission which I have sought from all of you and not yet received is a very minor one. Still, it's general effect is larger and I see that all have dodged the polynomial reach. - Tim
From: Marshall on 22 Jan 2010 15:25
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. 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. > Well I guess I'm not a Platonist. In my view, platonism is best understood as a metaphor. > I didn't declare jihad on the Cauchy boys. But these militant > platonists sure are annoying. That is a simple instantiation of a familiar theorem: for all x, militant x sure are annoying. Marshall |