Does inductive reasoning lead to knowledge?
in [Logic]
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From: Tim Golden BandTech.com on 22 Jan 2010 09:23 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. > > > 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 whereas for me the focus is on abstract algebra, and I am saying that abstract algebra contains a self-contradiction. > > 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. > > > 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. > > > 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. 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. 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 . Is your own belief system personified? Certainly it is. It is the duty of the mathematician to identify false or incomplete beliefs. Now you can harp on 'duty'. But please do not avoid the operator discussion. > > > Here we see a unique product > > operator in common use which goes ignored. > > Here? Where? b i . > > 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. In this way we may actually have a new product model. 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? > 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 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. I know that I have said this ad nauseum 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. > > > 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. Please see above. > > > 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. But here you are backtracking and perhaps you are mustering up support for Marshall's argument. Please do clarify your statement for it does not read cleanly. > > 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. 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. > > > 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. > > > is a > > relevant detail I believe, though I am not relying upon this within my > > argument. > > 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? > > [ Unless we're going polar. Then you'd multiply by ( 0, 90 ) to get > ( a, 90 ) ] > > > In the polynomial development we'll > > see the tuple used outside of the cartesian form. > > And with polar coordinates as well. > Or we could use a space-filling curve and shift down to one > dimension. We haven't discussed a topology to the complex numbers > yet. So dimensionality is up for grabs. > > But so what? Yes, I do think that this is pretty much where we are at. So, the subject of abstract algebra may have some holes and even conflicts in it. This is where we are at. > > > I do wish that this > > discussion could be taken up to that level, but without receiving > > verification at this simple level I see no point in attempting that > > discussion. > > Verification of _what_? You haven't made a single cogent point or > formulated a coherent argument yet that I've seen. Oh, I see we are back at this game again. It is like I'm talking with a jeckel and a hyde sort of character paragraph by paragraph. > > > You have weighed in with some nice content but have > > avoided the kernel of the discussion. > > Largely because I can't figure out what you're talking about. And > because you've resisted all opportunities to clarify. > > I weighed in with some trite and obvious observations because you had > managed to frustrate Marshall into making what seemed to me to be an > obviously false statement. I'd be frustrated too if I tried to > respond to each of your posts in a meaningful fashion. > > > Care to take a risk? Here you > > see I have no choice but to attempt to draw good minds into the fray, > > "fray". Not a good way to think about this. The goal should be a > meeting of the minds. OK, sincerely Briggs, I do appreciate your attitude in this one line above. I am comfortable with rhetoric and without it the topic would be drab. We have freedom on this medium and in no way do I want to curtail your freedom, but then, I must also take my own. > > > and the resistance to address the criticism is completely > > understandable from a humanistic standpoint. > > Be very careful. You could end up like James Harris. Never attribute > to [other people's] malice that which can be adequately explained by > [your own] stupidity. Ahhh... Now this is a fine piece of rhetoric, and highly unmathematical I might add. Still, based on your first post I suspect that your next post will be more mathematical Uncle Briggs. - Tim
From: Herman Jurjus on 22 Jan 2010 10:28 Tim Golden BandTech.com wrote: > 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. 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. Ever heard of 'implicit typecast'? -- Cheers, Herman Jurjus
From: Marshall on 22 Jan 2010 11:03 On Jan 22, 6:23 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 21, 1:27 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > I still remain open to the possibility that I am wrong. You say that, but it's not true. > However I am wrong then my language should be falsifiable. 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. > > 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. Thus: your points about complex numbers are merely about the notation. > Ahh. So maybe you do get it even while you deny getting it. > [...] > Jeeze, I guess you didn't get it. > [...] > Ahhh. So you do get it. What's also become abundantly clear is that you are incapable of distinguishing between someone disagreeing with you and someone who doesn't understand the point you are trying to make. The point you are trying to make is trivial, simplistic, and invalid. Everyone understands it; they just don't think it has any validity. Marshall
From: Alan Smaill on 22 Jan 2010 11:29 Marshall <marshall.spight(a)gmail.com> writes: > On Jan 22, 6:23�am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > On Jan 21, 1:27 pm, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > > > > I still remain open to the possibility that I am wrong. > > You say that, but it's not true. > > > > However I am wrong then my language should be falsifiable. > > 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? > > > Marshall -- Alan Smaill
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." |