Does inductive reasoning lead to knowledge?
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From: Aatu Koskensilta on 21 Jan 2010 08:42 jbriggs444 <jbriggs444(a)gmail.com> writes: > I was trained as a Dedekind cut guy. But I feel no need to declare > Jihad against the infidel Cauchy sequence dudes. Well, yes. Such matters turn on but uninteresting (from a general mathematical point of view) details of the set theoretic construction adopted, and are of no wider mathematical significance whatever. That a real number is a complex number is a wholly unobjectionable claim in ordinary mathematics. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: J. Clarke on 21 Jan 2010 09:45 Aatu Koskensilta wrote: > jbriggs444 <jbriggs444(a)gmail.com> writes: > >> I was trained as a Dedekind cut guy. But I feel no need to declare >> Jihad against the infidel Cauchy sequence dudes. > > Well, yes. Such matters turn on but uninteresting (from a general > mathematical point of view) details of the set theoretic construction > adopted, and are of no wider mathematical significance whatever. That > a real number is a complex number is a wholly unobjectionable claim in > ordinary mathematics. I think it really gets down to the oft-discussed concept of "mathematical maturity".
From: Aatu Koskensilta on 21 Jan 2010 10:03 "J. Clarke" <jclarke.usenet(a)cox.net> writes: > Aatu Koskensilta wrote: > >> Well, yes. Such matters turn on but uninteresting (from a general >> mathematical point of view) details of the set theoretic construction >> adopted, and are of no wider mathematical significance whatever. That >> a real number is a complex number is a wholly unobjectionable claim >> in ordinary mathematics. > > I think it really gets down to the oft-discussed concept of > "mathematical maturity". True. But this concept isn't as much oft-discussed as oft-mentioned. Saying anything insightful about "mathematical maturity" is surprisingly difficult. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tim Golden BandTech.com on 21 Jan 2010 11:07 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? Should I simply accept the constructions which have been repeated many times by accomplished minds? Here lays a humanistic problem where the mathematical cannot be separated. 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. Here we see a unique product operator in common use which goes ignored. The same offense is in use further along but with much more density of information which further clouds the discussion. The a+bi complex representation is a very simple instance. As you point out the taking of a cartesian tuple ( a, b ) and then pulling out the first component and calling it real is a relevant detail I believe, though I am not relying upon this within my argument. The value ( a, 0 ) is not a one dimensional entity. In the polynomial development we'll see the tuple used outside of the cartesian form. 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. You have weighed in with some nice content but have avoided the kernel of the discussion. Care to take a risk? Here you see I have no choice but to attempt to draw good minds into the fray, and the resistance to address the criticism is completely understandable from a humanistic standpoint. Still, aren't we doing mathematics? What better medium is there than this one for such a discussion? We are fortunate to be in the internet age. Thanks again for weighing in. - Tim
From: jbriggs444 on 21 Jan 2010 13:27
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? > 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. 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. > 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. > 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 we see a unique product > operator in common use which goes ignored. Here? Where? Unique product operator? What's that? And how does it tie into a 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. > 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. > 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. 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). > 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. > 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 ). [ 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? > 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. > 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. > 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. |