Does inductive reasoning lead to knowledge?
in [Logic]
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From: J. Clarke on 18 Jan 2010 13:18 Tim Golden BandTech.com wrote: > On Jan 16, 12:41 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >> Tim Golden BandTech.com wrote: >>> On Jan 14, 1:19 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>> Tim Golden BandTech.com wrote: >>>>> On Jan 14, 9:39 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>>>> Zinnic wrote: >>>>>>> On Jan 13, 11:56 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>>> BS (Big Snip) >>>>>>> He is working thru his boring posting algorithm again. His final >>>>>>> step will be to sick his 'patsy' onto you. He defecates. She >>>>>>> flushes. >> >>>>>> I suspect you're right. I would like to know where this >>>>>> "basketweaving class" is being held. I suspect that it contains >>>>>> some interesting "students". >> >>>>> J. Clarke speaks of rings above here eloquently. I wonder if you >>>>> would offer your criticism on the following: >> >>>>> The complex number >>>>> a + b i >>>>> are considered to be consistent with ring terminology, with a >>>>> product and sum being consistently defined and being algebraically >>>>> well behaved, yet within this number form itself >>>>> a + b i >>>>> we see one product >>>>> b i >>>>> and one sum >>>>> a + (bi) >>>>> which are inconsistent with the group and ring definitions since a >>>>> and b are real, and i is not real. Thus the very construction of >>>>> the complex number via its definition is not compatible with this >>>>> abstract algebraic form. >> >>>> Be kind enough to exactly state the definitions you are using. It >>>> is difficult to follow your argument if you are not clear in your >>>> definitions. And is your question whether the particular examples >>>> you use violate closure or do you have some other issue in mind? >> >>> As simply as I put it the ring (or group) definitions require >>> elements of the same set to operate upon and return a resultant in >>> that same set. The usage of the form >>> a + b i >>> is inherently conflicted. Now, I have read your link and do see that >>> you see the problem even while you deny seeing it, and I have to >>> admit that this is the most effective answer that I've received on >>> this focus. >> >> Uh, it's not me that sees the problem, it's Hamilton. I didn't >> write that book. >> >> How are you defining a complex number? > > I'm not redefining anything. I did not assert that you were. I asked you to state the definition that you are using. > I am using existing terminology. The > a + b i > form ought to be getting the scrutiny which I've given it in the AA > books. Please state your definition of "the a+bi form". > Clearly this sum operator does not coexist with the ring > definition, which would have us returning a single element from two > elements. This sum operator performs no such action. Please state your definition of "the sum operator". Please state "the ring definition". You seem to be assuming that something that you read in a text somewhere is known, in that exact form, to everyone in the world. Texts differ in rigor depending on their age, their authorship, and their target audience. You are arguing a point whose resolution requires a certain degree of rigor. I do not want to state the definitions I use and risk being accused of attempting to impose orthodoxy--I want to see the definitions that you are using. Once we have those definitions in hand then we can go on with some hope of a successful resolution. >> If you're defining it as being of >> the form "a+bi" then a rigorous treatment requires that you _always_ >> express it as "a+bi" even if a or b is zero. So you would not write >> "a", you would write "a+0i". You would not write "bi", you would >> write "0+bi". So for your above case, you would not write "a+(bi)", >> you would write "a+0i+(0+bi)". It's only when you attempt a >> rigorous analysis while using the shorthand nonrigorous notation >> that this particular ambiguity becomes an issue. > > This resolution which you've resorted to is actually worse, for now > upon expanding the complex value > 1.2 i > to > 0 + 1.2 i > I must then again allow for the expansion to > ( 0 + 0 i ) + ( 0 + 1.2 i ) That is not an "expansion", that is the summation of two complex numbers. > and so forth, resulting in the same nauseous effect that the ring > quotient has. What "nauseous effect" is this, and please be kind enough to define "the ring quotient". > If anything the spirit of the ring definition takes us > the other way... resolving all such operations and yielding a soliton > element. Please define "soliton element". >>> So the treatment of tuple is the best answer, as you've outlined >>> below. This crops up again at the usage of >>> "polynomial with real coefficients" >>> which is in use at the construction of higher order forms. >>> and I suppose you will support my accusation that this language >>> needs modification, for if it is to be compatible with your own sly >>> interpretation we will not cast the phrase "real coefficient" as >>> anything near to a real number coefficient, since a dimensional form >>> is implied. To what degree is this then a redefinition of the real >>> number to a high dimensional usage? Then, the recovery of the >>> complex number within the quotient ring is celebrated as yet >>> another feat. >> >> I honestly have no clue what you are on about here. Could you give >> some examples? > > The recovery of complex numbers from the ring quotient is stated with > terminology > "The quotient ring R[X]/(X^2 + 1) is naturally isomorphic to the > field of complex numbers C" > - http://en.wikipedia.org/wiki/Quotient_ring#Examples , third > example That doesn't say anything about "the ring quotient". It states that one can construct a quotient ring that is isomorphic to the complex numbers, which is interesting but I don't see where you are going with it. >>> You have legitimated my concern even while denying its legitimacy. I >>> would submit to you that such behavior is highly unmathematical. >> >> Definition of notation and the rigorous following of that notation >> is hardly "unmathematical". It's when you define notation and then >> abandon it in your argument that you become "unmathematical". >> >>> I do not wish to deny the unit vector form, but then too, the >>> freedom with which the two forms are taken as the same may be taken >>> as farcical. >> >> In what way? > > Again, this all goes back to the definition of ring. This definition > is granting us operators. This is how primitive the definition is. If > we use these operators too loosely then we are insincere in our usage > of this math. And yet with all this insistence on rigor you refuse to state the definition you are using for the elements of the ring and the definition you are using for the ring operators. > This is how I see the subject of abstract algebra (AA). > It goes to the trouble of making these pristine definitions and then > it ignores them when it wants to. Abstract algebra is to a substantial extent the study of what happens when one redefines operations such as "+". If you are going to use "the ring definition" then you must clearly state the definition you are using if you are to have any hope of having your argument understood by others. > Already I know that you have seen > what I am attempting to discuss with you, and the inability of a > modern mathematician to criticize the math is proof of my earlier > social theory statements on mimicry. I'll be happy to "criticize the math" if you will do me the kindness of DEFINING THE TERMS YOU ARE USING. But you have not done that and have rejected every well known definition that I have suggested. > If you go over this edge then you > have joined the crazies. Mathematics is a game. The rules are that you define your terms and state your axioms and then proceed in a logical manner from there. The only "crazies" are the ones who refuse to define their terms and state their axioms and yet demand that their views be understood and respected. If you don't want to define your terms and state your axioms then you are playing some game other than "mathematics". > Thus the establishment position is more than > we give it credit for. It has a social momentum which disallows > mathematical developments through an enforcement whose basis is > instinctual. That mathematicians are human is a factor which > mathematicians attempt to divorce themselves from. However, this > divorce is not actually possible. In other wordse you are oppressed by the "establishment" because it does not accept your unorthodox views? >> A complex number in one form may be uniquely mapped to one in >> the other form. The same is true of the operations defined on them. >> The two notations are equivalent in result, the difference between >> them is that one is more convenient to write than the other. >> >> Can you give an example of a theorem that is true in one notation >> and not in the other? > > I have given an axiom which is not compatible with both forms of > notation. Please restate this axiom as I seem to have missed it. > That said I do feel comfortable and even enjoy the unit > vector notation. Would you be kind enough to define this "unit vector notation"? > My attack is on the subject of abstract algebra as > incomplete and conflicted, whereby the conversation that I am > attempting to have with you does not register within that AA > framework. All mathematics is incomplete. If you can prove that any axiom of abstract algebra is in conflict with any other axiom of abstract algebra, the Fields Medal is almost certainly yours. But if you are going to prove that you need to learn to play the math game, and one of the rules of the math game is that you STATE YOUR DEFINITIONS AND YOUR AXIOMS and not vaguely throw around terms like "the ring definition". > I guess if you want a theorem the first thing that pops > into mind is on the square root of negative one. What about it? Do you have a theorem to state concerning it? > This value goes > undefined in the real numbers, and whether we perceive then a higher > dimensional form as inherently existent already within the real > numbers would be quite a discussion. I still don't see any theorem. > But I don't think that this is > going to be an effective answer to your question. Even when we > consider the tuple form > ( a0, 0, 0, 0 ) > why should we grant this to be a real value when we've spec'd four > dimensions of data? Who has asserted that it is "a real value"? You have not stated the context. > Again, no theorem here. This is true. > This issue is down in the > definitional guts. You will avoid my last post and the statements on > high dimensional data I suppose and so I should expect the same here. I don't see any point in changing the topic. >>> Is the real number inherently high dimensional? >> >> Please give an example that shows your argument. > > I just did. The example is the value > ( a0, 0, 0, 0 ) . What of it? What are you asserting concerning that expression and on what basis? Go step by step, pretend I'm stupid. >>> This would be quite a dramatic adjustment that abstract algebraists >>> have been drawing their students into. >> >> Again you've lost me. >> >>> This study returns one to >>> scrutinize the cartesian product itself. I have learned that the >>> usage of the cartesian product is optional in the construction of >>> higher order math. This does not necessarily contradict the >>> existing higher order math, but it does cause one to scrutinize it. >> >> What do you mean when you say "scrutinizing the cartesian product"? >> The Cartesian product simply defines all possible pairings of >> elements of two or more sets. It doesn't define any relation >> between the sets other than itself, it does not define any operation >> other than construction of the Cartesian product. > In your willingness to declare an equivalence between > ( a0, 0, 0, 0 ) > and > ( a0, 0 ) > and > ( a0 ) > and > a0 > I see that the cartesian product is being offended. Without this > offense abstract algebra will not stand up. The quotient ring will > fall. In what way is "the Cartesian product" "offended"? And "equivalence" is not something one "declares". Define the set of which those are elements and one can prove or disprove that they are equivalent depending on how that set is defined. As for abstract algebra not standing up, you've lost me completely. Abstract algebra deals in abstractions. Rings, groups, fields, operations, sets, all are arbitrary. Your arguments based on ordered tuples make no sense until you define the sets of which those tuples are a part and the operations defined on those sets. <earlier post snipped> >> I asked you specific questions, which you did not answer, instead >> just repeating what you had already stated. Please answer the >> questions because at the moment it is difficult to figure out >> what you are on about. > > I don't see that any of these answers are have provided much steerage > to the discussion. Try them and see what happens? I'm trying to understand your viewpoint but you are assuming that immediately grasp every term that you use and define it in the same way that you do. Without mutually agreed upon definitions the conversation goes at cross purposes. > I believe that there is a wall here of > impenetrability brought to you by your human instincts. Nope. The wall here is that you seem to be defining terms very vaguely and then getting confused when your vague definitions come into conflict. > You've > provided several resolutions to the problem while denying the > existence of the problem. You have not shown that the problem is in anything but your own understanding. If you can define your terms clearly and state your axioms clearly and the "problem" still exists then I will recant that statement. > Here should I insult you? No, I should > resign. How about trying playing the math game by the math rules? > Thank you for your patient responses. Perhaps eventually some > young mind will come accross the conversation here and see the issue. > If you choose to respond then I will carry on, but also I wish to > offer my resignation to this thread's tendril. I really do appreciate > your time and see this as a sliver of proof. Not much of one, but it > does not take much of one. Always the constructions will lay open to > future constructions, and getting back to the title of this thread I > declare it irrelevant, for knowledge is constructed by humans. We are > prisoners of reality with no direct access to its substrate. Hindsight > of construction guides us. This is a nonmathematical proof. It is the > basis of mathematics. Mathematics is a game played by stating definitions and axioms and deriving a logical structure from them. If your definitions and axioms are vaguely written or unclear then your logical structure will fall apart. The problem I see is that you without stating the definitions and axioms that you believe to be in place nonetheless attempt to attack them, and the result is not that you show flaws in mathematics but that you make yourself appear confused. So try stating, clearly, the definitions and axioms that you are using. If you don't like the way the complex numbers are defined, then define them however you wish and work out the body of the theory that results and present it to a mathematical journal and if what you have come up with is (a) free of logical errors and (b) sufficiently different from what has gone before to be considered "interesting", you too will be a respected mathematician. And if your approach is clearly "better" than any of the other several approaches then it will become the standard approach.
From: Tim Golden BandTech.com on 18 Jan 2010 17:21 On Jan 18, 1:18 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > Tim Golden BandTech.com wrote: > > On Jan 16, 12:41 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > >> Tim Golden BandTech.com wrote: > >>> On Jan 14, 1:19 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > >>>> Tim Golden BandTech.com wrote: > >>>>> On Jan 14, 9:39 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > >>>>> J. Clarke speaks of rings above here eloquently. I wonder if you > >>>>> would offer your criticism on the following: > > >>>>> The complex number > >>>>> a + b i > >>>>> are considered to be consistent with ring terminology, with a > >>>>> product and sum being consistently defined and being algebraically > >>>>> well behaved, yet within this number form itself > >>>>> a + b i > >>>>> we see one product > >>>>> b i > >>>>> and one sum > >>>>> a + (bi) > >>>>> which are inconsistent with the group and ring definitions since a > >>>>> and b are real, and i is not real. Thus the very construction of > >>>>> the complex number via its definition is not compatible with this > >>>>> abstract algebraic form. > > >>>> Be kind enough to exactly state the definitions you are using. It > >>>> is difficult to follow your argument if you are not clear in your > >>>> definitions. And is your question whether the particular examples > >>>> you use violate closure or do you have some other issue in mind? How about http://en.wikipedia.org/wiki/Ring_(mathematics)#Formal_definition Can we agree to this as a starting point? The closure requirements are so far all that I have relied upon to make my argument. I am sorry I misuse the terminology and where I have used "ring quotient" I should have used "quotient ring", though I see no way to confuse the statement. Again, applying the closure principles to the two operators in the complex value a + b i we see no agreement with the ring definition. It is this simple. b is real. i is not real. Therefore this product b i is incompatible with the ring definition's product. Further the sum will not resolve to a single element, where all sums will have two elements to operate upon. This is so simple that I cannot see how any confusion can creep in. The same concept can be reapplied to the polynomial with real coefficients. Are the coefficients truly real? Apply the operators of the ring and we see that the elements to which these real coefficients apply must also be real. Thus the entire sum within such polynomials must be real, by the same simple closure principle. I have made these statements now several times to you and you have offered up that the definition of a complex value is in tuple form (a,b). This does nothing to change my argument on the usual form a + b i and I readily admit that in the z form there can be little to argue over so long as we discuss in terms of z1 + z2, z3 z4 style sums and products. I got here back in time by studying the quotient ring in an attempt to understand some math work. The work particularly relies upon polynomials with real coefficients. This multiplication of a real valued coefficient to an X which has only vague meaning is beyond my ability to understand. And in going back to the definition of ring I see that this construction(the polynomial itself) is conflicted, as I have outlined. - Tim > > >>> As simply as I put it the ring (or group) definitions require > >>> elements of the same set to operate upon and return a resultant in > >>> that same set. The usage of the form > >>> a + b i > >>> is inherently conflicted. Now, I have read your link and do see that > >>> you see the problem even while you deny seeing it, and I have to > >>> admit that this is the most effective answer that I've received on > >>> this focus. > > >> Uh, it's not me that sees the problem, it's Hamilton. I didn't > >> write that book. > > >> How are you defining a complex number? > > > I'm not redefining anything. > > I did not assert that you were. I asked you to state the definition that > you are using. > > > I am using existing terminology. The > > a + b i > > form ought to be getting the scrutiny which I've given it in the AA > > books. > > Please state your definition of "the a+bi form". > > > Clearly this sum operator does not coexist with the ring > > definition, which would have us returning a single element from two > > elements. This sum operator performs no such action. > > Please state your definition of "the sum operator". > > Please state "the ring definition". > > You seem to be assuming that something that you read in a text somewhere is > known, in that exact form, to everyone in the world. Texts differ in rigor > depending on their age, their authorship, and their target audience. You > are arguing a point whose resolution requires a certain degree of rigor. I > do not want to state the definitions I use and risk being accused of > attempting to impose orthodoxy--I want to see the definitions that you are > using. Once we have those definitions in hand then we can go on with some > hope of a successful resolution. > > >> If you're defining it as being of > >> the form "a+bi" then a rigorous treatment requires that you _always_ > >> express it as "a+bi" even if a or b is zero. So you would not write > >> "a", you would write "a+0i". You would not write "bi", you would > >> write "0+bi". So for your above case, you would not write "a+(bi)", > >> you would write "a+0i+(0+bi)". It's only when you attempt a > >> rigorous analysis while using the shorthand nonrigorous notation > >> that this particular ambiguity becomes an issue. > > > This resolution which you've resorted to is actually worse, for now > > upon expanding the complex value > > 1.2 i > > to > > 0 + 1.2 i > > I must then again allow for the expansion to > > ( 0 + 0 i ) + ( 0 + 1.2 i ) > > That is not an "expansion", that is the summation of two complex numbers. > > > and so forth, resulting in the same nauseous effect that the ring > > quotient has. > > What "nauseous effect" is this, and please be kind enough to define "the > ring quotient". > > > If anything the spirit of the ring definition takes us > > the other way... resolving all such operations and yielding a soliton > > element. > > Please define "soliton element". > > >>> So the treatment of tuple is the best answer, as you've outlined > >>> below. This crops up again at the usage of > >>> "polynomial with real coefficients" > >>> which is in use at the construction of higher order forms. > >>> and I suppose you will support my accusation that this language > >>> needs modification, for if it is to be compatible with your own sly > >>> interpretation we will not cast the phrase "real coefficient" as > >>> anything near to a real number coefficient, since a dimensional form > >>> is implied. To what degree is this then a redefinition of the real > >>> number to a high dimensional usage? Then, the recovery of the > >>> complex number within the quotient ring is celebrated as yet > >>> another feat. > > >> I honestly have no clue what you are on about here. Could you give > >> some examples? > > > The recovery of complex numbers from the ring quotient is stated with > > terminology > > "The quotient ring R[X]/(X^2 + 1) is naturally isomorphic to the > > field of complex numbers C" > > -http://en.wikipedia.org/wiki/Quotient_ring#Examples, third > > example > > That doesn't say anything about "the ring quotient". It states that one can > construct a quotient ring that is isomorphic to the complex numbers, which > is interesting but I don't see where you are going with it. > > >>> You have legitimated my concern even while denying its legitimacy. I > >>> would submit to you that such behavior is highly unmathematical. > > >> Definition of notation and the rigorous following of that notation > >> is hardly "unmathematical". It's when you define notation and then > >> abandon it in your argument that you become "unmathematical". > > >>> I do not wish to deny the unit vector form, but then too, the > >>> freedom with which the two forms are taken as the same may be taken > >>> as farcical. > > >> In what way? > > > Again, this all goes back to the definition of ring. This definition > > is granting us operators. This is how primitive the definition is. If > > we use these operators too loosely then we are insincere in our usage > > of this math. > > And yet with all this insistence on rigor you refuse to state the definition > you are using for the elements of the ring and the definition you are using > for the ring operators. > > > This is how I see the subject of abstract algebra (AA). > > It goes to the trouble of making these pristine definitions and then > > it ignores them when it wants to. > > Abstract algebra is to a substantial extent the study of what happens when > one redefines operations such as "+". If you are going to use "the ring > definition" then you must clearly state the definition you are using if you > are to have any hope of having your argument understood by others. > > > Already I know that you have seen > > what I am attempting to discuss with you, and the inability of a > > modern mathematician to criticize the math is proof of my earlier > > social theory statements on mimicry. > > I'll be happy to "criticize the math" if you will do me the kindness of > DEFINING THE TERMS YOU ARE USING. > But you have not done that and have > rejected every well known definition that I have suggested. > > > If you go over this edge then you > > have joined the crazies. > > Mathematics is a game. The rules are that you define your terms and state > your axioms and then proceed in a logical manner from there. The only > "crazies" are the ones who refuse to define their terms and state their > axioms and yet demand that their views be understood and respected. > > If you don't want to define your terms and state your axioms then you are > playing some game other than "mathematics". > > > Thus the establishment position is more than > > we give it credit for. It has a social momentum which disallows > > mathematical developments through an enforcement whose basis is > > instinctual. That mathematicians are human is a factor which > > mathematicians attempt to divorce themselves from. However, this > > divorce is not actually possible. > > In other wordse you are oppressed by the "establishment" because it does not > accept your unorthodox views? > > >> A complex number in one form may be uniquely mapped to one in > >> the other form. The same is true of the operations defined on them. > >> The two notations are equivalent in result, the difference between > >> them is that one is more convenient to write than the other. > > >> Can you give an example of a theorem that is true in one notation > >> and not in the other? > > > I have given an axiom which is not compatible with both forms of > > notation. > > Please restate this axiom as I seem to have missed it. > > > That said I do feel comfortable and even enjoy the unit > > vector notation. > > Would you be kind enough to define this "unit vector notation"? > > > My attack is on the subject of abstract algebra as > > incomplete and conflicted, whereby the conversation that I am > > attempting to have with you does not register within that AA > > framework. > > All mathematics is incomplete. If you can prove that any axiom of abstract > algebra is in conflict with any other axiom of abstract algebra, the Fields > Medal is almost certainly yours. But if you are going to prove that you > need to learn to play the math game, and one of the rules of the math game > is that you STATE YOUR DEFINITIONS AND YOUR AXIOMS and not vaguely throw > around terms like "the ring definition". > > > I guess if you want a theorem the first thing that pops > > into mind is on the square root of negative one. > > What about it? Do you have a theorem to state concerning it? > > > This value goes > > undefined in the real numbers, and whether we perceive then a higher > > dimensional form as inherently existent already within the real > > numbers would be quite a discussion. > > I still don't see any theorem. > > > But I don't think that this is > > going to be an effective answer to your question. Even when we > > consider the tuple form > > ( a0, 0, 0, 0 ) > > why should we grant this to be a real value when we've spec'd four > > dimensions of data? > > Who has asserted that it is "a real value"? You have not stated the > context. > > > Again, no theorem here. > > This is true. > > > This issue is down in the > > definitional guts. You will avoid my last post and the statements on > > high dimensional data I suppose and so I should expect the same here. > > I don't see any point in changing the topic. > > >>> Is the real number inherently high dimensional? > > >> Please give an example that shows your argument. > > > I just did. The example is the value > > ( a0, 0, 0, 0 ) . > > What of it? What are you asserting concerning that expression and on what > basis? Go step by step, pretend I'm stupid. > > >>> This would be quite a dramatic adjustment that abstract algebraists > >>> have been drawing their students into. > > >> Again you've lost me. > > >>> This study returns one to > >>> scrutinize the cartesian product itself. I have learned that the > >>> usage of the cartesian product is optional in the construction of > >>> higher order math. This does not necessarily contradict the > >>> existing higher order math, but it does cause one to scrutinize it. > > >> What do you mean when you say "scrutinizing the cartesian product"? > >> The Cartesian product simply defines all possible pairings of > >> elements of two or more sets. It doesn't define any relation > >> between the sets other than itself, it does not define any operation > >> other than construction of the Cartesian product. > > In your willingness to declare an equivalence between > > ( a0, 0, 0, 0 ) > > and > > ( a0, 0 ) > > and > > ( a0 ) > > and > > a0 > > I see that the cartesian product is being offended. Without this > > offense abstract algebra will not stand up. The quotient ring will > > fall. > > In what way is "the Cartesian product" "offended"? And "equivalence" is not > something one "declares". Define the set of which those are elements and > one can prove or disprove that they are equivalent depending on how that set > is defined. > > As for abstract algebra not standing up, you've lost me completely. > Abstract algebra deals in abstractions. Rings, groups, fields, operations, > sets, all are arbitrary. Your arguments based on ordered tuples make no > sense until you define the sets of which those tuples are a part and the > operations defined on those sets. > > <earlier post snipped> > > >> I asked you specific questions, which you did not answer, instead > >> just repeating what you had already stated. Please answer the > >> questions because at the moment it is difficult to figure out > >> what you are on about. > > > I don't see that any of these answers are have provided much steerage > > to the discussion. > > Try them and see what happens? I'm trying to understand your viewpoint but > you are assuming that immediately grasp every term that you use and define > it in the same way that you do. Without mutually agreed upon definitions > the conversation goes at cross purposes. > > > I believe that there is a wall here of > > impenetrability brought to you by your human instincts. > > Nope. The wall here is that you seem to be defining terms very vaguely and > then getting confused when your vague definitions come into conflict. > > > You've > > provided several resolutions to the problem while denying the > > existence of the problem. > > You have not shown that the problem is in anything but your own > understanding. If you can define your terms clearly and state your axioms > clearly and the "problem" still exists then I will recant that statement. > > > Here should I insult you? No, I should > > resign. > > How about trying playing the math game by the math rules? > > > Thank you for your patient responses. Perhaps eventually some > > young mind will come accross the conversation here and see the issue. > > If you choose to respond then I will carry on, but also I wish to > > offer my resignation to this thread's tendril. I really do appreciate > > your time and see this as a sliver of proof. Not much of one, but it > > does not take much of one. Always the constructions will lay open to > > future constructions, and getting back to the title of this thread I > > declare it irrelevant, for knowledge is constructed by humans. We are > > prisoners of reality with no direct access to its substrate. Hindsight > > of construction guides us. This is a nonmathematical proof. It is the > > basis of mathematics. > > Mathematics is a game played by stating definitions and axioms and deriving > a logical structure from them. If your definitions and axioms are vaguely > written or unclear then your logical structure will fall apart. The problem > I see is that you without stating the definitions and axioms that you > believe to be in place nonetheless attempt to attack them, and the result is > not that you show flaws in mathematics but that you make yourself appear > confused. > > So try stating, clearly, the definitions and axioms that you are using. > > If you don't like the way the complex numbers are defined, then define them > however you wish and work out the body of the theory that results and > present it to a mathematical journal and if what you have come up with is > (a) free of logical errors and (b) sufficiently different from what has gone > before to be considered "interesting", you too will be a respected > mathematician. And if your approach is clearly "better" than any of the > other several approaches then it will become the standard approach.
From: Marshall on 18 Jan 2010 19:01 On Jan 18, 2:21 pm, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 18, 1:18 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > > > >>>> Be kind enough to exactly state the definitions you are using. It > > >>>> is difficult to follow your argument if you are not clear in your > > >>>> definitions. And is your question whether the particular examples > > >>>> you use violate closure or do you have some other issue in mind? > > How about > http://en.wikipedia.org/wiki/Ring_(mathematics)#Formal_definition > Can we agree to this as a starting point? > The closure requirements are so far all that I have relied upon to > make my argument. You seem to be under some misapprehensions about closure. > Again, applying the closure > principles to the two operators in the complex value > a + b i > we see no agreement with the ring definition. It is this simple. b is > real. i is not real. Therefore this product > b i > is incompatible with the ring definition's product. Which ring? It isn't the case that there's just "the one ring"; (this isn't Tolkien.) Furthermore, that product can be understood to be purely notational if it bothers you. Also, if b is a real, then b is also a complex. In which case the product works just fine as the complex product. There really isn't any way to look at it (as far as I know) where it doesn't work just as one would expect. > Further the sum > will not resolve to a single element, where all sums will have two > elements to operate upon. It resolves to a single element in the carrier set that is the set of complex numbers. It is not two elements. You can think of it as having two components to it, but that doesn't make it more than one element. Doesn't your argument work just as well with the rational numbers? Thus: Again, applying the closure principles to the "/" operator in the rational value a / b we see no agreement with the rational ring definition. It is this simple. a and b are integers. Therefore this quotient a/b is incompatible with the ring definition's product. Marshall
From: J. Clarke on 18 Jan 2010 21:27 Tim Golden BandTech.com wrote: > On Jan 18, 1:18 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >> Tim Golden BandTech.com wrote: >>> On Jan 16, 12:41 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>> Tim Golden BandTech.com wrote: >>>>> On Jan 14, 1:19 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>>>> Tim Golden BandTech.com wrote: >>>>>>> On Jan 14, 9:39 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >>>>>>> J. Clarke speaks of rings above here eloquently. I wonder if you >>>>>>> would offer your criticism on the following: >> >>>>>>> The complex number >>>>>>> a + b i >>>>>>> are considered to be consistent with ring terminology, with a >>>>>>> product and sum being consistently defined and being >>>>>>> algebraically well behaved, yet within this number form itself >>>>>>> a + b i >>>>>>> we see one product >>>>>>> b i >>>>>>> and one sum >>>>>>> a + (bi) >>>>>>> which are inconsistent with the group and ring definitions >>>>>>> since a and b are real, and i is not real. Thus the very >>>>>>> construction of the complex number via its definition is not >>>>>>> compatible with this abstract algebraic form. >> >>>>>> Be kind enough to exactly state the definitions you are using. >>>>>> It is difficult to follow your argument if you are not clear in >>>>>> your definitions. And is your question whether the particular >>>>>> examples you use violate closure or do you have some other issue >>>>>> in mind? > > > How about > http://en.wikipedia.org/wiki/Ring_(mathematics)#Formal_definition > Can we agree to this as a starting point? > The closure requirements are so far all that I have relied upon to > make my argument. I am sorry I misuse the terminology and where I have > used "ring quotient" I should have used "quotient ring", though I see > no way to confuse the statement. Again, applying the closure > principles to the two operators in the complex value > a + b i > we see no agreement with the ring definition. You are skipping steps here. You are rushing from the definition of a ring to the complex numbers. The complex numbers constitute one ring among many. You have not defined the set of complex numbers nor have you defined the operations on the complex numbers. > It is this simple. b is > real. i is not real. Therefore this product > b i > is incompatible with the ring definition's product. Further the sum > will not resolve to a single element, where all sums will have two > elements to operate upon. This is so simple that I cannot see how any > confusion can creep in. Since you have not stated the definition of "complex number" nor have you stated the definition of "product", there is no way to determine whether your above statement is valid by your definitions. > The same concept can be reapplied to the polynomial with real > coefficients. Once again you have not defined your terms. Define "polynomial with real coefficients" and we can go from there. > Are the coefficients truly real? Does your definition require that they be? > Apply the operators of > the ring Which operators? You have not defined any operators. > and we see that the elements to which these real coefficients > apply must also be real. Why must they be real? > Thus the entire sum within such polynomials > must be real, by the same simple closure principle. Again you have not defined any operations. Without defining your operations any discussion of closure is pointless. > I have made these statements now several times to you and you have > offered up that the definition of a complex value is in tuple form > (a,b). If you don't like that one then offer up another one. > This does nothing to change my argument on the usual form > a + b i > and I readily admit that in the z form there can be little to argue > over so long as we discuss in terms of > z1 + z2, z3 z4 > style sums and products. Rather than vaguely saying "the usual form", please state the definition of "the usual form" as you understand it. > I got here back in time by studying the quotient ring in an attempt to > understand some math work. Which particular quotient ring? There is no "the" quotient ring. > The work particularly relies upon > polynomials with real coefficients. Which work? > This multiplication of a real > valued coefficient to an X which has only vague meaning > is beyond my ability to understand. So define X. > And in going back to the > definition of ring I see that this construction(the polynomial itself) > is conflicted, as I have outlined. Which construction? To create a ring you must provide certain definitions. Once you have provided the definitions then you can discuss matters such as closure. You seem to be assuming that there is only one kind of ring, "the" ring. That is not the case. <snip>
From: Patricia Aldoraz on 18 Jan 2010 21:41
On Jan 19, 2:19 am, jmfbahciv <jmfbahciv(a)aol> wrote: > This topic might have been interesting. Even in the possible world envisaged, it would be no thanks to your contributions though. |