Does inductive reasoning lead to knowledge?
in [Logic]
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From: Tim Golden BandTech.com on 19 Jan 2010 08:17 On Jan 18, 7:01 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > 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. No Marshall. We can use one ring definition. This is how fundamental my complaint is. You attempt the same embedding argument that Mr. Clarke presented, but if we actually express each value that way you will have engaged a runaway system: b i = 0 + bi = 0 + 0 i + 0 + b i = ... I have seen such silliness with zeros up in purified polynomial defenses, but down at this level isn't this getting a bit carried away? No matter how many zeros we place here we still have the product b i which offends not just one implementation of a ring; it offends the crux or a principle of the ring, and so I refute your criticism. The tuple is a way out, but then the reals are not a subset. I should have asked Mr. Clarke for a definition of tuple, but then, I'm not a stifler. The tuple takes a confusing turn in polynomial land, where the confusion with the cartesian product is disconcerting. Consistency is only partial Marshall. - Tim > > 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: jmfbahciv on 19 Jan 2010 09:54 Patricia Aldoraz wrote: > 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. The thread drift, which you snipped so you could snipe, is way beyond the topic you didn't want to discuss. /BAH
From: Marshall on 19 Jan 2010 10:37 On Jan 19, 5:17 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> wrote: > On Jan 18, 7:01 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > 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. > > No Marshall. We can use one ring definition. This is how fundamental > my complaint is. > You attempt the same embedding argument that Mr. Clarke presented, but > if we actually express each value that way you will have engaged a > runaway system: > b i = 0 + bi = 0 + 0 i + 0 + b i = ... So? What's wrong with the equation(s) above? Nothing that I can see. Also note that: 5 = 0 + 5 = 0 + 0 + 5 = ... Don't you agree? Is addition of natural numbers a "runaway system?" I don't see how an infinite number of equations makes for any kind of problem. > I have seen such silliness with zeros up in purified polynomial > defenses, but down at this level isn't this getting a bit carried > away? No matter how many zeros we place here we still have the product > b i > which offends not just one implementation of a ring; it offends the > crux or a principle of the ring, and so I refute your criticism. Which principle is offended? You really need to get more explicit. And anyway, what does "offended" mean? If your argument relies on closure, you have to show that closure does not hold in some circumstances. It's not enough to claim that a crux is offended. And by the way, closure holds. > Consistency is only partial Marshall. I am afraid I don't agree. Also, you didn't address this: > > 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: Tim Golden BandTech.com on 19 Jan 2010 12:31 On Jan 19, 10:37 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jan 19, 5:17 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> > wrote: > > > > > On Jan 18, 7:01 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > 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. > > > No Marshall. We can use one ring definition. This is how fundamental > > my complaint is. > > You attempt the same embedding argument that Mr. Clarke presented, but > > if we actually express each value that way you will have engaged a > > runaway system: > > b i = 0 + bi = 0 + 0 i + 0 + b i = ... > > So? What's wrong with the equation(s) above? Nothing > that I can see. > > Also note that: > > 5 = 0 + 5 = 0 + 0 + 5 = ... > > Don't you agree? Is addition of natural numbers a > "runaway system?" I don't see how an infinite number > of equations makes for any kind of problem. Why did you propose this resolution in the first place? You are now defending it and I must resolve the issue by returning to the source of your argument. I admit that the zeros are harmless, but then are they doing anything at all? You built this, not me. Why did you build it? It does not resolve my initial complaint and so we must return to the initial point which caused this response. That is my own claim that the product b i of the complex representation a + b i where a and b are real and i is not real is not a ring product. Thus this form is incompatible with the ring terminology. I need nothing more than the closure requirement of the ring definition to prove this. While Clarke asks that I define the real number I can only see such a requirement as a stifling reaction. The discussion that I wish to have is right here in these simple terms. You've reacted by providing an argument of embedded zeros and I have now responded, and leave this to you for steerage, for it is your path of choice, not mine. > > > I have seen such silliness with zeros up in purified polynomial > > defenses, but down at this level isn't this getting a bit carried > > away? No matter how many zeros we place here we still have the product > > b i > > which offends not just one implementation of a ring; it offends the > > crux or a principle of the ring, and so I refute your criticism. > > Which principle is offended? You really need to get more > explicit. And anyway, what does "offended" mean? > > If your argument relies on closure, you have to show > that closure does not hold in some circumstances. > It's not enough to claim that a crux is offended. And > by the way, closure holds. > > > Consistency is only partial Marshall. > > I am afraid I don't agree. Also, you didn't address > this: > > > > 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 The operator is not associative. ( 4 / 2 ) / 2 = 4 / ( 2 / 2 ) fails the test with '/' as either operator of the ring definition according to http://en.wikipedia.org/wiki/Ring_mathematics#Formal_definition I fail to see any symmetry in your argument to mine. Here you are claiming that a broken system is broken, whereas I am claiming that there is a small conflict within a system which is considered to be working well. Are you quipping on inductive reasoning here? For you to have this discussion with Mr. Clarke will require first that you define everything even if it is commonly understood, and all the way down please. Don't just give the critical information. We can't trust anything until you've fully defined it. Just imagine how well the world will work when we all work to this degree of detail in every communication. And at the bottom will be just a bunch of assumptions footing the whole thing. The truly serious argument lays beyond this point, but the complex value a+bi is a simple form of the breakage of structural integrity, and by the very definition which claims to support it. By even attempting to resolve it you have admitted the existence of this conflict, and yet your mind will continue its denial, thus closing the conversation. For you it is as if it never happened. It can be viewed already in the dialog with Clarke and you are essentially repeating his steps, though the rational construction is a fresh attemp. I will try to stay openminded to your point there if you care to keep it going. Also I should point out that if I am wrong then there should be a logical conflict with my statement. This should be produced by you as an accusation rather than fraying off a resolution. It should be very direct but you see in the construction b i there is so little there to be considered that the statement is obviously true. This product is different than the product of the ring definition and it goes unappreciated. b and i are not of the same set, period. Just these few bits of information are needed to process this thought, not a laundry list of definitions. - Tim
From: J. Clarke on 19 Jan 2010 14:27
Tim Golden BandTech.com wrote: > On Jan 19, 10:37 am, Marshall <marshall.spi...(a)gmail.com> wrote: >> On Jan 19, 5:17 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com> >> wrote: >> >> >> >>> On Jan 18, 7:01 pm, Marshall <marshall.spi...(a)gmail.com> wrote: >> >>>>> 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. >> >>> No Marshall. We can use one ring definition. This is how fundamental >>> my complaint is. >>> You attempt the same embedding argument that Mr. Clarke presented, >>> but if we actually express each value that way you will have >>> engaged a runaway system: >>> b i = 0 + bi = 0 + 0 i + 0 + b i = ... >> >> So? What's wrong with the equation(s) above? Nothing >> that I can see. >> >> Also note that: >> >> 5 = 0 + 5 = 0 + 0 + 5 = ... >> >> Don't you agree? Is addition of natural numbers a >> "runaway system?" I don't see how an infinite number >> of equations makes for any kind of problem. > > Why did you propose this resolution in the first place? You are now > defending it and I must resolve the issue by returning to the source > of your argument. I admit that the zeros are harmless, but then are > they doing anything at all? You built this, not me. Why did you build > it? It does not resolve my initial complaint and so we must return to > the initial point which caused this response. That is my own claim > that the product > b i > of the complex representation > a + b i > where a and b are real and i is not real is not a ring product. Thus > this form is incompatible with the ring terminology. I need nothing > more than the closure requirement of the ring definition to prove > this. While Clarke asks that I define the real number I can only see > such a requirement as a stifling reaction. The discussion that I wish > to have is right here in these simple terms. You've reacted by > providing an argument of embedded zeros and I have now responded, and > leave this to you for steerage, for it is your path of choice, not > mine. > >> >>> I have seen such silliness with zeros up in purified polynomial >>> defenses, but down at this level isn't this getting a bit carried >>> away? No matter how many zeros we place here we still have the >>> product b i >>> which offends not just one implementation of a ring; it offends the >>> crux or a principle of the ring, and so I refute your criticism. >> >> Which principle is offended? You really need to get more >> explicit. And anyway, what does "offended" mean? >> >> If your argument relies on closure, you have to show >> that closure does not hold in some circumstances. >> It's not enough to claim that a crux is offended. And >> by the way, closure holds. >> >>> Consistency is only partial Marshall. >> >> I am afraid I don't agree. Also, you didn't address >> this: >> >>>> 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 > > The operator is not associative. > ( 4 / 2 ) / 2 = 4 / ( 2 / 2 ) > fails the test with '/' as either operator of the ring definition > according to > http://en.wikipedia.org/wiki/Ring_mathematics#Formal_definition > I fail to see any symmetry in your argument to mine. Here you are > claiming that a broken system is broken, whereas I am claiming that > there is a small conflict within a system which is considered to be > working well. Are you quipping on inductive reasoning here? > > For you to have this discussion with Mr. Clarke will require first > that you define everything even if it is commonly understood, and all > the way down please. Don't just give the critical information. We > can't trust anything until you've fully defined it. Just imagine how > well the world will work when we all work to this degree of detail in > every communication. And at the bottom will be just a bunch of > assumptions footing the whole thing. > > The truly serious argument lays beyond this point, but the complex > value a+bi is a simple form of the breakage of structural integrity, > and by the very definition which claims to support it. By even > attempting to resolve it you have admitted the existence of this > conflict, and yet your mind will continue its denial, thus closing the > conversation. For you it is as if it never happened. It can be viewed > already in the dialog with Clarke and you are essentially repeating > his steps, though the rational construction is a fresh attemp. I will > try to stay openminded to your point there if you care to keep it > going. Also I should point out that if I am wrong then there should be > a logical conflict with my statement. This should be produced by you > as an accusation rather than fraying off a resolution. It should be > very direct but you see in the construction > b i > there is so little there to be considered that the statement is > obviously true. This product is different than the product of the ring > definition and it goes unappreciated. > b and i are not of the same set, period. Just these few bits of > information are needed to process this thought, not a laundry list of > definitions. Interesting. You rail at mathematicians for not questioning their assumptions, and yet you when asked to state your assumptions at the risk of others questioning them you argue that you are being "stifled". |