What makes a trivial operator trivial ?
in [Math]
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From: T.H. Ray on 1 Feb 2010 00:49 Bacle wrote > > > > "Aage Andersen" <aaan(REMOVE)@email.dk> wrote in > > message > > > news:4b65a3d1$0$56792$edfadb0f(a)dtext02.news.tele.dk... > > > > > > > "Huang" > > >> Zero apples is the same thing as zero oranges. > It > > is nonsense to say > > >> that apples are oranges, except in such trivial > > instances as this. If > > >> zero was the only number you had at your > disposal, > > all you would be > > >> able to do is trivial and/or nonsensical > > procedures. > > > > > > I think it is a mistake to consider zero apples > the > > same as zero oranges. > > > > > Evidently you do not think. > > Not so, if he meant this: if one is to consider > er zero > apples (or a collection with zero apples) as an > n empty > set of apples, same for oranges. Then we cannot say > y that > the empty set of apples is the same as the empty set > t of > oranges, as there can be only one empty set (if not, > , one > empty set E contains an element not contained in an > n empty set E' ...) > > > > > > > > > > > > Unfortunately, Huang's sophist argument suffers from the same flawed reasoning by which one proves that 0 = 1. This reply of mine goes back to 24 December 2006, in which Huang put forth the same argument as in this thread: "And that, my friend, is precisely the logical equivalent of saying that all numbers are identical to zero. I am reminded of some dialogue in the movie National Lampoon's Christmas Vacation, between the Chevy Chase character and the little girl, something like: 'How do you feel about Christmas coming?' 'Shittin' bricks!' 'You shouldn't use language like that.' 'I'm sorry. Shittin' rocks.' The counting numbers (natural numbers) would remain the same whether one called them 0_1,0_2,0_3... or 1,2,3... That is why, in the Dedekind-Peano axioms of arithmetic, the terms 'number,' 'zero' and 'successor' are left undefined." Tom
From: Huang on 2 Feb 2010 23:30 On Feb 1, 9:17 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > Huang wrote: > > On Jan 31, 11:24 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote: > >> On Jan 31, 9:48 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > >>> If you consider the empty set by itself it is nonsensical. It needs > >>> set theory to make sense. Nullity is nonsensical when it stands > >>> alone without all of the other apparatus which allows it to make > >>> sense. It is very much like the number zero in many respects. > > >>> And the number zero would be nonsensical if it were considered > >>> alone, without all of the other mathematics which allows it to make > >>> sense. > > >>> Zero apples is the same thing as zero oranges. It is nonsense to say > >>> that apples are oranges, except in such trivial instances as this. > >>> If zero was the only number you had at your disposal, all you would > >>> be able to do is trivial and/or nonsensical procedures. > > >>> Mathematics has trivial numbers, and trivial sets, but no trivial > >>> operators. My position is that the reason that mathematics has no > >>> trivial operator is because the concept of operator is based on very > >>> old and time honored philosophical considerations, and > >>> mathematicians typically abhor philosophy. They prefer to prove > >>> things. > > >>> But can you prove this ? Can you prove that when an arbitrary > >>> quantity is lumped together with another arbitrary quantity that > >>> the result is the fusion of the two quantities into a single > >>> quantity and we should call this addition ? Can that really be > >>> proved ? > > >>> When an arbitrary quantity is lumped together with an arbitrary > >>> number of quantities of equal magnitude to form a single quantity, > >>> that the result is a process called multiplication and shall be > >>> regarded as an operator and signified by the symbol "X" ? Is this > >>> proveable ? > > >>> These things are not proveable, they are definitions. > > >>> And so we cannot prove that there is any such thing as a trivial > >>> operator, we simlpy have to define it. > > >>> The definition is similar to zero, or the empty set, you already > >>> know what it does, that is all. > > >> For logic to work all empty sets are equal. > > >> An empty apple barrel is not the same as an empty oil barrel. But > >> neither is an empty set. Each is a member of the set of barrels.- > >> Hide quoted text - > > >> - Show quoted text - > > > I dont know that you can turn that into a theorem. I dont think thats > > proveable. > > > I would agree that there is such a thing as zero oil, and there is > > such a thing as zero apples. But there is really no way to prove that > > they posses uniqueness. I feel that the number zero may be unique when > > regarded as such, but it's presence implies the property of anti- > > uniqueness. > > > Anti-uniqueness is really a strange thing from a philosophical > > standpoint. It is not surprising to me that you wont find it in any > > math book. But I think that it's very useful in understanding > > singularities or singular behaviour. > > > But lastly, there is a connection here to indeterminacy. The overall > > argument may seem rather juvenile or mathematically immature, but the > > mere fact that you can approach the concept of indeterminacy using a > > rather mechanical series of steps - to me is profound. > > If you don't define your terms then you end up with bullshit. And your > argument that "there is really no way to prove that they possess uniqueness" > demonstrates this. If you define them one way then they are unique, if you > define them another then they are not, if you are playing philosophy instead > of mathematics then you're in the wrong shop.- Hide quoted text - > > - Show quoted text - I would agree that it is bullshit IFF one is trying to frame this as orthodox mathematics. But I dont think that it is neccesarily BS. Mathematics is based on things which are well defined. If we start talking about anti-uniqueness we can get a better understanding of things which are "other than mathematics", but not neccesarily BS. If you insist that the whole approach is BS, I think that you are thinking in terms of existential dichotomy. The only possibilities are [1] that which exists and [2] that which does not, and there is no middle ground. If you allow existential indeterminacy, then you have to consider such things as quasi-defineability. I would argue that this is a valid alternative and that while it is formally not mathematics, it allows the creation of a tool which is consistent with mathematics and produces the same quantitative results.
From: J. Clarke on 2 Feb 2010 23:55 Huang wrote: > On Feb 1, 9:17 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >> Huang wrote: >>> On Jan 31, 11:24 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote: >>>> On Jan 31, 9:48 pm, Huang <huangxienc...(a)yahoo.com> wrote: >> >>>>> If you consider the empty set by itself it is nonsensical. It >>>>> needs set theory to make sense. Nullity is nonsensical when it >>>>> stands alone without all of the other apparatus which allows it >>>>> to make sense. It is very much like the number zero in many >>>>> respects. >> >>>>> And the number zero would be nonsensical if it were considered >>>>> alone, without all of the other mathematics which allows it to >>>>> make sense. >> >>>>> Zero apples is the same thing as zero oranges. It is nonsense to >>>>> say that apples are oranges, except in such trivial instances as >>>>> this. If zero was the only number you had at your disposal, all >>>>> you would be able to do is trivial and/or nonsensical procedures. >> >>>>> Mathematics has trivial numbers, and trivial sets, but no trivial >>>>> operators. My position is that the reason that mathematics has no >>>>> trivial operator is because the concept of operator is based on >>>>> very old and time honored philosophical considerations, and >>>>> mathematicians typically abhor philosophy. They prefer to prove >>>>> things. >> >>>>> But can you prove this ? Can you prove that when an arbitrary >>>>> quantity is lumped together with another arbitrary quantity that >>>>> the result is the fusion of the two quantities into a single >>>>> quantity and we should call this addition ? Can that really be >>>>> proved ? >> >>>>> When an arbitrary quantity is lumped together with an arbitrary >>>>> number of quantities of equal magnitude to form a single quantity, >>>>> that the result is a process called multiplication and shall be >>>>> regarded as an operator and signified by the symbol "X" ? Is this >>>>> proveable ? >> >>>>> These things are not proveable, they are definitions. >> >>>>> And so we cannot prove that there is any such thing as a trivial >>>>> operator, we simlpy have to define it. >> >>>>> The definition is similar to zero, or the empty set, you already >>>>> know what it does, that is all. >> >>>> For logic to work all empty sets are equal. >> >>>> An empty apple barrel is not the same as an empty oil barrel. But >>>> neither is an empty set. Each is a member of the set of barrels.- >>>> Hide quoted text - >> >>>> - Show quoted text - >> >>> I dont know that you can turn that into a theorem. I dont think >>> thats proveable. >> >>> I would agree that there is such a thing as zero oil, and there is >>> such a thing as zero apples. But there is really no way to prove >>> that they posses uniqueness. I feel that the number zero may be >>> unique when regarded as such, but it's presence implies the >>> property of anti- uniqueness. >> >>> Anti-uniqueness is really a strange thing from a philosophical >>> standpoint. It is not surprising to me that you wont find it in any >>> math book. But I think that it's very useful in understanding >>> singularities or singular behaviour. >> >>> But lastly, there is a connection here to indeterminacy. The overall >>> argument may seem rather juvenile or mathematically immature, but >>> the mere fact that you can approach the concept of indeterminacy >>> using a rather mechanical series of steps - to me is profound. >> >> If you don't define your terms then you end up with bullshit. And >> your argument that "there is really no way to prove that they >> possess uniqueness" demonstrates this. If you define them one way >> then they are unique, if you define them another then they are not, >> if you are playing philosophy instead of mathematics then you're in >> the wrong shop.- Hide quoted text - >> >> - Show quoted text - > > > I would agree that it is bullshit IFF one is trying to frame this as > orthodox mathematics. But I dont think that it is neccesarily BS. If it's not mathematics then it's off topic for either newsgroup to which it is posted. > Mathematics is based on things which are well defined. If we start > talking about anti-uniqueness we can get a better understanding of > things which are "other than mathematics", but not neccesarily BS. Why do you insist on discussiong things that are other than mathematics in an inappropriate forum? > If you insist that the whole approach is BS, I think that you are > thinking in terms of existential dichotomy. The only possibilities are > [1] that which exists and [2] that which does not, and there is no > middle ground. I am thinking that your existential dichotomy is neither physics nor mathematics. > If you allow existential indeterminacy, then you have to consider such > things as quasi-defineability. I would argue that this is a valid > alternative and that while it is formally not mathematics, it allows > the creation of a tool which is consistent with mathematics and > produces the same quantitative results. Gotcha. It's not math and you know it's not math but you're going to argue it on a math newsgroup anyway. <plonk>
From: Huang on 3 Feb 2010 00:45 On Feb 2, 10:55 pm, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > Huang wrote: > > On Feb 1, 9:17 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > >> Huang wrote: > >>> On Jan 31, 11:24 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote: > >>>> On Jan 31, 9:48 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > >>>>> If you consider the empty set by itself it is nonsensical. It > >>>>> needs set theory to make sense. Nullity is nonsensical when it > >>>>> stands alone without all of the other apparatus which allows it > >>>>> to make sense. It is very much like the number zero in many > >>>>> respects. > > >>>>> And the number zero would be nonsensical if it were considered > >>>>> alone, without all of the other mathematics which allows it to > >>>>> make sense. > > >>>>> Zero apples is the same thing as zero oranges. It is nonsense to > >>>>> say that apples are oranges, except in such trivial instances as > >>>>> this. If zero was the only number you had at your disposal, all > >>>>> you would be able to do is trivial and/or nonsensical procedures. > > >>>>> Mathematics has trivial numbers, and trivial sets, but no trivial > >>>>> operators. My position is that the reason that mathematics has no > >>>>> trivial operator is because the concept of operator is based on > >>>>> very old and time honored philosophical considerations, and > >>>>> mathematicians typically abhor philosophy. They prefer to prove > >>>>> things. > > >>>>> But can you prove this ? Can you prove that when an arbitrary > >>>>> quantity is lumped together with another arbitrary quantity that > >>>>> the result is the fusion of the two quantities into a single > >>>>> quantity and we should call this addition ? Can that really be > >>>>> proved ? > > >>>>> When an arbitrary quantity is lumped together with an arbitrary > >>>>> number of quantities of equal magnitude to form a single quantity, > >>>>> that the result is a process called multiplication and shall be > >>>>> regarded as an operator and signified by the symbol "X" ? Is this > >>>>> proveable ? > > >>>>> These things are not proveable, they are definitions. > > >>>>> And so we cannot prove that there is any such thing as a trivial > >>>>> operator, we simlpy have to define it. > > >>>>> The definition is similar to zero, or the empty set, you already > >>>>> know what it does, that is all. > > >>>> For logic to work all empty sets are equal. > > >>>> An empty apple barrel is not the same as an empty oil barrel. But > >>>> neither is an empty set. Each is a member of the set of barrels.- > >>>> Hide quoted text - > > >>>> - Show quoted text - > > >>> I dont know that you can turn that into a theorem. I dont think > >>> thats proveable. > > >>> I would agree that there is such a thing as zero oil, and there is > >>> such a thing as zero apples. But there is really no way to prove > >>> that they posses uniqueness. I feel that the number zero may be > >>> unique when regarded as such, but it's presence implies the > >>> property of anti- uniqueness. > > >>> Anti-uniqueness is really a strange thing from a philosophical > >>> standpoint. It is not surprising to me that you wont find it in any > >>> math book. But I think that it's very useful in understanding > >>> singularities or singular behaviour. > > >>> But lastly, there is a connection here to indeterminacy. The overall > >>> argument may seem rather juvenile or mathematically immature, but > >>> the mere fact that you can approach the concept of indeterminacy > >>> using a rather mechanical series of steps - to me is profound. > > >> If you don't define your terms then you end up with bullshit. And > >> your argument that "there is really no way to prove that they > >> possess uniqueness" demonstrates this. If you define them one way > >> then they are unique, if you define them another then they are not, > >> if you are playing philosophy instead of mathematics then you're in > >> the wrong shop.- Hide quoted text - > > >> - Show quoted text - > > > I would agree that it is bullshit IFF one is trying to frame this as > > orthodox mathematics. But I dont think that it is neccesarily BS. > > If it's not mathematics then it's off topic for either newsgroup to which it > is posted. > > > Mathematics is based on things which are well defined. If we start > > talking about anti-uniqueness we can get a better understanding of > > things which are "other than mathematics", but not neccesarily BS. > > Why do you insist on discussiong things that are other than mathematics in > an inappropriate forum? > > > If you insist that the whole approach is BS, I think that you are > > thinking in terms of existential dichotomy. The only possibilities are > > [1] that which exists and [2] that which does not, and there is no > > middle ground. > > I am thinking that your existential dichotomy is neither physics nor > mathematics. > > > If you allow existential indeterminacy, then you have to consider such > > things as quasi-defineability. I would argue that this is a valid > > alternative and that while it is formally not mathematics, it allows > > the creation of a tool which is consistent with mathematics and > > produces the same quantitative results. > > Gotcha. It's not math and you know it's not math but you're going to argue > it on a math newsgroup anyway. > > <plonk>- Hide quoted text - > > - Show quoted text - You can plonk me if you wish, but you cannot dispute the fact that what I am talking about is a tool which is directly consistent with mathematics. I would even say that it is "derivable" from mathematics. And so by virtue of that, it is directly related to mathematics, and relevant to mathematics. I would insist that indeed it is not math, but disagree that it is BS. Far from it. Mathematics and Conjectural modelling may be regarded as being deriveable from each other. If you insist that mathematics is superior to conjecture, then all you have done is to announce that you have a bias in favor or traditional deterministic modelling. My position is that we cannot possibly know whether anything is deterministic or not. Mathematics and Conjecture are "equivalent" in the sense of Einstein, and whether one models using math or conjeture is irrelevant because the quantitative results are identical, the only difference being the philosophical consideration of precisely what is the difference between that which is "given" and that which is "presumed".
From: Huang on 3 Feb 2010 07:44
> > > If you allow existential indeterminacy, then you have to consider such > > > things as quasi-defineability. I would argue that this is a valid > > > alternative and that while it is formally not mathematics, it allows > > > the creation of a tool which is consistent with mathematics and > > > produces the same quantitative results. > > > Gotcha. It's not math and you know it's not math but you're going to argue > > it on a math newsgroup anyway. > > > <plonk>- Hide quoted text - > > > - Show quoted text - > > You can plonk me if you wish, but you cannot dispute the fact that > what I am talking about is a tool which is directly consistent with > mathematics. I would even say that it is "derivable" from mathematics. > > And so by virtue of that, it is directly related to mathematics, and > relevant to mathematics. > > I would insist that indeed it is not math, but disagree that it is BS. > Far from it. > > Mathematics and Conjectural modelling may be regarded as being > deriveable from each other. If you insist that mathematics is superior > to conjecture, then all you have done is to announce that you have a > bias in favor or traditional deterministic modelling. My position is > that we cannot possibly know whether anything is deterministic or not. > Mathematics and Conjecture are "equivalent" in the sense of Einstein, > and whether one models using math or conjeture is irrelevant because > the quantitative results are identical, the only difference being the > philosophical consideration of precisely what is the difference > between that which is "given" and that which is "presumed".- Hide quoted text - > > - Show quoted text - I should add a small correction. I wrote that ".......the only difference being the philosophical consideration of precisely what is the difference between that which is "given" and that which is "presumed".- .........." This could be worded a little better. It is better to say that there is a difference between "that which is a given", and "that which might be a given". And that is heart of the matter. Your position is that you will make models _only_ with "givens", and I would allow modelling of "might be givens". That is the difference. There is no numerical or quantitative difference in the results which would derive from either approach - none. And this fact makes the two approaches completely, totally and seamlessly interchangeable. One direct result of this is that it becomes a formal scientific fact that : "It cannot be known whether the universe is deterministic or not." This is a very strange and useful thing. Consider - some hold the view the G_d is law. Some say G_d is love, but others say G_d is "law". G_d's law may be seen as being manifest in nature by virtue of the fact that nature is modellable with mathematics. A very beautiful idea when you think about it, but without getting too deep into religious views - let me say that if physics is unifiable using math and math alone, then G_d will have revealed himself to man in the form of this system of laws. But G_d is under no obligation to do so. G_d is under no obligation to reveal himself to man. The universe may be modelled as being based exclusively on the "law" of mathematics, or it may be modellable using Conjecture which is completely "lawless". Both approaches work. G_d remains a mystery to man, and the question between agnosticism and ultra orthodoxy will not be resolveable by any logical or science based formalism. Do you see where this is going ? |