What makes a trivial operator trivial ?
in [Math]
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From: Frisbieinstein on 1 Feb 2010 00:24 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.
From: Androcles on 1 Feb 2010 03:22 "Frisbieinstein" <patmpowers(a)gmail.com> wrote in message news:6359a960-c4fa-4c0e-a759-b9d72c31ec17(a)u19g2000prh.googlegroups.com... 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. ========================================== An empty oil barrel is NOT a dog's breakfast. An empty oil barrel is NOT the 8:30 to London on platform 2. An empty oil barrel is NOT an apple. One can construct an infinite list of what things are not and the list contains (useful information) NOT. Tell us what an empty oil barrel IS and how to distinguish it from an empty apple barrel. An empty apple barrel IS the same as an empty oil barrel until you write "Apples" on the staves to make it different.
From: Huang on 1 Feb 2010 08:24 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.
From: Dan Cass on 31 Jan 2010 22:37 > > "Frisbieinstein" <patmpowers(a)gmail.com> wrote in > message > news:6359a960-c4fa-4c0e-a759-b9d72c31ec17(a)u19g2000prh. > googlegroups.com... > 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. > > ========================================== > An empty oil barrel is NOT a dog's breakfast. > An empty oil barrel is NOT the 8:30 to London on > platform 2. > An empty oil barrel is NOT an apple. > One can construct an infinite list of what things are > not and the list > contains (useful information) NOT. > Tell us what an empty oil barrel IS and how to > distinguish it from > an empty apple barrel. > > An empty apple barrel IS the same as an empty oil > barrel until > you write "Apples" on the staves to make it > different. > > > > But an empty oil barrel smells worse than an empty apple barrel.
From: J. Clarke on 1 Feb 2010 10:17
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. |