What makes a trivial operator trivial ?
in [Math]
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From: Huang on 31 Jan 2010 08:48 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.
From: Androcles on 31 Jan 2010 09:43 "Huang" <huangxienchen(a)yahoo.com> wrote in message news:f0df5ca3-4dbb-4c7c-91e9-e1f1adbc4b99(a)a32g2000yqm.googlegroups.com... > 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. False. The word "Philosophy" literally means "love of wisdom". Mathematicians typically adore philosophy. > They prefer to prove things. It would not be wise to accept opinion without proof, no self- respecting philosopher would do so. I reject your opinion (or "position", as you call it) and ask you to prove "mathematicians typically abhor philosophy" or retract your claim. > 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. Minus (-) is a unary operator, the binary operator is +. Trivially, 3-1 = 3 + (-1). A computer, whether human or artificial, will evaluate a+b even when a and/or b are less than zero. Prove or disprove that i (symbol for sqrt(-1)) is a unary operator.
From: Aage Andersen on 31 Jan 2010 10:38 "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. Aage
From: Huang on 31 Jan 2010 10:44 On Jan 31, 9:38 am, "Aage Andersen" <aaan(REMOVE)@email.dk> wrote: > "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. > > Aage They may be regarded as being identical, or non-identical. I think it gets into the area of proveability, but also seems intrinsically indeterminate. So indeterminacy seems to be related to these extremely simplistic things.
From: Aage Andersen on 31 Jan 2010 11:18
"Huang" "Aage Andersen" > "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. They may be regarded as being identical, or non-identical. I think it gets into the area of proveability, but also seems intrinsically indeterminate. So indeterminacy seems to be related to these extremely simplistic things. -------------------------------------------- -2 oranges /= -2 apples -1 oranges /= -1 apples 0 oranges ?= 0 apples 1 oranges /= 1 apples 2 oranges /= 2 apples in general n oranges /= n apples, if n /= 0 Why this discontinuity for n = 0? I prefer 0 oranges /= 0 apples. Perhaps we could DEFINE: 0 oranges /= 0 apples ? Aage |