The Identity Crisis of i: Is i truly unique compared to -i?

[arildno]

other than that they are quantitatively opposite,

That is why they are not equivalent, qualitatively or something else.

there is no difference between i and -i. they both have equal claim to squaring to be -1. and there is little else you can say about their properties

So?
Existence of some sort of sameness does not imply equivalence or identity.

 

[jambaugh]

That is why they are not equivalent, qualitatively or something else.

So?
Existence of some sort of sameness does not imply equivalence or identity.

"Equivalence" and identity are not the same thing. Equivalence is "the existence of some sort of sameness" i.e. the existence of an isomorphism. Hence we speak of equivalence relations (between different objects) and equivalence classes. It has a formal meaning in mathematics and that meaning does indeed apply to the conjugation isomorphism in the complex division algebra.

But this argument is silly ****Because**** you and the others have not established within which context you are referring to i and complex numbers. Until you carefully establish such a context questions of "what ____ really means" or "really is defined to be" are subject to reinterpretation and you can argue in circles, never getting anywhere.

 

[arildno]

But this argument is silly ****Because**** you and the others have not established within which context you are referring to i and complex numbers.

I did that in my first post.

 

[disregardthat]

"Equivalence" and identity are not the same thing. Equivalence is "the existence of some sort of sameness"

Isn't "sameness" a bit ambiguous? I prefer the following formal definitions of equivalence and equality: Equivalence is used when objects are equivalent with respect to some defined equivalence relation, may it be an isomorphism, modulo class, etc. Equality however in set theory is set equality, and thus equality is an example of an equivalence. E.g. two numbers are equal because equality refers to some construction of the numbers in set theory. (1=1 because {{}}={{}} )

 

[arildno]

Equivalence is "the existence of some sort of sameness" i.e. the existence of an isomorphism. Hence we speak of equivalence relations (between different objects) and equivalence classes.

Yawn.

Take a field A , like the complex numbers and another one B, for simplicity its mirror image, so that a mapping exists between a+ib in A to a-ib in B.

Here, "i" in A is equivalent to "-i" in B, but "i" and "-i" are not "qualitatively equivalent" within anyone of the systems.

It is precisely because in BOTH fields A and B that elements within each of them retain their uniqueness from other elements that there exists a bijection from A on B or vice versa.

 

[rbj]

"Equivalence" and identity are not the same thing. Equivalence is "the existence of some sort of sameness" i.e. the existence of an isomorphism.

Yawn.

ya know, you can sort of arrogantly dismiss jambaugh, but that posturing does you no good. he is correct (almost, i might have a semantic bone to pick) and you are wrong.

you may hide behind your Science Adviser badge, if you want (it's no big deal, many of us have been there), but you're still wrong.

Take a field A , like the complex numbers and another one B, for simplicity its mirror image, so that a mapping exists between a+ib in A to a-ib in B.

Here, "i" in A is equivalent to "-i" in B, but "i" and "-i" are not "qualitatively equivalent" within anyone of the systems.

yes, they are. they are qualitatively equivalent. they have equivalent quality. there is no quality in which they differ. and, since they are not the same number, they are not quantitatively equivalent. seems like you could make use of a dictionary.

It is precisely because in BOTH fields A and B that elements within each of them retain their uniqueness from other elements that there exists a bijection from A on B or vice versa.

sure, and it's a simple 1-to-1 mapping. big deal.

doesn't say anything about the qualitative properties of +i and -i. you name a single property (using words) that +i has that -i does not also have (or vise-versa). conventions like "clockwise" or "left" do not count. except for biological accidents of nature (or some QM issue regarding parity violation and the weak interaction) "left" has the same quality as "right", although they point in opposite directions. their only difference is one of convention.

name one qualitative property that differs between +i and -i, smarty pants.

 

[disregardthat]

doesn't say anything about the qualitative properties of +i and -i. you name a single property (using words) that +i has that -i does not also have (or vise-versa). conventions like "clockwise" or "left" do not count. except for biological accidents of nature (or some QM issue regarding parity violation and the weak interaction) "left" has the same quality as "right", although they point in opposite directions. their only difference is one of convention.

name one qualitative property that differs between +i and -i, smarty pants.

You name a bunch of this which are allowed and a bunch of things which are disallowed, the problem here is that the "sameness" between i and -i is ill-defined. You must in mathematical terms define what you mean by "qualitative property", and what it means that i and -i share it.

 

[Hurkyl]

Sigh, I was hoping you would all work this out amongst yourselves.

You must in mathematical terms define what you mean by "qualitative property", and what it means that i and -i share it.

I second this. I hereby forbid any further discussion in this thread trying to make assertions about ill-defined terminology.

 

[rbj]

i'm still waiting for a single property (described with words and without arbitrary convention) that +i has and -i does not.

off hand i can name two properties that they have in common:

1) they are both outside the set of real numbers.
2) squaring either gets you -1.

i want to hear someone identify (with words) that "+i has this [such-and-such] mathematical property (as opposed to a notational property) and -i does not."

ill-defined, Hurk?

 

[Hurkyl]

i is the only solution for x in the equation x=i.

 

[SkyWatcher]

-i is the only solution for x in the equation x=-i.

Please note:
In mathematics, you don't have to initially understand anything, you first just "get used to it."

 

[disregardthat]

i want to hear someone identify (with words) that "+i has this [such-and-such] mathematical property (as opposed to a notational property) and -i does not."

A notational property is a mathematical property. I'd even argue they are equivalent.

Anyway, 1 = 1 is surely a notational and a mathematical property of 1.

The whole "equivalence" between i and -i can be summed up in the trivial assertion that R[-i] and R are isomorphic as fields. But this is all relative; sqrt(2) and -sqrt(2) are similarly "equivalent" in, say, Q[sqrt(2)]. I can argue all day long that there is no "qualitative" difference between sqrt(2) and -sqrt(2), but as you can see it is entirely meaningless.

 

[Hurkyl]

i want to hear someone identify (with words) that "+i has this [such-and-such] mathematical property (as opposed to a notational property) and -i does not."

That will have to wait until you explain what you mean by "mathematical property" and by "notational property".

 

[arildno]

A notational property is a mathematical property. I'd even argue they are equivalent.

Anyway, 1 = 1 is surely a notational and a mathematical property of 1.

The whole "equivalence" between i and -i can be summed up in the trivial assertion that R[-i] and R are isomorphic as fields. But this is all relative; sqrt(2) and -sqrt(2) are similarly "equivalent" in, say, Q[sqrt(2)]. I can argue all day long that there is no "qualitative" difference between sqrt(2) and -sqrt(2), but as you can see it is entirely meaningless.

Correct.

If this was the only basis for the asserted concept of "qualitatively equivalent", then all numbers are qualitatively equivalent to each other, notationally, mathematically, and decidedly propertarially so.

And therefore, the whole concept can be dismissed as..meaningless.

The defenders of the concept have yet to bring some solid mental shape to this amorphous beast.

 

[Char. Limit]

i'm still waiting for a single property (described with words and without arbitrary convention) that +i has and -i does not.

Here's one:

\sqrt{i} = \frac{1+i}{\sqrt{2}}

\sqrt{-i} = \frac{1-i}{\sqrt{2}}

\sqrt{i} - \sqrt{-i} = \sqrt{2}i

That last fact wouldn't be true if i and -i were no different.

 

[rbj]

Here's one:

\sqrt{i} = \frac{1+i}{\sqrt{2}}

\sqrt{-i} = \frac{1-i}{\sqrt{2}}

\sqrt{i} - \sqrt{-i} = \sqrt{2}i

That last fact wouldn't be true if i and -i were no different.

i am taking \sqrt{2} to be 0 < \sqrt{2}

this is true for +i :

\frac{1+i}{\sqrt{2}} - \frac{1-i}{\sqrt{2}} = \sqrt{2}(i)

are you saying that if you were to swap every occurrence of +i with -i and vise versa that this isn't true?

\frac{1-i}{\sqrt{2}} - \frac{1+i}{\sqrt{2}} = \sqrt{2}(-i)

i don't get your point.

 

[jambaugh]

Yawn.

Take a field A , like the complex numbers and another one B, for simplicity its mirror image, so that a mapping exists between a+ib in A to a-ib in B.

Here, "i" in A is equivalent to "-i" in B, but "i" and "-i" are not "qualitatively equivalent" within anyone of the systems.

It is precisely because in BOTH fields A and B that elements within each of them retain their uniqueness from other elements that there exists a bijection from A on B or vice versa.

The conjugation involution is an automorphism of the complex number field. (That's a mapping from A to A if you like).

Re: the other post. You did not "establish a context" in your first post. You mentioned Hamiltonian formalisms and proceeded with one of the many equivalent constructions of the complex number field. There are others and the Hamiltonian formulation doesn't care which. By context I mean the epistemological context by which you can determine if a claim (made in this thread) is true or false.

For example in a physical context there ain't no "i" as such. QM is a real theory and you can carry out the whole of it without ever mentioning complex numbers. They are just a handy shortcut through many a tedious definition and calculation. Classical physics likewise. They provide insufficient context to answer these questions.

Now within mathematics one can (again I mention this) consider i in the context of a group, a vector (e.g. an ordered pair), a field, a division ring, a clifford algebra,... in all those contexts i and -i are equivalent (in the mathematical sense of the existence of an automorphic mapping between them, namely *). Of course they are not equal as that would imply i = 0 and I think it is clear no one here things they are the same identical object (in whatever context).

[To others invoking radical notation]
People write \sqrt{-1} as if it settles the matter but the notation has built into it a convention. The "principle square root" is one of the square roots picked by convention to be "the principle one". One may change the convention so what we formerly though of as i is identical with -\sqrt{-1}.

 

[jambaugh]

i'm still waiting for a single property (described with words and without arbitrary convention) that +i has and -i does not.

And you will wait until doomsday as they are isomorphic under * so there is none. That fact settles this particular matter entirely.