Measuring Complex Numbers: Is it Possible?

[Hurkyl]

and the arguments you have made still appear to be sophistry.

Out of curiousity, just what do you think is the point I'm trying to convey?

 

[rbj]

Out of curiousity, just what do you think is the point I'm trying to convey?

i've been trying to crap out on this thread, but you sucked me back in. the salient point I've been thinking you're trying to convey is:

Real numbers aren't any more real than imaginary numbers.

to which i disagree.

i know that the formal discussion of what is truly real and what is not is "ontology" (venesch's usage, not yours), but as soon as i hear (or read) that word in the context of physical science or engineering or similar, i usually run away. i am assuming we're getting past some of these deeper philosophical issues like Descartes or whether or not I'm in the Truman Show or am just a brain in some mad scientiest's laboratory and he (or the purple blob or the "god") is just applying stimulus and i am (more accurately, my disembodied brain is) reacting to this stimulus.

if we get well past that, i really think, and have defended, that fundamental physical quantities that we perceive or measure (which is just an extension of precision of what we perceive) are real numbers. (our measurements are rational, too, but i am not saying that the actual physical quanities the measurements are meant to measure are, themselves, rational.) additionally, if there is some hypothetical and "hidden" physical quantity, like a deBroglie wave, that is, in our physical equations, a complex quantity, the actual effect of such hypothetical quantity actually manifests itself as real (probability and expectation values) and such equations can be restated, albeit less elegantly, as pairs or sets of equations of real variables. I'm not recommending that we do physics that way, we should use the elegant and concise and well-established quantitative laws having complex variables, but when you apply such to a real physical context (an experiment), it is still real quantities that you are looking for. (the complex results you might ultimately view are constructed from real measurements of real numbered quantities.)

real numbers have properties that do make them more "real", or congruent to reality (without getting too philosophical about what we mean by "reality"), than what they've been calling "imaginary numbers".

 

[Crosson]

the actual physical quanities the measurements are meant to measure

Do you think that these quantities are tabulated in some book? In what sense do these "actual" quantities exist, other then in our imagination?

 

[vanesch]

i know that the formal discussion of what is truly real and what is not is "ontology" (venesch's usage, not yours), but as soon as i hear (or read) that word in the context of physical science or engineering or similar, i usually run away. i am assuming we're getting past some of these deeper philosophical issues like Descartes or whether or not I'm in the Truman Show or am just a brain in some mad scientiest's laboratory and he (or the purple blob or the "god") is just applying stimulus and i am (more accurately, my disembodied brain is) reacting to this stimulus.

Ok, if we take on this hypothesis, then what we take for ontologically real must be the formal elements of our theory that are supposed to be charged with a physical meaning, right ?

if we get well past that, i really think, and have defended, that fundamental physical quantities that we perceive or measure (which is just an extension of precision of what we perceive) are real numbers. (our measurements are rational, too, but i am not saying that the actual physical quanities the measurements are meant to measure are, themselves, rational.)

You are jumping back and fro again. What do you consider real now ? Only *observable* things (that is, observations), or "the formal thing these observations try to approach" ?
Because in the first case, we are NOT in the paradigm you previously seemed to accept (namely that there is a genuine real world out there that is more or less correctly described by our theories), and you only accept "observations" without any "background to them". In that case, we are - as you accept - not ever measuring genuine real numbers, but at best are counting.

However, if you accept the "reality of the formal entity behind the measurement", then you have to accept, well, the formal entity as it is, in its most observer-independent notion as it is present in the theory. In other words, you cannot accept the coordinate system, and reject the manifold !

additionally, if there is some hypothetical and "hidden" physical quantity, like a deBroglie wave, that is, in our physical equations, a complex quantity, the actual effect of such hypothetical quantity actually manifests itself as real (probability and expectation values) and such equations can be restated, albeit less elegantly, as pairs or sets of equations of real variables. I'm not recommending that we do physics that way, we should use the elegant and concise and well-established quantitative laws having complex variables, but when you apply such to a real physical context (an experiment), it is still real quantities that you are looking for. (the complex results you might ultimately view are constructed from real measurements of real numbered quantities.)

I think you've just DEFINED that what you consider as real, are formal elements that are represented by real numbers. That definition doesn't hold any water in the case that the natural formal structure is not the real numbers system, such as a vector or a complex quantity.

real numbers have properties that do make them more "real", or congruent to reality (without getting too philosophical about what we mean by "reality"), than what they've been calling "imaginary numbers".

You are just *using* real numbers as your DEFINITION of what you take to be "ontological reality", thereby removing from the latter term its philosophical meaning. In that case, Hurkyl is as entitled as you to also redefine it, and take imaginary numbers.

Redefining a term is not the same as giving an argument of why the original definition of the word should correspond to the concept of your definition !

 

[Hurkyl]

Hrm. You're almost right, rbj. I was worried you were much further off -- thinking that I was arguing the reals are not real, or that the complexes are real.

There's an additional nuance: I'm trying to argue that our scientific theories do not provide evidence the reals are more real than the complexes.


I make the following assertion:

there exists a mathematical theory satisfying:
(1) Interpreted as a physical theory, it is empirically indistinguishable from Newtonian mechanics.
(2) All distances are imaginary.


Question 1: If my assertion happens to be correct, do you agree that it demonstrates that the use of real numbers to quantify things in physical theories is merely a convention?

Question 2: Do you believe my assertion?

 

[rbj]

Do you think that these quantities are tabulated in some book? In what sense do these "actual" quantities exist, other then in our imagination?

no, it's the actual amount of force that causes something to move, or the actual amount of time (in someone's frame of reference) that elapsed while this something moved from some given position to another.

i think we're going to start going around the maypole again here.

 

[rbj]

venesch, i don't know how to begin to respond to your last thing. i need to read/parse/understand it more.

Hrm. You're almost right, rbj. I was worried you were much further off -- thinking that I was arguing the reals are not real, or that the complexes are real.

There's an additional nuance: I'm trying to argue that our scientific theories do not provide evidence the reals are more real than the complexes.

i might agree with that. alls I'm saying is that we don't measure anything fundamental as real (and rational) and the only reason the additional qualifier "rational" is put in there is because of the nature of finite precision of our measurement on a quantity that might have an irrational quantitative value. we will never know, from the POV of measurement, but sometimes can expect, from a theoretical description, that a fundmental quantity somewhere (a length or something) would have a quantity that is actually irrational, but we measure a rational estimate of it.

I make the following assertion:

there exists a mathematical theory satisfying:
(1) Interpreted as a physical theory, it is empirically indistinguishable from Newtonian mechanics.
(2) All distances are imaginary.Question 1: If my assertion happens to be correct, do you agree that it demonstrates that the use of real numbers to quantify things in physical theories is merely a convention?

Question 2: Do you believe my assertion?

A1: i have to wait and see.

A2: i have to wait and see.

i have my doubts. don't know how you deal with the concept of pressure when your measure of area are always negative numbers.