I don't see complex quantities as very much different from other
abstractions that people seem to be far more comfortable with.

There's no difficulty. The abstraction simplifies the conception. There
is a real danger -- hard to avoid -- that the abstraction replaces in
one's mind the abstracted reality.

Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.

Yes it is all phase shifts, but just saying that leaves a lot out.

glen herrmannsfeldt wrote:

...

Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.

Yes it is all phase shifts, but just saying that leaves a lot out.

There's no doubt that "real and imaginary" is a good way to describe the
phenomena. We gain understanding with the compact, high-level,
abstraction, but we lose some understanding by taking the mathematical
description to be the actual phenomenon.

Jerry

On 10/30/2009 3:09 PM, Jerry Avins wrote:
glen herrmannsfeldt wrote:

...

Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.

Yes it is all phase shifts, but just saying that leaves a lot out.

There's no doubt that "real and imaginary" is a good way to describe the
phenomena. We gain understanding with the compact, high-level,
abstraction, but we lose some understanding by taking the mathematical
description to be the actual phenomenon.

Jerry

Is the level of abstraction the difficulty? Numbers are a means to
abstract quantities, fractions to abstract ratios, real numbers to
further abstract quantities, and algebra is a means to abstract numbers.
Is it really so much different to abstract complex values?

I don't see complex quantities as very much different from other
abstractions that people seem to be far more comfortable with.

glen herrmannsfeldt wrote:

...

Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.

Yes it is all phase shifts, but just saying that leaves a lot out.

There's no doubt that "real and imaginary" is a good way to describe the
phenomena. We gain understanding with the compact, high-level,
abstraction, but we lose some understanding by taking the mathematical
description to be the actual phenomenon.

Jerry Avins <jya at (no spam) ieee.org> wrote:
(snip, someone wrote)

I don't see complex quantities as very much different from other
abstractions that people seem to be far more comfortable with.

There's no difficulty. The abstraction simplifies the conception. There
is a real danger -- hard to avoid -- that the abstraction replaces in
one's mind the abstracted reality.

Another place where complex numbers seem more than just an abstraction
is the evanescent wave:

http://en.wikipedia.org/wiki/Evanescent_wave

Jerry Avins wrote:
glen herrmannsfeldt wrote:

...

Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.

Yes it is all phase shifts, but just saying that leaves a lot out.

There's no doubt that "real and imaginary" is a good way to describe
the phenomena. We gain understanding with the compact, high-level,
abstraction, but we lose some understanding by taking the mathematical
description to be the actual phenomenon.


Isn't this kind of stating the obvious? Come to it, even "real numbers"
represented in a computer are really combinations of integers under the
hood (and below that, merely aggregations of binary states), so they are
no more real than imaginary numbers. All numbers are ultimately
mythological; the rest is simply counting, and very few things in the
physical world can be measured ~exactly~. Sometimes that matters - the
butterfly effect!

how about net force. pick any single dimension (so the vectors are
scalers) and for a body with two equal and opposite forces applied, it
does not accelerate and the net force is zero. picking which
direction is a convention, but some positive force got something added
to it, and the result is zero. what would you call that other force?
non-real?

might be a convention and the aliens on the planet Zog might have been
smart enough to adopt a convention that those muck less massive little
things "flying about" the nucleus are the positive ones, making what
we would call protons negatively charged. at least they would have to
learn that the electron flow is in the opposite direction of the
current they solved for with Kirchoff's (or Zog's) laws. but either
way, there are physical quantities that can only be represented as
quantitatively negative.

[...]
Complex numbers are a clever and useful way to represent those
quantities in cases where phase shift has meaning.

Compact notations expand out ability to comprehend (same root as in
prehensile) complicated things. Vector analysis, quaternions, complex
numbers, matrix algebra .. without them, we'd be hard pressed to
express, let alone understand, some phenomena as
concepts. Nevertheless, being built up from simpler stuff (Shall we go
back to Peano's axioms?) thay constitute the HLLs of math.

Jerry Avins <jya at (no spam) ieee.org> wrote:
(snip, I wrote)

Another place where complex numbers seem more than just an abstraction
is the evanescent wave:

http://en.wikipedia.org/wiki/Evanescent_wave

I think you confound the explanation for the explained phenomenon.

Maybe.

I completely agree that for problems where complex numbers are
used to represent phase, for example voltages and currents, that
they are just to simplify the math, and that voltages and currents
are real. I is slightly less obvious in the case of lumped
impedances, where one can still separate out the real (R) from
the imaginary (L, C). One can write the solution to the differential
equation as a complex exponential or a decaying exponential multiplied
by a sinusoid.

With less ideal L, R, C, and even more in the case of spatially
distributed L, R, C, and then to the case of an EM wave going
through a lossy dielectric it is much less obvious. The EM wave
is interacting with the electrons according to some differential
equations. The solutions to those equations have complex exponential
in them related to the exp(iwt) term in the EM wave. They still
do even if the EM wave is cos(wt).

Even more, consider a material whose properties change continuously
in space. First, the analytical solution gets much harder, but
then you have to switch over somewhere from the cos(kx) solution
to the exp(-ax) solution. Even more, the solution can be different
in different directions. In the evanescent wave case, it is a decaying
exponential along one axis, but not the other axis.

Calling a voltage or current complex is somewhat strange.
Calling an impedance, dielectric constant, or index of refraction
complex much less strange. For one, voltages and currents and
electric fields are measured (more or less) directly.
Impedances and dielectric constants by their effect on currents,
voltages, and electric fields. Voltages and currents have phase,
but impedances don't.

Jerry Avins <jya at (no spam) ieee.org> writes:
[...]
Complex numbers are a clever and useful way to represent those
quantities in cases where phase shift has meaning.

Compact notations expand out ability to comprehend (same root as in
prehensile) complicated things. Vector analysis, quaternions, complex
numbers, matrix algebra .. without them, we'd be hard pressed to
express, let alone understand, some phenomena as
concepts. Nevertheless, being built up from simpler stuff (Shall we go
back to Peano's axioms?) thay constitute the HLLs of math.

Jerry,

First of all, Peano's axioms have nothing to do with complex, or even
real, numbers. They are another way to get, eventually, to the ring of
integers (other than the standard group-based development) and integral
domains.

And once again I must point out that your intimations are, at a minimum,
at odds with the field of mathematics (namely, abstract algebra) that
has proven extremely useful (re: the realization that no formulas exist
for roots of equations greater than order 4). Under this system of
mathematics, the complex numbers are NOT just a notational
convenience. Rather, they are a field without which the solution of
arbitrary algebraic equations over the reals cannot be determined.

You have argued many times that they are just a "notational
convenience." I posit that the extremely rich utility in viewing
algebraic systems through the lens of two operations (addition and
multiplication), i.e., as in the basic ring definition, makes this
assertion patently false.

[...]
I wouldn't argue even about promoting "notational convenience" to
"notational necessity". I argue that the notation and the phenomenon
described by it are not the same thing. As Korzybski wrote, "The map
is not the territory it represents". Granted, he went on, "but if
correct, it has a similar structure to the territory, which accounts
for its usefulness". Still, I think it is important to distinguish
between "similar" and "same".

On Oct 30, 7:29 pm, Richard Dobson<richarddob... at (no spam) blueyonder.co.uk
wrote:
Jerry Avins wrote:
glen herrmannsfeldt wrote:
...
Which in the end describes the interdependence between the real and
imaginary parts of the dielectric constant and/or index of refraction.
Yes it is all phase shifts, but just saying that leaves a lot out.
There's no doubt that "real and imaginary" is a good way to describe the
phenomena. We gain understanding with the compact, high-level,
abstraction, but we lose some understanding by taking the mathematical
description to be the actual phenomenon.
Isn't this kind of stating the obvious? Come to it, even "real numbers"
represented in a computer are really combinations of integers under the
hood (and below that, merely aggregations of binary states),

exactly! we are getting closer to the physical issue. it's not so
much that numbers are being manipulated in the computer, but it's
binary states according to some combinatorial logic that is
implemented with MOSFETs acting as switches which is a consequence of
physical interactions happening inside of these devices.

so they are
no more real than imaginary numbers. All numbers are ultimately
mythological;

oh, i dunno if i should do this at 2 in the morning.

okay, numbers themselves are a human construct, but i am convinced
that another intelligent life being (if SETI ever stumbles upon one)
that is sophisticated enough to transmit messages that we happen upon
will also have the concept of numbers, and will, in fact discover and
formalize the same objective facts as derivable theorems (in whatever
symbology/convention they used) that we do (maybe they would prove
Fermat's Last Theorem "before" we had). so numbers have a broader
existence than just a human construct. indeed some birds and chimps
think in terms of small integers.

but, the issue of reality is this: whether or not "numbers", real or
imaginary, exist as a real thing or just in the imagination of a
person or computer is the phenomena we call "physical quantity".
those things are real. physical quantity is a real thing that really
exists whether there were intelligent beings aware of it (and
quantifying it with these made-up numbers).

now the http://nrich.maths.org/5961 link posted by Rich was
interesting in comparing to the concept of negative numbers, whether
they be truly "real" or not. when they first started popping up was
to represent debt as a "negative fortune" so that if it added to a
fortune of equal size it adds to zero - the two parties don't owe each
other a thing. but, it is argued, that money and debt are human
social constructs. material itself (and the value of a service) which
are traded between peoples still are non-negatively valued.

are *differences* between commensurable physical quantities,
themselves physical quantities? then potential energy is a physical
quantity that can have negative value, so then negative numbers are
real. but maybe the "difference" between two physical quantities is
just a human construct, and not directly existing.

how about net force. pick any single dimension (so the vectors are
scalers) and for a body with two equal and opposite forces applied, it
does not accelerate and the net force is zero. picking which
direction is a convention, but some positive force got something added
to it, and the result is zero. what would you call that other force?
non-real?

now, what we believe to be true about many various kinds of quantities
or real stuff in physical reality is that when objects or particles or
"things" are collected or aggregated, the quantitative physical
property that they have in common adds according to the model or
meaning of addition we have in arithmetic. so, whether or not certain
kinds of numbers are "real" in that there are real things out there
that have those numbers as properties, what *is* real in physical
reality are physical quantities; mass, charge, energy, and physical
interactions (fields, forces, probabilities of existence, etc.). and
some of these real things add in their effect on other real things.

finally, if that does not persuade someone that negative quantities do
not exist for real in physical reality, then consider charged
particles. whether they be electrons and positrons (-1 and +1) or the
quarks (-1/3 and 2/3) and some of these things combine and>>you get
no charge<<. so it looks like that not only negative integers are
real things, so are positive and negative rational numbers real. at
least they describe or model physical quantities in reality
accurately. now picking which is "negative" and which is "positive"
might be a convention and the aliens on the planet Zog might have been
smart enough to adopt a convention that those muck less massive little
things "flying about" the nucleus are the positive ones, making what
we would call protons negatively charged. at least they would have to
learn that the electron flow is in the opposite direction of the
current they solved for with Kirchoff's (or Zog's) laws. but either
way, there are physical quantities that can only be represented as
quantitatively negative.

negative numbers are real.

now do the same for complex or imaginary numbers. every fundamental
particle property or quantitative interaction you have (including QM,
because the rubber does not yet hit the road until you compute
expectation values) is real. every quantitative law of interaction
that you will find in the physics book can be expressed as a more
complicated set of equations involving only real quantities. you will
never put a pair of leads on a voltage at a particular instance of
time and have the meter tell you that it is "3+4i volts". or that
3+4i electrons passed a defined boundary in a given time period.

the rest is simply counting, and very few things in the
physical world can be measured ~exactly~.

only things that exist only as integer quantities. nearly everything
(but not everything, masses of particles is an example) can be boiled
down to integer quantities. but this still says nothing about if
physical quantities that really exist, exist in imaginary quantities.

so far, *every* argument that has insisted some do have been
sophistic. Jerry has recognized that.

r b-j

Randy Yates wrote:
Jerry Avins <jya at (no spam) ieee.org> writes:
[...]
Complex numbers are a clever and useful way to represent those
quantities in cases where phase shift has meaning.

Compact notations expand out ability to comprehend (same root as in
prehensile) complicated things. Vector analysis, quaternions, complex
numbers, matrix algebra .. without them, we'd be hard pressed to
express, let alone understand, some phenomena as
concepts. Nevertheless, being built up from simpler stuff (Shall we go
back to Peano's axioms?) thay constitute the HLLs of math.

Jerry,

First of all, Peano's axioms have nothing to do with complex, or even
real, numbers. They are another way to get, eventually, to the ring of
integers (other than the standard group-based development) and integral
domains.

And once again I must point out that your intimations are, at a minimum,
at odds with the field of mathematics (namely, abstract algebra) that
has proven extremely useful (re: the realization that no formulas exist
for roots of equations greater than order 4). Under this system of
mathematics, the complex numbers are NOT just a notational
convenience. Rather, they are a field without which the solution of
arbitrary algebraic equations over the reals cannot be determined.

You have argued many times that they are just a "notational
convenience." I posit that the extremely rich utility in viewing
algebraic systems through the lens of two operations (addition and
multiplication), i.e., as in the basic ring definition, makes this
assertion patently false.

I wouldn't argue even about promoting "notational convenience" to
"notational necessity". I argue that the notation and the phenomenon
described by it are not the same thing. As Korzybski wrote, "The map is
not the territory it represents". Granted, he went on, "but if correct,
it has a similar structure to the territory, which accounts for its
usefulness". Still, I think it is important to distinguish between
"similar" and "same".

Jerry

Jerry Avins <jya at (no spam) ieee.org> writes:
[...]
I wouldn't argue even about promoting "notational convenience" to
"notational necessity". I argue that the notation and the phenomenon
described by it are not the same thing. As Korzybski wrote, "The map
is not the territory it represents". Granted, he went on, "but if
correct, it has a similar structure to the territory, which accounts
for its usefulness". Still, I think it is important to distinguish
between "similar" and "same".

Jerry,

This is not a matter of mistaking the map for reality, but rather of
arguing whether or not the map accurately portrays reality.

Since neither one of us knows with absolute certainty and complete
characterization what the "phenomenom" (i.e., some aspect of reality)
is, neither one of us can argue that it "is" or "is not" correctly
portrayed by a certain "map."

However, the longer, more extensively, and more repeatedly a "map"
demonstrates itself by experience and experiment to accurately
represent, and even to predict, that phenomenom - that aspect of
reality - the less likely it is that the map is wrong.

I believe that is the case with complex numbers. Arguing against their
existence is, in my opinion, similar to arguing against the statement
that you have four grandchildren (e.g.) on the basis that the concept of
"4" is a map and not necessarily representative of reality.