Measuring Complex Numbers: Is it Possible?

[Swapnil]

Yeah yeah, complex/imaginary numbers are as *real* as real numbers. Just like irrational numbers are every bit *rational* as rational numbers.

OK, so complex numbers are useful--very useful--but I think that a lot of people, including myself, still don't take them as a part of physical reality; they see them only as a mathematical abstraction. I mean, can you really make measurements in complex numbers? An ammeter can show that you have -1.333 amps flowing through your circuit, but it will never show that you have 8.2 - j3.2 amps flowing through your circuit.

Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

I'd love to hear you guy's opinion on this and bring your perspectives on the table.

 

[Gokul43201]

Ever used a digital lock-in amplifier? It displays real and imaginary parts of any signal with respect to a reference signal.

 

[neutrino]

Y
Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

Even if such a machine existed, and as Gokul has pointed out that it does, the output is only going to be a pair of real numbers, because every complex number is an ordered pair of real numbers.

 

[cesiumfrog]

Worse, the output is a pair of rational numbers..

 

[Hurkyl]

Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

Most measuring devices do display quantities in complex numbers. It's just that they usually restrict themselves to the purely real subset of the complexes.

 

[rbj]

an instrument that compared two sinusoids of the same frequency could display the "gain" of one sinusoid over the other as a complex number. but it's still an abstraction. i know others have objected to it, but i think that "imaginary" is an appropriate term for any number that, when squared, becomes negative. those numbers ain't "real", but we can imagine them. sorta.

 

[Hurkyl]

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

 

[rbj]

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

that is a philosophical statement, not a scientific one, that i, as well as a lot of knowledgeable people (some with PhDs) disagree with. and, even though I'm not a physicist (electrical engineer who does signal processing), i think i can hold my own with this, Hurk. you have all sorts of issues to get through, like defining/agreeing on what is real. I'm equating the concepts of "real numbers" to "quantities of physical stuff" and there is no case where some quantity of physical stuff (not an abstraction like \Psi(x,y,z) ) is measured as an imaginary or complex number. fundamentally, when we measure or perceive the amount of something, it a real number. because of our limited precision, we also measure it as a rational number, but we know that, when limited precision is not an issue, that there are ideal ratios of quantity that are irrational numbers.

 

[Hurkyl]

that is a philosophical statement, not a scientific one

Sure. And it was made in reply to a philosophical statement.

Hurk. you have all sorts of issues to get through, like defining/agreeing on what is real.

Not necessarily -- the crucial part of my position operates on the formal level: that there is no scientific reason to give the real numbers a priviliged status. I think the crux of my position is probably best expressed via a parody of your argument:

I'm equating the concept of "3-dimensional real vector space" to "relative displacements between physical locations", and there is no case where a relative displacement is measured as an real number, or even a complex number. Fundamentally, when we measure or perceive a displacement between things, it is a vector.​

What you presented is the usual argument that the reals are somehow special -- but all it does is pass the buck: it makes no attempt to explain why "quantity of physical stuff" is more real than things like "position", "angle", "constellation point", "quantum state", or "gravitational field".

Our measuring devices measure in whatever mathematical structure they're built to measure. If we build a device that gives a complex number to something called "gain", then by golly, the result of that measurement is a complex number. If I have a protractor that measures in multiples of pi radians, then by golly, any measurement I make with it is an irrational number. Digital devices measure in rational numbers because it's convenient -- not because it's a fundamental physical restriction.

And we don't perceive things in numbers -- we perceive things in, for example, visual stimuli. Counting the objects we see is an abstract process we've trained ourselves to do. (In fact, identifying objects in our field of vision is a fairly abstract process too!) The only reason you don't think of it as such is because you learned that particular abstraction when you were very young.


We use the real numbers to describe "quantities of physical stuff" because the real numbers are a mathematical structure that has the properties we would like to ascribe to "quantities of physical stuff". Not because the reals are somehow the only "real" number system.



The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.


(Ironically, I would have thought an electrical engineer would be more likey than most to treat the reals and complexes as being on equal footing!)

 

[vanesch]

The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.

Amen to that

Just to see at what point the real numbers were perceived as bizarre, think of the shock the Pythagoreans underwent when they discovered that simple geometrical lengths could not be constructed using rationals alone...

 

[rbj]

Not necessarily -- the crucial part of my position operates on the formal level: that there is no scientific reason to give the real numbers a priviliged status.

dunno what you mean by "priviliged status". certainly numbers are a sort of "human" invention (but chimpanzees and parrots have been shown to be able to count and i think an intelligent alien life would come to know nearly the same mathematical concepts we do, in due time) but the concept of "quantity of stuff" trancends human invention. before humans were around to count, the quantities of physical stuff existed and the quantitative interaction (that we presently describe with physical law) between that physical stuff existed. all of that quantity corresponds to real numbers.

I think the crux of my position is probably best expressed via a parody of your argument:

I'm equating the concept of "3-dimensional real vector space" to "relative displacements between physical locations", and there is no case where a relative displacement is measured as an real number, or even a complex number. Fundamentally, when we measure or perceive a displacement between things, it is a vector.​

What you presented is the usual argument that the reals are somehow special -- but all it does is pass the buck: it makes no attempt to explain why "quantity of physical stuff" is more real than things like "position", "angle", "constellation point", "quantum state", or "gravitational field".

maybe I'm dense, but i don't see the connection in your parody.

anyway, I'm making no attempt to explain "why" about quantity of stuff, only the observation of physical stuff and how much. that "how much" is invariably a real number.

Our measuring devices measure in whatever mathematical structure they're built to measure. If we build a device that gives a complex number to something called "gain", then by golly, the result of that measurement is a complex number. If I have a protractor that measures in multiples of pi radians, then by golly, any measurement I make with it is an irrational number. Digital devices measure in rational numbers because it's convenient -- not because it's a fundamental physical restriction.

i'm not saying there's a fundamental physical restriction against irrational quantities, only imaginary quantities. no physical quantity is fundamentally measured as "imaginary". we can abstract description of physical situations by use of these mathematical constructs called "imaginary" or "complex", but the quantities that we really measure are real. and, because the finite precision of our means of measuring, the raw quantities are also rational, but we might convert the raw measured quantity to another (say, by multiplying by \pi or \sqrt{2} and that result can be irrational, but the irrationality of that is not meaningful because there is a raw quantity arbitrarily close to the one we measured that, when converted the same way, would turn out to be rational.

And we don't perceive things in numbers -- we perceive things in, for example, visual stimuli. Counting the objects we see is an abstract process we've trained ourselves to do. (In fact, identifying objects in our field of vision is a fairly abstract process too!) The only reason you don't think of it as such is because you learned that particular abstraction when you were very young.

we perceive things quantitatively, even if only approximate. there was this tribe in the Amazon that had terms for "none", "one", "two", "three", or "many". even that crude calculus was quantitative to some extent. nonetheless physical quantity exists, whether we perceive it quantitatively or not.

We use the real numbers to describe "quantities of physical stuff" because the real numbers are a mathematical structure that has the properties we would like to ascribe to "quantities of physical stuff". Not because the reals are somehow the only "real" number system.

but that's exactly what makes them real. because quantities of real or actual physical stuff are measured, in terms of whatever unit (even natural units) as real numbers. real numbers are about quantities of stuff that is really there. that really exists.

The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.

i am not advocating Abstractophobia. but i do subscribe to the Einsteinian notion that things should be described as simply as possible, but no simpler. abstraction can help simplify physics or signal processing or any esoteric discipline. complex numbers, matrices, metric spaces (hilbert) and functional analysis can help simplify conceptualizing some real system and I'm all for that. but I'm against forgetting what the real quantities were to begin with.

(Ironically, I would have thought an electrical engineer would be more likey than most to treat the reals and complexes as being on equal footing!)

i generally do. when i think of driving a linear, time-invariant system with a sinusoid, i don't think of

x(t) = A \cos(\omega t + \phi)

i think of

x(t) = \left( A e^{i \phi} \right) e^{i \omega t} = \mathbf{X} e^{i \omega t}

it makes my life much easier. but i remember that this is but one component of the real signal and that there is another implied component:

\left( A e^{-i \phi} \right) e^{-i \omega t} = \mathbf{X}^{*} e^{-i \omega t}

and i know that a linear, time-invariant system made with real components (there ain't any real resistors with an impedance of (40 + i 30) \Omega) will act on that complex conjugate component exactly as it did with the first complex compoent, except with all +i replaced with -i and the output will be exactly the same as the output of the first component except it will be conjugated. then the real or actual or true (whatever adjective you want) result will, again, be a quantity measured as a real number, if i were to put this on a scope or voltmeter or whatever instrument to measure it.

complex (and imaginary) numbers are useful in dealing with reality, but only if you remember that they are imaginary.

 

[MeJennifer]

Yeah yeah, complex/imaginary numbers are as *real* as real numbers. Just like irrational numbers are every bit *rational* as rational numbers.

OK, so complex numbers are useful--very useful--but I think that a lot of people, including myself, still don't take them as a part of physical reality; they see them only as a mathematical abstraction. I mean, can you really make measurements in complex numbers? An ammeter can show that you have -1.333 amps flowing through your circuit, but it will never show that you have 8.2 - j3.2 amps flowing through your circuit.

Do you guys think that this uneasiness we feel using complex numbers would be better resolved if we would start using instruments that display measured quantities in complex numbers? Is such a thing even makes sense? It is possible? How much potential does this idea have?

I'd love to hear you guy's opinion on this and bring your perspectives on the table.

Since this is in the physics and not the philosophy section I like to point out that numbers are not measured neither real nor complex!
Numbers are simply mathematical abstractions.

In physics there is use for both real and complex numbers. And by the way real numbers are actually a subset of complex numbers.

 

[vanesch]

anyway, I'm making no attempt to explain "why" about quantity of stuff, only the observation of physical stuff and how much. that "how much" is invariably a real number.

Actually, not. "How much" is in fact always natural number. Any measurement is "a number of times a smallest unit of measurement". An ADC in a measurement system "measures" a number of times the smallest step in the voltage to be quantized. Of course, from this natural number of times a unit measurement, one can easily abstract away to a rational number of bigger units: A measurement can result in n times the unit, plus m times a tenth of the unit, plus k times a hundreth of a unit etc... So in a way, measurements can also be considered as rational numbers. But that's about it.

At no point you measure a REAL number. Real numbers are a mathematical abstraction, following from the closure of the rational numbers, and in fact most real numbers cannot even be written down in any way.

Once we are using abstractions of this kind, we can go on to other mathematical structures, such as vectors, complex numbers etc...

 

[russ_watters]

The reason I care about this topic is a practical one -- all the alledgedly "abstract" mathematical structures are created precisely because they capture some specific, useful, and often quite concrete notion. Abstractophobia is, IMHO, very harmful to one's ability and education.

This topic comes up from time to time and it is just plain absurd. It really gets on my nerves. All any mathematical device is is a tool for describing something real. No one device is any more or less real than any other.

The term "imaginary number" is simply a victim of its own poorly chosen name.

 

[russ_watters]

dunno what you mean by "priviliged status". certainly numbers are a sort of "human" invention (but chimpanzees and parrots have been shown to be able to count and i think an intelligent alien life would come to know nearly the same mathematical concepts we do, in due time) but the concept of "quantity of stuff" trancends human invention. before humans were around to count, the quantities of physical stuff existed and the quantitative interaction (that we presently describe with physical law) between that physical stuff existed. all of that quantity corresponds to real numbers.

Is the square root of 2 any less real because you can't count it on your fingers? I mean seriously - you're suggesting that only things that monkeys can do with math count as real!

Btw, an alien we are likely to meet would certainly understand our system of math. It is for all intents and purposes a universal language.

 

[rbj]

This topic comes up from time to time and it is just plain absurd. It really gets on my nerves. All any mathematical device is is a tool for describing something real.

that's not true. it might be said for "Applied Mathematics", but not always for the extremely abstract pure mathematics that these academics do.

The term "imaginary number" is simply a victim of its own poorly chosen name.

also quite debatable.

Is the square root of 2 any less real because you can't count it on your fingers? I mean seriously - you're suggesting that only things that monkeys can do with math count as real!

i didn't suggest that at all. i was only suggesting that the concept of "quantity" is not merely a human construct. other living beings have that concept, intelligent aliens have that concept, and, in fact, the concept applies to physical reality by use of quantitative physical law even if there weren't living beings around to contemplate or measure it. and with respect to physical reality, it's only the real quantities that apply even though we living beings might be about contemplate "imaginary numbers" as an abstraction.

Btw, an alien we are likely to meet would certainly understand our system of math. It is for all intents and purposes a universal language.

i would agree. they probably would have a concept of "complex numbers" with whatever name and they probably would not expect their voltmeters to measure a complex or imaginary quantity either.

Actually, not. "How much" is in fact always natural number. Any measurement is "a number of times a smallest unit of measurement". An ADC in a measurement system "measures" a number of times the smallest step in the voltage to be quantized. Of course, from this natural number of times a unit measurement, one can easily abstract away to a rational number of bigger units: A measurement can result in n times the unit, plus m times a tenth of the unit, plus k times a hundreth of a unit etc... So in a way, measurements can also be considered as rational numbers. But that's about it.

i completely agree with you, vanesch, but i do not see how that refutes anything i (as well as quite a few academics) have said about this.

this nature of metrology is something that i have been interested in a long time which is why I'm a proponent of Natural Units (Planck Units or a variant) for use in thinking about this stuff (even if they are impractical) and why I'm a big cheerleader for the likes of Michael Duff whenever he takes on a proponent of VSL or some other varying dimensionful universal "constant". whenever you talk about the amount of some dimensionful physical quantity, you have to also ask "with respect to what?".

At no point you measure a REAL number.

no, we measure quantities that are real and rational even if there is, conceivably, a physical quantity that is irrational (that we can only determine to a finite precision or as rational). but we don't measure physical quantities that are imaginary, because there aren't any.

there are physical quantities that are natural or whole numbers - these quantities are counted when measured. there are physical quantities that are rational and we might have a prayer to measure those exactly. there are even conceivable physical quantities that are irrational, like the distance between opposite corners of a perfect square of 1 unit length on a side - we can't measure it exactly but we can get closer and closer to the limit of precision of our ruler or tape-measure. there are even physical quantities that are intrinsically negative (or bi-polar). whether you assign "negative" to protons or electrons, one of them got to be negative because you can add these physical quantities and get zero. so it makes sense to have a measure of charge that is intrinsically negative. but there are no physical quantities that we can measure that are imaginary or complex, only real. and while the result of our measurement must also be rational (due to inherent error or limitations of measurement), the quantity being measured might, as far as we know, be irrational.

Real numbers are a mathematical abstraction, following from the closure of the rational numbers, and in fact most real numbers cannot even be written down in any way.

yes. there are an uncountably infinite number of real numbers that cannot be expressed exactly in any manner. all rational numbers can and some of these irrational numbers can have an abstract expression (like \sqrt{2} or \pi) that has exact meaning even ef we can't get to an exact value)

Once we are using abstractions of this kind, we can go on to other mathematical structures, such as vectors, complex numbers etc...

fine. i agree with that also.

 

[Hurkyl]

no, we measure quantities that are real and rational

Here is a picture of my ruler. It measures in centimeters.

It sure looks like I'm getting an irrational number when I use it to measure a length.

And you've already stated that measuring devices can measure complex quantities:

an instrument that compared two sinusoids of the same frequency could display the "gain" of one sinusoid over the other as a complex number.

The instrument measured gain. Gain is a physical quantity. Gain is complex. Ergo, the instrument measured a complex quantity.

(What is "gain" in this context? phase difference? Phase difference certainly isn't a real quantity)


Incidentally...

Why do you think physical quantities even can be expressed as real numbers? One could develop all of classical mechanics based on, for example, the hyperreal numbers, and it would make all the same predictions. There is, classically, absolutely no way to tell if the universe has "real" lengths, or "hyperreal" lengths. (Or something entirely different)


Furthermore, why do you think physical quantity can only be expressed as a real number? The imaginary axis of the complexes is exactly as good as the real axis for, for example, measuring lengths. Here's a picture of another of my rulers: it measures in imcentimeters. (The conversion between centimeters and imcentimeters is that 1 cm = i imcm)

yes. there are an uncountably infinite number of real numbers that cannot be expressed exactly in any manner.

Actually, I believe that statement cannot be proven. I don't think you can even prove that there exists a single real number that cannot be expressed exactly! There are some very subtle logical issues here -- for a bit of related flavor, see Skolem's paradox.

i'm a proponent of Natural Units

One important thing to note: for example, if there "exist", two incommensurate lengths (their ratio is not rational), then it is mathematically impossible to represent length as natural number multiples of some fundamental length.

 

[vanesch]

Actually, I believe that statement cannot be proven. I don't think you can even prove that there exists a single real number that cannot be expressed exactly!

Here I don't follow you. Given that you can only write a countable number of mathematical expressions (finite sequences of symbols), and given that the real axis contains a continuously uncountable number of reals, it should be obvious that not every real number has a formal expression to it, no ?

 

[vanesch]

i completely agree with you, vanesch, but i do not see how that refutes anything i (as well as quite a few academics) have said about this.

What I meant, was: the only "quantities" we know about "for real" are measurements, which are essentially natural numbers: we count stuff. A first step of "modelisation" is to use rational numbers, but that's just a trick to do "unit conversion". However, the quantities we measure have only an indirect relationship with what might "be out there". It is not because, when we observe that a meter stick goes 3 times in the length of a pole, plus 6 times a tenth of the meter stick, plus 2 times a tenth of a tenth of a meter stick, that there is anything "real" about this number. It is just an observation we made.

Apart from that, we have abstract formal theories which are our "physical theories", which use more sophisticated mathematical constructions, and according to one's tastes, one takes some of these formal elements "for real". However, that's just an option, a hypothesis. In classical mechanics, we hence imagine that we live in some kind of 3-dim Euclidean space, with "points" in space. But that's on one hand just a mathematical formal tool, and the hypothesis of its reality is only that: a hypothesis.
You can now discuss exactly what variant of mathematically equivalent structures is "the real structure" of space: are it 3-tuples of real numbers, or are it points on a 3-dim affine space with distance (nothing to do with numbers!) or even something else ? But that's a discussion which looks a lot like finding out the number of angels dancing on the head of a pin.
As such, if complex numbers are part of a mathematical structure of a physical theory, it is entirely up to you to decide in what way this complex structure is "real out there" and in which way it is just a way to tie up two real numbers. All of this is hypothetical. What comes out of a good physical theory is a number which will, in the end, predict what will be the approximate rational ratio between "a meter stick" and a "measurement".

Personally, I tend to take the stance that it is very practical to accept the hypothesis that the formal elements of a physical theory are "real", with the caveat that this is in any case not anything more than a working hypothesis.

 

[Swapnil]

I am just curious, is it possible to normalize some fundamental physical constant to 1 so that all measurements would be rational? For example, if we set some fundamental physical constant like Planck's constant or something to 1, then is it possible that all the physical quantities we measure like electric field, gravitational force, etc would become would become rational?

Also, I think that the idea of devices not being able to measure certain things precisely or whatever is a rather tricky problem because the devices we use to measure the "physical reality" are not independent of "mathematical abstractions." I think we usually design devices under the assumption of some mathematical model.

 

[Hurkyl]

Here I don't follow you. Given that you can only write a countable number of mathematical expressions (finite sequences of symbols), and given that the real axis contains a continuously uncountable number of reals, it should be obvious that not every real number has a formal expression to it, no ?

That's why it's a (pseudo)paradox. And I confess that I've been guilty of perpetrating that myth.

I pointed at Skolem's (pseudo)paradox -- he proved that there exists a model of set theory in which there are only countably many real numbers. (Despite the fact you can prove, in the model, that there are uncountably many real numbers) There's an internal vs external thing going on -- internally the naturals and reals can never be in bijection, but it's possible for an external bijectino to exist.

I can try and explain it in depth, but not right at this moment. (and this probably isn't the right thread for it anyways) Prompt me for it somehow if you want to hear it.

 

[vanesch]

That's why it's a (pseudo)paradox. And I confess that I've been guilty of perpetrating that myth.

I pointed at Skolem's (pseudo)paradox -- he proved that there exists a model of set theory in which there are only countably many real numbers. (Despite the fact you can prove, in the model, that there are uncountably many real numbers) There's an internal vs external thing going on -- internally the naturals and reals can never be in bijection, but it's possible for an external bijectino to exist.

I can try and explain it in depth, but not right at this moment. (and this probably isn't the right thread for it anyways) Prompt me for it somehow if you want to hear it.

You have teached me something totally new I was absolutely not aware of Never heard of that. Must be a mathematician's secret

 

[Swapnil]

I am just curious, is it possible to normalize some fundamental physical constant to 1 so that all measurements would be rational? For example, if we set some fundamental physical constant like Planck's constant or something to 1, then is it possible that all the physical quantities we measure like electric field, gravitational force, etc would become would become rational?

Also, I think that the idea of devices not being able to measure certain things precisely or whatever is a rather tricky problem because the devices we use to measure the "physical reality" are not independent of "mathematical abstractions." I think we usually design devices under the assumption of some mathematical model.

What about the above question?

 

[Crosson]

Natural numbers 1,2,3... are "natural" because they can be used to count.

Rational numbers are rational because they represent parts of wholes; any sub set of a finite set is a fraction of the original set.

Real numbers are introduced because our geometric intuition suggest that objects such as right triangles exists, and we believe that lines can have any length, and not all such inuitively "real" elements of a geometry diagram are fractional parts of each other.

The real numbers are the unique complete totally ordered field. The rationals are a totally ordered field (totally ordered captures their linearity, and calling them a field means they have all the familiar aspects of multiplication and addition). Completeness (not continuity) is the property of "having no holes" i.e. every convergent sequence of lines converges to something that is a line!
Viewed in this respect, the set of real numbers is perhaps too large a domain for physical measurements.

In summary, the reason people reject imaginary numbers is because they do not measure lengths, and all of our units are reduced to measurements of mass, length and time.

 

[Hurkyl]

In summary, the reason people reject imaginary numbers is because they do not measure lengths, and all of our units are reduced to measurements of mass, length and time.

Doesn't this very same argument suggest that you reject the notion of, say, momentum for precisely the same reason? (Because it can be reduced to mass, length, and time)

My point is that this is a pretty crummy reason -- assuming that you can reduce all measurements to mass, length, and time, that doesn't make it a good idea.

 

[Crosson]

Hurkyl -- Imaginary numbers are not necessary to measure length, mass and time. Therefore imaginary numbers are not necessary to measure (classical) momentum.

 

[Hurkyl]

Hurkyl -- Imaginary numbers are not necessary to measure length, mass and time. Therefore imaginary numbers are not necessary to measure (classical) momentum.

So? Complex numbers are necessary to measure, for example, impedence.

Incidentally, let's look at the momentum example, for fun. Momentum is a 3-vector. (let's stick to nonrelativistic, for simplicity) Your argument in post #24 says that because 3-vectors don't measure mass, length, or time, we should reject 3-vectors. Did you really mean to reject vector calculus?

 

[rbj]

So? Complex numbers are necessary to measure, for example, impedence.

complex impedence is a contruct, a human abstraction that allows us to more easily think about what these reactive elements do. you can construct equations similar to Kirchoff's Laws with complex sinusoids (Ae^{i \omega t}) and complex impedances, but the real expressions of what is physically going on would be Kirchoff's Laws with the integral and differential volt-amp characteristics of the reactive elements.

Hurkyl, nothing you have said so far (and i don't presume to understand every word, but many words i do and the arguments are less than weak) has made your case. not at all.

real (as in "reality") physical quantities are expressible as real numbers (with units, if not in Natural units) and are measured as rational numbers. pretty obvious despite how many angels might dance on the head of a pin.

 

[rbj]

What I meant, was: the only "quantities" we know about "for real" are measurements, which are essentially natural numbers: we count stuff.

i agree with that.

A first step of "modelisation" is to use rational numbers, but that's just a trick to do "unit conversion". However, the quantities we measure have only an indirect relationship with what might "be out there". It is not because, when we observe that a meter stick goes 3 times in the length of a pole, plus 6 times a tenth of the meter stick, plus 2 times a tenth of a tenth of a meter stick, that there is anything "real" about this number. It is just an observation we made.

we could get into some "real" sophistry with this. i could say that i am just some brain in a mad scientists laboratory that is receiving stimulus that only appears to be real and isn't really real (or "truly" real). my observation of my computer screen and the words from you are "just" an observation that is not about something necessarily real. assuming we don't go down that road, then i would have to disagree with the notion that our measurements (whether with a meter stick or something more sophisticated) isn't necessarily about anything "real". they most certainly are, but there is both finite precision (meaning that even if our measuring instruments had perfect "linearity", there would still be the integer measurement with respect to a unit or standard of measurement) and non-zero error (the stochastic noise and deterministic but unknown non-linearity in our measurement mapping function). but the measurement is about a quantity in reality, assuming we are real and really looking at (observing) some real physical process happening somewhere and somewhen. we we do such observation, we are measuring rational approximations to some real quantity that may or may not be rational, but we will not know nor can find out since any rational or irrational quantity can be approximated to any non-zero precision with another rational quantity. unless we are counting whole objects (whether balls or elementary charges) we cannot know any measurement we make is exact.

Apart from that, we have abstract formal theories which are our "physical theories", which use more sophisticated mathematical constructions, and according to one's tastes, one takes some of these formal elements "for real".

i agree with that.

However, that's just an option, a hypothesis. In classical mechanics, we hence imagine that we live in some kind of 3-dim Euclidean space, with "points" in space. But that's on one hand just a mathematical formal tool, and the hypothesis of its reality is only that: a hypothesis.
You can now discuss exactly what variant of mathematically equivalent structures is "the real structure" of space: are it 3-tuples of real numbers, or are it points on a 3-dim affine space with distance (nothing to do with numbers!) or even something else ? But that's a discussion which looks a lot like finding out the number of angels dancing on the head of a pin.

i agree with that, but do not understand the concept of distance without numbers.

As such, if complex numbers are part of a mathematical structure of a physical theory, it is entirely up to you to decide in what way this complex structure is "real out there" and in which way it is just a way to tie up two real numbers. All of this is hypothetical. What comes out of a good physical theory is a number which will, in the end, predict what will be the approximate rational ratio between "a meter stick" and a "measurement".

Personally, I tend to take the stance that it is very practical to accept the hypothesis that the formal elements of a physical theory are "real", with the caveat that this is in any case not anything more than a working hypothesis.

fine. you can express physical law nicely and compactly sometimes with complex variables. i think Schrodinger's Eq. is a good example (but, to make it meaningful, you have to put in the magnitude square and expectation which results in real numbers).

but you can also express the same law as a pair of laws simultaneously acting on a pair of quantities (\mathrm{Re}(\Psi) and \mathrm{Im}(\Psi) also expressing probability density and expectations or moments as real functions of those two quantities. neither our measurement nor the physical law must be expressed in complex form, even though i understand that it is convenient to.

 

[Hurkyl]

Hurkyl, nothing you have said so far (and i don't presume to understand every word, but many words i do and the arguments are less than weak) has made your case. not at all.

real (as in "reality") physical quantities are expressible as real numbers (with units, if not in Natural units) and are measured as rational numbers. pretty obvious despite how many angels might dance on the head of a pin.

You know, I really don't appreciate the "You're wrong, I'm right." style of debate.

I have a lot to say -- maybe it will be clearer if I actually try to organize it into sections.


1. Real numbers "real"?
Just to be sure we're on the same page, recall that my point here is that our scientific theories do not support the notion that real numbers are "real", nor do they support the notion that complex numbers are not "real".

1.1. Reduction
One of the main themes in your argument is the notion of reduction. You look at a concept like impedance, which our physical theories describe with complex numbers, and choose to reduce it into different concepts which can be described with real numbers. You look at a concept like momentum, which our physical theories describe with vectors, and choose to reduce it to a triple of real numbers. You even look at the complex numbers themselves, and choose to reduce them into pairs of real numbers.

But none of these support your point. These explain how a person who rejects the complex numbers and vectors could still manage to do physics despite that handicap. They do not explain why the real numbers should be considered "real", and why complex numbers and vectors should not be considered "real".

In fact, there is a simple reductio ad absurdum argument to demonstrate that reduction cannot be used to support your assertions. Suppose that the form of argument

Anything we can do with X, we can do with Y. Therefore, Y is more "real" than X.​

is a valid one. Then, we would be able to argue

Anything we can do with complex numbers, we can do with real numbers. Therefore, the reals are more "real" than the complexes.​

But, we would also be able to argue

Anything we can do with real numbers, we can do with complex numbers. Therefore, the complexes are more "real" than the reals.​

Voilà: a contradiction. There are, in fact, many different structures to which one can reduce the reals.

1.2. Abstraction
Everything notion in a physical theory is an abstract. This follows directly from the very definition of abstract. Impedance, momentum, length, time, mass; these are all abstract notions, constructed by humans. Complex number, real number, natural number: also abstracts.

Ugh, I have more to say, so I'm going to have to cut this short. I'll post what I have, and leave some short parting comments:


You can see I've attached another picture of a ruler.

You tried to argue that measuring in rational multiples of pi is not measuring an irrational number. (Which is silly, because a rational multiple of pi is an irrational number) Well, I've attached another example of a ruler that measures in the irrationals to block that particular argument.


Also, when you talk about finite precision implying we measure in rational numbers -- that's not because you think significant figures are meaningful, is it?

 

[rbj]

You know, I really don't appreciate the "You're wrong, I'm right." style of debate.

Hurkyl, I'm not trying to get in a sort of demeaning "your argument is stupid" debate and, to be clear, i respect you and the other physicists here greatly. don't get me (or my attitude) wrong about this. but i mean what i said: except for concepts that i have never learned or previously heard of (and i took quite a bit of applied math inc. Real Analysis, Complex Analysis, Functional Analysis, etc. as an undergrad and graduate student of EE, 30 years ago) such as "hyperreal numbers", every other part of your argument is, essentially, sophistry. utterly unpersuasive. i'll touch on some (somewhat re-ordered):

Why do you think physical quantities even can be expressed as real numbers?

uh, because they have been.

we count out "9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom" and call that period of time a "second". during that same period of time, we measure the distance, in terms of the spacing between two little scratch marks on a special piece of platinum/iridium that we had been calling a "meter", that light travels in vacuo and that distance is 299792458 of those meters. (later we redefine what we mean by "meter" to guarantee that same number.) we measure electrical current by counting electrons passing some surface boundary (not really the right way, but they may redefine the kilogram so that this could be the right way). we measure electrical voltage by measuring in some manner how much mean energy these electrons have at different places. we measure how much energy by how much they move something (perhaps themselves) against a force. we measure force by ...

these are the physical quantities that we measure, not complex impedance. we construct conceptually a "measure" of complex impedance of something by making real measurements of real quantites. but there is no (instantaneous or mean) voltage or energy or number of electrons that was measured to be complex. all of these measurements were real, and, because of finite precision, rational.

Here is a picture of my ruler. It measures in centimeters.

View attachment 8707

It sure looks like I'm getting an irrational number when I use it to measure a length.

this doesn't say anything.

Furthermore, why do you think physical quantity can only be expressed as a real number? The imaginary axis of the complexes is exactly as good as the real axis for, for example, measuring lengths. Here's a picture of another of my rulers: it measures in imcentimeters. (The conversion between centimeters and imcentimeters is that 1 cm = i imcm)

View attachment 8708

no, you don't have such a ruler, you have a ruler of real length with markings on it that you apparently want to be interpreted as imaginary numbers. but there are no imaginary lengths. if you use that ruler to measure real lengths (since it is a real ruler) you will get real results, despite the markings.

One important thing to note: for example, if there "exist", two incommensurate lengths (their ratio is not rational), then it is mathematically impossible to represent length as natural number multiples of some fundamental length.

well, sure, i guess. you can't represent it exactly. but neither can you measure (or perceive) it exactly. that's why our measurement (or perception, which is just measurement with the devices inherent in our senses) of physical quantity are rational numbers times some reference quantity. but even though i can't measure the diagonal of a perfect square (of unit length on a side) to be exactly the \sqrt{2}, i can accurately theorize that such would be the length and confirm with an actual ("real") measurement that results in a rational approximation to the "theoretical value" to as close as the technology allows.

And you've already stated that measuring devices can measure complex quantities:

i didn't say that. i said that fundamentally measuring devices measure real quantities.

The instrument measured gain.

naw, it measures input and output voltages and computes gain.

Gain is a physical quantity.

not in the sense we are debating. "Gain", as a magnitude, is a ratio of physical quantities. the complex gain that we signal processing engineers talk about is an abstraction that incorporates the real magnitude gain along with phase shift for a restricted class of input and output signals we call sinusoids. this complex gain, as a number, has no meaning for more general signals. for an LTI (linear and time-invariant) system a complex gain that is a function of frequency can be used to relate the general input to output.

Gain is complex. Ergo, the instrument measured a complex quantity.

(What is "gain" in this context? phase difference? Phase difference certainly isn't a real quantity)

time difference is. that's all phase difference is, normalized to the period of the sinusoidal function, another real quantity. so how is phase difference not real?

(back to the recent post...)

1. Real numbers "real"?
Just to be sure we're on the same page, recall that my point here is that our scientific theories do not support the notion that real numbers are "real", nor do they support the notion that complex numbers are not "real".

i think i agree with you that this is your point and, in that sense, we're on the same page. but i disagree with that point.

1.1. Reduction
One of the main themes in your argument is the notion of reduction. You look at a concept like impedance, which our physical theories describe with complex numbers, and choose to reduce it into different concepts which can be described with real numbers. You look at a concept like momentum, which our physical theories describe with vectors, and choose to reduce it to a triple of real numbers. You even look at the complex numbers themselves, and choose to reduce them into pairs of real numbers.

But none of these support your point. These explain how a person who rejects the complex numbers and vectors could still manage to do physics despite that handicap. They do not explain why the real numbers should be considered "real", and why complex numbers and vectors should not be considered "real".

physical law ultimately comes from empirical observation of "real" events, of reality. we see certain events happen in such a repeatable and systematic manner that we infer a law that, hopefully in a simpler language, describes these events consistently to within the degree of accuracy that we observe or measure. we rely on that law until we find exceptions to what the law says, and then look for a better law (and hope that correspondance principle is satisfied). the real numbers are considered "real" because those are the numbers we fundamentally use to measure and describe, quantitatively, the physical phenonmena we are observing and codifying into laws. now, for simplicity and elegance, the law may preferably be stated in such a way that vectors and/or complex numbers represent quantites that the law refers to, but when that law is used to make a prediction that we hope to verify in experiment, a reduction to real quantities is made (and is part of the law, such as expectations or probability densities in the case of Schrodinger, or real changes in magnitude and phases for a sinusoidal electrical system) and measurements of the corresponding quantities in the real physical world are made and compared. perhaps those real measurements or real quantities are combined in some manner to create complex or vector quantities for use in our conceptualization (BTW, i didn't say vectors with real components aren't real, and i don't think you can successful use that as evidence of an inconsistancy - vectors are different than complex numbers) .

In fact, there is a simple reductio ad absurdum argument to demonstrate that reduction cannot be used to support your assertions. Suppose that the form of argument

Anything we can do with X, we can do with Y. Therefore, Y is more "real" than X.​

is a valid one. Then, we would be able to argue

Anything we can do with complex numbers, we can do with real numbers. Therefore, the reals are more "real" than the complexes.​

But, we would also be able to argue

Anything we can do with real numbers, we can do with complex numbers. Therefore, the complexes are more "real" than the reals.​

Voilà: a contradiction. There are, in fact, many different structures to which one can reduce the reals.

utterly unpersuasive. this doesn't say anything are physical quantity.

1.2. Abstraction
Everything notion in a physical theory is an abstract. This follows directly from the very definition of abstract. Impedance, momentum, length, time, mass; these are all abstract notions, constructed by humans.

impedance, perhaps. but not length or time or mass or charge. those are real things. they're real stuff. you can go to Stuff-Mart and buy them. i do not concede that these things are mere constucts of humans (unless we're in The Truman Show). stars are spaced apart by real distances (that change in time) and have real mass. this stuff is real and the kind of numbers we use to measure it and describe it are real numbers.

Complex number, real number, natural number: also abstracts.

the numbers and strict meaning to us for what they represent are abtracts, but the physical quantities that we measure are real things and the kind of numbers that are used to measure these real things are real numbers (actually a subset).

You can see I've attached another picture of a ruler.

You tried to argue that measuring in rational multiples of pi is not measuring an irrational number. (Which is silly, because a rational multiple of pi is an irrational number)

but the mutliple of \pi is not. the number multiplying it is rational (by axiom).

Well, I've attached another example of a ruler that measures in the irrationals to block that particular argument.

this doesn't convince at all. it says nothing.

Also, when you talk about finite precision implying we measure in rational numbers -- that's not because you think significant figures are meaningful, is it?

there are degrees of meaningfulness. but this is off-topic.

Hurkyl, you still haven't gone beyond sophistry here. Real numbers are what we use to measure and describe fundamental physical quantity in reality. Imaginary and Complex numbers are not used for such. Real numbers are qualitatively different. the unit in the real number line (i.e. the number "1") is the multiplicative identity and is qualitatively different than its negative "-1". but the unit in the imaginary line, the imaginary unit that we conceive and label i is qualitatively no different than its negative -i. both have equal claim to square to be -1. if every textbook and math/science/technical paper in the world was rewritten and -i was substituted in for i (which would also have the effect of replacing every occurace of -i with +i), all facts and theorems would remain equally true. the same cannot be said for +1 and -1. they are not interchangable.

 

[Crosson]

It is not that difficult to see why people question the need for complex numbers in physics. The SI base units are sufficient for expressing all known measurable physical quantities, and these units always (Kilograms, meters, coulombs, etc) refer to non-imaginary quantities.

My response is that imaginary numbers are not ever needed to discuss the results of measurements, but in some cases they are needed to discuss the dynamics that underlie the observations.

 

[rbj]

It is not that difficult to see why people question the need for complex numbers in physics. The SI base units are sufficient for expressing all known measurable physical quantities, and these units always (Kilograms, meters, coulombs, etc) refer to non-imaginary quantities.

My response is that imaginary numbers are not ever needed to discuss the results of measurements, but in some cases they are needed to discuss the dynamics that underlie the observations.

i think that complex numbers and variables and techniques (integration on paths, residues, etc.) are very, very, very useful (to within \epsilon of "needed") for description of the dynamics, but since every complex number can be split into its representation of two real numbers and every complex equation can be split into two equations equating the real and imaginary parts, complex numbers aren't absolutely needed, but expressing such equations in the complex form is far more compact and elucidating and easier to solve than than the counterpart pairs of real equations.

 

[Swapnil]

since every complex number can be split into its representation of two real numbers and every complex equation can be split into two equations equating the real and imaginary parts, complex numbers aren't absolutely needed, but expressing such equations in the complex form is far more compact and elucidating and easier to solve than than the counterpart pairs of real equations.

But can't you give the same argument for real numbers. If you forget about irratinal numbers, then real numbers can be written as a pair of natural numbers to give you rational numbers (for example, (a,b) = a/b ).
(Don't know what we'll do about irrational numbers or negative numbers though...)

naw, it measures input and output voltages and computes gain.

I agree. But HOW does it measure input and output voltages? Perhaps there are another set of quantities that are really "measured" and used to compute input and output voltages...

 

[rbj]

But can't you give the same argument for real numbers. If you forget about irratinal numbers, then real numbers can be written as a pair of natural numbers to give you rational numbers (for example, (a,b) = a/b ).

i s'pose but i don't see what you mean by making the "same argument". might be OT.

I agree. But HOW does it measure input and output voltages? Perhaps there are another set of quantities that are really "measured" and used to compute input and output voltages...

how gets to be more and more technical and messy, but ultimately it gets down to counting charge accumulating somewhere and knowing (quantiatively) the nature of materials that result in a property we call "resistivity" or "conductivity" as well as some geometric properties of parts (we call "resistors").

 

[Hurkyl]

BTW, i didn't say vectors with real components aren't real, and i don't think you can successful use that as evidence of an inconsistancy - vectors are different than complex numbers

Okay, then can explain to me why your argument that the complexes are not "real" fails to apply just as well to vectors?

(This question, incidentally, is the one I'm most interested in hearing answered)

so how is phase difference not real?

Phase is circular. The reals are linear. Circles are not lines. 0 and 360 are different real numbers, but 0° and 360° are identical phase differences.

Real numbers are qualitatively different. the unit in the real number line (i.e. the number "1") is the multiplicative identity and is qualitatively different than its negative "-1". but the unit in the imaginary line, the imaginary unit that we conceive and label i is qualitatively no different than its negative -i. both have equal claim to square to be -1. if every textbook and math/science/technical paper in the world was rewritten and -i was substituted in for i (which would also have the effect of replacing every occurace of -i with +i), all facts and theorems would remain equally true. the same cannot be said for +1 and -1. they are not

What "real" meaning does the unit have? I can't think of a "physical quantity" where +1 has an absolute meaning: the + is simply denoting an orientation with respect to a chosen convention, and the 1 is simply denoting a scale with respect to a chosen unit.

Multiplication does not have an absolute physical meaning: every physical law involving a multiplication also includes a physical constant that modifies the equation so that the scale and orientation turn out correct.

If we change the units on certain measurements, the physical constants adjust to compensate.

If we change our sign convention, the physical constants adjust to compensate.

We can even change quantities from purely real to purely imaginary, and the physical constants adjust to compensate.

(Algebraically, all three of these amount to exactly the same operation)

e.g. I could select three oriented lengths, one along the up-down axis, one along the East-West axis, and one along the North-South axis. I could then declare that the first one is +1, the second is -3, and the third is 3+2i, and all of physics still works, all physical laws retain their form: the physical constants simply take on values with respect to these new units.


(In summary... of course you can swap -1 and +1! I can also replace +1 with +2 and -1 with -2. I can even replace +1 with +i and -1 with -i, if I choose. It's just that these symmetries are a little bit more involved than that of swapping +i and -i at every single occurence)

 

[Swapnil]

i s'pose but i don't see what you mean by making the "same argument". might be OT.

since every real number can be split into its representation of two natural numbers* and every real equation can be split into two equations equating the denominator and numerator, real numbers aren't absolutely needed, but expressing such equations in the real form is far more compact and elucidating and easier to solve than than the counterpart pairs of natural equations.

*I know I am ignoring two important subsets of real numbers -- the irrationals and negative numbers. So my argument doesn't do much unless there is a way to represent irrational numbers and negative numbers as a pair of natural numbers.

 

[rbj]

since every real number can be split into its representation of two natural numbers*

no, only rationals. not every real.

and every real equation can be split into two equations equating the denominator and numerator,

that's not true either. the fact is that

z_1 = z_2

where these are possibly complex, this is the same as

\mathrm{Re}(z_1) = \mathrm{Re}(z_2)
and
\mathrm{Im}(z_1) = \mathrm{Im}(z_2)

if the top equation is true, the bottom pair must be. and vice versa.

but it's not true that:

\frac{y_1}{x_1} = \frac{y_2}{x_2}

means that:

y_1 = y_2
and
x_1 = x_2

it could be true, but it wouldn't have to be. but the converse is true.

so i don't see the parallel here. this is about different stuff.

real numbers aren't absolutely needed, but expressing such equations in the real form is far more compact and elucidating and easier to solve than than the counterpart pairs of natural equations.

*I know I am ignoring two important subsets of real numbers -- the irrationals and negative numbers. So my argument doesn't do much unless there is a way to represent irrational numbers and negative numbers as a pair of natural numbers.

it's not the same.

 

[Manchot]

There are many instruments that display complex quantities. Network analyzers, for one.

 

[rbj]

There are many instruments that display complex quantities. Network analyzers, for one.

surely. still these complex quantities are caculated from a series (at least a pair) of real measurements. there is no real place were you clip on a pair leads onto an unknown voltage and read out, from a single instantaneous measurement, a voltage that is complex.

but with concepts like the Hilbert Transform, we can certainly construct, in the mind of a computer using the mathematical rules of complex numbers (which are just the reasonable analytic extensions of the same rules for reals), create a "complex" quantity and treat it as such.

 

[LeonhardEuler]

Here is an example that I think might help. It's been said here that only natural numbers can be used to count things, but this is not true. If you want a number system to be used only for the purpose of counting things, the integer multiples of imaginary numbers will do. This is because counting requires only a notion of addition and not of multiplication and the natural multiples of imaginaries will behave in every way exactly like the naturals themselves for the purposes of addition. . So while it may seem natural in our culture to say that you have 7 apples and absurd to say that you have 7i apples, it is only a matter of convention. If you have 7i apples and take 2i away, you have 5i apples. The only reason you wouldn't normally say this is that we have grown up so accustomed to representing quantities of things in natural numbers that the fact that this itself is merely an abstraction is lost on us. We think there really are 7 apples, when in fact the quantity of apples is just represented by the number 7 in our convention.

 

[Hurkyl]

Here is an example that I think might help.
...So while it may seem natural in our culture to say that you have 7 apples and absurd to say that you have 7i apples, it is only a matter of convention. If you have 7i apples and take 2i away, you have 5i apples.
...

Actually, it is perfectly accurate to say that what you describe actually is a system of natural numbers: if we define the successor of x as x+i, then it satisfies Peano's axioms.

(And happily, you show that the addition of purely imaginary numbers agrees with the usual addition operation defined on a system of natural numbers)

 

[LeonhardEuler]

Actually, it is perfectly accurate to say that what you describe actually is a system of natural numbers: if we define the successor of x as x+i, then it satisfies Peano's axioms.

(And happily, you show that the addition of purely imaginary numbers agrees with the usual addition operation defined on a system of natural numbers)

Are you sure? To be clear, even though multiplication is not used in the application I'm talking about, I'm still assuming these are ordinary imaginary numbers with multiplication defined in the usual way so that i*i=-1, which means that this system is not closed under multiplication. I'm not sure if the set I described actually is the natural numbers because mathworld defines them as the set of positive integers, clearly ruling out the possibility of this set being the natural numbers, but wikipedia (which first of all includes 0 as a natural number) gives a construction in which only the properties of addition are assumed, and then multiplication is defined in a way that gives the normal properties of the natural numbers, from which it is possible to derive statements like i*i=i in this case.

Regardless, though, this does show an example of counting things with non-real numbers.

 

[vanesch]

surely. still these complex quantities are caculated from a series (at least a pair) of real measurements.

And, as I argued earlier, these "real" measurements are in fact calculated from naturals (like ADC outputs and the like) - and even not naturals, but a finite subset of it. So, all we're doing when we are measuring, is *counting*. All the rest is abstraction and modeling, whether or not we are aware of it. Real numbers are abstractions, vectors are abstractions, complex numbers are abstractions, manifolds are abstractions. The only thing we do is counting, in that we compare integer multiples of reference concepts with to-be-measured concepts. In many cases, we tend to believe that we are *really* measuring some genuine quantity, but that is because we take our abstract models "for real" - which can be a good thing, but then we should take also our other abstractions (such as complex numbers, vectors,...) for real.
There is no manner in which the abstraction of the real numbers is in any way "more real" than other abstractions. Now, it is true that other abstractions often have the structure of a differentiable manifold, and hence are describable by (sets of) real numbers (which is then called a coordinate system), but there's 1) no reason why this description has any more reality to it than the structure itself (rather the opposite: usually we give more "geometrical existence" to the manifold structure than to its coordinates, which have an arbitrariness to it) 2) no reason why all mathematical structures we need or will need in physics will be of the differentiable manifold kind.

There's another way to show why "real numbers" are "less real" than abstract manifold structures: consider change of units!
The real number "1" has a special meaning in the structure of real numbers: it is the unit element of multiplication. There's no other real number satisfying this property. Given that "1" is special, this would mean that, if there were genuine physical reality to real numbers, there would be a special physical meaning to a physical quantity which corresponds to the real number "1". But if we measure something to be 1 meter long, then this simply means that, compared to our measuring reference (meter), we find that the thing we measure seems to be "equal" concerning the observation of length. If we change our reference to "feet", we get another number (of the order of 3 something). It is not "1" anymore. So there's nothing special to "1" in our measurements, while the REAL NUMBER 1 is special and unique.
So the actual mathematical object we are using to model our measurements with, is not "the real numbers", but a 1-dimensional linear differentiable manifold, over which we can define different coordinate systems (corresponding to different units), consisting of mappings from the real numbers into our "abstract but real" manifold. As such, we see that, if we want to assign some "reality" to some mathematical structure, it is not the set of real numbers itself, but rather the differentiable manifold itself.
But if we go through this abstraction, then there's no reason to prefer one-dimensional manifolds over multidimensional manifolds. Some of those can have extra structure, such as distance, or curvature, or complex or symplectic structure or whatever. There's then no reason to give more reality to a single real coordinate over that structure, than to the structure itself.

 

[rbj]

And, as I argued earlier, these "real" measurements are in fact calculated from naturals (like ADC outputs and the like) - and even not naturals, but a finite subset of it. So, all we're doing when we are measuring, is *counting*. All the rest is abstraction and modeling, whether or not we are aware of it. Real numbers are abstractions, vectors are abstractions, complex numbers are abstractions, manifolds are abstractions. The only thing we do is counting, in that we compare integer multiples of reference concepts with to-be-measured concepts. In many cases, we tend to believe that we are *really* measuring some genuine quantity, but that is because we take our abstract models "for real" - which can be a good thing, but then we should take also our other abstractions (such as complex numbers, vectors,...) for real.

i think i agree with that, but...

There is no manner in which the abstraction of the real numbers is in any way "more real" than other abstractions.

i don't agree with this. a counter-example would have to do with a real and idealized geometry in 2-space. a perfect square of unit length on the side. even though i measure the length of the diagonal to be rational (my finite precision measuring instrument only has integer values of the tiny ticks at the limit of its precision), i know that, in the ideal, these rational measurents of the diagonal will only approach this theoretical irrational value as my precision gets better and better. they will not be approaching a rational value in the idealized limit.

but this is not the case for complex or imaginary numbers. there ain't no square out there with the length of diagonal of 5 i using any units of reality.

i'll try to crack the remaining point you made.

 

[vanesch]

Proof by color

i don't agree with this. a counter-example would have to do with a real and idealized geometry in 2-space. a perfect square of unit length on the side. even though i measure the length of the diagonal to be rational (my finite precision measuring instrument only has integer values of the tiny ticks at the limit of its precision), i know that, in the ideal, these rational measurents of the diagonal will only approach this theoretical irrational value as my precision gets better and better.

Clearly you are setting up a theoretical idealized mathematical entity which you give the status of "real", but you have absolutely no guarantee that your idealized concept of "perfect square" is any more real than the other hypotheses you made. There is absolutely no requirement for perfect squares to have any fundamental reality, any more than any other theoretical and formal concept (such as Newton's equation of motion or the like).
You have already "taken for granted" that Euclidean space is somehow "reality" before you can even consider the "reality" of a perfect square.
And then we're back to square ( ) one: IF you assign (by hypothesis) some reality to the Euclidean space, then you should do so too to any other theoretical/formal construction of the physical theory at hand ; if that theory uses vectors, or manifolds, or complex numbers, then that's also just as much part of it, and hence of its (hypothetical) "reality".

I know that it is difficult to let go, but "Euclidean space" is just as well a hypothetical and formal construction as any other part of a physical theory, and it is only within such a formal structure that a concept like "perfect square" makes sense. As such, the irrationality of the diagonal is only a product of our formal structure, which is the Euclidean space, and there is no way to verify this empirically. To illustrate this, consider the possibility that what we call space, is actually a discrete structure of some kind (a finite set of things we could call "points"), but which is so fine-grained, that for all practical purposes, every Euclidean manipulation is correct up to, say, 50 decimals. We can make very good approximations to a square in such a structure, but the concept of "perfect square" as in Euclidean space, is meaningless: it would be just a "set of 4 elements of the finite set of points".
In such a view, there would be no problem of "the irrationality of the diagonal". It is only in the formal model of Euclidean geometry that such a thing arises. It's a good formal model, which makes good predictions for measurements of what we call "distance", with say, 30 decimals. But it is only that: a formal model.

but this is not the case for complex or imaginary numbers. there ain't no square out there with the length of diagonal of 5 i using any units of reality.

In the same way as there are no distances equal to -5.6 meters. And if you are going to tell me that we CAN have that in a coordinate system, then I can say that we CAN have "distances" of "3.0 + 5.2 i" meters in the Euclidean plane, according to a very similar construction.

"distances" are not "more or less real" than other theoretical constructions in physical theories. Turns out that Euclidean geometry is very closely related, as a formal structure, to R^3. You can call Euclidean geometry "the theory of measurements of distances and angles".

But other physical theories use other formal structures, and "the theory of electrical networks" for instance, uses complex numbers in a natural way.

 

[rbj]

vanesch (and Hurkl), i won't be able to keep up because some of what you guys say needs more digestion before response which i may end up not having time for. but, i'll try to respond to the "easy" stuff.

In the same way as there are no distances equal to -5.6 meters.

sure, distances or positions are relative. and a distance metric in a metric space is always real and non-negative. and we always measure it to be real, rational, and non-negative. but negative quantities do exist in nature. electric charge is one. i don't care which you choose as "positive" charge (proton vs. electron), but whichever you choose as positive, the other must be negative. negative quantities of some "stuff" are real things in nature. we can construct other quantities, like rate of change, which will necessarily have negative values. it's perfectly appropriate to include the negative numbers with "real" numbers.

And if you are going to tell me that we CAN have that in a coordinate system, then I can say that we CAN have "distances" of "3.0 + 5.2 i" meters in the Euclidean plane, according to a very similar construction.

you can say it, but it's not true. not in reality. there is no qualitative difference in the dimension that you're defining as "imaginary" and the dimension with real coordinates. but there is a qualitative difference between imaginary quantities and real quantities. square a real quantity and you have similar real animal. square an imaginary quantity, and the result is no longer imaginary.

"distances" are not "more or less real" than other theoretical constructions in physical theories.

but they are or may be in reality. a voltage of 5 volts is a real physical thing (that can be measured to some precision), but a "phasor" (the EE kind, not the Star Trek weapon) of 4 + 3i is not a real physical thing, but an abstraction that makes our life easier.

Turns out that Euclidean geometry is very closely related, as a formal structure, to R^3. You can call Euclidean geometry "the theory of measurements of distances and angles".

that's pretty damn closely related to the 3-dimensional space we, and the milky way galaxy, live in.

But other physical theories use other formal structures, and "the theory of electrical networks" for instance, uses complex numbers in a natural way.

i'm quite familiar with the theory of electrical networks. i do DSP for a living and deal with complex quantities or variables all the time. they're abstrations. they have some correspondance to "real" quantities (quantities we perceive or measure in reality), but they are not the real quantities.

 

[Crosson]

I am not sure there is anyone in this thread is an opponent of complex numbers, but I noticed some arguments in favor that are choir-preaching-to-choir type i.e. "if we demand closure under this algebraic/analytical/geometrical operation, why not this one?"

Most opponents to complex numbers are coming from an anthropocentric perspective, they are only motivated to learn things that immediately relate to humans. The only way to show them that mathematics provides another type of motivation is to put their perspective in context: talk to students about why they "believe" in various number systems (whole, integer, rational, real) so that they see that the reasons for believing in complex numbers are equally compelling, from a particular point of view.

 

[rbj]

just to be clear: i am no opponent to complex numbers (to the concept of their existence, or to their use either in theory or practice). i am a big proponent of complex numbers.

i am only an opponent to the concepts that quantities of raw physical stuff is measured as complex (we construct complex parameters from multiple, at least 2, "real" measurements) or that what we call "imaginary numbers" are every bit as "real" (meaning having to do with actual quantities of stuff in physical reality) as what we call "real numbers".

but, when doing math, i treat complex/imaginary numbers with every amount of care that i treat real numbers. it's just that i think the terms "real numbers" and "imaginary numbers" are appropriate terms because fundamental physical quantities in reality are real numbers. so what better term can you think of for those other kinds of numbers of values that no basic physical quantities would take on, but we can imagine that when squared, result in a negative (real) number?

 

[Hurkyl]

there is no qualitative difference in the dimension that you're defining as "imaginary" and the dimension with real coordinates.

Doesn't "realness" count as a quality?

a voltage of 5 volts is a real physical thing

This surprises me even more than you thinking that 3-vectors are "real". I honestly cannot see any consistent pattern to what you believe is "real" and what is not "real".

Actually, that's not entirely true -- there is the obvious pattern that you label something "real" if and only if it doesn't invoke complex numbers. But I'm assuming that is merely a correlation and not causation.

I'm still looking forward to an explanation of why 3-vectors are "real", and now I'm even more looking forward to an explanation of why electric potential is "real".

But now I'm curious as to if there are any quantities measured by real numbers that you would consider not "real".