[feoren]

> complex numbers are a promise to square something at a later date and recover a real number

Except, most complex numbers don't square to a real number. Only those lying along the complex or real axes square to a real number; everything else just squares to another (non-real) complex number. In what way do complex numbers represent a "promise" to square it later and recover a real number? Who is making this promise? I feel like this is falling into the same trap of believing that complex numbers are not allowed to simply exist on their own merit.

I think it's quite serendipitous that the number system designed to algebraically close the reals to include roots of polynomials like x^4 + 1 happens to also cleanly describe so much of physics. There happens to be a lot of physics that boils down to "magnitude and phase" where those quantities interact in the same way complex numbers do, but it's not a-priori obvious that electromagnetism shouldn't need some third quantity as well, nor that we shouldn't be using quaternions instead, nor some other algebraic structure defined over 2D or 3D or 4D vectors.

Indeed, as you point out, there are plenty of more complicated mathematical structures that are best for describing other parts of physics, like spinors, Lie groups, and special unitary groups. It's not a-priori obvious that Lie groups should be so important to physics either. But neither should anyone protest their use as somehow not "really existing". It is true that complex numbers do not physically exist -- neither do Lie groups, and neither does the number 7. We got lucky that mathematicians had already explored an algebra that turned out to be perfect for "magnitude-and-phase" physics, but it doesn't seem like "squaring to a real number" had anything to do with why they are useful. Real numbers have no stronger claim to truly representing physics than complex numbers, spinors, or Lie groups do.

[rsfern]

I think this is just loose terminology, instead of squaring they should have said “multiply by the complex conjugate”, which is what you do to quantum mechanical wavefunctions to obtain real-valued probability amplitudes

[nathan_compton]

> Real numbers have no stronger claim to truly representing physics than complex numbers, spinors, or Lie groups do.

Eh, call me when your detector gives you back a complex number. Measurements return real numbers. I've never known one to return a complex valued one. Probabilities are real numbers. I feel this puts real numbers in a privileged position. If you ever wrote a theory that suggested that you lay a ruler against an object and measure a complex value, you'd be in trouble.

[feoren]

> Eh, call me when your detector gives you back a complex number. Measurements return real numbers.

There are an uncountably infinite number of real numbers. 100% of them (but not all) are not computable, and cannot be written down or described. Measurements do not return "true" real numbers. Measurements return whatever the detector is designed to return. Digital measurements return binary floating-point, fixed-point, or integer numbers. Some measurements return "red" vs. "blue". Pregnancy detectors return "1 line" or "2 lines". All it would take for a detector to give a complex number is to design one that measures something that can be described as a complex number, and return it as a complex number. For example, a phasor measurement unit:

https://en.wikipedia.org/wiki/Phasor_measurement_unit

Should I call you?

[nathan_compton]

I think there is a credible case to be made that all we ever actually measure is relative displacements in space. We design objects (physical or mathematical) to convert these displacements into quantities or units of interest and might even decorate such with some additional structure beyond the reals, but in the end, we are measuring distances relative to a standard. This account becomes somewhat tricky when digital and/or electronic measurements are taken into account, but goes through, I believe.

When I say measurements are real I mean that displacements between objects in space are represented with real numbers.

You make a good and interesting point as to whether the actual structure of the reals, which is, as you say, pretty strange, meaningfully corresponds to relative displacements in space, but this is a separate point. If we wished to be finitist, we could argue that we measure over a sparse subset of the reals or something like that or we could define various methods of putting the rational numbers to use for this purpose. But my larger point is that in the end the physical world appears to be entirely sensible to us only as relative displacements of objects in space and these appear to map to something very like the real numbers.

In fact, physics at its most basic is encoded in these terms as well, where any system is conceptualized as being encoded by its q's, which correspond to relative displacements, and the generators of their motion, roughly speaking either the time derivatives of the q's or their conjugate momenta. But the q's are what we have to work with. At any instant in time we must lay our rulers out, one way or another, and then construct any other physical property of interest in terms of those displacements.

[feoren]

A "measurement" in the general sense (not quantum) is about amplifying or isolating some signal out of all the noise. I simply don't agree that all measurements boil down to relative displacements. Did a chemical reaction occur or not? Is it red or is it blue? Sure, in many of these measurements, something tangentially related to distance (like light wavelength) is involved, but lots of things are involved: measurements are inherently "macro" phenomena, which means many processes will be involved, not the least of which is electrochemical signaling in the brain of the measurer. It's just far too reductive to say that every quantity we measure is actually relative displacement. I could just as justifiably (or more so) say that every measurement is actually just measuring the strength of an electromagnetic field. Why is this helping? What does this have to do with which numbers "truly exist" or not?

[nathan_compton]

I guess we should both read the SEP page on measurement.

https://plato.stanford.edu/entries/measurement-science/

Note that Bertrand Russel is on "team Nathan" in the sense that he thinks measurements relate to (at most) the reals.

I don't think anyone ever really measures the elecromagnetic field. We might, for example, measure the displacement of a charged object attached to a spring in an electromagnetic field. Or, if the field is changing in time we measure that displacement as a function of the position of the hands on a clock. But it is very hard for me to think of a situation where we measure something other than a distance at its most basic level. Even in a DAC we measure voltage relative to some calibrated voltage which we measured using a voltmeter which shows us our answer as a deflection in a meter.

This is particularly relevant in QM because in fact all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.

[feoren]

> I don't think anyone ever really measures the elecromagnetic field. We might, for example, measure the displacement of a charged object attached to a spring in an electromagnetic field.

I don't think anyone ever really measures the displacement of a charged object attached to a spring in an electromagnetic field. We might, for example, measure the difference in strength of neuronal activation in our brain from neurotransmitters emitted in a chain traveling from the photoreceptor cells in our fovea, which are responding to differing quantities of photoisomers that have had their shape altered by the absorption of different frequencies of photons reflected differently off the needle of a gauge in the display of an instrument which is measuring the displacement of a charged object attached to a spring in an electromagnetic field.

This is what I mean when I say it's overly reductive to say a measurement is necessarily a displacement. A measurement is lots of things, and not all of them can be represented as a spatial displacement unless you really shoehorn it.

> all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.

Is that even true? Spin is a measurable quantity, and you cannot possibly get a spin of 0.2. Most measurable quantum numbers are essentially integers (integer multiples of some conventional base). Remember this discussion is about whether complex numbers are "lower-class citizens" than real numbers in physics. If you're going for measurable quantum numbers, these are almost all counterexamples to the idea that real numbers are special, and instead hint that it's simply integers that are the only first-class citizens in physics. I also fundamentally disagree that the possible eigenvalues of hermitian operators are the sole criteria we should be using to "rank" the truthiness or realness of mathematical structures.

I deeply do not care what philosophy has to say about this, either. Philosophy as a discipline is completely incapable of determining the truth, because it is unwilling to ever reject a single idea; all it is is a giant collection of shower thoughts. Every philosophy page, including the one you linked, somewhere includes "According to Aristotle ...". If you're trying to learn evolutionary biology and you read "according to Lamarck ...", then you are not learning science, you're reading science history. Yet this is all philosophy ever can say about anything. Science curates ideas; philosophy hoards them.