Real numbers are not “real”. (Inspired by Imaginary numbers are not real, an elementary introduction to Clifford Algebra I came across a long time ago).
I find it a bit funny that people tend to think of real numbers as “real” numbers, as opposed to, say, imaginary numbers, which are not only not real, but also not “real” in a way a Realist would use the word. The paper above even takes pride in not using i in calculations. There is also an occasional discussion in philosophy papers and online of the wave function in QM not being “real” because it uses imaginary numbers.
I find it funny because real numbers are no more “real” than any other numbers. Even the set of all integers is not very “real”, as basically everything in the Universe is finite, due to the cut-offs at various scales, such as the Planck scale and the age of the Universe, and whenever you try to disregard these cut-offs, things tend to blow up in your face.
One can potentially consider finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate. The rest are simply useful mathematical abstractions.
One would think that, given that many useful numbers like e and pi are no more “real” than i or infinity, people would get a clue and stop arguing, but no.
I’ve never thought of real numbers as any more real (in the non-mathematical sense of the word) than other numbers, and I’ve been peeved by popularizations which use “real” and “imaginary” without making it clear that they’re using them with a specific technical meaning (e.g. stuff like “special relativity has shown that if space is real time must be imaginary, and vice versa”—yeah, they do have squares with opposite signs (though modern notation uses real 4-vectors and a non-positive-definite metric), but that’s not how a reader would be most likely to interpret that sentence).
Agreed. I have no beef with the term ‘complex’ for the complex numbers. It’s the ‘real’ for the others, and the ‘imaginary’ for the new stuff, that I mind.
I wonder if a very short treatment of abstract algebra should be given in high school, right before you get to complex numbers. Might reduce the number chauvinism and help with the illusion of number realism.
I wonder if a very short treatment of abstract algebra should be given in high school
Maybe in an AP-level course? The high-school math is pretty instrumental, focused on solving problems and passing tests. Actually, I think this is probably best covered in a relevant college-level philosophy course.
I’ve taught a few people about the complex numbers, by stepping through expanding the naturals with the introduction of negatives to make integers, fractions to make rationals, irrationals to make reals, and finally (the ‘novel’ stage for my audience) imaginary numbers to make the complex numbers.
I emphasise the point that the new system always seems weird and confusing at first to the people who aren’t used to it, and sometimes gets given a nasty name in contrast to the nice name of the old system (especially ‘imaginary’ vs ‘real’ and ‘irrational’ vs ‘rational’) but the new numbers are never more or less worthwhile than the old system—they’re just different, and useful in new ways.
One can potentially consider finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate.
Lots of numbers correspond to things. It seems arbitrary to say that integers are real numbers, because they correspond to discrete objects we see, but rotations aren’t numbers, even though they correspond to transformations we can make on an object.
I more or less agree with you that the terms are inappropriate and often actively misleading. In relation to this, I’ve often wondered about the usual justification for the postulate that observables in quantum mechanics must be represented by Hermitian operators. The justification is that the expectation values of our observables must be real numbers. Doesn’t this sort of justification stem from an intuition that real numbers are somehow more real than imaginary numbers? The attitude seems to be: Imaginary numbers can certainly play a theoretical role in physics, but when it comes to describing our immediate empirical situation, quantification must be in terms of real numbers—it makes no sense for a measurement outcome to be described by a complex number.
Maybe this doesn’t really represent the crux of the justfication. The argument might be that quantum mechanics needs to approximate classical mechanics at the macroscopic level, and classical mechanics only involves real numbers. Since observables correspond to measurement procedures, and measurement procedures are macroscopic, observables must have real expectation values. This argument seems fine if you genuinely think of observables as representing measurement procedures, but if you think of observables as representing measurable properties (this seems to be a common perspective) then these properties need not be macroscopic, in which case it is unclear why they are constrained by a requirement to replicate classical mechanics.
I’m not questioning the claim that observables must be Hermitian, just suggesting that the usual justification isn’t really that great. Maybe we should provide other arguments for this claim when teaching QM, in order to avoid entrenching the “real numbers are more real” fallacy. For instance, one could argue that the quantities we measure in QM experiments (energy, momentum, spin) are quantities that are conserved if certain symmetries hold (I think one might be able to construct an argument that any quantity that is properly considered measurable must satisfy this constraint). Symmetry transformations are represented by unitary operators, so measurable quantities must be represented by generators of such transformations, which must be Hermitian operators. I haven’t thought through this argument carefully, but my intuition is that something of this sort is a better justification for the preference for Hermitian operators than the constraint that expectation values must be real.
I’d work backwards. As I said, “finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate”. If you accept that, instrumentally, we can only ever see a count of something, like the number of notches on a ruler, or marks to the left of the needle of some gauge, or the number of clock ticks before a certain event, then you can extrapolate backwards in your logic to include negative numbers, fractions, rationals, reals and complex numbers as convenient abstractions. But you have to leave a path forward in your models to eventually end up with the number of notches on a dial. If your observables are not Hermitian, you cut off this path from models to counts.
If that’s your view, though, I don’t see why you think real expectation values are better than complex ones. If our measurements have the structure of integers, and our expectation values are real, then in our model of measurement we’re going to have a map from the real line (or some segment thereof) representing possible expectation values to the integers representing notches on your dial. This map will presumably associate an element of some partition of the real line into finite regions with each integer. So this map must be part of your model of measurement in order to connect your integer measurements with the continuous space of expectation values allowed by the theory.
But if you’re relying on some such map anyway, why can’t the expectation values be complex? You could construct a map from a partition of the complex plane to the integers. More specifically, your claim that the ultimate measurement results must be integers doesn’t rule out the possibility that each measurement result gives a pair of integers (instead of a dial moving along a single dimension, think of one that can move in two dimensions). For such measurements, it may even be more natural to think of a map from the complex plane to the measurement outcome, with one of the integers associated with the real part and the other with the imaginary part. As long as you allow for this possibility, I don’t see the motivation for restricting observables to Hermitian operators. Surely it isn’t a requirement of instrumentalism that the measurements we make must only be one-dimensional.
Surely it isn’t a requirement of instrumentalism that the measurements we make must only be one-dimensional.
Yeah, that would be a dangling requirement.
More specifically, your claim that the ultimate measurement results must be integers doesn’t rule out the possibility that each measurement result gives a pair of integers (instead of a dial moving along a single dimension, think of one that can move in two dimensions).
That would be an interesting counterfactual universe, where you need at least two numbers to make sense of anything. I cannot quite imagine how this would work, but maybe this is a limit of my imagination. In the Universe we are currently stuck with, with the classical world described by classical mechanics based on real numbers, and the quantum world manifesting itself only through classical measurements, any useful model must ultimately lead to a set of readings off some dial, each of which should make sense separately.
One can potentially consider finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate. The rest are simply useful mathematical abstractions.
Real numbers are not “real”. (Inspired by Imaginary numbers are not real, an elementary introduction to Clifford Algebra I came across a long time ago).
I find it a bit funny that people tend to think of real numbers as “real” numbers, as opposed to, say, imaginary numbers, which are not only not real, but also not “real” in a way a Realist would use the word. The paper above even takes pride in not using i in calculations. There is also an occasional discussion in philosophy papers and online of the wave function in QM not being “real” because it uses imaginary numbers.
I find it funny because real numbers are no more “real” than any other numbers. Even the set of all integers is not very “real”, as basically everything in the Universe is finite, due to the cut-offs at various scales, such as the Planck scale and the age of the Universe, and whenever you try to disregard these cut-offs, things tend to blow up in your face.
One can potentially consider finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate. The rest are simply useful mathematical abstractions.
One would think that, given that many useful numbers like e and pi are no more “real” than i or infinity, people would get a clue and stop arguing, but no.
I’ve never thought of real numbers as any more real (in the non-mathematical sense of the word) than other numbers, and I’ve been peeved by popularizations which use “real” and “imaginary” without making it clear that they’re using them with a specific technical meaning (e.g. stuff like “special relativity has shown that if space is real time must be imaginary, and vice versa”—yeah, they do have squares with opposite signs (though modern notation uses real 4-vectors and a non-positive-definite metric), but that’s not how a reader would be most likely to interpret that sentence).
Agreed. I have no beef with the term ‘complex’ for the complex numbers. It’s the ‘real’ for the others, and the ‘imaginary’ for the new stuff, that I mind.
I wonder if a very short treatment of abstract algebra should be given in high school, right before you get to complex numbers. Might reduce the number chauvinism and help with the illusion of number realism.
Never heard this term before :)
Maybe in an AP-level course? The high-school math is pretty instrumental, focused on solving problems and passing tests. Actually, I think this is probably best covered in a relevant college-level philosophy course.
I’ve taught a few people about the complex numbers, by stepping through expanding the naturals with the introduction of negatives to make integers, fractions to make rationals, irrationals to make reals, and finally (the ‘novel’ stage for my audience) imaginary numbers to make the complex numbers.
I emphasise the point that the new system always seems weird and confusing at first to the people who aren’t used to it, and sometimes gets given a nasty name in contrast to the nice name of the old system (especially ‘imaginary’ vs ‘real’ and ‘irrational’ vs ‘rational’) but the new numbers are never more or less worthwhile than the old system—they’re just different, and useful in new ways.
Lots of numbers correspond to things. It seems arbitrary to say that integers are real numbers, because they correspond to discrete objects we see, but rotations aren’t numbers, even though they correspond to transformations we can make on an object.
I more or less agree with you that the terms are inappropriate and often actively misleading. In relation to this, I’ve often wondered about the usual justification for the postulate that observables in quantum mechanics must be represented by Hermitian operators. The justification is that the expectation values of our observables must be real numbers. Doesn’t this sort of justification stem from an intuition that real numbers are somehow more real than imaginary numbers? The attitude seems to be: Imaginary numbers can certainly play a theoretical role in physics, but when it comes to describing our immediate empirical situation, quantification must be in terms of real numbers—it makes no sense for a measurement outcome to be described by a complex number.
Maybe this doesn’t really represent the crux of the justfication. The argument might be that quantum mechanics needs to approximate classical mechanics at the macroscopic level, and classical mechanics only involves real numbers. Since observables correspond to measurement procedures, and measurement procedures are macroscopic, observables must have real expectation values. This argument seems fine if you genuinely think of observables as representing measurement procedures, but if you think of observables as representing measurable properties (this seems to be a common perspective) then these properties need not be macroscopic, in which case it is unclear why they are constrained by a requirement to replicate classical mechanics.
I’m not questioning the claim that observables must be Hermitian, just suggesting that the usual justification isn’t really that great. Maybe we should provide other arguments for this claim when teaching QM, in order to avoid entrenching the “real numbers are more real” fallacy. For instance, one could argue that the quantities we measure in QM experiments (energy, momentum, spin) are quantities that are conserved if certain symmetries hold (I think one might be able to construct an argument that any quantity that is properly considered measurable must satisfy this constraint). Symmetry transformations are represented by unitary operators, so measurable quantities must be represented by generators of such transformations, which must be Hermitian operators. I haven’t thought through this argument carefully, but my intuition is that something of this sort is a better justification for the preference for Hermitian operators than the constraint that expectation values must be real.
I’d work backwards. As I said, “finite integers as the most “real”, given that they correspond to discrete objects we can see, count and calculate”. If you accept that, instrumentally, we can only ever see a count of something, like the number of notches on a ruler, or marks to the left of the needle of some gauge, or the number of clock ticks before a certain event, then you can extrapolate backwards in your logic to include negative numbers, fractions, rationals, reals and complex numbers as convenient abstractions. But you have to leave a path forward in your models to eventually end up with the number of notches on a dial. If your observables are not Hermitian, you cut off this path from models to counts.
If that’s your view, though, I don’t see why you think real expectation values are better than complex ones. If our measurements have the structure of integers, and our expectation values are real, then in our model of measurement we’re going to have a map from the real line (or some segment thereof) representing possible expectation values to the integers representing notches on your dial. This map will presumably associate an element of some partition of the real line into finite regions with each integer. So this map must be part of your model of measurement in order to connect your integer measurements with the continuous space of expectation values allowed by the theory.
But if you’re relying on some such map anyway, why can’t the expectation values be complex? You could construct a map from a partition of the complex plane to the integers. More specifically, your claim that the ultimate measurement results must be integers doesn’t rule out the possibility that each measurement result gives a pair of integers (instead of a dial moving along a single dimension, think of one that can move in two dimensions). For such measurements, it may even be more natural to think of a map from the complex plane to the measurement outcome, with one of the integers associated with the real part and the other with the imaginary part. As long as you allow for this possibility, I don’t see the motivation for restricting observables to Hermitian operators. Surely it isn’t a requirement of instrumentalism that the measurements we make must only be one-dimensional.
Yeah, that would be a dangling requirement.
That would be an interesting counterfactual universe, where you need at least two numbers to make sense of anything. I cannot quite imagine how this would work, but maybe this is a limit of my imagination. In the Universe we are currently stuck with, with the classical world described by classical mechanics based on real numbers, and the quantum world manifesting itself only through classical measurements, any useful model must ultimately lead to a set of readings off some dial, each of which should make sense separately.
I fail to see the contrast.