7 ms·

One could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex n

One could argue that all numbers are an abstraction; they don't "exist" in any physical form, even integers. Therefore, why is it such a surprise that complex numbers are sometimes also needed to model reality?

deleted 6y ago
[deleted]
j-pb 6y ago
Interestingly enough it would be "especially" integers that don't exist physically in mosts scales of reality if physics is to have any mathematical coherence.
- zeroonetwothree 6y ago
  Eh, because of quantisation you could argue only integers exist ;)
  - eigenket 6y ago
    Quantised is not synonymous with discretised. The fact that (for example) energy in a simple harmonic oscillator comes in discrete packets doesn't mean that the kinetic energy of a free particle is. Some quantities in quantum mechanics are as continuous as their classical analogues.
  - GoblinSlayer 6y ago
    Then you learn about Zeeman effect.
- jMyles 6y ago
  The phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized. It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force. And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
  - eigenket 6y ago
    Quantum mechanics is not a discrete theory in the sense you seem to be suggesting. There is this idea that is is, which comes from the discrete energy levels it predicts for a simple harmonic oscillator, or the hydrogen atom, but quantities like the momentum, kinetic energy, position etc of a quantum particle are completely allowed to vary continuously. In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
    - jMyles 6y ago
      I didn't mean to paint it as a discrete theory (and in fact I added the shoutout to the Loop Quantum Gravity stuff at the end in part because I was concerned about this interpretation). However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
      - eigenket 6y ago
        > However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right? I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
        foxes 6y ago
        To add, perhaps in a very general hand wavy way "quantisation" is something like replacing a commutative algebra A, with a non commutative algebra A_h. I do not necessarily think there is something discrete in this process. In physics functions get sent to operators. States get sent to rays. It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
- abdullahkhalids 6y ago
  One of the most influential physicists of all time, John Wheeler, was of the philosophical opinion that physical laws must only be describable in terms of a finite integers [1]. This is an ideal that at least some other physicists hope for. Recently, the "It from Qubit" program in physics [2], attempts to understand reality by reducing everything to computation on qubits, and has led to some interesting insights. [1] Information, physics, quantum: The search for links, John Wheeler 1990 [[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-sciences/it-from-qubit/ https://www.simonsfoundation.org/mathematics-physical-scienc... Edit: fixed citation 1. And removed reference to infinities, as indicated below.
  - GoblinSlayer 6y ago
    Continuum is not infinity.
    - abdullahkhalids 6y ago
      You are correct. I only had distant memories. I have edited.
  - j-pb 6y ago
    Do you know if these efforts also only limit the formula describing these Qbit to computable reals? I think it's a quite interesting crossroad. Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven. Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
    - GoblinSlayer 6y ago
CyberRabbi 6y ago
Makes me wonder about all the other things that don’t materially exist yet are nonetheless true and not arbitrary and would be discovered even if humans did not exist.
101008 6y ago
If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2. Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).
- random314 6y ago
  "God made the integers; all else is the work of man." Kronecker
- germanier 6y ago
  (I assume you meant natural numbers, as "-1" is not a useful cardinality for a set) The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
  - 101008 6y ago
    Yes, sorry, you are right. Natural numbers, not integeres. My bad.
- GoblinSlayer 6y ago
  If you split one apple in two, you get two apples?
  - viklove 6y ago
    Technically you have 0 apples, and you have 2 halves of an apple.
  - emergie 6y ago
abdullahkhalids 6y ago
Perhaps, you would appreciate a similar question. Why does the buck stop at complex numbers, and we don't need quaternions or some other "weirder" field to describe reality? The answer will have to grapple with both the philosophy of physics and the properties of these various fields.
- Twisol 6y ago
  Another commenter linked this post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021 https://www.scottaaronson.com/blog/?p=4021 The money quote (which leads a few more paragraphs on the details) is this: > But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.
  - phkahler 6y ago
    But elegance is irrelevant. Imaginary numbers are a subset of the quaternions, just as reals are a subset of the imaginaries. The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
    - Twisol 6y ago
      The referenced argument rests on superluminal communication, not elegance. The quaternions are not even sufficient, unless we believe that superluminal signaling is physical.
bloak 6y ago
I tend to agree with you, despite the famous quotation: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man")
- michelpp 6y ago
  Imagine going back to Kronecker and their contemporaries and telling them that a spinning top weighs more than a top at rest. They would have laughed you out of the room, but a few years later the arrogant scientific confidence of the 19th century was completely crushed by quantum mechanics and general relativity. Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful. "Clouds are not spheres, Mountains are not cones" - Mandelbrot
  - EthanHeilman 6y ago
    A spinning top weighs more than a top at rest? I get that mass increases with speed, but that works in a rotating system as well as a linear system?
    - skzv 6y ago
      At any given instant, velocity is linear. That is, if you take a snapshot of the top, you can draw a straight vector for each particle of the top, indicating the direction of velocity. And, if the top suddenly flew apart, each piece would continue in a straight line, following its velocity at that instant.
      - EthanHeilman 6y ago
        Can velocity exist in a snapshot?
        skzv 6y ago
        Or rephrased: is instantaneous velocity real? Are limits real, or just a mathematical invention? Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.
ko27 6y ago
Gödel, Escher, Bach (book by Douglas Hofstadter) actually touches on this topic. They are an abstraction, but they are isomorphic to reality. The big news here is that we thought complex numbers have no isomorphism with reality (physics) that couldn't be explained with real numbers only. https://en.wikipedia.org/wiki/Isomorphism https://en.wikipedia.org/wiki/Isomorphism
- GoblinSlayer 6y ago
  Complex numbers themselves can be explained with real numbers only.
  - ko27 6y ago
    You didn't define what you meant by "explained". You are more then welcome to explain the results of the linked study (once its released) with real numbers only.
    - GoblinSlayer 6y ago
      https://news.ycombinator.com/item?id=26329883 https://news.ycombinator.com/item?id=26329883 - like this. AIU the article disproves a specific theory that threw away the imaginary part of wave function or something like that, so the result is kinda expected, but it doesn't mean that complex numbers are indispensable.
      - ko27 6y ago
        That doesn't explain complex numbers with real numbers (matrices are not real numbers). So the point of the article, and my comment still stands.
        GoblinSlayer 6y ago
        You can say that about anything, like real quantum mechanics isn't just real numbers, but also functions, equations, operators and a lot of stuff like that.

deleted 6y ago

[deleted]

j-pb 6y ago

Interestingly enough it would be "especially" integers that don't exist physically in mosts scales of reality if physics is to have any mathematical coherence.

zeroonetwothree 6y ago
Eh, because of quantisation you could argue only integers exist ;)
- eigenket 6y ago
  Quantised is not synonymous with discretised. The fact that (for example) energy in a simple harmonic oscillator comes in discrete packets doesn't mean that the kinetic energy of a free particle is. Some quantities in quantum mechanics are as continuous as their classical analogues.
- GoblinSlayer 6y ago
  Then you learn about Zeeman effect.
jMyles 6y ago
The phrase "mosts [sic] scales of reality" is doing an incredible amount of heavy lifting here, though, since the most reliable mathematics describing three of what appear to be the four fundamental forces of nature are indeed quantized. It's more accurate to say that only integers appear to be relevant in describing the nature of the nuclear forces and electromagnetism/electroweak force. And it seems that, at least for the moment, we're headed in a direction of quantization of gravity and a Planck unit of time as well.
- eigenket 6y ago
  Quantum mechanics is not a discrete theory in the sense you seem to be suggesting. There is this idea that is is, which comes from the discrete energy levels it predicts for a simple harmonic oscillator, or the hydrogen atom, but quantities like the momentum, kinetic energy, position etc of a quantum particle are completely allowed to vary continuously. In some sense (because of the wavefunction) quantum mechanics is less discrete than classical mechanics, a qubit system which would be classicaly described by a single bit is described by a state that sits on a continuous manifold (the Bloch sphere).
  - jMyles 6y ago
    I didn't mean to paint it as a discrete theory (and in fact I added the shoutout to the Loop Quantum Gravity stuff at the end in part because I was concerned about this interpretation). However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right?
    - eigenket 6y ago
      > However, across quantum mechanics, there is at least unity on the 'quantum' part. And that's really the point of contention here in terms of integer values, right? I don't know what this means, but there is this idea that quantum mechanics leads to integer valued stuff because the first models studied were things like the hydrogen atom and the simple harmonic oscillator which both have a natural integer valued observable (the energy). This is not generally the case, however and in quantum mechanics there are many observables which are not integer valued, for example the kinetic energy of a free particle can be any non-negative real.
      - foxes 6y ago
        To add, perhaps in a very general hand wavy way "quantisation" is something like replacing a commutative algebra A, with a non commutative algebra A_h. I do not necessarily think there is something discrete in this process. In physics functions get sent to operators. States get sent to rays. It turns out that a lot of interesting physics these operators have physically interesting discrete spectrum.
abdullahkhalids 6y ago
One of the most influential physicists of all time, John Wheeler, was of the philosophical opinion that physical laws must only be describable in terms of a finite integers [1]. This is an ideal that at least some other physicists hope for. Recently, the "It from Qubit" program in physics [2], attempts to understand reality by reducing everything to computation on qubits, and has led to some interesting insights. [1] Information, physics, quantum: The search for links, John Wheeler 1990 [[2] The Simon's institute has formally funded such a program, though the movement within physics is much larger https://www.simonsfoundation.org/mathematics-physical-sciences/it-from-qubit/ https://www.simonsfoundation.org/mathematics-physical-scienc... Edit: fixed citation 1. And removed reference to infinities, as indicated below.
- GoblinSlayer 6y ago
  Continuum is not infinity.
  - abdullahkhalids 6y ago
    You are correct. I only had distant memories. I have edited.
- j-pb 6y ago
  Do you know if these efforts also only limit the formula describing these Qbit to computable reals? I think it's a quite interesting crossroad. Either reals don't exist, and everything is at most turing complete, in which case the universe is detemernistic and predictable, yet there are very hard boundaries to what can be known and proven. Or reals DO exist in which case the church turing thesis is false, as the universe (with it's operations on reals) is a reified example of a super turing system, in which case we can know a lot more things than in the discrete case, but some of that knowledge can only be grasped intuitively, because the moment you try to articulate it into formulas, it's discretized and thus limited to your sub-universe/brain powerfull turing machine.
  - GoblinSlayer 6y ago

CyberRabbi 6y ago

Makes me wonder about all the other things that don’t materially exist yet are nonetheless true and not arbitrary and would be discovered even if humans did not exist.

101008 6y ago

If you follow the construction of numbers, integers are defined as how many elements are in a set. So integers kind of exist: if you have an empty bag, and add an apple, 1 is the cardinality of that bag. If you add another apple, then the cardinality is 2. Then you define all the operations between integers as operations of cardinality between set. I really enjoyed that semester when I was getting my degree on Maths, by the way, although I use the knowledge I got there as a curiosity at parties, because it didn't prove useful (not that it is bad).

random314 6y ago
"God made the integers; all else is the work of man." Kronecker
germanier 6y ago
(I assume you meant natural numbers, as "-1" is not a useful cardinality for a set) The construction you refer to is due to Frege and Russell but it has its own kinds of subtleties. In particular it does not work in ZFC which we often base upon, so we usually use Peano's axioms instead to define them (depending on your context of course). While they are not all too surprising, I wouldn't call them absolutely obvious.
- 101008 6y ago
  Yes, sorry, you are right. Natural numbers, not integeres. My bad.
GoblinSlayer 6y ago
If you split one apple in two, you get two apples?
- viklove 6y ago
  Technically you have 0 apples, and you have 2 halves of an apple.
- emergie 6y ago

abdullahkhalids 6y ago

Perhaps, you would appreciate a similar question. Why does the buck stop at complex numbers, and we don't need quaternions or some other "weirder" field to describe reality? The answer will have to grapple with both the philosophy of physics and the properties of these various fields.

Twisol 6y ago
Another commenter linked this post by Scott Aaronson: https://www.scottaaronson.com/blog/?p=4021 https://www.scottaaronson.com/blog/?p=4021 The money quote (which leads a few more paragraphs on the details) is this: > But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe.
- phkahler 6y ago
  But elegance is irrelevant. Imaginary numbers are a subset of the quaternions, just as reals are a subset of the imaginaries. The proper question is not "do I like it" but is it necessary. The answer apparently is No for the quaternions.
  - Twisol 6y ago
    The referenced argument rests on superluminal communication, not elegance. The quaternions are not even sufficient, unless we believe that superluminal signaling is physical.

bloak 6y ago

I tend to agree with you, despite the famous quotation: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man")

michelpp 6y ago
Imagine going back to Kronecker and their contemporaries and telling them that a spinning top weighs more than a top at rest. They would have laughed you out of the room, but a few years later the arrogant scientific confidence of the 19th century was completely crushed by quantum mechanics and general relativity. Science is not a body of knowledge, it is a frontier of ignorance. I like to "feel" that complex numbers are part of the fabric of the universe, if only because it would be more beautiful. "Clouds are not spheres, Mountains are not cones" - Mandelbrot
- EthanHeilman 6y ago
  A spinning top weighs more than a top at rest? I get that mass increases with speed, but that works in a rotating system as well as a linear system?
  - skzv 6y ago
    At any given instant, velocity is linear. That is, if you take a snapshot of the top, you can draw a straight vector for each particle of the top, indicating the direction of velocity. And, if the top suddenly flew apart, each piece would continue in a straight line, following its velocity at that instant.
    - EthanHeilman 6y ago
      Can velocity exist in a snapshot?
      - skzv 6y ago
        Or rephrased: is instantaneous velocity real? Are limits real, or just a mathematical invention? Well, I think instantaneous velocity is real, and I think I have a convincing argument. Imagine you have a piece of gum stuck to your bike tire. As your tire spins, the piece of gum spins with it, and its velocity is constantly changing. But when the piece of gum flies off your tire, it flies off in a straight line, following the instantaneous velocity at the moment of separation. Seeing as the gum can fly off in a straight line, its instantaneous velocity at that moment must be well-defined.

ko27 6y ago

Gödel, Escher, Bach (book by Douglas Hofstadter) actually touches on this topic. They are an abstraction, but they are isomorphic to reality. The big news here is that we thought complex numbers have no isomorphism with reality (physics) that couldn't be explained with real numbers only. https://en.wikipedia.org/wiki/Isomorphism https://en.wikipedia.org/wiki/Isomorphism

GoblinSlayer 6y ago
Complex numbers themselves can be explained with real numbers only.
- ko27 6y ago
  You didn't define what you meant by "explained". You are more then welcome to explain the results of the linked study (once its released) with real numbers only.
  - GoblinSlayer 6y ago
    https://news.ycombinator.com/item?id=26329883 https://news.ycombinator.com/item?id=26329883 - like this. AIU the article disproves a specific theory that threw away the imaginary part of wave function or something like that, so the result is kinda expected, but it doesn't mean that complex numbers are indispensable.
    - ko27 6y ago
      That doesn't explain complex numbers with real numbers (matrices are not real numbers). So the point of the article, and my comment still stands.
      - GoblinSlayer 6y ago
        You can say that about anything, like real quantum mechanics isn't just real numbers, but also functions, equations, operators and a lot of stuff like that.