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Interesting post. Can you please flesh out the following? I don’t quite follow it.. > having infinitely many states would require them to be infinitesimally s

Interesting post. Can you please flesh out the following? I don’t quite follow it.. > having infinitely many states would require them to be infinitesimally similar, and hence indistinguishable from each other within a finite amount of time

chriswarbo 5y ago
Sure. Let's take a finite region, like a room. We don't want to get too bogged down in the details of human psychology, anatomy, etc. so we'll stick to something very low-level like particle physics (which, in principle, can describe a person in exact detail). One state of this region is that it's completely empty (maybe quantum effects rule that out, but we can still include it if we're being conservative ;) ). Another state could have, say, a single electron, at some position we'll call (0, 0, 0). Another could have a single electron at position (0, 0, 1). Another has an electron at (25, 5, 3) and a proton at position (0, 0, 0). And so on. Let's consider the states which just contain a single electron, somewhere in the room (I'm not bothering with quantum effects, but we would do the same thing just in Hilbert space instead of 3D space). The region is finite, so the coordinates of this electron are bounded: if the boundaries are at, say, 0 and 100 in each axis, then putting the electron at position (101, 0, 0) would be the same state as the empty one we started with (since the electron is outside the region we care about, and there's nothing else). Now let's say we do some measurement of this electron's position, which is only accurate to whole number coordinates. That gives us 100^3 = 1,000,000 distinguishable states with a single electron. We might imagine all sorts of different states 'in between', but they can't affect that measurement due to its limited resolution. If we increase the resolution of our measurement by 10, we get 10^3 times as many distinguishable states; another factor of 10 gives 10^3 times as many states again; and so on. However, regardless of how finely we can resolve the electron's position, we can only distinguish between finitely-many states. Any differences finer than that limit are irrelevant to the measurement, and hence to any behaviour that depends on that measurement. If we could resolve between infinitely-many positions for the electron, that would require distinguishing between states which are infinitesimally-close together, i.e. on an infinitely-fine grid; this would require distinguishing between two numbers whose decimals only differ after infinitely many digits. This doesn't seem like a reasonable capability. The same principle applies when we have two electrons in the room, or trillions, or any combination of electrons, protons, photons, etc. in any arrangement we like. A similar thing happens when we reach really large numbers too, e.g. trying to put as many particles in the region as we can: at some point the relative difference between, say, a googolplex and five particles versus a googolplex and six particles becomes too small to distinguish. There will eventually be a limit. (This is like the resolution problem, but instead of adding decimal places to the right of each number, we're adding factors of 10 to their left) One way to get around this problem of limited resolution is to avoid an explicit 'measurement', and instead have the region itself behave differently for different states. A good example is chaotic behaviour, where tiny changes in an initial state will grow exponentially until those differences eventually become distinguishable. However, the infinitesimally-similar states described above will remain infinitesimally close for all finite lengths of time; in order to distinguish between them, we would need to wait an infinite amount of time.
- lurquer 5y ago
  Ah. I hadn’t thought of that. So putting aside Plank and all the experimental stuff, even on a philosophical level there are problems with thinking of space as continuous. For, that with which we measure space is necessarily discrete and the time-frames in which we measure are necessarily finite. So, even if space was continuous, two ‘states’ that are infinitesimally similar could for ALL practical purposes be considered identical. Is that it?
  - chriswarbo 5y ago
    Pretty much. As a far less formal analogy, it's similar to how people complain that digital audio in general has lower quality than analogue, just from a discrete vs continuous standpoint. When in fact there's no limit to the quality of a digital representation (if we liked, we could store a gigabyte per sample, or a petabyte, or whatever). Yet we can always draw the line somewhere, beyond which there are no meaningful distinctions.

chriswarbo 5y ago

Sure. Let's take a finite region, like a room. We don't want to get too bogged down in the details of human psychology, anatomy, etc. so we'll stick to something very low-level like particle physics (which, in principle, can describe a person in exact detail). One state of this region is that it's completely empty (maybe quantum effects rule that out, but we can still include it if we're being conservative ;) ). Another state could have, say, a single electron, at some position we'll call (0, 0, 0). Another could have a single electron at position (0, 0, 1). Another has an electron at (25, 5, 3) and a proton at position (0, 0, 0). And so on. Let's consider the states which just contain a single electron, somewhere in the room (I'm not bothering with quantum effects, but we would do the same thing just in Hilbert space instead of 3D space). The region is finite, so the coordinates of this electron are bounded: if the boundaries are at, say, 0 and 100 in each axis, then putting the electron at position (101, 0, 0) would be the same state as the empty one we started with (since the electron is outside the region we care about, and there's nothing else). Now let's say we do some measurement of this electron's position, which is only accurate to whole number coordinates. That gives us 100^3 = 1,000,000 distinguishable states with a single electron. We might imagine all sorts of different states 'in between', but they can't affect that measurement due to its limited resolution. If we increase the resolution of our measurement by 10, we get 10^3 times as many distinguishable states; another factor of 10 gives 10^3 times as many states again; and so on. However, regardless of how finely we can resolve the electron's position, we can only distinguish between finitely-many states. Any differences finer than that limit are irrelevant to the measurement, and hence to any behaviour that depends on that measurement. If we could resolve between infinitely-many positions for the electron, that would require distinguishing between states which are infinitesimally-close together, i.e. on an infinitely-fine grid; this would require distinguishing between two numbers whose decimals only differ after infinitely many digits. This doesn't seem like a reasonable capability. The same principle applies when we have two electrons in the room, or trillions, or any combination of electrons, protons, photons, etc. in any arrangement we like. A similar thing happens when we reach really large numbers too, e.g. trying to put as many particles in the region as we can: at some point the relative difference between, say, a googolplex and five particles versus a googolplex and six particles becomes too small to distinguish. There will eventually be a limit. (This is like the resolution problem, but instead of adding decimal places to the right of each number, we're adding factors of 10 to their left) One way to get around this problem of limited resolution is to avoid an explicit 'measurement', and instead have the region itself behave differently for different states. A good example is chaotic behaviour, where tiny changes in an initial state will grow exponentially until those differences eventually become distinguishable. However, the infinitesimally-similar states described above will remain infinitesimally close for all finite lengths of time; in order to distinguish between them, we would need to wait an infinite amount of time.

lurquer 5y ago
Ah. I hadn’t thought of that. So putting aside Plank and all the experimental stuff, even on a philosophical level there are problems with thinking of space as continuous. For, that with which we measure space is necessarily discrete and the time-frames in which we measure are necessarily finite. So, even if space was continuous, two ‘states’ that are infinitesimally similar could for ALL practical purposes be considered identical. Is that it?
- chriswarbo 5y ago
  Pretty much. As a far less formal analogy, it's similar to how people complain that digital audio in general has lower quality than analogue, just from a discrete vs continuous standpoint. When in fact there's no limit to the quality of a digital representation (if we liked, we could store a gigabyte per sample, or a petabyte, or whatever). Yet we can always draw the line somewhere, beyond which there are no meaningful distinctions.