It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
That works if you're fixed to a grid, but the argument is much more general: it allows symbols to be of arbitrary size (as long as they're still within the unit square), in any orientation, formed by continuous or (not-too-pathological) discontinuous penstrokes, etc.
It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
Yes it's easier and not too different from Turing's argument. However, Turing made this proof in 1936, you know, before the concept of pixels exists.
Is that counterintuitive? There's a finite number of electrons in a human brain (which fits in a space of size 1m^3), and information takes time to propagate across any distance, and humans die before 150 years; this all gestures at there being not only finitely many personality types and ways of thinking, but finitely many human mind states .
I guess that means reincarnation is real.
Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known before you can apply it; and even if the pigeonhole principle did apply, you still can't know by that argument whether any given brain-state is repeated (and if so, how many times). Things which you can deduce using only mathematics, and no physical data, are necessarily extremely weak statements about reality.
It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
That works if you're fixed to a grid, but the argument is much more general: it allows symbols to be of arbitrary size (as long as they're still within the unit square), in any orientation, formed by continuous or (not-too-pathological) discontinuous penstrokes, etc.
It doesn't, however, allow for non-closed symbols. I can imagine a spiralling brushstroke that gets fainter as it approaches the centre. Maybe the proof can be strengthened, and it certainly won't pass the distinguishability criterion, but we must be rigorous, here if anywhere.
It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
Yes it's easier and not too different from Turing's argument. However, Turing made this proof in 1936, you know, before the concept of pixels exists.
Not far off though, "pix" was in use by at least 1932, and "pixel" by 1956[0].
Interesting argument. This assumes that cognition must also be happening on a compact manifold which seems like a reasonable assumption but the conclusion is somewhat counterintuitive because it means there are only finitely many personality types and ways of thinking.
Is that counterintuitive? There's a finite number of electrons in a human brain (which fits in a space of size 1m^3), and information takes time to propagate across any distance, and humans die before 150 years; this all gestures at there being not only finitely many personality types and ways of thinking, but finitely many human mind states .
But there may yet be more brain states than number of possible thoughts by the finite humans.
It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
Yes it's easier and not too different from Turing's argument. However, Turing made this proof in 1936, you know, before the concept of pixels exists.
nyquist's sampling theorem is from 01915. western union started offering wirephoto service in 01921, and the associated press started mass distribution of news photos in 01935. these aren't discrete pixels but they did demonstrate that nyquist's sampling theorem applies to images too
Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known be…
If you are a mental clone but uncausally linked then is that still reincarnation?
Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known be…
The universe is mathematical, all we can ever know about it is mathematical.
It is even easier to just say, given a fixed area of paper, and a finite printing resolution, there is a finite number of symbols possible? Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
Yes it's easier and not too different from Turing's argument. However, Turing made this proof in 1936, you know, before the concept of pixels exists.
Approximating an arbitrary shape by covering it with tiny squares is not exactly new. The method of exhaustion goes back to 200BC.