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When you talk of "all brains" do you mean, literally "all brains" or just HUMAN brains? Does my dog have this level of complexity? Did Koko the Gorilla? Did Alex the parrot?
Although I don't really know what he's ultimately trying to say, I wouldn't try and argue against any sort of implication of the correlation of mathematics being constant. You're familiar with science, you must have dealt with infinities in mathematics. Aren't there technically an infinite amount of boxes in every integral even if different integrals have bigger or smaller ranges? From what I understand, that's likely analogous to one of points he's making. In the same way, every fractal has an infinite amount of complexity. Realistically a physical fractal can't exist because of the quantinization of matter and energy, but he seems to in a way be suggesting "conscious part of the brain = component of mathematical fractal, therefore = infinitely complex", which means any brain has an infinite amount of complexity if all consciousness' does actually follow that pattern, even if the brains they inhabit have different ranges like for instance cognitive abilities. How on Earth could you know that consciousness is in any way related to a fractal? I really don't know.
Physics has a rather dicey relationship with "infinity."
On the one hand, problems like The Grand Hotel Cigar Mystery and Banach–Tarski paradox seem obviously physically impossible. On the other hand, Calculus simply can't work without infinity, and we really need Calculus to do physics.
Well, when you are integrating, you aren't using infinite squares or anything. You are just using small enough ones that the function you are integrating over behaves linearly at that length scale. Otherwise, there wouldnt be that non integratable set of functions that involve fractals.
Well, when you are integrating, you aren't using infinite squares or anything. You are just using small enough ones that the function you are integrating over behaves linearly at that length scale. Otherwise, there wouldnt be that non integratable set of functions that involve fractals.
Well, can you explain to me how the sides of a square go from 2 to 21/2 without infinity?
If you find the length of a diagonal between two sides of a square of length unity, you get 21/2. Yet, if you take a step-wise function of the same line, you get 2. My understanding is only through infinity is this resolved, or that Prognathous was wrong.
Well, when you are integrating, you aren't using infinite squares or anything. You are just using small enough ones that the function you are integrating over behaves linearly at that length scale. Otherwise, there wouldnt be that non integratable set of functions that involve fractals.
Well, can you explain to me how the sides of a square go from 2 to 21/2 without infinity?
If you find the length of a diagonal between two sides of a square of length unity, you get 21/2. Yet, if you take a step-wise function of the same line, you get 2. My understanding is only through infinity is this resolved, or that Prognathous was wrong.
Is there some other way to resolve this problem?
This doesn't work, since in the end you'll not have a straight line, but 2*infinite infinitely small lines which all are 1/infinite is length. Infinity is funny like that.
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We have laboured long to build a heaven, only to find it populated with horrors.
This doesn't work, since in the end you'll not have a straight line, but 2*infinite infinitely small lines which all are 1/infinite is length. Infinity is funny like that.
Ummm... that was my point about Calculus, infinity, and physics. So, I'm not sure how your statement refutes what I was saying. If anything, it seems to strengthen it.
Well, can you explain to me how the sides of a square go from 2 to 21/2 without infinity?
If you find the length of a diagonal between two sides of a square of length unity, you get 21/2. Yet, if you take a step-wise function of the same line, you get 2. My understanding is only through infinity is this resolved, or that Prognathous was wrong.
Is there some other way to resolve this problem?
...
This is an optical illusion. What's going on here is that the picture leads one to mistakenly believe that successive subdivisions of the "staircase" approximate the diagonal line -- which they do in area beneath but not in arc length. An error analysis reveals that the sequence of approximation errors is actually constant (E_n = 2 - sqrt(2)) and so even an infinite iteration of the subdivision will not converge on the diagonal line with respect to arc length. The supposed "approximation process" is actually a sham -- the subdivisions aren't getting you any closer to the correct answer.
So it's not the case that "only through infinity is this resolved." Even with infinity, it's not resolved. It can be resolved by ensuring that the approximation process involved is actually approximating the thing it's supposed to be approximating. In this case, that's not happening, because taxicab arc length does not approximate euclidean arc length.
An error analysis reveals that the sequence of approximation errors is actually constant (E_n = 2 - sqrt(2)) and so even an infinite iteration of the subdivision will not converge on the diagonal line with respect to arc length.
I actually remember hearing that somewhere, but I guess I forgot. Even as I typed my question I felt like something was off. Thank you for setting me straight.
What's your take--anyway--Crashing00, about the physical significance of infinity?
What's your take--anyway--Crashing00, about the physical significance of infinity?
Well, I suppose it depends on the context in which the infinity arises. There seems to be a marked difference between the way physicists treat infinities that represent unboundedly large quantities of "actual stuff" and the way they treat infinities that represent, say, a continuum of points or a high-cardinality topological space as a background against which the "actual stuff" is happening.
On the matter of "unbounded quantities of actual stuff": I'm not a physicist, but observing from 1000 feet, it seems to me that physicists are assiduous in their avoidance of these infinities. Generally when an infinity shows up in the mathematics of a physical theory, the resulting prediction of the theory is apocalyptic at best and gibberish at worst. 20th century quantum theorists in the intellectual lineage of Feynman invented entirely new techniques for dodging infinity, and when they still got infinities when adding gravity to the mix, they invented entirely new and speculative theories -- not to explain any experimental result, but for the sole purpose of getting rid of the infinities!
So physicists abhor infinity so much that they seem to be willing to discard the core tenet of science -- "only experiment can arbitrate theory" -- in order to avoid it. I'm not sure that means that physics abhors infinities, but since physicsists sometimes do use this business to explain actual experimental results when they're not busy inventing new ways to avoid infinity, I'm willing to take it as positive evidence.
On the matter of continua and high-cardinality spaces: This seems more controversial. The best-accepted physical theories treat space as a continuum and rely heavily on the mathematics of Hilbert spaces, which have enormous infinite cardinalities. On the other hand, I know that there are discretized approaches, both theoretical (loop quantum gravity) and practical (lattice chromodynamics), but they are either explicitly designed to be approximate or completely speculative.
So I don't tend to weigh in on the continuum issue since it's really not decidable yet as far as I can see. For instance, you mentioned the Planck length: I don't immediately buy into the claim that the existence of the Planck scale makes space discrete, or even counts as evidence of the discreteness of space. First of all, all of the mathematics that tells us the Planck length is special presupposes a continuum of space. Second, a breakdown in an abstraction only licenses us to discard that particular abstraction, not to replace it with a particular substitute. Third, as far as I understand them (which I admit is very little), some speculative string theory models are capable of coherently dealing with the Planck scale without actually discretizing space.
So physicists abhor infinity so much that they seem to be willing to discard the core tenet of science -- "only experiment can arbitrate theory" -- in order to avoid it. I'm not sure that means that physics abhors infinities, but since physicsists sometimes do use this business to explain actual experimental results when they're not busy inventing new ways to avoid infinity, I'm willing to take it as positive evidence.
For instance, you mentioned the Planck length: I don't immediately buy into the claim that the existence of the Planck scale makes space discrete, or even counts as evidence of the discreteness of space.
Nor were you meant to. It's just something I've come to believe while doing my time in the muck.
What's your take--anyway--Crashing00, about the physical significance of infinity?
Well, I suppose it depends on the context in which the infinity arises. There seems to be a marked difference between the way physicists treat infinities that represent unboundedly large quantities of "actual stuff" and the way they treat infinities that represent, say, a continuum of points or a high-cardinality topological space as a background against which the "actual stuff" is happening.
On the matter of "unbounded quantities of actual stuff": I'm not a physicist, but observing from 1000 feet, it seems to me that physicists are assiduous in their avoidance of these infinities. Generally when an infinity shows up in the mathematics of a physical theory, the resulting prediction of the theory is apocalyptic at best and gibberish at worst. 20th century quantum theorists in the intellectual lineage of Feynman invented entirely new techniques for dodging infinity, and when they still got infinities when adding gravity to the mix, they invented entirely new and speculative theories -- not to explain any experimental result, but for the sole purpose of getting rid of the infinities!
So physicists abhor infinity so much that they seem to be willing to discard the core tenet of science -- "only experiment can arbitrate theory" -- in order to avoid it. I'm not sure that means that physics abhors infinities, but since physicsists sometimes do use this business to explain actual experimental results when they're not busy inventing new ways to avoid infinity, I'm willing to take it as positive evidence.
On the matter of continua and high-cardinality spaces: This seems more controversial. The best-accepted physical theories treat space as a continuum and rely heavily on the mathematics of Hilbert spaces, which have enormous infinite cardinalities. On the other hand, I know that there are discretized approaches, both theoretical (loop quantum gravity) and practical (lattice chromodynamics), but they are either explicitly designed to be approximate or completely speculative.
So I don't tend to weigh in on the continuum issue since it's really not decidable yet as far as I can see. For instance, you mentioned the Planck length: I don't immediately buy into the claim that the existence of the Planck scale makes space discrete, or even counts as evidence of the discreteness of space. First of all, all of the mathematics that tells us the Planck length is special presupposes a continuum of space. Second, a breakdown in an abstraction only licenses us to discard that particular abstraction, not to replace it with a particular substitute. Third, as far as I understand them (which I admit is very little), some speculative string theory models are capable of coherently dealing with the Planck scale without actually discretizing space.
What if there simply physically was an "infinite" amount? Let's say for instance, there is literally a never ending amount of space no matter how far you go, there will always be more in flat space, would you consider that assuming we have a way to prove there will always be more space that flat space is infinitely large? Infinity isn't a value, it's more of a concept, so if the limit of something isn't a real number but keeps getting indefinitely bigger, isn't it infinity?
It's worth noting part of that is a question regarding the "recording and storage medium" used for the information. Things can appear infinite which in fact aren't.
I often have debated with my science minded friends regarding the mechanics and reasonable believability of Simulism, for example.(not an attempt to derail the conversation, but I think it's relevant)
The largest argument I've seen against simulism is that the difficulty of simulating a universe requires a prohibitively large amount of computational power.
A common counterargument is that our system need only be smaller than the containing system, and that the physics of our universe are not a valid comparison for an external containing universe's physics. Personally I don't like relying on this as I feel it avoids the issue rather than addressing it.
Instead it is worth noting ways of conserving computational power, especially when viewed against the backdrop of how humans handle computing limitations.
We have the observable universe, which regardless of the unobservable universe's size, is finite. Then we have the currently observed parts of the observable universe, which is drastically more finite. Graphical renderings of 3D locations handle this by having a finite draw distance and only rendering what is "observable". This drastically reduces computational needs. Humans(and other animals capable of processing visual and tactile stimuli) have limits to the information they can observe. Not all of it is being accessed at one time and thus "loaded into active memory" of the universe.
Waveform Collapse is another interesting mechanic of the universe. If it is actually a real mechanic and not simply a result of Quantum Physics being an incomplete approximation of the actual mechanics of the universe, it suggests that information is "approximated", in other words, the universe only calculates the information that is needed, when it is needed, like a computer calling functions and modules.
Admittedly this is kind of an Anthropocentric view, but in the event that the Earth(not necessarily humans, just life on Earth itself and/or this solar system) is the focus of said calculations, that drastically reduces the computational needs for a simulation to something extravagant but a lot more plausible than the idea of actually actively simulating a universe.
Even in the case that Simulism is incorrect, it still supports the idea that the universe itself functions as a computer with a storage medium and processing/memory limitations utilizing shortcuts and diminished stored variables/minimum active memory usage.
So even if the universe isn't, say, a simulation actively created on a computer, it certainly acts as if it is a computer of some sort, or alternately as if computers themselves are simplified versions of the universe's functionality on a fundamental level instead of just a practical level.
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Although I don't really know what he's ultimately trying to say, I wouldn't try and argue against any sort of implication of the correlation of mathematics being constant. You're familiar with science, you must have dealt with infinities in mathematics. Aren't there technically an infinite amount of boxes in every integral even if different integrals have bigger or smaller ranges? From what I understand, that's likely analogous to one of points he's making. In the same way, every fractal has an infinite amount of complexity. Realistically a physical fractal can't exist because of the quantinization of matter and energy, but he seems to in a way be suggesting "conscious part of the brain = component of mathematical fractal, therefore = infinitely complex", which means any brain has an infinite amount of complexity if all consciousness' does actually follow that pattern, even if the brains they inhabit have different ranges like for instance cognitive abilities. How on Earth could you know that consciousness is in any way related to a fractal? I really don't know.
On the one hand, problems like The Grand Hotel Cigar Mystery and Banach–Tarski paradox seem obviously physically impossible. On the other hand, Calculus simply can't work without infinity, and we really need Calculus to do physics.
So,
Well, can you explain to me how the sides of a square go from 2 to 21/2 without infinity?
If you find the length of a diagonal between two sides of a square of length unity, you get 21/2. Yet, if you take a step-wise function of the same line, you get 2. My understanding is only through infinity is this resolved, or that Prognathous was wrong.
Is there some other way to resolve this problem?
This doesn't work, since in the end you'll not have a straight line, but 2*infinite infinitely small lines which all are 1/infinite is length. Infinity is funny like that.
Ummm... that was my point about Calculus, infinity, and physics. So, I'm not sure how your statement refutes what I was saying. If anything, it seems to strengthen it.
...
This is an optical illusion. What's going on here is that the picture leads one to mistakenly believe that successive subdivisions of the "staircase" approximate the diagonal line -- which they do in area beneath but not in arc length. An error analysis reveals that the sequence of approximation errors is actually constant (E_n = 2 - sqrt(2)) and so even an infinite iteration of the subdivision will not converge on the diagonal line with respect to arc length. The supposed "approximation process" is actually a sham -- the subdivisions aren't getting you any closer to the correct answer.
So it's not the case that "only through infinity is this resolved." Even with infinity, it's not resolved. It can be resolved by ensuring that the approximation process involved is actually approximating the thing it's supposed to be approximating. In this case, that's not happening, because taxicab arc length does not approximate euclidean arc length.
Which if thou dost not use for clearing away the clouds from thy mind
It will go and thou wilt go, never to return.
I actually remember hearing that somewhere, but I guess I forgot. Even as I typed my question I felt like something was off. Thank you for setting me straight.
What's your take--anyway--Crashing00, about the physical significance of infinity?
Well, I suppose it depends on the context in which the infinity arises. There seems to be a marked difference between the way physicists treat infinities that represent unboundedly large quantities of "actual stuff" and the way they treat infinities that represent, say, a continuum of points or a high-cardinality topological space as a background against which the "actual stuff" is happening.
On the matter of "unbounded quantities of actual stuff": I'm not a physicist, but observing from 1000 feet, it seems to me that physicists are assiduous in their avoidance of these infinities. Generally when an infinity shows up in the mathematics of a physical theory, the resulting prediction of the theory is apocalyptic at best and gibberish at worst. 20th century quantum theorists in the intellectual lineage of Feynman invented entirely new techniques for dodging infinity, and when they still got infinities when adding gravity to the mix, they invented entirely new and speculative theories -- not to explain any experimental result, but for the sole purpose of getting rid of the infinities!
So physicists abhor infinity so much that they seem to be willing to discard the core tenet of science -- "only experiment can arbitrate theory" -- in order to avoid it. I'm not sure that means that physics abhors infinities, but since physicsists sometimes do use this business to explain actual experimental results when they're not busy inventing new ways to avoid infinity, I'm willing to take it as positive evidence.
On the matter of continua and high-cardinality spaces: This seems more controversial. The best-accepted physical theories treat space as a continuum and rely heavily on the mathematics of Hilbert spaces, which have enormous infinite cardinalities. On the other hand, I know that there are discretized approaches, both theoretical (loop quantum gravity) and practical (lattice chromodynamics), but they are either explicitly designed to be approximate or completely speculative.
So I don't tend to weigh in on the continuum issue since it's really not decidable yet as far as I can see. For instance, you mentioned the Planck length: I don't immediately buy into the claim that the existence of the Planck scale makes space discrete, or even counts as evidence of the discreteness of space. First of all, all of the mathematics that tells us the Planck length is special presupposes a continuum of space. Second, a breakdown in an abstraction only licenses us to discard that particular abstraction, not to replace it with a particular substitute. Third, as far as I understand them (which I admit is very little), some speculative string theory models are capable of coherently dealing with the Planck scale without actually discretizing space.
Which if thou dost not use for clearing away the clouds from thy mind
It will go and thou wilt go, never to return.
Sounds about right.
Nor were you meant to. It's just something I've come to believe while doing my time in the muck.
But I also propose even distribution of number of cards in each rarity: Large set: 60 c, 60 u, 60 r, 60 m.
Probabilities of particular cards: Common 7/60, Uncommon 1/12, Rare 1/20, Mythic 1/60.
What if there simply physically was an "infinite" amount? Let's say for instance, there is literally a never ending amount of space no matter how far you go, there will always be more in flat space, would you consider that assuming we have a way to prove there will always be more space that flat space is infinitely large? Infinity isn't a value, it's more of a concept, so if the limit of something isn't a real number but keeps getting indefinitely bigger, isn't it infinity?
I often have debated with my science minded friends regarding the mechanics and reasonable believability of Simulism, for example.(not an attempt to derail the conversation, but I think it's relevant)
The largest argument I've seen against simulism is that the difficulty of simulating a universe requires a prohibitively large amount of computational power.
A common counterargument is that our system need only be smaller than the containing system, and that the physics of our universe are not a valid comparison for an external containing universe's physics. Personally I don't like relying on this as I feel it avoids the issue rather than addressing it.
Instead it is worth noting ways of conserving computational power, especially when viewed against the backdrop of how humans handle computing limitations.
We have the observable universe, which regardless of the unobservable universe's size, is finite. Then we have the currently observed parts of the observable universe, which is drastically more finite. Graphical renderings of 3D locations handle this by having a finite draw distance and only rendering what is "observable". This drastically reduces computational needs. Humans(and other animals capable of processing visual and tactile stimuli) have limits to the information they can observe. Not all of it is being accessed at one time and thus "loaded into active memory" of the universe.
Waveform Collapse is another interesting mechanic of the universe. If it is actually a real mechanic and not simply a result of Quantum Physics being an incomplete approximation of the actual mechanics of the universe, it suggests that information is "approximated", in other words, the universe only calculates the information that is needed, when it is needed, like a computer calling functions and modules.
These things would also be supported by the lack of apparent extraterrestrial activity http://en.wikipedia.org/wiki/Fermi_paradox
Admittedly this is kind of an Anthropocentric view, but in the event that the Earth(not necessarily humans, just life on Earth itself and/or this solar system) is the focus of said calculations, that drastically reduces the computational needs for a simulation to something extravagant but a lot more plausible than the idea of actually actively simulating a universe.
Even in the case that Simulism is incorrect, it still supports the idea that the universe itself functions as a computer with a storage medium and processing/memory limitations utilizing shortcuts and diminished stored variables/minimum active memory usage.
So even if the universe isn't, say, a simulation actively created on a computer, it certainly acts as if it is a computer of some sort, or alternately as if computers themselves are simplified versions of the universe's functionality on a fundamental level instead of just a practical level.