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Author Topic:   Assumption of smoothness

icosahedron posted 06-18-99 10:50 AM ET
Most people, including most physicists, assume that reality is smoothly continuous.
However, I have never had a smooth experience (except in the context of bowel pressure relief). In other words, my experiences seem discrete and digital in time and space, whether they are real or imagined.
From speaking with others, I get the sense that we all think, and hence experience, in a discrete, digital fashion. Certainly it seems that communication, and the closely related phenomenon of perception, are both digital in nature.
So what's with this standard assumption of smoothness?
Perhaps there is no difference between discrete and smoothly continuous frameworks, they approximate one another. But in fact this is certainly not true. If reality is discrete, then there are only a finite number of paths between any two points. There are qualitative differences.
I wonder what others think?
- icosahedron

Saras posted 06-18-99 10:56 AM ET
Fnord!
Saras posted 06-18-99 10:56 AM ET
Azugal!
Saras posted 06-18-99 10:58 AM ET
Traspreambulation of pseudo-cosmic time-space dilation.
Resource Consumer posted 06-18-99 10:58 AM ET
It all depends on the frequency of the discrete events, I think. If the sampling rate is very high then they approximate a continuous flow.
How do we up the sampling rate to match the theory? Either we live faster or convince ourselves that the world moves faster than it is.
Solution and slogan.
Do drugs for theory
Resource Consumer
- consuming -
Saras posted 06-18-99 11:00 AM ET
We have tried to present a comprehensive and didactic account of both the principles and methods used to value and hedge variance swaps. We have explained both the intuitive and the rigorous approach to replication.
In markets with a volatility skew (the real world for most swaps of interest), the intuitive approach loses its footing. Here, using the rigorous approach, one can still value variance swaps by replication. Remarkably, we have succeeded in deriving analytic approximations that work well for the swap value under commonly used skew
parameterizations.
Octopus posted 06-18-99 11:12 AM ET
I practical terms, there is very little difference between infinity and a very large finite number. The number of possible games of chess that exist is probably finite because of the rules of the game, but that number is so high that it would be extraordinarily unlikely for you to ever see the same game played twice.
What observations lead you to believe in discrete time-space? Can you propose any experiments to detect the difference between a discrete time-space and a continuous one?
jig posted 06-18-99 11:28 AM ET
I just can't comprehend how time-space could be discrete.
icosahedron posted 06-18-99 01:31 PM ET
Resource consumer: A high-frequency discrete space is still qualitatively different from a smooth, continuous space. There is a definite progression of point density.
The simplest case is a discrete lattice. For example, the set of points (a,b,c) where a, b, and c are integers. The frequency of such a space can be chosen arbitrarily small, yet there will always be spans which contain no point of the space; there do not exist points between adjacent points.
The next case is when a, b, and c are rational numbers. In this case, there is a point between any two points, and we might say that this space is dense. Yet, it can be mapped into a sequence, so it does have holes in some sense.
Finally, we reach the smooth, continuous space where a, b, and c are real numbers, which is the conventional "holodeck" for most spatial rumination.
The qualitative differences between these three cases are quite large. In reality, we can only use the first case (or finitely many superpositions of the first case) for represenational purposes, e.g., in computer screens.
My question is whether the higher order cases are even appropriate except as approximations.
- icosahedron
icosahedron posted 06-18-99 01:33 PM ET
Saras, every space associated with financial markets is finite and discrete (prices, volume, etc.).
- icosahedron
icosahedron posted 06-18-99 01:42 PM ET
Octopus, the difference between infinity and a very large number is ... infinite!
I am not just being trivially cute. Infinity is not really a number nor a state, and cannot really be sensically talked about or represented. In reality, ou can't go to infinity, count to infinity, take anything to infinity, or cut something into infinitessimal pieces. Infinity is non-conceptual.
If space is discrete, then looking at it as continuous will introduce "rounding error" at every corner. This is not insignificant in general, though in many simple applications it is negligible. But physical theory relies on the ability to manipulate numbers with impunity as abstract concepts with no error, and in this case the rounding error could accumulate to swamp the truth so that we don't see the real answer.
As for observations that point to a discrete spacetime, I note in particular that energy comes in little packets, energetic interactions occur discretely, and there are finitely many energetic events in any finite volume of space.
We cannot detect anything continuous. Our methods are wholly discrete, because that is the way we think -- digitally (not linearly, like a computer, but more like a hologram -- digitally).
- icosahedron
icosahedron posted 06-18-99 01:43 PM ET
I just cannot comprehend how spacetime could be anything other than discrete, given that my perceptions are discrete.
- icosahedron
Khan Singh posted 06-20-99 12:57 PM ET
Your perceptions aren't discrete. They are only discrete close to the limits of your perception. Everything is quantum above a certain sampling frequency.
GP posted 06-20-99 04:15 PM ET
tune a radio station using a dial or even just adjust the volume with a dial. The experience is analog. You don't detect sudden jumps in sound right? it's a smooth sound as far as you can tell. Maybe at some fundamental pohysics level it's unsmooth. although i don't know the rules of physics for phonons.
CrackGenius posted 06-20-99 07:14 PM ET
icosahedron I've thought about the same issue (though not extensively). I generally agree with you about the non-smoothness of reality. I think that since we accept Planck's propositions about quanta then simultaneously we should accept digital reality coming or 'happening' in discrete amounts.
I'm not sure why you think that scientists believe the smoothness assumption.
CrackGenius
Not a physicist, but a good approximation nevertheless .
Khan Singh posted 06-20-99 08:34 PM ET
It is a mistake to think that the smooth classical world is "composed" of smaller dicrete particles that act in a non-smooth way. Smoothness is instead composed of the statistical sum of the quantum particles. Smoothness consists not of the particles but of the statistics themselves. This is a strongly counter-intuitive notion, but if you want to look at the world precisely (ie mathematically) then you have to follow that method to its logical conclusion.
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Author	Topic: Assumption of smoothness
icosahedron	posted 06-18-99 10:50 AM ET Most people, including most physicists, assume that reality is smoothly continuous. However, I have never had a smooth experience (except in the context of bowel pressure relief). In other words, my experiences seem discrete and digital in time and space, whether they are real or imagined. From speaking with others, I get the sense that we all think, and hence experience, in a discrete, digital fashion. Certainly it seems that communication, and the closely related phenomenon of perception, are both digital in nature. So what's with this standard assumption of smoothness? Perhaps there is no difference between discrete and smoothly continuous frameworks, they approximate one another. But in fact this is certainly not true. If reality is discrete, then there are only a finite number of paths between any two points. There are qualitative differences. I wonder what others think? - icosahedron
Saras	posted 06-18-99 10:56 AM ET Fnord!
Saras	posted 06-18-99 10:56 AM ET Azugal!
Saras	posted 06-18-99 10:58 AM ET Traspreambulation of pseudo-cosmic time-space dilation.
Resource Consumer	posted 06-18-99 10:58 AM ET It all depends on the frequency of the discrete events, I think. If the sampling rate is very high then they approximate a continuous flow. How do we up the sampling rate to match the theory? Either we live faster or convince ourselves that the world moves faster than it is. Solution and slogan. Do drugs for theory Resource Consumer - consuming -
Saras	posted 06-18-99 11:00 AM ET We have tried to present a comprehensive and didactic account of both the principles and methods used to value and hedge variance swaps. We have explained both the intuitive and the rigorous approach to replication. In markets with a volatility skew (the real world for most swaps of interest), the intuitive approach loses its footing. Here, using the rigorous approach, one can still value variance swaps by replication. Remarkably, we have succeeded in deriving analytic approximations that work well for the swap value under commonly used skew parameterizations.
Octopus	posted 06-18-99 11:12 AM ET I practical terms, there is very little difference between infinity and a very large finite number. The number of possible games of chess that exist is probably finite because of the rules of the game, but that number is so high that it would be extraordinarily unlikely for you to ever see the same game played twice. What observations lead you to believe in discrete time-space? Can you propose any experiments to detect the difference between a discrete time-space and a continuous one?
jig	posted 06-18-99 11:28 AM ET I just can't comprehend how time-space could be discrete.
icosahedron	posted 06-18-99 01:31 PM ET Resource consumer: A high-frequency discrete space is still qualitatively different from a smooth, continuous space. There is a definite progression of point density. The simplest case is a discrete lattice. For example, the set of points (a,b,c) where a, b, and c are integers. The frequency of such a space can be chosen arbitrarily small, yet there will always be spans which contain no point of the space; there do not exist points between adjacent points. The next case is when a, b, and c are rational numbers. In this case, there is a point between any two points, and we might say that this space is dense. Yet, it can be mapped into a sequence, so it does have holes in some sense. Finally, we reach the smooth, continuous space where a, b, and c are real numbers, which is the conventional "holodeck" for most spatial rumination. The qualitative differences between these three cases are quite large. In reality, we can only use the first case (or finitely many superpositions of the first case) for represenational purposes, e.g., in computer screens. My question is whether the higher order cases are even appropriate except as approximations. - icosahedron
icosahedron	posted 06-18-99 01:33 PM ET Saras, every space associated with financial markets is finite and discrete (prices, volume, etc.). - icosahedron
icosahedron	posted 06-18-99 01:42 PM ET Octopus, the difference between infinity and a very large number is ... infinite! I am not just being trivially cute. Infinity is not really a number nor a state, and cannot really be sensically talked about or represented. In reality, ou can't go to infinity, count to infinity, take anything to infinity, or cut something into infinitessimal pieces. Infinity is non-conceptual. If space is discrete, then looking at it as continuous will introduce "rounding error" at every corner. This is not insignificant in general, though in many simple applications it is negligible. But physical theory relies on the ability to manipulate numbers with impunity as abstract concepts with no error, and in this case the rounding error could accumulate to swamp the truth so that we don't see the real answer. As for observations that point to a discrete spacetime, I note in particular that energy comes in little packets, energetic interactions occur discretely, and there are finitely many energetic events in any finite volume of space. We cannot detect anything continuous. Our methods are wholly discrete, because that is the way we think -- digitally (not linearly, like a computer, but more like a hologram -- digitally). - icosahedron
icosahedron	posted 06-18-99 01:43 PM ET I just cannot comprehend how spacetime could be anything other than discrete, given that my perceptions are discrete. - icosahedron
Khan Singh	posted 06-20-99 12:57 PM ET Your perceptions aren't discrete. They are only discrete close to the limits of your perception. Everything is quantum above a certain sampling frequency.
GP	posted 06-20-99 04:15 PM ET tune a radio station using a dial or even just adjust the volume with a dial. The experience is analog. You don't detect sudden jumps in sound right? it's a smooth sound as far as you can tell. Maybe at some fundamental pohysics level it's unsmooth. although i don't know the rules of physics for phonons.
CrackGenius	posted 06-20-99 07:14 PM ET icosahedron I've thought about the same issue (though not extensively). I generally agree with you about the non-smoothness of reality. I think that since we accept Planck's propositions about quanta then simultaneously we should accept digital reality coming or 'happening' in discrete amounts. I'm not sure why you think that scientists believe the smoothness assumption. CrackGenius Not a physicist, but a good approximation nevertheless .
Khan Singh	posted 06-20-99 08:34 PM ET It is a mistake to think that the smooth classical world is "composed" of smaller dicrete particles that act in a non-smooth way. Smoothness is instead composed of the statistical sum of the quantum particles. Smoothness consists not of the particles but of the statistics themselves. This is a strongly counter-intuitive notion, but if you want to look at the world precisely (ie mathematically) then you have to follow that method to its logical conclusion.