Continuity - Revision history

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 ==Problems==
 * Particles in physics are discreet, but almost every physical equation is expressed assuming continuous domains at least for one variable to range over; that is, the real and complex number sets are ubiquitously used to model physical parameters of almost any conceivable system. And that model works extremely well, except for a few cases, like Superposition where particles are [[Entangled]].
-* As the incredible difficulty to finding solutions to the three dimensional Ising model proves beyond doubt, discrete problems are fundamentally just as hard as continuum problems, in many cases probably even more so. More interestingly, physically relevant solutions of the higher dimensional Ising model correspond to modes of certain continuous equations, and the actual "beef" of the theory is in finding these intricate correspondences between systems that look very different on the surface, and yet, share very fundamental properties.
+* As the incredible difficulty to finding solutions to the three dimensional Ising model proves beyond doubt, discrete problems are fundamentally just as hard as continuum problems, in many cases probably even more so. More interestingly, physically relevant solutions of the higher dimensional Ising model correspond to modes of certain continuous equations, and the actual "beef" of the theory is in finding these intricate correspondences between systems that look very different on the surface, and yet, share very fundamental properties. See the wiki [[Ising Machine]] for was to solve such a model.
 Nevertheless, even if from this point of view the continuum seems to be a core, essential part of physical theories, it's a well-known property that almost all of its members (i.e. except for a set of zero Lebesgue measure) are uncomputable. That means that every set of real numbers that can be coded to compute with -as a description of a set of boundary conditions, for example-, is nothing but a zero-measure element of the continuum.

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 ==Context==
 * [[Continuity]] and the Infinite have been argued since the time of Zeno's paradox about Achilles. Some of the paradox were eliminated by the use of Wiener and Lebesgue Measure. See the wiki page on the [[Axiom of Choice]].
 * Continuous spaces were the topic of Euclid's (and Riemann's) geometries.
 * Continuous time is necessary to keep up with continuous spaces.
 * Roughness can be viewed as a Hausdorff dimension applied to lines and surfaces.
 ==Problems==

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 ==References==
 [[Category: Mathematics]]
 [[Category: Physics]]

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 * Continuous spaces were the topic of Euclid's (and Riemann's) geometries.
 * Continuous time is necessary to keep up with continuous spaces.
 ==Problems==

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 ==Context==
 * [[Continuity]] and the Infinite have been argued since the time of Zeno's paradox about Achilles. Some of the paradox were eliminated by the use of Wiener and Lebesgue Measure. See the wiki page on the [[Axiom of Choice]].
-* Continuous spaces were the topic of Euclid's (and Reiman's) geometries.
+* Continuous spaces were the topic of Euclid's (and Riemann's) geometries.
 * Continuous time is necessary to keep up with continuous spaces.

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 ==Context==
 ==Problems==
-* Particles in physics are discreet, but almost every physical equation is expressed assuming continuous domains at least for one variable to range over; that is, the real and complex number sets are ubiquitously used to model physical parameters of almost any conceivable system.
+* Particles in physics are discreet, but almost every physical equation is expressed assuming continuous domains at least for one variable to range over; that is, the real and complex number sets are ubiquitously used to model physical parameters of almost any conceivable system. And that model works extremely well, except for a few cases, like [[Superposition]]
 Nevertheless, even if from this point of view the continuum seems to be a core, essential part of physical theories, it's a well-known property that almost all of its members (i.e. except for a set of zero Lebesgue measure) are uncomputable. That means that every set of real numbers that can be coded to compute with -as a description of a set of boundary conditions, for example-, is nothing but a zero-measure element of the continuum.
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 Can this continuum-dependent approach to physics be replaced by the strict use of completely countable formalisms, in a language which assumes and talks of no more than discrete structures. What I'm asking is if the fact that we can only deal with discrete quantities, may be embedded in physical theories themselves from their conception and nothing more being allowed to sneak in -explicitly excluding uncomputable stuff; or if, on the other hand, there are some fundamental reasons to keep holding on to continuous structures in physics.
 Think of it as a case for Occam's Razor: why would we assert that such an infinite amount of points "are out there", if we can't possibly refer to them? We have no intuition about, and no possible way to reach the full continuum. Even if I'm taking too much a philosophical stance with this, it makes me think of "here be dragons" warnings at the corners of Middle Age's maps.
 ==Solutions==
 Firstly, username Yrogirg makes some good points in that many mathematical theories which are modeled using the set of reals (and I think I even used these terms in the technical sense here) entail many facettes which are really agebraic in nature. You can easily implement derivations by representing the quantities of interest as distinct symbols and translating the associated abstract rules of computation to manipulation of these symbols.