L. Edgar Otto 11 May, 2012
So we have three shells and we know the first two are shown empty. What is the probability there is a pea under the third shell, and so on...
So, 2 + 2 = 5 is nonsense, and this is not a question of higher logic for the great bulk of the tree of mathematics of which it, like Euclidean geometry is a matter of certainty.
But other than some form of probability measure for things like the values in physics how do we measure where logic is in the main certainty the nothingness to consider in the computation to include the empty state of the shell?
This is a question of existence or non existence- and could this not be the case of the universe in the first place... as Sheldon brags in the sit com The Big Bang... "I exist therefore that I do not is not a possibility."? Matter in the gross sense is the common saying that when someone says matter is an illusion ask him about the brick falling on his head.
With a wider view we can metaphorically suggest, and it could be an actuality, that the vacuum has structure in that it is not nothingness, it could indeed have vibrational structure at least by the physical and mathematical properties in it. Is this any more metaphysical than a vacuum can be said to be full as in negative values predicting particles like positrons? Or that if things we expect are stretched out, separated, a photon that is exchanged jumps as if it covered and empty distance.
Of course there are errors of intuition- but in the non necessity, the quasi-certainty of it all, there are errors of theory at least to the extensions it perceives and encompasses. We cannot choose, well, in a series of things if the chances are true or false, a long chain of the that reaches each step and the last as the certainty of the whole and the goal if our truth in the steps is but a matter of probability. Not to say that on some level fuzzy logic will not work in its application where the redundancy in non existent or minimized in the non-linear chaos that mechanical systems even if working perfectly may experience.
If there is a structure of the vacuum there are methods to treat these with signs and symbols or geometry in that they are so intelligible as to lead to a clearer use of mathematical analysis.
Position space and phase space are not two separate disciplines for they can be viewed as one- it is not just a matter of things, spheres, that vibrate to the wave equations or that are stretched out as exponential, sines, and the like even if we assert for one of these forms as metaphor and the other a simile, the particle or space does not distinguish which on the ground of perceptual limits which it is.
Is it not enough to understand that where vectors are orthogonal in Euclidean space, quantum or in the relativity, that the matrices as four space (even with tensor methods or the methods of partial differentials- moreover the determinants in relation to matrices over the negative saddle space) that for electromagnetic or gravitational forces it is the same haunting pattern that by itself proves light as invariant?
Now being in the general logic the geometries, Euclidean and not, as real, all stand or fall together, at least in the vertical direction a spinning sphere or hyperboloid of one sheet in the z axis is invariant.
We could and do make it an axiom of the relativity by virtue that the wave from viewed from an observer is a sphere and relative to someone in that center of a sphere he sees such a wave front. So in all directions from the center the condition is fulfilled that the distance is unity and not a negation or subtraction in any of the space group directions. But this is also as but a postulate really left open to interpretation including experiments where the application of the analysis logically stands or fall together and does not stray far from a distance once given as if on a cylinder to which on either side we can image the real or imaginary mirror. So in the empty shell, with the considerations of existence in the omega or universe of discourse where probability is defined as rather rigorously, we should add the obvious and wave front invariance (with respect to coordinates) general lack of rigid distinction between what is inside or outside the euclidean spaces between the sphere or hyperboloid as if an Euclidean cylinder.
In structural counts we do imagine that there are compliments to some count of orthogonal subspaces but also we have to assert in the simplex spaces (triangles) that part of the sub-cell count is a distinguished null element (yet abstractly not the indistinguishable null space.)
In the game of logic if we can count beyond three or four, in a sense 2 + 2 can equal 5 of which the last element is not seen as if a null element of rest perhaps in the totality yet not the null non existent general space or that of wide but a measureless or ultimately irrelevant expansion. What would be a criticism of someones logic, as this is also the intrinsic source or observation of abstract motion, is that the pink unicorn of 2 + 2 = 5 (and all such question if the division where the ghost of departed quantities such as momentum is indeterminate, by especially zero) would be that one does not understand that such non entities make the old gray mare more real, logically. What does one gain when even we know that simply saying "it is not the case that..." especially if we imagine distinction of similarity if by only a difference in time by the truth of this secret logicians have to ignore as a working principle?
It is also not clear that the metalanguage based on primes where they may be forever hidden from us as pink unicorns in the methods of Godel for uniqueness, that numbers are not thereby all the more real- for this extensions into a wider continuum not provable in that system (nor unprovable) makes of any unity or prime in the abstract null space of our conception as if a complimentary universe of discourse, say one over Omega as if compressed or reduced into some unclear direction, even if this grounds the dimensionless values we find in observation or pure counting (come on, you have to realize to solve some problems of balance where say weights are identical one must assign them a label, not the quantity in itself will always give an answer.) if in a sense a number itself as say a prime within itself remains in some local or existential point of theory independent of the idea that a any context may make it a non-necessary prime.
Here the various ideas of the transfinite meet in the physicality the idea of bifurcation of paths, of chance or choice, and what sort of phenomena does the choosing. Nature is quite adept at "magician's choice" wherein it appears to one who guesses which of two cups with handles pointing right or left the magician claims it was the other one of his own- and apparently an empty and reversed direction of his choosing relative to the mark making the choices of what he thinks is a symmetric state of some extended fact.
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