I worked on this FQXI essay contest because it suggested to me the idea of an information channel from the physical world into some minimum mathematical model, a channel through which information is carried and conserved by means of "infomorphisms." (The references for this term are in my paper for the contest: "Simple math for questions to physicists," which I'll abbreviate as SMQP in the rest of this post.) Long after the contest has ended, this is my final post in this thread.
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"Dark matter is a hypothetical kind of matter that cannot be seen with telescopes..." From the Wikipedia entry on dark matter.
Dark matter seems like a puzzle about what exists. Because to interact with light and be seen with a telescope, it seems an object must occupy some finite region of space-- the particle must *exist* in space.
So if dark matter does Not interact with light in this manner-- which would require an existence in some finite region of space-- then a particle of dark matter must not exist in any finite region of space. Somehow the particle of dark matter must exist only in a finite region of time.
Hypothesis: A particle of dark matter has a center of gravity but no extension in space. A particle of dark matter exists Not in space, but only in time.
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"In the standard model of the evolution of the universe, galactic filaments form along and follow web-like strings of dark matter." From the Wikipedia entry on Galaxy Filament.
It seems there are regions of the Universe with no matter, regions with dark matter, and regions with both dark and ordinary matter.
Hypothesis: Where a particle of ordinary matter exists there exists a particle of dark matter. Where a particle of dark matter exists there exists the possibility of a particle of ordinary matter.
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"In physical cosmology and astronomy, dark energy is an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe." From the Wikipedia entry on Dark Energy.
Hypothesis: It requires work to expand something that exists only in time into something that exists in space. Dark energy fuels this expansion of possibilities. But thermodynamic work is inefficient. So dark matter disperses into the Universe from some kind of thermodynamic inefficiency.
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The mathematics in SMQP seems like a minimum model for these hypotheses, which also supports the attached diagram about what seems to be a kind of thermodynamics.
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A cycle produces the stream of time.
For the above hypotheses-- rather than a field in space-- a "non-wellfounded set" would be the preferred mathematical object for modeling a particle of dark matter that has extension only in time, since the non-wellfounded set is useful for expressing a stream in time that is associated with a repeating cycle. (I learned what I know about non-wellfounded sets from the book by Barwise and Moss, "Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena.")
For example, the experience of a individual existing in proper time can be expressed by using such a set used to model such a cycle:
1. individual_existence = (changes, individual_existence)
Every instant of time in an individual's existence is the same "now"; Everyday the individual existence-- in human terms, the "self"-- is constant. As long as individual existence lasts, it's the same individual existence or in the case of a human being, it's the same "self."
By repeatedly applying equation 1, we get a model of individual existence associated with a stream of changes:
individual_existence = (changes, (changes, individual_existence))
individual_existence = (changes, (changes, (changes, individual_existence)))
individual_existence = (changes, (changes, (changes, (changes, individual_existence))))
And so on.
The non-wellfounded set can also model that individual existence in time goes through cycles of destruction of the old and creation of the new. Since the particle of dark matter is assumed by these hypotheses to exist only in time, the standard idea of time as a point moving on a line would be the wrong kind of circularity, because "moving" in this idea implies that, in some interval of time, space is traversed. But it seems that space should be removed from the concepts used to model existence of something that only exists in time.
Here is how the syntax of the non-wellfounded sets models an individual existence using-- not the motion of a point along a line in space-- but instead, a cycle of destroying the old and creating the new:
"-" means "destroying the old"
"" means "creating the new"p
A - B = C
C D = A
A. "individual_existence = ( changes, individual_existence)"
Minus (destroying the old) B. "(changes, )"
Equals C. "individual_existence = individual_existence"
Adding (creating the new) D. "(changes, )"
Equals A. "individual_existence = ( changes, individual_existence)"
And so on.
In this way the idea is modeled that the stream of changes associated with a constant individual existence and its stream of time of proper time arise from a cycle of destroying the old and creating the new.
Therefore for these hypotheses, the particle of dark matter is (I think) minimally modeled as a non-wellfounded set which, through a cycle of destroying the old and creating the new produces the stream of changes associated with individual particle existence, while maintaining constant individual existence.
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"Gearing" dark matter to the standard model of time, as modeled by the non-wellfounded set.
2. time = (real_number, time)
Equation 2 uses the non-wellfounded set to model the stream of real numbers measuring time. Notice that unlike the model of a point moving on a line, there is no need to assume the previous existence of a line with one part past and another part future, with the location of the moving point being the "now."
The idea is first to "gear" this real number to the non-wellfounded set used to model the particle of dark matter, thus modeling that the stream of real numbers in proper time of an individual existence is actuated by a particle of dark matter.
3. time = (nonstandard_monad, time)
Equation 3 models a stream of nonstandard monads, each monad containing in its standard part the real number in equation 2. (References are in SMQP.) The halo of nonstandard points around the standard point in the monad comprise the "nonstandard past" and the "nonstandard future," both infinitesimal in the mathematical model of nonstandard analysis.
With this step in modeling the particle of dark matter, room is made available for the particle of dark matter to exist outside of space but still extended in time. The idea modeled is that the particle of dark matter exists extended in time in the nonstandard past and the nonstandard future, while space exists only in the standard part of the monad, the standard "now." The standard "now" is actuated by the particle of dark matter which exists around it in the nonstandard past and nonstandard future.
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An iterated transformation powers the cycle.
Having introduced a monad that is a system of the nonstandard future, the nonstandard past, and the standard present, all inside a non-wellfounded set that models a particle of dark matter, the question becomes how this system actuates proper time of a particle of ordinary matter.
In SMQP, this is where some mathematics of information enters the picture, specifically the mathematical theory of information called Situation Theory. Inside the particle of dark matter (inside the non-wellfounded set) the monad starts to carry information "about" the particle of ordinary matter for which it actuates proper time. The idea being modeled by the mathematics is that this must be powered by dark energy.
Without the fuel of dark energy, the particle of dark matter is "about" nothing. It exists by itself in the vacuum. But given the fuel of dark energy, it becomes "about" something-- the particle of ordinary matter for which it carries the proper time. In SMQP this addition of ordinary matter is modeled by a complex number (which is supported as information through situation theory by the nonstandard future in the monad inside the non-wellfounded set) becoming non zero. When this complex number is zero, the particle of dark matter exists by itself without being associated with a particle of ordinary matter. But when this complex number becomes non zero, the particle of dark matter is associated with a particle of ordinary matter. The idea being modeled is that when this complex number is zero, there is no dark energy fueling the particle of dark matter. And when this complex number becomes non zero, there is dark energy fueling the particle of dark matter, in which case there now exists a particle of ordinary matter, about which the complex number, as modeled by situation theory, then carries information, and for which the non-well founded set actuates the stream of proper time.
In SMQP the Born rule is incorporated into a "Born infomorphism," a transmission of information from the nonstandard future to the nonstandard past which carries information about the particle of ordinary matter for which the particle of dark matter actuates proper time. The idea being modeled is that here is a kind of thermodynamics.
The Born informorphism has the same mathematical form as the game of Probability Learning, much studied in laboratories and in SMQP, modeled using the mathematics of Shannon's theory of information. The idea being modeled is that in Probability Learning, "fuel" is fed into a system. But not all the fuel is consumed. In the Born infomorphism, this fuel would escape into the Universe at large, thus dispersing dark energy throughout and allowing it to increase possibilities in the form of space as it increased possibilities (from zero to non zero) in the Born infomorphism.
The limitations of the essay contest to nine pages kept me from writing too much about this in SMQP.Attachment #1: 5_image.jpg