Notes
[Sunday 8 June 2008 - Saturday 14 June 2008]
[Notebook: DB 64 Gravitation]
Sunday 8 June 2008
Monday 9 June 2008
[page 55]
Tuesday 10 June 2008
Confidence in the view inspires the work to make it clear, like demonstrating the existence of Bose-Einstein condensates of integral spin atoms.
The first step in science is recording the data ie rendering permanent records of ephemeral events, recording where a photon with a certain energy was absorbed at a certain time.
The superposition of the Universe is two dimensional and its properties are captured (apparently perfectly) by the addition of complex numbers. Temporal sequences are represented by temporal sequences of complex numbers constrained
[page 56]
by unitarity.
The global source is constrained by conservation of probability/energy which translates into unitary evolution in a complex Hilbert space.
Reality is local and has no symmetries, though symmetries can be detected within it, ultimately every event (ie every message) is unique.
Wednesday 11 June 2008
Why is the Universe quantized? The purpose of this paper is to explore the hypothesis that the observable Universe is isomorphic to mathematics. We may divide mathematics into discrete and continuous. In its current incarnation, physics uses continuous mathematics to describe the statistics of discrete events through quantum mechanics and quantum field theory. Quantum mechanics is reversible with respect to time, whereas quantum field theory introduces causation via the finite fixed velocity of light, stipulating that [within the limits if uncertainty] no event can be influenced by events outside its past light cone.
Mathematics might also be divided into interesting and boring. This is a very subjective division, but let us agree here that mathematics is interesting if it models some aspect of our experience. (We find, for instance, that if we use pythagoras' theorem to lay out the walls of a building, we end up making the building square).
Formalist mathematics is self contained in that it defines certain rules for manipulating symbols and uses these rules to
[page 57]
establish correspondences between various strings of symbols such as 2 + 2 = 4, where = signifies a correspondence, that is one may be substituted for the other in the domain of the particular correspondence, in this case cardinal arithmetic. This is a scalar correspondence. There are also ordinal correspondences, in which the corresponding entities have many corresponding degrees of freedom rather than one.
Mathematics itself proves that the limit to the computation of correspondences is set by a universal Turing machines, and so any correspondence that cannot be determined by a deterministic machine is called incomputable.
The observable world is quantized. All of experimental physics thus comes down to counting events. There are a large number of different events to be counted and so the counting process requires some sort of classification of events before they are counted. This classification may range from non-classification where we count every event to such a fine division of events that every event is considered unique, with a maximum count of one. This method of counting is reflected by Fermi statistics, where the maximum occupation number of any state is 1. Fermi-Dirac statistics - Wikipedia
Physicists label states using either integral or continuous numbers. The quantum no cloning theorem tells us that we cannot duplicate a state and so we conclude that quantum mechanically, every state is unique. No cloning theorem - Wikipedia
Quantum states may be represented mathematically in Hilbert space.
[page 58]
A phase space for mathematics is the Cantor Universe which we construct using set theory, beginning with the set of natural numbers and the axiom of the power set. Jech Given this axiom, it is formally true that given set containing any cardinal number of elements we can always find a set with a greater cardinal.
Here we construct such a Universe using permutations of ordered sets rather than subsets. Given the peculiarities of transfinite arithmetic, both constructions yield sets with the same [transfinite] cardinal, but ordering and permutation yield a group structure which is used in physics. The set of natural numbers is an example of the first transfinite cardinal. The cardinal of the set of permutations of the first transfinite cardinal is the second transfinite cardinal, and so on.
We imagine computable transformations in this phase space to be executed by Turing machines. Given the nature of permutations, permutations of transfinite length can nevertheless [be effected] by a finite machine which is capable of executing finite cycles within the complete set of permutations.
Since the space of permutations is complete, it has the group property so that changing one permutation will create another member of the set. In order to prevent the duplication of permutations, we establish the convention that a permutation can only change by communicating with another so that any transformation is accompanied by its inverse and unicity is preserved.
[page 59]
Quantum theory predicts reproductive success, ie rate of communication into the future. 'We measured reproductive success by counting ookinetes (zygotes at 18 hours of age)'. [ref?]
We can establish a one-one correspondence between Hilbert space and the Cantor Universe, since they are both recursively defined function spaces.
Every permutation can be viewed as a function whose domain is the set of initial permutations and range is the set of final permutations. There are ℵ1 possible permutations of the ℵ0 natural numbers, but since there are only ℵ0 different Turing machines, we are led to believe that the bulk of permutations are incomputable.
Let us guess that the computable subset of these permutations corresponds in some way to the subset of eigenfunctions of the matrices which represent permissible discrete observations and the incomputable subset corresponds to continuum states.
Mathematical discussions of infinity are coded in finite discrete conversations and journal articles. Physical reality appears to follow the same course: the infinite and continuous processes postulated to explain the physical world communicate with physicists and eachother in discrete quanta.
Thursday 12 June 2008
The world is dirty in the sense that every message is unique, so taken as a whole there is no symmetry to
[page 60]
behold. Symmetry only appears on analysis and we see that each unique point is defined by a number of intersecting symmetries, ie a spacelike pint in the Universe.
MESSAGE = SPACELIKE
The classical God is the ultimate spacelike slice tota simul, a superposition of all possible states of existence which we may see as a superposition of all the distinct strings in the symmetric Universe. Aquinas 45
On the final causes
Final causes are the visions of the future; past causes are the constraints arising from the past. Future causes and past causes are linked by insight and [?}
. . .
