Re:

As you may have noticed, my whole train of thought on computationalismris based on the rediscovery of intutionist mathematics under the name &quo=;computation".

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The difference between classical math and computation is that classica=ly, a function has a value as soon as it is defined, but in the computatio=al paradigm, it has to be actually computed, using some generator. This al=o applies for functions that designate truth. For something to be true in =ntuitionist mathematics, you will always have to show the money: you have =o demonstrate that you know how to make a process that can actually perfor= the necessary steps.

This has some interesting implication: computation cannot be paradoxic=l. In the computational framework, there can be no set of all sets that do=s not contain itself. Instead, you'd have to define functions that add=and remove sets from each other, and as a result, you might up with some p=riodic fluctuation, but not with an illegal state.

Intuitionist math fits together with automata theory. It turns out tha= there is a universal computer, i.e. a function that can itself compute al= computable functions (Turing completeness). All functions that implement =he universal computer can effectively compute the same set of functions, b=t they may differ in how efficiently they can do it. Efficiency relates to=computational complexity classes.

The simplest universal computers known are some cellular automata,

wit= Minsky and Wolfram arguing about who found the shortest one.

Boolean alge=ra is Turing complete, too, as is the NAND gate, the

lambda calculus, and =lmost all programming languages. The Church

Turing thesis says that all un=versal computers can compute each

other, and therefore have the same power=

I suspect that it is possible that the Church Turing thesis is also a =hysical law, i.e. it is impossible to build physical computer that can cal=ulate more than a Turing machine. However, that conflicts with the traditi=nal intuitions of most of physics: that the universe is geometric, i.e. hy=ercomputational. The fact that we cannot construct a hypercomputer, not ju=t not in physics, but also not mathematically (where we take its existence=as given when we perform geometry), makes me suspect that perhaps even God=cannot make a true geometric universe.

How can we recover continuous space from discrete computation? Well, s=acetime is the set of all locations that can store information, and the se= of all trajectories along which this information can flow, as seen from t=e perspective of an observer. We can get such an arrangement from a flat l=ttice (i.e. a graph) that is approximately regular and fine grained enough= If we disturb the lattice structure by adding more links, we get nonlocal=ty (i.e. some information appears in distant lattice positions), and if we=remove links, we get spatial superposition (some locations are not danglin=, so we cannot project them to a single coordinate any more, but must proj=ct them into a region).

On the elementary level, we can define a space by using a set of objec=s, and a bijective function that maps a scalar value to a subset of these =bjects. The easiest way of doing might be to define a typed relationship t=at orders each pair of objects, and differences in the scalar are mapped t= the number of successive links of that relationship type. We can use mult=ple relationship types to obtain multiple dimensions, and if we choose the=relationships suitably we may also construct operators that relate the dim=nsions to each other via translation, rotation and nesting, so we derive t=e properties of Euclidean spaces.

To get to relativistic space, we need to first think about how informa=ion might travel through a lattice. If we just equalize value differential= at neighboring locations, we will see that the information dissipates qui=kly and won't travel very far. To transmit information over large dist=nces in a lattice, it must be packaged in a way that preserves the value a=d a momentum (in the sense of direction), so we can discern its origin. A =ood toy model might be the Game of Life automaton, which operates on a reg=lar two dimensional lattice and allows the construction of stable, traveli=g oscillators (gliders). In Game of life, only the immediate neighbor loca=ions are involved, so gliders can only travel in very few directions. A mo=e fine grained momentum requires that the oscillator occupies a large set =f adjacent lattice locations. SmoothLife is a variant of Game of Life that=uses very large neighborhoods and indeed delivers stable oscillators that =an travel in arbitrary directions.

I think I have some idea how to extend this toy model towards oscillat=rs with variable speed and more than two dimensions. It may also possible =o show that there are reasons why stable traveling oscillators can exist i= id, 2d and 3d but not in 4d, for similar reasons why stable planetary orb=ts only work in 3d.

To give a brief intution about a traveling oscillator as a wavelet: Th=nk of a wavelet as two concentric circles, one representing the deviation =bove zero, the other one the deviation below zero. They try to equalize, b=t because the catch up is not immediately, they just switch their value in=tead. (This is the discretized simplification.) Now displace the inner cir=le with respect to the outer one: the arrangement starts to travel. Making=the pattern stable requires
distorting the circles, and probably relaxing =he discretization by increasing the resolution. The frequency of the wavel=t

oscillation is inversely related to how fast it can travel.

You can also think of a wavelet as a vortex in a traveling liquid.

The vortex is entirely generated by the molecular dynamics within the liquid (=hich are our discrete lattice computations), and it does not dissolve beca=se it is a stable oscillator. The vortex can travel perpendicular to the d=rection of the fluid, which is equivalent to traveling in space. It cannot=go arbitrarily fast: the progression of the liquid defines a lightcone in =hich each molecule can influence other molecules, and which limits the tra=el of every possible vortex.

Also, the faster the vortex moves sideways, t=e slower it must oscillate, because the both translation and state change =epend on sharing the same underlying computation. It will also have to con=tract in the direction of movement to remain stable, and it will be maximal=y contracted at the border of the light cone. (The contraction of a vortex=is equivalent to giving it a momentum.)

An observer will always have to be implemented as a stable system capa=le of state change, i.e. as a system of vortices that interact in such a w=y that they form a multistable oscillator that can travel in unison. From =he perspective of the observer, time is observed rate of state change in i=s environment, and it depends on its own rate of change, which in turn dep=nds on the speed of the observer. This gives rise to relativistic time. Al=o, the observer does not perceive itself as being distorted, but it will n=rmalize itself, and instead perceive its environment around itself as bein= distorted. As a result, the observer will always have the impression to t=avel exactly in the middle of its light cone. This model seems to recover =orentz invariance, but with a slight catch: it seems to me that while spee= of light is constant and there is no preferred frame of reference wrt acc=leration, the resolution of the universe changes with the speed of the obs=rver. No idea if this is a bug or a feature, or if it will be neutralized =y something I cannot see yet before I have a proper simulation.

Obviously, all of the above is just a conjecture. I can make a convinc=ng looking animation, and I am confident that many features like simultane=ty etc. will work out, but I don't yet know if a proper numeric simula=ion will indeed work as neatly as I imagine.

Energy? Unlimited? Equal per computation ? Non local ? =AO Two

places at once? Distribution s. Field effects time to compute=/ all

the same time ? Synchronized

In the computational oscillator universe, energy ha= two forms: there is the information contained in the oscillator pattern i=self, which to me looks like its mass: how much information fluctuates in =ach step? (Mass is basically displacement of information in time.) And there is momentum, which is the amount of information that gets translated alo=g the computational graph. (Momentum is displacement of information in spa=e.)

If we look at the relationship between the locus of computation and the glo=al state, a number of variants are possible: - global calculation advances all bits in the state vector at the same time=br> - single bit local calculation advances one one bit at a time

• multi-local calculation has a number of individual "read/write heads=quot; that weave simultaneously

All variants can be realized so that the resulting dynamics are the same, w=ich means that they would be independent from the perspective of an observ=r. However, variants B and C could also be implemented in such a way that =he outcome of the computation depends on the order in which locations of t=e universe are touched. I doubt that this is the case, because it might ma=e the universe look for stochastic than it does.

Energy comes from?