Computability, Life, Entropy and Arrow of Time in a Mathematical Universe

7/31/2019 Computability, Life, Entropy and Arrow of Time in a Mathematical Universe

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Computability, life, Entropy and Arrow of Time in

a Mathematical universe

Alberto Gmez Corona, agocorona(at)gmail.com 2006

Keywords: Entrophy, Arrow of time, computability, Natural Selection, Grand Unification

There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it ishere, it will instantly disappear and be replaced by something even more bizarre and inexplicable. Thereis another theory which states that this has already happened. Douglas Adams

An universe made of Mathematics?

Once, upon a time (3 May 2003, Scientifica american) An article appeared in ScientificAmerican named Paralel univesesbyMax Tegmark. A more extended version of the article can be

downloadedhere. This article examines how the distribution of cosmic background radiation suggest thatthe universe is infinite. The Inflation theories and the Superstring theories suggest that different parameter

values give different kinds of universes. There are many nondimensional parameters whose value seemsfinely tuned for the existence of life. non dimensional parameters are constants of nature whose value is nondependent on the units of measure. For example, the number PI is a non dimensional parameter. Somemagnitudes, like he masses and charges of particles, the relative strenght between the four basic interactionsof matter etc can be expresses in the form of non-dimensional constants. The variation of a few percent canmake the universe inhabitable. Why this fine tuning must exist and no others?. This and otherconsiderations makes natural to think that there are such inhabited universes. But this is nothing but one ofmany levels in which the experimental evidence and the lack of reasonable constraints compels to thinkthere are different levels of parallel universes. And this is simply because it is the simpler explanation,

because other alternatives demand unreasonable hypothesis. The mathematical universes hypothesis, is alogical conclusion of all of this stuff. The perceptible worl obey mathematical laws at the fundamental level,

and all the macroscopic phenomena, including life, are a consequence of such laws.

According with the mathematical universe hypothesis, each mathematical equation describes an universe,from the equation of a circle to the equation that describes our physical "real" universe. Roughly speaking,each mathematical equation describes a n-dimensionalmanifold. A manifold is the multidimensional figurethat contains all the possible paraameters that the free parameters in the formula can take. For example, forthe formula x2+y2=C with C a constant describes a circle. Is a uni-dimensional figure. taking C as a variable,the manifold is a two-dimensional cone in a three-dimensional space. Under this view, our lifes are set ofpoint that describe trajectories in the manifold along the time dimension. Time is seen from thismathematical point of view, a subjetive perception
http://www.sciam.com/article.cfm?chanID=sa006&articleID=000F1EDD-B48A-1E90-8EA5809EC5880000http://space.mit.edu/home/tegmark/http://space.mit.edu/home/tegmark/http://arxiv.org/abs/astro-ph/0302131http://arxiv.org/abs/astro-ph/0302131http://arxiv.org/abs/astro-ph/0302131http://arxiv.org/abs/0704.0646http://arxiv.org/abs/0704.0646http://arxiv.org/abs/0704.0646http://en.wikipedia.org/wiki/Manifoldhttp://en.wikipedia.org/wiki/Manifoldhttp://en.wikipedia.org/wiki/Manifoldhttp://www.sciam.com/article.cfm?chanID=sa006&articleID=000F1EDD-B48A-1E90-8EA5809EC5880000http://space.mit.edu/home/tegmark/http://space.mit.edu/home/tegmark/http://arxiv.org/abs/astro-ph/0302131http://arxiv.org/abs/0704.0646http://arxiv.org/abs/0704.0646http://arxiv.org/abs/0704.0646http://arxiv.org/abs/0704.0646http://en.wikipedia.org/wiki/Manifoldhttp://www.sciam.com/article.cfm?chanID=sa006&articleID=000F1EDD-B48A-1E90-8EA5809EC5880000


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It seems that the universe in which we live has a relatively simple mathematical description in comparisonwith what would be expected. Why in the first place the universe has to be mathematical?. Mathematical ina broad sense means "absence of contradictions". My hypothesis is that the mathematicity and simplicity issomething necessary for the computational constraints that life imposes. Thus, my hypothesis is that everypossible universe exist, but life and Natural Selection needs simplicity, continuity and absence ofcontradictions in the laws of the universe.

What restrictions impose the life to any habitable universe

For a self aware structure to emerge in a universe in which previously didnt, the only known mechanism isDarwinian evolution and his generalization: the genetic algorithm. But evolution where? The objective is toestablish the necessary conditions that a universe must have for evolution to exist.

The evolution impose a set of constraints over the universe, highlighted by Max Tegmark in his TOE articleIs ``the theory of everything'' merely the ultimate ensemble theory? .These constrains comes from thefollowing pre-requisites:

1 predictability2 stability3 complexity

From these requirements, Tegmark deduces some consequences about a inhabitable mathematicaluniverses. If we are completely agnostics and we dont event start from formal mathematics but from anyrational or irrational form of universe, it may be seen that the existence of life and intelligence imposes strictconstraints applicable to any world, being mathematical or not, one of them is that it has to have a lot ofmathematics inside, as I will show.

If we take a look at the three prerequisites, ln the light of evolutionary biology and his generalization, thegenetic algorithms, we can extract some considerations:

1 Predictability is a constraint introduced by the fact that life, in a way or other, is composed of a set ofbehaviors in order adapt to repeated sequences of events that appear in the environment. Intelligenceappears in life to predict complex situations in the future in order to make the organism to be ready for itand to choose the adequate behaviour. Predictability is the seed of Reason and intelligence. If for example,the being inhabits a universe in which Boolean Algebra is not valid, the organism has no such basic tool topredict the future and to accomplish accordingly, so no intelligence can arise in this case. More basic thanthat, and appearing at an earlier time in evolution, if there is no similar thing like fluid dynamics, and fluidsare ever chaotic, the being will never be capable to evolve to swing. That kind of reason can be applied to anyphysical aspect in which the being has to evolve.

Predictability, under my point of view, rules out any non-mathematical world, and seriously impairs anyuniverse in which there is a paradox, although slight paradoxes which can marginally affect evolution canexist. If we consider that, I may guess, and it is only a speculation, the Russell paradox may be slightmathematical failures in the world we live, although these marginal incoherences does not precludes we

believe- intelligent life.

2 Stability is related to predictability. Stability is necessary because, for an evolutionary process to achieveresults, it is necessary a stable environment along many generations (or iterations, in algorithmic terms).Taking together stability with predictability we can obtain another implication for the mathematicalmanifold we inhabit and to other possible inhabitable universes:
http://en.wikipedia.org/wiki/Russell's_paradoxhttp://en.wikipedia.org/wiki/Russell's_paradoxhttp://en.wikipedia.org/wiki/Russell's_paradoxhttp://en.wikipedia.org/wiki/Russell's_paradox


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it is necessary a sort of local causality that permits that the most recent and near events are the ones whomost affect the immediate future, while the farther event in time and space produce less events locally inspace and time. This requirement of smoothness comes from the fact that for life and intelligence evolution,due to the iterative nature of it, and therefore the implication of relative short life and short reach for anyevolvable life form, the most relevant events for the immediate future of the life form have to be in theorganism life experience, that is, if the far space and time events produces frequent events locally, they will

be seen as random under the limited experience of the being. Although the innate instincts deal withunknown events and may evolve and generate complex behaviours if the previous generations were exposedto these events, it is necessary a local event that trigger in order to activate this behaviour. This non localcausality doesnt favour evolution. Local causality does.

Also, it is doubtful that intelligence can appear if there is non local causality, since the world would be plentyof random events for which the reason will be irrelevant, and because that, the reason and intelligence

would be non-adaptive in these conditions, and, therefore, would never may appear in that kind ofuniverse.

For example, some conditions coming from the far past may be, in the context of a earth like planet, themountains, due to billions of years of plate tectonics and other long term geological events, but themountains are environmental conditions, not in any sense events, if we understand an event as somethingappearing at some instant of time. An earthquake, in the contrary, may be an event due to the same

geological reasons. But intelligence -and even life-, can be guessed, may not appear in a world plenty ofearthquakes, meteoric impacts and so on. Not to mention elephants appearing and disappearing.

The stability and predictability constraints has far reaching implications: a inhabitable universe must haveclear rules (by 1) and these rules have to have local causation (by 2). Obviously it is necessary to elaboratemore these implications.

3 complexity-simplicity: complexity is, at first sight, a requirement, since the life and intelligence that weknow demand complicated physical and computational architectures. But how complicated the physical-mathematical laws may be? The answer, as we extract from evolutionary theory is: Enough to support life,

but not more. Why?. There are two reasons:

a) because high forms of life , candidates to be intelligent beings, needs heavy neural computations foranything: from sound and image recognition, displacement, trajectory calculus, body functions, resourceoptimizations etc. And, more strongly, intelligence needs a lot of other neural processing, closely associatedto certain mathematics. Life evolution is, in a certain way, the instinctive discovery of the -local form- of themathematical rules of the corresponding universe. If the rules of the universe are too complex, it is moredifficult that a being can evolve to master them in order to survive and even have intelligence. If, forexample, the rules are plenty of complicated exceptions with implications in the life of the living structure,in a way similar to if-else rule in a simulation, or non derivable singularities in a mathematical manifold, theliving being is not capable of "discover" them by evolution in order to perform the life activities like theabove mentioned.

b) And this goes to the second reason for simplicity: exceptions not only make the evolution more complex,but hardly achievable, since the evolution process is parsimonious: For example, there is no animal withwheels, because a wheel can never be "invented" by a progressive adaptation. Once the Mutationist school ofthe Evolutionary Biology was discredited, it is accepted that life forms only progress trough continuous

adaptation to the changing environment, so big changes necessary to adapt to environmental complexitiesare not permitted or very unlikely to be realistic. There is a very highly unlikely that a organism develops anadaptation to a very complex environment without following the intermediate path. That reinforces theconclusions of 3a).

The conclusions advocates for a (1) predictable (mathematical, mostly free of contradictions) (2) with localcausality and (3) as simple and smooth as possible universe. So evolution favour the simplest mathematicaluniverse capable of support it.


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The best way to refine these conclusions is to study, formally, the restrictions of the mathematicalenvironments considered as possible locations in the corresponding universes which, applying the geneticalgorithm, results in a certain kind of evolution, besides that this universe, as a pre-condition, must allowreplicable structures.

Computer simulations:

What is said about the primary problem: the origin of everything can be applied to the, for the secondaryproblem: the universe generated by an existing intelligence, that is, a computer simulation.

Why an universe with life needs the Ockham Razor

Why? Why the word has such bias? I think that I have an answer:

Im interested both in machine learning trough genetic algorithms and in the reason why the realityappears to adopt the most simple mathematical law.

My conjecture link both aspects : By the anthropic principle and the multiverse hypothesis, It appears thatthe universe in which we live is the most simple possible because for biological organism to "learn"instinctivelely trough variation and selection (and , thus, to learn the world trough genetic algorithms), it isa requisite that the fitness landscape, the world, must obey simple, lineal and, smooth laws at themacroscopic scale most of the time for most of the environments.

A more complex universe may need much more time for life to evolve, and this time could be more that thelife span of the entire universe.

Chaotic and non lineal phenomena must be marginal effects of underlying microscopic lineal laws that alsodescribe the rest of the world. (for example a local turbulence of the water obey the same simplehydrodynamic laws that a laminar flow)

This conjecture simply says that, among the universes complex enough, the simplest ones are more probableto harbor complex life.

A more rigorous explanation in mathematical and physical terms need to introduce the concept offitnesslandscape
http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8http://www.google.com/search?q=fitness+landscape&sourceid=ie7&rls=com.microsoft:en-US&ie=utf8&oe=utf8


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A fitness landscape (FL) is a N-dimensional surface,each coordinate contains one parameter that defines an living being. One parameter could be the length ofthe legs, another could be the mean intake of food of this animal and so on. The vertical axis is the fitness ofthis combination of parameters. Along the surface, certain combinations of parameters produce morefitness that others.

The FL uses to be represented as a two dimensional surface, but the real FL includes all the relevantparameters. Dont get overwhelmed by the complexity of this. Just lets mentally take two of the Nparameters (a projection on two axis). For every two arbitrary parameters, there must be a two dimensionalsurface like the one in the figure.

Natural selection by definition, try to find more optimal design starting from less optimal ones. Bydefinition, the combination of animal parameters will describe, trough evolution time, generation togeneration, a line in the fitness landscape that goes from less to more values of fitness.

This is a cross-cut, in the fitness landscape around alocal maximum; To this maximum converges different lines of evolution trough natural selection, becausemore appropriate combinations of parameters gives more fitness, more offspring that continues thetendency to climb to the maximum. We see two of such lines in this crosscut.

It can be demonstrated that every small change in the parameter values towards the optimum value mustcorrespond with an increase of fitness for the evolution to progress towards the maximum. That is , the firstderivative of the curve must be positive when ascending from the left side. There can be relatively big or

small changes in fitness for a small change, but the fitness change must grow towards the maximum.otherwise, the evolution will stop at every local maximum it encounters (for example, A). Suboptimalmutations can move down from a local maximum A to the bottom of the valley so that the offspring of theapparently less adapted living being can start to climb towards a higher maximum B than A. However, alandscape plagued with similar local maximums will make evolution towards higher maximums verydifficult.


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The worst case is a random fitness landscape wherepeaks and valleys are close together. There is no defined direction of evolution in this case. Selection can not

work.

The shape of the fitness landscape is closely related with the laws of physics. For example, there is la lineal,

non chaotic relationship between the length of the leg of antelope and the maximum speed that he canachieve with other parameters staying constant. This is because the laws of forces, both for lineal andcircular movements are constants. The relation between force and acceleration, the relation betweenangular momentum and angular acceleration are expressed in simple lineal functions that are analyticallycontinuous. Because this linearity, the muscular and nervous system must not be complex in order tocontrol for more or less velocity. Simple systems are more easily "discoverable" by natural selection.

Moreover, as I said before, selection operates better with simple laws because non complex curves with a lotof local maximums impede a rapid progress towards higher maximums. If , for example, laws of circularmomentum were non lineal perhaps for legs of 25, 50 and 60 cm would have had local maximums. This

would impede a rapid progress towards the global maximum of 1,30 m for example. Fractal or chaoticrelationships between physical parameters would make evolution impossible, because the fitness landscape

would be random. No life could evolve in such universe.

To summarize:

-Complicated laws produce complicated phenomena

-In some cases, linear laws produce linear environments (with smooth fitness landscapes) where life canevolve.

These are the strong points in my argumentation.it follows that we must live in a certain universe where macroscopic phenomena must obey smooth,continuous, and parsimonious laws for the fitness landscape to be that way; that is, to permit life.

It is our universe, the thing that has been selected for life, just because it is simple. Therefore we, livingbeings, succeed when we try to explain real phenomena through smooth, simple, continuous, andparsimonious laws with little assumptions (some of them, that does not work always). Our universe has the

bias (at least withing the limits of sizes, energies and so on withing which life evolve). It is not by chance.Other universes have not such bias, but they are empty of observers. The so called kolmogorovcomplexityofthe everyday world is a minimum.My arguments advocate for continuity and simplicity applied to the physical laws relevant for the living

beings. This is not the case of the extreme small and the extreme big.(In this case the Ockham razor shouldnot apply for these strange cases, according with my conjecture).
http://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_enhttp://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_enhttp://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_enhttp://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_enhttp://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_enhttp://www.google.com/search?hl=es&rls=com.microsoft%3Aen-US&q=kolmogorov+complexity&lr=lang_en


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For example if a world like ours to exist, it is necessary the quantization of energy, plus fermionic particles.Without these ingredients, atoms would immediately collapse if the classical macroscopic laws were alsovalid for the subatomic scales. The only kind of matter existent would have been black holes!.A quantization of energy in the microscopic world is necessary for the existence of the atoms, chemistry,organic chemistry and life.

it seems like our ordinary environment is an oasis of simplicity for which the more complicated laws of thevery small and the very big conspirates to produce indeed.

The arrow of time is the easiest computational direction for lifein the manifold

The Arrow of time is, at last, a common subjective experience experimented by living beings. According withthe laws of physics, time is just a dimension.. or not even that . The solutions of the equations of GeneralRelativity are quasi-arbitrary four dimensional manifolds.where time is just a local dimension. That is, can

be approximately supposed that in a certain point, the time could be considered as a dimension, but thedirection of this dimension can change from point to point. Superstring theory suggest even more bizarre 11-dimensional, geometrical figures. No intuition about the arrow of time or any other subjective experiencecan be extracted from physical theories.

The only natural law that links with subjective experience is the anthropic principle applied to life ingeneral. Life imposes strong restrictions in the particular form that our Universe (or portion of Universe)has in the infinite sea of optional solutions of General Relativity and String Theory. These restrictions alsoapplies to the initial conditions. of this universe. The observed increase of entropy with time in our visibleuniverse means that it started with a very improbable configuration. But this reasoning is circular: If we tryto elucidate what really is the arrow of time, we can not use concepts that presuppose a certain direction ofthe arrow of time !!

The question, reformulated in strict physical terms is as such: In the four or eleven-dimensional manifolddescribed by the equations of relativity or the superstring theory respectively, why our lifes follows a linefrom less to more probable configurations of matter, that is, from less to more entropy, that is, from lessmicro-states to more micro-states for each observable state?. Why ?

This last view in terms of micro-states is the key for the explanation: causes are in the side of less micro-states. Effects are in the side of more micro-states, because there are less causes than effects. In the otherside, life is all about prediction of the future. An organism can not make use of the environment for its ownends if he can not predict what will follow at the chemical, biological, instinctive or rational levels.Computationally, it is much more easy to simulate the evolution of a system where entropy increase than inthe opposite direction. The precision demanded for a reverse simulation is much higher: The calculus of thefragmentation of a glass is not very difficult. Essentially, the results are not very different using a precision

or another. But a reverse simulation from a broken glass never will reach the re-composition of the glass, nomatter how precise the measure of the real position of the pieces from a real case were introduced in thesimulation.

But the fact is that in our universe, glasses do recompose themselves, the flame of the candles dorecombines liberating oxygen and make grow the candle, objects lighter than water sink. Why? becausethese events exist in our space time; Just go in the reverse time dimension in our space-time manifold to seethem. The laws of physics permits them. They are just reversible chemical reactions, reversible objectcollisions at the particle or macroscopic level.


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In terms of our perception of time, the outcomes we see happens just because they are quasi-infinitelyprobable and the reverse counterparts, quasi infinitely improbable. But, that is also an illusion of the arrowof time, because , In terms of time-agnostic space-time manifold reasoning, our life vector in space-time goalong the increase of entropy, not the other way around. That is: the outcomes of probability laws are aconsequence of our trajectory in space time. Why our life follow this direction?. The reason iscomputational, as I said before.

The essence of life is to identify risks and opportunities, that is, to identify causes to react accordingly inorder to achieve effects that permit survival and reproduction, while maintaining the internal disordercontrolled. This happens at all levels of life. From the chemical to the neuronal level, the living beings lookat the present to predict the effects in the physical direction of entropy increase. life can not operate in other

but in such direction, because this is the computable direction.

Computation is done at all levels. chemically a cell computes by generating proteins triggered by anstimulus, these proteins will aid the cell to cope with what will come nest, either engulf food in theprotoplasm and digest it or to expel toxins to scape from a predator. For this reason life chooses the easypath in the universe-manifold: because computationally it is easier

he key here is that neither our universe is simulated nor time has meaning outside our psychology. Thereare simulation, but this simulation is carried out by us, the living beings. We are the ones that simulate in

advance the events along a direction of the manifold in order to advance actions for the next point in thiscoordinate. Why? because we need to plain further actions in order to grow and reproduce in successiveprogression in this direction. This progression along this direction is what we perceive as time.

Because living beings are the ones that must simulate in advance what comes next, at the chemical,instinctive, rational level, this imposes very serious computational restrictions to the direction of time.Simply, the reverse simulation , along the entropy increase demands infinite or near infinite resourcescomputational resources. It is also possible simulate in any lateral direction, any direction in the manifold,

but I hypothesize that they are also very heavy to calculate. Thinking in terms of Natural Selection: Theliving beings that tried to progress along other directions are extinct, They needed too much computationalresources!!!. Or even never appeared in the first place!.

For this reason, the perception of time, entropy and probability, and the initial conditions of the universeare a byproduct of this restriction of computability in living beings.

living beings are like fractals that grow, reproduce and die along the temporal coordinate. There are twoways to express a trajectory , and here I use the Max Tegmark coined frog/bird view; The frog view is timedependent and uses the input of the previous step, which is the view of computers, living beings and us, andthe other is the bird view that contemplates the manifold or part of a manifold . The first type of beingsstruggle for anticipating the next step,. The second see the enlarged Mandelbrot figure of all our life, andthis figure is part of the manifold described maybe by a single formula The M formula or whatever may bethe grand unification theory.

This is not so difficult to understand: This is the difference between the expression of a integral inmathematical terms versus the step that a computer must perform along the integration coordinate tocalculate its numerical value.

A single macroscopic state (you in your room reading this) has many microscopic states (the possiblecombinations of positions and velocities of all the particles in the room). A macrostate has more entropythan other when the number of possible combinations of microscopic states is greater. A ninja in the groundafter the forward jump and the surrounding hot air has more entropy than the ninja in the top of the

building, because higer temperature has more possible combinations of particle speeds.

If I were living in the reverse arrow of time, to predict that the ninja in the ground flexing his legs will jumpto a three floor building I have to track the concrete microscopic state of the ninja, that is the positions and

velocities of all the particles in the legs of the ninja plus the surrounding air molecules


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In the right arrow of time, I can do an educated guess of where the ninja might land by knowing only hismacrostate, that is, his position on the top of the building.

computational needs for survival in forward and backward arrow of time

This presentation has more on this subject:

Arrow of time determined by life s easier direction for computation

References:

Evolution ofBiological Information

Why intelligent life needs an universe with a limited set of mathematicalstructures

Intelligent life needs its own harware, that is, brains. brains execute algorithms. Some results of somealgoritms are certain laws and conclussions that predict with more or less accuracy the outcome of things inthe reallty. It happens that many laws or models apply to different scales of the reality; Thus, the process ofdiscovery of the reality is possible. Reality happens to accept a limited set of mathematical structures.Some of them are ubicuous. The simplest example is, 2+2=4 whatever kind of objects we apply to. If thiiswere not the case, rhe evolution would not have the opportunity to create brains that infer things based onthe previously accumulaed knowledge.
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Computability, Life, Entropy and Arrow of Time in a Mathematical Universe

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