Is the Future Pre-determined?

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.

Einstein & Infeld ( 1938 )

The assumption of an absolute determinism is the essential foundation of every scientific inquiry.

Max Planck ( 1958 )

As I understand the question of whether the physical universe is deterministic or not is profound both in philosophy and religion. The aim in this essay is not to attempt to answer the question. I only present facts as well as observations of my own.

Determinism has fundamental implications amongst others for questions such as whether a person has a “free will” and directly related to this in religion the Christian beliefs in “conversion” as opposed to that of “the elect.”

In science and in reality determinism is intimately linked to concepts such as “chance”, “risk”, “coincidence”, “luck”, “cause and effect”.

For practical purposes it decides the question of fate, or then, of destiny.

The narrative is loose and the essay informal. I will digress on issues in natural science and explain some ideas in mathematics relevant in an indirect way. My intention really is more to entertain and stimulate the imagination.

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If one were to define chance as the outcome of a random movement which interlocks with no causes, I should maintain that it does not exist at all, that it is a wholly empty term denoting nothing substantial.

Boethius ( c.480 – c.524/5 )

One may understand the universe and the laws for its “time-evolution” (change / development of its physical state with time) as either a “semigroup” or as a “dynamical system”. In contrast to a semigroup a dynamical system inherently asserts that future time as well as past time is infinite, time has no beginning and no end. However as a semi-group there must be a start; a time equal to zero. The Big-Bang, or the moment of Creation.

To explain what is meant by a “phase space” as a term used in mathematics and in physics, without going into any technical detail and without considerations of individual models, I will attempt to convey the idea in familiar language and within common experience.

A “point” in the phase space specifies the whole configuration, the complete physical state of a system. In mechanics for instance it gives the position, mass, velocity and acceleration and all relevant information of each individual particle (eg at some moment in time).

The phase space is then the totality, the collection of all possible and all conceivable states (points) for that system. It should not be confused with our familiar two-dimensional plane and three-dimensional space. Mathematically the phase space is normally of infinite dimension. It is definitely not a space as we are familiar with this word.

In a way we may think of a point in phase space as a frozen frame in some video film capturing all information as at that instant. Some further imagination applied from science-fiction stories and films could help! Given some specified fixed initial state x, a “trajectory” or “orbit” is then such a time curve, from history to future.

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The defining properties of a semigroup are very reasonable if one thinks of time passing by.

If we start in Cape Town and drive to Durban and then from there to Pretoria, we end up in the same place as when starting in Cape Town and driving straight to Pretoria.

The most important rule would assert that if time starts now this instant, and we land up somewhere C (in phase space!) in exactly three hours; then we must land up in this very place C when starting now and are at point B in one hour, and after starting again at B, two hours pass.

Thinking of daily experience all is self-evident and this is an essential property when experiencing time. This was of course not a new insight at all- the real achievement and actual significance is to be able to define this “causality” in a mathematical statement, an equation of evolution.

For interest’s sake, our causality is formulated very simply and concisely in the equation:

E(t +s) = E(t) E(s)

If in addition we can “return” from any given moment in time backward to any previous time our system is invertible and is called a dynamical system. It has this additional property, an additional constraint, a kind of “perfect memory.”

An interesting observation: If the invertability condition is dropped so that we have a semigroup then then it is possible to arrive at the same state x following different, distinct, trajectory.

It means you cannot know your history. No-one can. As a manner of speaking: I know where I am, but I don’t know how I got here..

In scientific modelling the laws for time-evolution of a system are encapsulated in some equations: initial conditions (time zero), and a system of partial differential equations. Different situations could apply to either, and the tipe of time-evolution approach that is used has far reaching consequences.

One might say: A trajectory in phase space is the way reality flows with time governed by the divine laws of nature.

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Past time is finite, future time is infinite.

Edwin Hubble ( 1937 )

Chaos theory is usually studied in a probabilistic framework. However it can be applied in deterministic systems.

One example is the dyadic transformation. It is a very simple process: If I start with a given number x between 0 and 2, and I double it, 2x, on taking the “fractional part” for my next (new!) x then I have done one step. I repeat this any number of times.

Symbolically: x(n+1) = 2 x(n) mod 1

This is in fact a deterministic process but since you are working with finite precision, on a computer, you are making round-off errors. When experimenting on a PC you will find that you lose accuracy rapidly. So soon you cannot have any clue where you should be with the given initial x, the first x, your “seed” number. It seems that the number is now random. It is not. Your “errors” compound radically. The limitations are on your side.

The same happens when want to predict the weather. I can predict one day ahead quite well. Another day? A week? No clue. Your guess is as good as mine. Plus in this case you have the sensitivity to data of an initial state, in other words your physical measurements available right now.

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Regarding the Solar system and the calculating of the relative positions of planets, going back in time, I am not familiar with the numerical methods nor the modelling used “backwards” in time. But it can only be a hopeless exercise and for those same exact reasons. For a start the differential equations are totally nonlinear not even remotely approachable by linear systems and on top of that you are calculating backward in time. It makes “closed form” ( “analytical” ) solutions impossible (with any known techniques). Your only recourse is number crunching.

Secondly your data simply has to be hopelessly inadequate. Added to that there are unavoidable intrinsic limitations on numerical precision of the machines. For the reason that even the slightest perturbation compounds very rapidly .

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These in my own mind brings no confidence in the proposed theories of human activity’s influence on climate change- global warming then. The “events” have been throughout the aeons and characterised by greatly varying time scales, some geological, others brief, all varying in severity, some of gradual onset, some of sudden onset. There is indisputable geographical evidence and even recorded history of these fluctuations. There is enough evidence to know that volcanic activity plays a role. However perturbations of the orbit of Earth clearly is the principle cause. These in turn easily can be explained by gravitational effects of the orbits and relative location in space, of the massive planets in particular Saturn, Jupiter and Uranus, the gas giants.

The fact is one cannot know the effect of orientations of planets in the past since you cannot know their positions, and therefore it would not be possible to have an accurate idea of the influence today. So one cannot be sure whether pollution or gravitational effects cause global warming.

Furthermore most of the physical evidence of human influence presented seems to me a matter of “putting the cart before the horse”. With this I mean that if some or other coincidental observations would now ascribe global warming to atmospheric pollution, then those observations are in fact themselves the result of the climate change.

And even, is it necessarily so that industrial (human) activity by itself does increase the amount of carbon dioxide significantly?

Personally I believe for the currently warming climate this is what the cause is: Perturbations of the Solar orbit of planet Earth, and the burning of fossil fuels contributes very little.

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If added the simple time “invertability” axiom makes a semigroup a dynamical system and results in unavoidable, surprising, even upsetting consequences.

Fig’s 1 & 2 relate to the recurrence theorem for dynamical systems as proved first by Poincarè, his theorem is of a very general nature there are very few restrictions. It has startling consequences, it states that an initial state of the phase space is always revisited, and in finite time. The diagrams should give one some understanding of the concept. One could interpret this amongst others to mean that the whole universe as it is at this instant will at some future time recur, exactly as it is now, and then of course again! Infinitely many times!

In the first diagram two balls are taken at random at every step, one from each box, and they are swopped around. It makes sense that at some time one should eventually revisit the starting point and be in that same position again.

The same should apply to the second example where two gases in separate halves of a tank mix together the instant a partition is removed. In my model the two examples are identical save that in the second the number of “balls” is of magnitudes higher. However, one should know that the expected return time in even a very small model as this is astronomical, of the same order as the estimated age of the universe.

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Every phenomenon, however trifling it be, has a cause, and a mind infinitely powerful, and infinitely well informed concerning the laws of nature could have foreseen it from the beginning of the ages. If a being with such a mind existed, we could play no game of chance with him; we should always lose.

Poincarè ( 1908 )

In order to make Leibniz’s idea of a “Monad” plausible I shall briefly explain how such behaviour is indeed found both in mathematics and in physically observed reality, and even everyday life.

The paradox “Achilles and the tortoise” originates from ancient Greece (Zeno). The idea is simple but it definitely is a non-trivial question.

A very ordinary event: a door is open and one wants to close it. One must first close it halfway. Then ¾ way, or, half of that which was left. Again, the new halfway mark must be reached and this continues indefinitely. The argument then is that one cannot close the door and this is the paradox.

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A much more interesting example is that of a ball that bounces until it comes to rest.

Fig 4 should make clear the idea of how a sum of infinitely many, non-zero and positive numbers, can be a finite (bounded / fixed) number. The diagrams should be self-explanatory.

This is not so spectacular, so let us instead consider the ball that hops. The ball is modelled as having a elasticity coefficient of k < 1. It would mean that with each hop kinetic energy is lost, in proportion to k.

The formulas needed and my calculations are on the diagrams and the claims may be verified.

Successive times for bounces also decrease in a geometric fashion so that the infinite sum converges and the stopping time T is finite. Note that there are infinitely many bounces, but the ball does stop.

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A few very typical properties of fractals are illustrated well with a (simple) geometric figure I now describe:

As in the illustration the geometric construction is done in steps:

1. We start with a isosceles triangle, each side 1 unit in length.

2. The midpoints of all three the sides are joined for the inscribed isosceles triangle, dividing the triangle into the four smaller triangles.

3. The procedure is then repeated, on each of these smaller triangles.

And this is repeated indefinitely, to infinity.

Our construction is a kind of triangular mesh, it has surprising properties:

For example although the combined areas stays exactly the same and equal to at the start (due to the fact that a line has no area) the sum of the lengths all line segments is infinite, this isn’t hard to confirm.

The really important thing for us is that the construction is “self-similar” and by that I mean if any little triangle, very tiny tiny is chosen, then the geometric figure obtained is identical to the original construction, it is a scaled version and in every other aspect it is an identical copy. This means that it by itself contains all information that our construction had.

One can make the same construction on regular polygons: a square, a pentagon, hexagon etc.

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There are analogies in physical reality for instance holograms. I am told that when a hologram is burnt in a crystal and the crystal is cut into similar pieces the hologram again appears exactly the same in each piece. We may compare this with a glass window. Through my window I see the whole garden. When ¾ of the window is covered, again I can see the whole of the garden by going closer to the window.

A further observation: in any theory in physics the assumption has to be made that the natural laws stay unchanged where-ever we are. For every single point.

So that the concept of a “Monad” is indeed quite realistic.

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It is fascinating how all this ties up with the empathy concept which was discovered by Prof Niko Sauer. The Empathy causality is a generalisation of that of a semigroup.

S(t +s)= S(t) E(s)

Again the idea wasn’t really so novel but the insight was on formulating this “dual” causality in a mathematical expression.

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Gottfried Wilhelm Leibniz

These principles have given me a way of explaining naturally the union or rather mutual agreement [conformitè] of the soul and the organic body. The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other by virtue of the pre-established harmony between all substances, since they are representations of one and the same universe.

The Monadology and Other Philosophical Writings ( 1714 ), trans. Robert Latta ( 1898 )