INFINITAS – The Concept of Infinity

The purpose of this collegium is to establish a group for those interested in ancient philosophy and a place where philosophical discussion and study may take place. Join at: http://romanrepublic.org/civitas/joint_ ... sophiae/42

Moderators: Marca Marcia, Gaius Flavius Aetius, Paullus Aemilius Gallus, Aula Flavia Philippa

INFINITAS – The Concept of Infinity

Postby Gaius Florius Lupus » Sun Nov 26, 2017 6:06 pm

Salvete amici!

In this thread I would like to discuss the concept of infinity and the profound implications, if we accept that the world is infinite in space and time. Both was suggested by Epicurus, but is today rejected by some modern scientists, either due to some unproven theories or due to an incomplete understanding of the known laws of nature (Cosmic Inflation would contradict Special Relativity). While Epicurus only used empirical observation and logical deduction, modern science is unfortunately increasingly based on arguments from authority, id est argumenta ad verecundiam (exempli gratia »Einstein said…«, »Stephen Hawking said…«,, »It is the consensus of the scientific community…«), or the Christian creation myth (Big Bang theory, singularities et cetera).
Since I am apparently defending Epicurus‘ views here, I would be happy, if someone else could try to disprove Epicurus based on the current Standard Model of Cosmology.

The following is a quote from Epicurus’ letter to Herodotus:

First of all, we must admit that nothing can come from that which does not exist. Were this fact otherwise, then everything would be produced from everything, and there would be no need of any first beginnings, or seeds.
Second, if that which disappeared were so absolutely destroyed as to become non-existent, then everything would soon perish, as the things into which they would be dissolved would have no existence.
As a result of these first two principles, we conclude that the universe as a whole has always been in existence as it now is and always will be in existence. For there is nothing else into which the universal whole can change, and there is nothing beyond the universe which can penetrate into it and produce anything new.

Epicurus establishes here, what is today known as the first Law of Thermodynamics, the conservation of energy. From there he concludes the eternity of existence, id est the universe, since nothing can spontaneously come into existence or cease to exist due to the laws of causality.
Today we know that this is not entirely true for event on a quantum scale, where we have no causality and events happen indeed spontaneously. However on a macroscopic level we have the laws of statistics that bring order into the randomness of quantum processes and make our world quite predictable and ruled by causality.

Additional to the infinity in time, Epicurus also proves in his letter to Herodotus that existence must be infinite in space.
The universe is boundless. We know this because that which is bounded has an extreme point, and that which has an extreme point is looked at in relationship to something else. Consequently, that which does not have an extreme point has no boundary. And if it has no boundary, it must be infinite, and not terminated by any limit. The universe then is infinite, both in regard to the quantity of matter from which it is made up, and to the magnitude of the void. For if the void were infinite, and the amount of matter was finite, then the matter would not be able to rest in any place. The elements of the universe would be transported about, scattered across the infinite void, lacking any power to steady itself, or to keep one another in their places by mutual repulsion. If, on the other hand, the void were finite, while the amount of the matter was infinite, then the elements clearly could never be contained in the void.

Since the totality of everything that exists (Epicurus calls this the universe.) cannot be distinguished from anything else, it cannot have a boundary and must therefore be infinite. And if the spacial extension of the universe is infinite, the totality of energy/matter must also be infinite.

Now such a statement does not sit well with modern cosmology, which is based on the Christian belief of a beginning and end of creation. The Catholic Priest and physicist Georges Lemaître came up with the theory of the Big Bang, the moment of creation, which defines the beginning of the universe. Later atheist physicists managed to get the Christian god out of the equation by assuming that the universe was at the beginning condensed in such a small space that it was subject to quantum effects, which are random and can explain a spontaneous creation.
However this theory of the Big Bang has always had serious problems that it cannot explain. Where is the anti-matter in the universe? Why are distant parts of the universe that had never any causal connection with each other so homogenous? How could the universe overcome the initial gravity whose escape velocity would be faster than light? The only way to explain some of these questions was assuming a movement faster than light (Inflation theory) or an unproven force called »dark energy« that explains the accelerated expansion of the universe.
The Big Bang theory has so far not been able to make any prediction which was later confirmed, which would be one of the requirements for a scientific model. All observations on a cosmic scale however contradicted the predictions of the Big Bang theory and the theory had later to be fixed by introducing new variables and arbitrary constants like »dark matter«.

Nevertheless the Standard Model of Cosmology assumes that the universe is not eternal, but began around 13.8 billion years ago. From this point on it expanded in an accelerated speed. It is not infinite in expansion, but space is curved in itself and its total size is big, but still finite. Depending on the amount of matter it is either curved convex, so that gravity will eventually in a far future reverse its expansion and cause a Big Crunch when it collapses again or it is curved concave, so that it will overcome gravity and expand infinitely until all matter loses its cohesion and the universe vaporizes into nothing. This means the universe would be finite in time.
Today the first model has been discarded, since it was observed that the universe seemed to expand at an accelerated speed.

Such a curved universe would require a non-Euclidean geometry. This means the sum of the inner angles of a triangle in a non-Euclidean space would be more than 180 degrees in a convex universe (relative density of matter and energy Ω>1) or less than 180 degrees in a concavely shaped universe (Ω<1). Furthermore two parallel lines in a convex universe would eventually meet, while they would diverge in a concave universe.
413px-End_of_universe.jpg
413px-End_of_universe.jpg (41 KiB) Viewed 1486 times

Therefore measurements were done to prove that the universe was not infinite but curved with a non-Euclidean geometry. However all measurements showed with all today’s possible accuracy that the universe was entirely flat (Ω=1) to an error margin of 1% (https://en.wikipedia.org/wiki/Flatness_ ... e-wmap3-11), id est the universe is Euclidean. Space has no curvature with all precision that we can measure today and seems to be infinite. This observation confirms Epicurus’ predictions and contradicts again the Big Bang theory with its universe that would have a finite size.

We have therefore to conclude that Epicurus’ initial assessment that the totality of existence, be it a universe or a multiverse, is infinite in space and time.
This has some serious implications that Epicurus already hinted at, and that I would like to explain a little bit further in later posts.
For now I would like to hear some objections against this assessment based on the Standard Model of modern cosmology. Or you might even have your own arguments against it. However we have to clarify this point before we move on.

Valete!
C. Florius Lupus
User avatar
Gaius Florius Lupus
 
Posts: 594
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna

Re: INFINITAS – The Concept of Infinity

Postby Gaius Florius Lupus » Sat Dec 02, 2017 5:50 pm

Salvete collegae!

Since no objections were brought forth against the infinity of all that exists, I will assume that we have agreed on it. We can now continue to analyze the logical implications of this premise.

Epicurus explains the following in his letter to Herodotus:
In fact, it is evident that these particles, though innumerable, have some kind of dimension or other. Whatever this dimension may be in other respects, the objects which are composed of them will have an infinite magnitude. As we examine forms which are determined and limits which are perceived by the senses, however, one conceives easily without it being necessary to study the question that objects composed of infinite material would be infinite in size. Consequently, one must come to look at every object composed of a finite number of elemental particles.

due to the quantum nature of the universe Epicurus concludes that every object is formed by a finite number of particles.
And in the same letter he writes:
The particles within the combined bodies, and these solid elements from which the combined bodies come, and into which they resolve themselves, assume an incalculable variety of forms.  This must be so because the numerous differences which the bodies present to us cannot possibly result from an aggregate of the same forms. Each type of form that we observe contains an innumerable number of particles, but there is not for that reason an infinity of particles; it is only that their number is beyond all calculation.

Epicurus concludes that while the number of particles in the universe is infinite the possible combinations of particles into combined bodies is finite (incalculable for practical reasons, but not infinite).
He continues:
Further, the number of worlds are also infinite, whether they resemble this one of ours or whether they are different from it.  For the particles are, as to their number, infinite, as I have shown above, and they necessarily move about at immense distances.  The infinite multitude of particles of the universe from which this world is formed, or by which it is produced, could not be entirely absorbed by one single world, nor even by any finite number of worlds, whether we suppose all those worlds to be like our own, or different form it.  We see that there is therefore no fact inconsistent with there being an infinity of worlds.

In summary Epicurus concludes that a combined body of an infinite number of particles would be infinite in size. However all objects that we can perceive have a finite size. This means they must be composed of a finite number of particles. Therefore there must be only finite number of possible combinations of particles that form objects or even worlds.
If the possible variations of combined bodies is finite, the whole of existence however is infinite, then every possible combination of particles must exist an infinite number of times.
Because if we divide infinity by a finite number, the result will always be infinite.
∞/n = ∞

This has very serious consequences for our understanding of the universe and ourselves that many people and even scholars are unaware of.
Epicurus summarizes this as the conclusion that there must be an infinite number of worlds, some different from ours, some similar, and some absolutely identical.

And it means more than that. It means that there is no principle difference between possible and existing. For mathematical reasons everything that is possible, must also exist in an infinite universe. The probability of its existence is a finite number; and no matter how small its probability may be, multiplied with an infinite number it will always become infinite. (1/n ˣ ∞ = ∞) We can therefore conclude that everything that is possible does not only exist with absolute certainty somewhere at some time in the universe, it exists an infinite number of times.
Things are either impossible or they exist an infinite number of times. There is nothing in between. Nothing can be unique. Nothing can exist only a finite number of times.
By this conclusion we have by the way disproved monotheism. Because if a being like a god is possible once, there is nothing to prevent its existence a second time or a third time or even an infinite number of times. If a god is possible, he exists an infinite number of times.

But this has even more serious consequences. It also means that we ourselves exist an infinite number of times. Our own consciousness is not unique but exists infinitely in the universe. Even if we die, there is an infinite number of copies of us in the universe left. However since they are far away from us beyond the horizon of the visible universe, they are not causally connected with us. This means we cannot place them in any chronological relation to us, because time depends on causality and they are too far away so that events there can never reach us.
And if there is no causal connection and no chronological order, we cannot say that this copy of us is earlier or later or simultaneously with us. Its existence is totally parallel and unrelated to us.
We cannot even say that the consciousness of these copies of us is identical or different from us, because both statements are indistinguishable and have no distinct consequences. And due to the second axiom of classical logic (Axioma Exclusi Tertii = law of the excluded middle), a meaningful statement cannot be true while its negation is also true. In this case it would be a pointless statement.
The statement that our consciousness exists in an infinite number of copies or that these copies are all different consciousnesses would be pointless. Both statements cannot be distinguished. They do not result in different consequences.

In fact causal connection is extremely important. The universe may be infinite, but due to its size not all is causally connected with us. Most of it is beyond our causal event horizon. Therefore the things that exist there beyond our horizon are meaningless to us. Only what is within the horizon of the observable universe is meaningful to us.
I will therefore call this »real to us« and distinguish it from »existing«. Things can exist in the infinite universe, but they might not be real to us within our finite observable universe.
The number of existing worlds is infinite, but the number of real (observable) worlds is finite.
If something is possible, it exists; but it is not necessarily real, id est within our observable horizon.

With this I want to conclude this post and provide the opportunity for discussion and objections.

After that I would like to discuss, whether or not Epicurus' infinite number of worlds might in some way be related to the Many-Worlds interpretation of quantum mechanics.

Valete!
C. Florius Lupus
User avatar
Gaius Florius Lupus
 
Posts: 594
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna

Re: INFINITAS – The Concept of Infinity

Postby Gaius Florius Lupus » Wed Dec 13, 2017 2:36 am

Salvete amici!

When we discuss the concept of infinity, there is one question left to ask: Is the infinite number of worlds as suggested by Epicurus the same as the worlds in the Many-Worlds interpretation of modern quantum mechanics?

Epicurus taught that an infinite number of worlds exist and that therefore any possible world must also exist in an infinite universe, even if it is beyond our event horizon and therefore not real for our consciousness since it is beyond the perception of our senses.

Now let us look at a short summary of the Many-Worlds interpretation of quantum mechanics:
From quantum physics we have learned that the world can be described by the Schroedinger equation, which is a wave function. This means it provides only probabilities for the possible states of a particle including its location and impulse. When we observe the particle, then it has a random state that cannot be predicted by any laws of causality. However the particle behave statistically, as if their states were determined by waves. According to the Many-Worlds interpretation in each case of a non-deterministic event, this means when the wave function of the Schrödinger equation offers more than one possible outcome, our world splits in distinct parallel worlds. In the case of the double-slit experiment the world would split in one world where the particle passes through the left slit and another one where the particle passes through the right slit. For a short period of time the two worlds would still be interconnected and could interact with each other, so that we would get the resulting interference pattern on the screen. But then the worlds would be irreversibly separated and continue their own path of events.
Since such non-deterministic events occur in an incredible number every millisecond, the universe would permanently split in an uncountable number of parallel universes where every possible outcome of events occurs. Everything, which is only remotely possible, would therefore happen somewhere in some of the uncountable number of alternative universes.

From this description we can already see a difference. While Epicurus talks about different worlds in one universe, the Many-Worlds interpretation assumes entirely different universes.
But the question is not that easily answered. We have to ask first, what “identical” and "different" means.

For this purpose we have to consider the second axiom of Classical Logic, the Law of the Excluded Middle (Axioma Exclusi Tertii, everything is either A or not A). If A and not A is the same, then the term A is meaningless and its use not logically valid. In the empiric world this means that an abstract concept must have distinct consequences from its negation. If a concept and its negation have the same consequences, then the concept is meaningless.

In our example we have to ask: Does a different world in the same infinite universe (but beyond the observable horizon) have any distinct consequence from a different world in a different universe?
The answer is: No, because both are not causally connected to us and therefore have no consequence for us. Different worlds in different universes are no distinct concept from different worlds in the same infinite universe. These parallel universes would be contained in our infinite universe. The use of the word “parallel” in this context makes no sense. It is an identical concept, because it has no distinct consequences.

The Many-Worlds interpretation of quantum mechanics however assumes that the parallel universes are shortly superpositioned to each other before they separate permanently and that they can shortly affect each other by interference.
Epicurus’ infinite worlds would not necessarily interfere with each other. This means a remote world that differs only in the position of one particle from our world would not necessarily have any connection with our world that would allow an interference as described by the Schroedinger equation. Nevertheless we have not excluded this possibility by any experiment. Neither has the Many-Worlds interpretation and the temporary superimposition of parallel universes been proved by any experiment. This is one of the reasons why most physicists prefer the Copenhagen interpretation over the Many-Worlds interpretation.
Actually the Many-World interpretation has exactly the same problem as the assumption that similar remote worlds in one infinite universe have a non-local connection with each other. If the one proves false, the other one would prove false too. We know that non-local connections can exist in quantum physics where particles are connected regardless of time and space from experiments. It is called “quantum entanglement”. Therefore we cannot discard the possibility that two remote places are somehow connected with each other. We can only discard the possibility that information or causality can travel that fast.

So what can we finally conclude from the fact that the universe is infinite?
The answer is, we know for sure that in an infinite universe similar worlds to ours must exist, just as described by the Many-Worlds interpretation of quantum mechanics. We do not know, whether or not the Many-Worlds interpretation is correct regarding a temporary interference between them in order to explain the wave function of the Schroedinger equation.

As we have seen the concept of infinity may appear simple and Epicurus’ conclusions may seem trivial to us today, but in fact they have consequences and implications that are relevant even in the face of the latest discoveries of quantum physics.
The main problem is that most of us believe that we have understood what infinity means while in fact we have not.

Valete!
C. Florius Lupus
User avatar
Gaius Florius Lupus
 
Posts: 594
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna


Return to Collegium Philosophicum

cron