Ok, let´s imagine that the Universe is finite (even being so enormous). At the border of the 'end', there will be Nothing.

But Nothing can not exist physically, since a single sub-atomic particle is enough for transforming Nothing into Universe.

Perhaps a theory about Nothing could explain the rapid expansion of the Universe: Nothing acts like a vaccuum, a pump sucking the Universe hence it expands more and more and more.

From a strict epistemiological point of view, we can not know if Universe is finite or not; all we can say is that we can receive data from a specific big ray of distance, but it does not mean we can reach everything.

I have no interest in metaphysics. Therefore, answers that deal with the Ultimate Shape of the Universe that goes beyond the observable universe are to me similar to, as well as trivial and irrelevant, as the answers to the question, "how many angles can dance on the head of a pin?" The question is related to the Observable Universe Only. The experiments done, so far, using the CMB show that the Observable Universe is Euclidean. The question is, "is there a way to show that this Observable universe is both Finite and Unbounded". thanks.

The observable universe is of course finite and bounded by our cosmological horizon, we can only see a small portion of the universe.

Your question on whether the universe is finite or infinite can only relate to the whole, not just the observable portion and Ivan's answer is correct and quite comprehensive.

Conventional cosmology doesn't assume the universe (as a whole) is finite but it allows for that possibility, if the curvature is positive as you say, it is finite, if it is zero or negative as is possible then it is probably infinite, but a combination of flat (zero overall curvature) but finite is possible through the 3-torus topology.

The measurement by the Planck mission was that ΩK=0.0±0.005 allowing for all the possibilities.

George Dishman : the fact that the observable universe is finite does not necessarily mean it is also bounded. This is the crux of the question because I can easily say that there is nothing beyond the observable universe. I am only interested in what we know and not in what may be possible. Such metaphysical possibilities belong in philosophy and not science until we find factual evidence for them. Thanks.

Dear Mustafa Ali Khan-Hussaini,

There is nothing metaphysical or philosophical about my reply, it is purely mathematical and the mathematical modelling of nature is the basis of science.

If the universe is finite, the simplest topology is that it has the same geometry as the surface of a 3-sphere, also called a glome. That is unbounded because there is no edge to the surface but if that was all there was when we look out in one direction and compare that with what we see diametrically opposite, the views should be a mirror image, the same part of the CMB seen from opposite sides. Searches have been done looking for such patterns but none have been found.

Second, if we live in such a universe, the radius of curvature is given by R=LH/√ΩK where LH is the Hubble Length, about 14.4 billion light years, and ΩK is the curvature density measured by Planck as ΩK=0.0±0.005. That suggests R is greater than 210 billion light years so the circumference is at least 1320 billion light years. We can only see 46 billion light years in any direction so we have very good observational (factual) evidence that there is a lot more out there than we will ever be able to see.

The specific question you asked was "is it possible for the universe to be finite and unbounded and yet be Euclidean?" and the answer to that is yes, but only if the topology is that of a 3-torus rather than a 3-sphere.

George Dishman : 1) "If the universe is finite, the simplest topology is that it has the same geometry as the surface of a 3-sphere, also called a glome". The question is precisely about the possibility of the universe being finite and yet not be a 3-sphere! 2) You talk about the curvature of this glome and yet say that the curvature is zero with error margin of 0.005. how is this possible? we take the p-value of < 0.05 to be sufficient for a measurement to be acceptable and yet you give a p-value of 0.005 for the curvature and still not accept that the Measured curvature is not Actually zero. I do not see any reason not to accept the curvature as being zero. 3) the rest of your calculations on radius and circumference are irrelevant if the curvature is zero. 4) Okay, so we can see 46 billion light years away. why cannot we take that as the whole universe?

There is a reason behind me putting this question on RG. so far the answers I got are not something that I am not already aware of and see them all over the cosmology literature. thanks.

Dear Mustafa Ali Khan-Hussaini,

GD: 1) "If the universe is finite, the simplest topology is that it has the same geometry as the surface of a 3-sphere, also called a glome".

MAK: The question is precisely about the possibility of the universe being finite and yet not be a 3-sphere!

The answer to that is "yes, but the topology has to be more complex, such as a 3-torus" as I said at the end of my post and as Ivan Gutierrez-Sagredo said in the very first reply. What more do you want to know?

MAK: 2) You talk about the curvature of this glome and yet say that the curvature is zero with error margin of 0.005. how is this possible? we take the p-value of < 0.05 to be sufficient for a measurement to be acceptable and yet you give a p-value of 0.005 for the curvature and still not accept that the Measured curvature is not Actually zero.

Current models suggest that there was a period called "inflation" in the very early universe. If that is correct, any finite positive curvature would have stayed positive, it couldn't become zero, but its magnitude would be greatly reduced, so the actual value might be for example +10-20. The instruments we have cannot measure with that accuracy, and local variations would have a larger Gaussian spread anyway.

MAK: I do not see any reason not to accept the curvature as being zero.

That is a common assumption, in which case the simplest topology is a flat plane and the universe should be expected to be infinite. Whether it is actually infinite or just very much larger than the observable part is not something we can distinguish.

MAK: 3) the rest of your calculations on radius and circumference are irrelevant if the curvature is zero.

The calculation gives us a minimum (at the one sigma confidence level) based on actual measurement. If the curvature is zero, we should assume the universe is infinite (I think most cosmologists do just that).

4) Okay, so we can see 46 billion light years away. why cannot we take that as the whole universe?

a) Because it is unscientific to assume a value that is less than the calculated minimum based on actual observation.

b) Because tests looking for repeated patterns in the CMB have found none.

George Dishman : I think we are talking past each other, so let me put it this way, "can one find a way that shows that, 1) the observable universe is the whole universe, 2) it is flat but it is neither infinite nor a 3-torus?". I say that it is not only possible but that IS the topology of our universe. this is the reason I put the question on this forum. thanks.

Dear Mustafa Ali Khan-Hussaini,

MAK: I think we are talking past each other, so let me put it this way, "can one find a way that shows that, 1) the observable universe is the whole universe ...

Let me be clear in my answer: no, that conflicts with scientific observation.

MAK: 2) it is flat but it is neither infinite nor a 3-torus?

Then you would need to look for some other form of even stranger topology, I've never heard of any other possibilities.

MAK: I say that it is not only possible but that IS the topology of our universe.

If you prefer to hold that as a religious belief, I will support your right to believe anything you wish. I prefer not to go down that metaphysical route and instead base my opinion on the observed facts.

MAK: this is the reason I put the question on this forum. thanks.

The purpose of a question should be to learn about something that is puzzling you, not to preach your own beliefs under the pretence of asking others to share their information.

George Dishman : 1) "Let me be clear in my answer: no, that conflicts with scientific observation". So, you are telling me that the question of the universe being more than what is observable is settled and is a fact that all cosmologists accept without question just like the expansion of the universe or that the sun is the center of our solar system! 2) "If you prefer to hold that as a religious belief, I will support your right to believe anything you wish. I prefer not to go down that metaphysical route and instead base my opinion on the observed facts". Which is more metaphysical, a) the observable universe is the whole universe or b) that the universe is a 3-torus? Just because you have "never heard of any other possibilities" does not mean that it does not exist or is not possible. 3) Having a strong "gut" feeling or a "hunch" is not equal to holding "a religious belief". the history of science is proof of the fact that many great discoveries started as a "hunch" or a "gut feeling". to believe strongly in this "hunch" is not same as holding a religious belief. If I believed in this "gut feeling" the same way as I hold my religious beliefs, then I would not have asked the question at all! Finally, 4)" The purpose of a question should be to learn about something that is puzzling you, not to preach your own beliefs under the pretence of asking others to share their information". This statement is ad homonymic. I have asked a question, which is certainly scientific, and is something for which I am trying to find an answer. do not I have the right to question your answer? is this not the scientific process? how did you conclude that these are my beliefs and this question is a "pretense" to "preach" them? do you know my beliefs and, more importantly, do you know me so well as to know my intentions? are these kinds of assumptions about people, you hardly know, scientific? how does one get information from others? is it not through questions? I do not know you, but I will not stop from questioning the answers of anyone, whoever it may be. thanks.

I think it will be better for me to rephrase the question in a way that will be, hopefully, clearer in it's meaning.

Let us assume, 1) that the observable universe is the whole universe and 2) that it's curvature is zero (or is Euclidean). Given these, can we find a way that shows, 1) that it is finite and 2) that it is unbounded without being a 3-torus or some other topology with a non-zero curvature.

I have a hunch that we can show this is possible if we can broaden our thinking horizon. thanks.

Mustafa, you should add to the reasoning the problem of possibility of an infinite unitary evolution. For an infinite (Euclidean) space, this is a logical necessity. The initial point of existence is incompatible with an infinite connected space, while the evolutionary (exponential) process always passes through 1. Of course, there is always the possibility to add spontaneity to space, which automatically calls into question the very possibility of the existence of conservation laws in the universe under consideration - compromise here is not rational.

Vasyl Komarov : Great! these are the kind of answers I am looking for. creative, unconventional, unique. these answers will help me in my own thinking about this question. the bookish answers I can read on my own and have done it. I got this idea and wanted to share with others and see what they think. thanks.

Physics, mathematics, theology and conventional philosophy, which are based on the world view of causality (or what Hegel called the “view of understanding”) cannot comprehend the Infinite and hence is incapable of giving an answer to this question one way or the other. Physicist Einstein (like others of the above mentioned disciplines) assumed a finite universe to formulate his theories of relativity. Modern physicists like Max Tegmark of Harvard thinks that the word “infinite” as a scientific idea is “ready for retirement") ! https://www.theguardian.com/science/2014/jan/12/what-scientific-idea-is-ready-for-retirement-edge-org

For modern mathematics, the Infinite is a "dangerous idea", because it is the “twin of Zero or vice versa” (Charles Seife)! Giordano Bruno was burnt on the Stake by the Inquisition; for insisting that the universe is infinite etc., etc.

But long before Professors Einstein, Tegmark et al., Immanuel Kant based on the world view of causality came across this question of finite or infinite universe as one of his famous antinomies. He showed conclusively (based on causality) that both sides of this antinomy can be showed to be equally true and equally false through logical argument alone! So, causality based natural science, mathematics, philosophy, theology etc., can keep on arguing forever (as is foolishly done in modern theoretical physics in relation to the various aspects of the theories of relativity, questions of multi-verses etc.), without coming to any conclusion!

Only a dialectical world view can give an understanding of the finite and the infinite as a contradiction (of the “unity of the opposites”); and the continuous and never-ending resolution of this contradiction through evolution giving rise to the phenomenology of the Infinite universe. But this means ditching all causality and mathematical idealism based Einsteinian concept of a created universe, Big Bang origin, “inflation”, “expansion” etc. in the dust bin of history, because in an infinite universe these terms of modern physics has no meaning!

The key to the understanding of the Infinite came with the profound idea of Spinoza about “limit”, which helped Hegel to formulate the dialectical view - the contradiction of the unity of the opposite of the finite and the Infinite. Spinoza formulated the principle that “all determination is negation”. To determine a thing is to cut it off (to negate it) from other things and so to limit it. To define a thing is to set boundary for it. The Infinite in this view is therefore “undetermined”or “undefined”, but in another sense is “self-determined”or “self-defined”, meaning that the Infinite is determined only by itself. This means that the Infinite sees only itself (and its dialectical opposite finite) beyond any arbitrary limit or boundary! Please see: Article The Infinite - As a Hegelian Philosophical Category and Its ...

We have no clue on what a cosmology can be.

Our models are naive hypotheses based on theories that have a limited range of application and after having extreme inductions.

• An example is the way relativistic formulas are derived: by using Lorentz force- which is applicable on charged bodies - and then a generalization is done for all kind of bodies..
• The concept of space-time does not exist, except from our dashboards: whenever we find an integral in a theory we are not allowed to define a relevant "space" just for our convenience
• All exotic object have not been observed unambiguously, just because they don't exist
• Enormous amount of money are spent in accelerating charged particle just to convince us for the orthodox theory (CERN)
• Almost a billion dollars was spent to convince us for the general orthodox theory (LIGO)

Personally I declare my ignorance and try to define some fundamental concepts here:

Article A new cosmological paradigm: universal locality

As for the main question:

The concept of a boundary is 100% anthropocentric, forget universe, it is not your garden!

Demetris Christopoulos : "The concept of a boundary is 100% anthropocentric, forget universe, it is not your garden!". But, what i am interested in is to find a way such that the observable universe does not have a boundary, and on the contrary it is unbounded even though it is finite! to put it in another way, "if we assume that the universe is Euclidean, does it necessarily follow that it has to have 1) a boundary or 2) it has to be infinite? Is it not possible that it can be neither, i.e. finite and unbounded? Abdul Malek brought up the Hegelian dialecticism showing the contradiction of the synthesis of finite and infinite as a contradiction. But, what about the synthesis of finitism and unboundedness (which is not the same as infinite). in this case the Hegelian dialecticism does not apply since the two are not opposites and therefore their synthesis is not a contradiction. one way that comes to my mind is to use probability and make it a core or basic characteristic of the universe at all levels and not just at the quantum level. with this we can make the boundary of the universe both finite and unbounded. it will be similar to the "boundary" between my "garden" and that of my neighbor's where they merge into each other as there is no fence that separates the two. both myself and the neighbor can only guess where our gardens end and the other's begin. in this situation we both know that our gardens, despite being flat/Euclidean, are finite but unbounded. thanks.

Mustafa Ali Khan-Hussaini : Your term “unbounded” has no meaning and is arising from confusion - very common, if you see things from the world view of causality. The premise of the Einsteinian view that the universe is finite and “unbounded” has no meaning at all and is contradictory (not in the sense of dialectical contradiction). It is an opportunistic and mystical idea, precisely meant to be confusing to cover-up the vacuity of that premise! Your term “probability” is also meaningless in the same way as “unbounded”.

Nothing is unbounded! If you posit a “being” or a "thing", it means it has identity and is determined (and negated, according to Spinoza) and hence must be bounded by something. Even in Hegel’s dialectics, the Infinite is bounded by itself!

In dialectics, any “existence” at all (of material or thought objects) is a contradiction of the “unity of the opposites” (meaning they have both unity and opposition at the same time!) and is mutually bounded by its opposite. This is a view that is impossible for formal logic of causality, which follows Aristotle’s principle, “Unity, Opposition and the excluded Middle”; meaning that a “being” can exist alone by itself, is given once for all (perfect in itself) by an outside power or God of theology and has no contradiction.

Causality works only for everyday life experience, classical mechanics, or simple systems of Newtonian physics where the cause and its effect can be clearly determined; but beyond these, as in the complex systems like biology, cosmology and more dramatically in the quantum phenomena causality get entangled into irresolvable contradictions, and an appeal to mystery, wonder, awe and in the last resort to theological God, the ultimate “first cause” and “the unmoved mover”!

For dialectics, because any existence is a contradiction and because reason cannot rest in a contradiction, this contradiction has be resolved through the mediation of chance and necessity (and not probability alone!"). But the resolved “being” is itself a contradiction with newly added content and must undergo resolution itself and so on (“negation of the negation”) without any end; giving rise to the phenomenology of the universe. Hence for dialectics, the universe must be Infinite, Eternal and Ever-Changing!

So for dialectics the impetus for motion, change, development, evolution etc., comes from within the thing itself, but for causality it must come from outside in every instance and in the last analysis from an omnipotent and omniscient being, meaning God. A fate that causality based modern physics now faces!

Please see the references I cited (and others in my RG profile) for further details. Sorry, if I do not find opportunity to respond to any further comments from you directed to me. Dialectics is un-intuitive (as quantum physics is!) for most people who are not used with this kind of reasoning. But I can assure anyone that it is a more powerful tool than causality for any kind of epistemology. I am using this tool in my practice of physics in the realm of cosmology and the quantum microcosm, and also in biology where causality becomes severely limited. Regards.

The answer is no. The universe is not Euclidean in large scales and intermediate scales, only approximate. So the conditionals do not rescue the question. Infinities are not found in nature, only in the abstractions that aspire to describe it. The universe is in some way finite. We don't observe boundaries except in the horizon limits of long ago and far away. Limits are one way that infinities don't happen.

The only possible boundary would be the set of extreme distances beyond which results of the big bang have not had time to travel. It presumes a larger space in which the universe resides as a part of the larger cosmos.

Unbounded universe still allows expansion so the exact same events are not likely to occur periodically.

If you consider that big bang of our universe occurred as the destruction of a super massive black hole during collapse of a false vacuum then your question is drawing more from abstract mathematics than from a physical description.

There is a competing view that space and time were created spontaneously along with the universe springing from nothing and with no cause other than chance in a big bang, giving rise to your question in the physical context. My objection is that spontaneous creation implies possibility of spontaneous destruction with nothing remaining, not even time and space.

Jerry Decker : "The universe is not Euclidean in large scales and intermediate scales, only approximate". we have a margin of error of 0.5% for the curvature when an error margin of < 5% is considered to be sufficient for the accuracy of a measurement to be acceptable! The possibility of spontaneous creation gave rise to a universe that was very different than what it is now. therefore, the spontaneous creation does not necessarily imply spontaneous destruction. even if spontaneous creation implies spontaneous destruction, why is that not acceptable? even if the universe is a 3-torus, it will not protect it from spontaneous destruction. thanks.

previously I used probability as a possible solution to the question. there is another way we can use. in this we make information as the Ultimate Reality. this means that matter, energy, space, time and so on are nothing but sets of elements which are pieces of information. the universe is thus one big set of information elements. we measure the various pieces of information when we study an "object", but no matter how accurate our measurements are, there is an Absolute, Irreducible Margin of Error that exists. it is this error that brought the universe into existence through the process of Big Bang. also, it is this margin of error that makes the observable universe both finite and unbounded even though it is Euclidean. it is quite possible to formulate a mathematical theory using these sets that is as logically consistent as any physical theory using matter, energy, space, time and so on. it is easily seen that the theory using probability and the theory using sets of information are equivalent. thanks.

Abdul, causal invariance is perfectly combined with logic, self-reference, evolution and infinity. But it, unequivocally, not friends with spontaneity, Euclidean space and, accordingly, with quantum physics, which, because of methodology, does not try to find out the causes of statistical data. The problem is not in quantum physics, it is a global problem of demarcation. Spontaneity really should be regarded as a null hypothesis or as an indicator that a system of the right scale is not included in the consideration.

Try to get rid of self-referencing in logic at first... it is the Science is a "world view of causality" in it's essence.

By the way, dialectics is causally invariant in principle.

Igael, it is a question of terminology - the concept of the "universe" is transformed along with the expanding horizon (since the time of geocentric views it repeatedly tipped over prefix "multi").

But unlike Mustafa, I can not exclude metaphysics, it's a matter of "existence", literally (-; And it dramatically affects the reasoning.

Dear Mustafa,

What I am trying to say is that all of our cosmological constants are taken from our own constraints since we are material beings.

All kind of questions:

• what was before? (why should exist before? because we are born someday?)
• is it finite? (why should it be classified such a way? because we understand it?)
• is it Euclidean? (why? in order to fit to our measuring procedures?)

are simple generalizations of our neighborhood characteristics.

Sorry, I don't know how our local universe is (that will be answered by Physics some day).

Vasyl Komarov

VFK> “Try to get rid of self-referencing in logic at first... it is the Science is a "world view of causality" in it's essence.”

Vasyl, As I already explained to Dr. Khan, that my participation in this forum unfortunately would to be limited for unavoidable reason, also I think we discussed this issue before and I don’t think further discussion here would be of any benefit. But I would like to respond to the two points you raised in your quote above, because both are true; but I have very good reasons for both and I am sensitive to any accusations about those points.

To take the second part first, I have to say that yes, natural science (so far) selectively used the world view of causality, even though it is a poorer tool of epistemology than the world view of dialectics and there is a perfectly good reason/explanation for this. In fact this is the primary point that I always emphasize in my discussion on modern natural science, specially the theories of relativity. The world view of dialectics and causality arose in early Greek philosophy more or less simultaneously (as dialectical unity of the opposites!) with (near) contemporaries like Heraclitus (544 – 483 B.C.) on the side of dialectics and Pythagoras of Samoa (580 (?) – 520 B.C.) and Permenides (515 – 450 B.C.) on the side of causality. While Heraclitus claimed “change due to inner conflict – (the brilliant germ of dialectics!”) as the primary attribute of the universe, the other two on the contrary emphasized stasis and claimed that change, motion are only illusions and not real.

In the evolution of life, of history and of human thought, development, change, or progress makes its appearance by the negation (the “negation of the negation” and the Portentous Power of the Negative” for Hegel) or destruction of what exists. Of necessity, and because of their very nature as the conservative, the resisting, the preserving side of what exists, the world view of causalty always sided with the established order of the time, while dialectics represented the revolutionary side, because dialectics denies the stability or the permanence of what exists. It is a standard Marxist contention that, “The ruling ideas of an epoch are the ideas of its ruling class”. Present parasitic monopoly capitalism has made the mathematical idealism based theories of relativity as the ruling ideas as a substitute of God of equally parasitic feudalism. Monopoly capitalism with the help of its “scientist serfs” is forcing physics to preach theology!

Please see the RG forum: https://www.researchgate.net/post/Am_I_the_only_one_that_is_doubtful_of_LIGOs_detection_of_gravitational_wave_GW150914

To be continued later on!

MLK: " ...the Einsteinian view that the universe is finite and “unbounded” has no meaning at all and is contradictory ..."

This is simply not true. Einstein postulated that the universe is finite in time (bounded at the big bang singularity), and unbounded in space -- i.e., if one could travel the four dimensional universe in a geodesic one would not find a boundary, and would end up at the starting point, just like the path of a geodesic on the 3-dimension Earth.

What if the world, though, is actually finite in space and unbounded in time. Would Einstein's GR equations suffer? I think not. In fact, there may be a cosmological advantage in a model of nonlinear time.

Mutafa Ali Khan you wrote " matter, energy, space, time and so on are nothing but sets of elements which are pieces of information"

Pieces of information describe physical components incompletely. The information is not the physical quantity, and is always found to be missing something important.

Uncertainty of 5% is allowed in single controlled science experiments that do not impact health or safety. Otherwise 0.5% is applied as the uncertainty limit for single controlled experiments affecting health and safety of just one person. These standards are not appropriate for a cosmology.

Earth gravity is only 3 parts per billion different from flat space. No one is trying to say it doesn't exist based on argument of insufficient percentage. Where gravity is considered Euclidian geometry is only approximate. The difference is found in passing of light and in orbit of planets, movement of galaxies and larger collections.

You wrote "there is an Absolute, Irreducible Margin of Error that exists. it is this error that brought the universe into existence through the process of Big Bang." I guess you are saying it happened by chance. I prefer the representation of Roger Penrose in cycles of time and recent speeches, giving causes for the universe to begin and end and begin again. I differ from him in that I allow some mass to pass from one cycle to the next if it is moving at extreme high speed.

Yes, Abdul Malek, we discussed this, including the topic "Is Any Effective Refutation of Einstein’s Theories of Relativity Possible?" you created, where just in june we discussed the (historical) long-term influence of dialectics and the integrated contradiction of the scientific method, so I will not bore you here again with logical paradoxes provoked by self-reference.

I see well in which aspects my and your interpretations converge. I sometimes react to your statements, because the contradictions in them help to think and transform my own thoughts, for which I am grateful, of course.

Dear Mustafa Ali Khan-Hussaini

George Dishman : 1) "Let me be clear in my answer: no, that conflicts with scientific observation".

MAK: So, you are telling me that the question of the universe being more than what is observable is settled and is a fact that all cosmologists accept without question just like the expansion of the universe or that the sun is the center of our solar system!

Most scientists accept that those are accurate but a few still try to find possible alternatives but within the bounds imposed by observation, that is the key aspect of science that separates it from religion and philosophy.

GD: 2) "If you prefer to hold that as a religious belief, I will support your right to believe anything you wish. I prefer not to go down that metaphysical route and instead base my opinion on the observed facts".

MAK: Which is more metaphysical, a) the observable universe is the whole universe or b) that the universe is a 3-torus?

That the the observable universe is the whole universe is contrary to observation. That the universe is a 3-torus would be a reasonable possibility but cannot be proven. It is however a solution to the question you asked, could the universe be both flat and finite.

Science tends to go with the simplest solution though, essentially based on "Ockam's Razor" or the Principle of Parsimony, there is no need to introduce complexity beyond what is essential within the constraints of observation. As such, if we observe zero curvature, the simplest topology is that the universe is Euclidean and therefore infinite.

MAK: Just because you have "never heard of any other possibilities" does not mean that it does not exist or is not possible.

Correct, but it also doesn't mean that such a possibility exists and whether the possibility exists or not doesn't alter the fact that there is no objective evidence for anything other than a simple Euclidean (flat) and infinite universe.

MAK: 3) Having a strong "gut" feeling or a "hunch" is not equal to holding "a religious belief ...

No, but it is also not scientific. Al science must be free of personal bias so the models should be derived purely from observation and experiment.

GD: The purpose of a question should be to learn about something that is puzzling you, not to preach your own beliefs under the pretence of asking others to share their information"

MAK: I have asked a question, which is certainly scientific, and is something for which I am trying to find an answer.

Yes, you did and I answered that with information to help you on your search, that was all perfectly reasonable. When it changed was where you said:

MAK: "I say that it is not only possible but that IS the topology of our universe."

That is not a question, it is a statement and it is contrary to the science. It is your personal belief.

MAK: how does one get information from others? is it not through questions?

Yes, and your initial questions were fine and they got you relevant and applicable information.

MAK: I think it will be better for me to rephrase the question in a way that will be, hopefully, clearer in it's meaning.

MAK: Let us assume, 1) that the observable universe is the whole universe

As has been said, that is not what observation tells us, you have gone outside science with that assumption.

MAK: 2) that it's curvature is zero (or is Euclidean). Given these, can we find a way that shows, 1) that it is finite and 2) that it is unbounded without being a 3-torus or some other topology with a non-zero curvature.

If the curvature is zero, it should be infinite. A two-dimensional slice through the universe would be a flat plane. To make it finite, you have to somehow fold it round so that it repeats like the surface of the Earth. The simplest topology that meets your request is the 3-torus. As you say, there may be other topologies of which I am not aware that also do this, but they cannot be simpler than the three-dimensional equivalent of a flat plane. Basically you are asking for the impossible, because it cannot be flat and not a plane and not wrapped round.

Perhaps it would help if I asked you some questions:

1) What evidence do you have that the universe is not larger than what is within our horizon?

2) What evidence do you have that the universe does not have positive curvature at a level less than the limit set by the Planck observation?

3) What evidence do you have that the universe, if flat, is not infinite?

P.S. Let me add a simple question to see if I can understand your view.

4) If there were ocean around Mount Everest, I could stand on the summit and my horizon would be 340km away. I could measure that, work out that the radius of the Earth was 6370km and that the circumference was around 40000km so I could see only 1.5% of the way round. Why do you think the same reasoning doesn't apply to the observable universe?

Jerry Decker : "Pieces of information describe physical components incompletely". But we use the binary system in IT do describe the physical characteristics. cosmologists use computer modeling to understand the universe or the formation of the solar system.

George Dishman : 1) just the use of Occam's Razor, 2) again Occam's razor and the error margin of 0.5%, 3) an infinite universe will not be compatible with the Big Bang Theory and the CMB. In such a universe the temperature of the CMB will be zero.

thanks.

Dear Mustafa,

GD: 1) What evidence do you have that the universe is not larger than what is within our horizon?

MAK: 1) just the use of Occam's Razor,

Okham's razor says the minimum size based on the Planck mission measurement is many times larger than we can see.

GD: 2) What evidence do you have that the universe does not have positive curvature at a level less than the limit set by the Planck observation?

MAK: 2) again Occam's razor and the error margin of 0.5%,

That margin allows for any value of density in the range 100±0.5% times the critical density, it doesn't narrow it down any more than that. Observations consistent with inflation says it should be much larger, for argument say around 1+10-30, times the critical density, so that would be the guess based on Okham's Razor, the universe might be 1015 times larger than our horizon.

MAK: 3) an infinite universe will not be compatible with the Big Bang Theory and the CMB. In such a universe the temperature of the CMB will be zero.

Not at all, the standard or "concordance" BB model because it is in accord with all observations, is flat and infinite and 13.8 billion years old. You seem to have a misapprehension about the model.

What about question (4) ?

Vasyl Komarov

Dear Vasyl, To continue my comment on “self-referencing":

So far as I am aware of, I am the only one who is trying to extend materialist dialectics to cosmology and quantum physics, in my various published books and journal articles. Examples are:

1. Cosmology:Article THE CONCEPTUAL DEFECT OF THE LAW OF UNIVERSAL GRAVITATION OR...

2. Quantum Physics:Article Real/Virtual Exchange of Quantum Particles as a Basis for th...

So, I can only cite myself in my various writings and comments! I continue to follow Frederick Engels who first formally initiated the application of materialist dialectics in natural science as a division of labour with Karl Marx in their extensively shared collaboration. After Engels’ death only (except from Lenin’s “Materialism and Empirio-Criticism) some Japanese physicists like Shoichi Sakata, Nambu et al. continued Engels’ approach significantly, in physics for a while in the thirties; but nothing else was done after them. In biology of course, Engels’ approach continues; starting from J.B.S. Haldane, I. Oparin. J.D. Bernal et al and continues till to day with Levin, Lewontin et al..

But in physics I see none and the simple answer why I have to reference my own works! I very often refer to Engels' work though, because I depend mostly on his lead. In my efforts to extend dialectics to new and revolutionary developments in physics after Marx and Engels; I find it necessary to extend (though controversial to some people I know) the scope of materialist dialectics itself! Please see (if you wish) my book, "The Philosophy of Space-Time: Whence Cometh Matter and Motion?", at the following link:

https://www.amazon.com/Philosophy-Space-Time-Whence-Cometh-Matter/dp/984041884X

Mustafa Ali Khan, binary number system does not give any guarantee of accuracy or completeness. Much of science is trying to say that math is the exact science, but the math is always an approximation for any number system.

Yes, the flat three-torus is finite and Euclidean.

Abdul, dialectics occurs naturally. In retrospect it is not difficult to observe it everywhere, what Thomas Kuhn wrote in "the structure of revolutions". Sometimes it is useful to try on the skin of the "inductive donkey".

Vasyl Komarov

VFK> “Abdul, dialectics occurs naturally. In retrospect it is not difficult to observe it everywhere”

Dear Vasyl,

You have a correct description of dialectics and in fact it is its great merit as a science of all sciences. Scientifically we can know only the past, the reality that has already happened or experienced. Idle dreaming and speculation, predictability and so on of the future without a solid basis in the past may please one’s ego and psyche; but is of little importance for acquiring positive knowledge. Trying on the skin of the ‘inductive donkey’ does not turn you into a “Superman” to have a vision of the future and be a good “Fortune Teller”; it can only turn you into a deceiver!

In view of your quote from Thomas Kuhn, I provide two below from the masters of dialectics:

“Philosophy, as the thought of the world, does not appear until reality has completed its formative process, and made itself ready. History thus corroborates the teaching of the conception that only in the maturity of reality does the ideal appear as counterpart to the real, apprehends the real world in its substance, and shapes it into an intellectual kingdom. When philosophy paints its grey in grey, one form of life has become old, and by means of grey it cannot be rejuvenated, but only known. The owl of Minerva, takes its flight only when the shades of night are gathering.” Hegel, “Philosophy of Right”

“We know only a single science, the science of history. One can look at history from two sides and divide it into the history of nature and the history of men. The two sides are, however, inseparable; the history of nature and the history of men are dependent on each other so long as men exist.” Marx and Engels, “The German Ideology”.

George Dishman : 1) "Okham's razor says the minimum size based on the Planck mission measurement is many times larger than we can see". How? Quite the opposite. Occam's razor will say, based on the Planck measurement, that the universe is observable and finite. 2) "That margin allows for any value of density in the range 100±0.5% times the critical density, it doesn't narrow it down any more than that. Observations consistent with inflation says it should be much larger". " the standard or "concordance" BB model because it is in accord with all observations, is flat and infinite". you are using the first statement to say the universe is flat and finite, a 3-torus, may be. with the second statement you are contradicting your first statement and saying flat and infinite! you need to choose one or the other. cannot have both. 3) "Why do you think the same reasoning doesn't apply to the observable universe?" Because we know the Earth has a positive curvature unlike the universe. thanks.

[Edit: I have updated all the values based on the final Planck report released this July.]

GD : 1) "Okham's razor says the minimum size based on the Planck mission measurement is many times larger than we can see".

MAK: How?

There are three possibilities of the curvature, it might be positive, zero or negative.

https://wmap.gsfc.nasa.gov/universe/uni_shape.html

If we lay aside the 3-torus for a moment, the three corresponding simple topologies are a 3-sphere, a flat universe or a hyperbolic universe. The first attached image shows those. The Planck mission result for mean curvature density is 0.0007±0.0019 corresponding to a total energy density of 99.93% ± 0.19% times the critical density (positive equivalent density implies negative curvature). If the curvature density is more than 100% then you get positive curvature, less than 100% gives negative curvature and exactly 100% is flat. Both the flat and negative curvature cases are spatially infinite so for the minimum possible size, we consider the case of the 3-sphere.

Now look at the second diagram. It shows a circle (nearly horizontal, dotted on the far side) drawn on a sphere. In this image, we would be located at the top at the centre of the red arc. That arc shows the region that is observable, it reaches 46.3 billion light years from us in every direction, you have to image the red arc being a diameter and although it's only drawn as a line, in reality it represents any line drawn through our location.

The radius marked R is the radius of curvature of the 3-sphere and it can be calculated as R = LHub/√ΩK.

where LHub is the Hubble Length and has a value of about 14.4 billion light years.

The smallest value of R corresponds to the highest value of ΩK. which is -0.0012 (that's 0.0007-0.0019) at one deviation. However, scientists usually demand that a robust result should be at least 5 sigma (0.0007-5*0.0019=-0.0088) so we can calculate both of those.

At one sigma, ΩK = -0.0012 and we can say R>415 billion light years (BLY) and to observe the whole universe we would need to be able to see half the circumference of the 3-sphere in each direction. That is simply πR > 1306 BLY.

At five sigma, we would use ΩK = -0.0088 and that means R>153 billion light years (BLY) and half the circumference is πR > 482 BLY.

Our horizon is 46.3 BLY which is a lot less than 482 BLY which is the smallest credible "5 sigma" value so that says there is definitely a lot more beyond our horizon.

MAK: Quite the opposite. Occam's razor will say, based on the Planck measurement, that the universe is observable and finite.

That is not correct as you can see above. If you disagree, show me your calculation.

MAK: you are using the first statement to say the universe is flat and finite, a 3-torus, may be. with the second statement you are contradicting your first statement and saying flat and infinite! you need to choose one or the other. cannot have both.

The measured values do not allow us to say which it is, what I am saying is that it might be finite or it might be infinite. However, even if it is finite, the minimum size for the maximum credible curvature is much more than the observable universe.

GD: 3) "Why do you think the same reasoning doesn't apply to the observable universe?"

MAK: Because we know the Earth has a positive curvature unlike the universe. thanks.

If the universe is finite, it must wrap round with positive curvature, just like the surface of the Earth. In the attached diagram of a circle on a sphere, the red arc would be the fraction visible from high altitude on Earth. The corresponding limit in the universe is the distance light has travelled since the universe became transparent and the CMB was released.

If the universe doesn't have positive curvature, Okham's Razor says it must be infinite as shown in the flat and hyperbolic ("saddle" shaped) graphics in the first attachment.

The only alternative I know of, as has been said a number of times, is if the topology is that of the surface of a 3-torus in which case it could be flat and finite, but a 3-torus is not what Okham's Razor would consider parsimonious.

George Dishman : If you were unaware, then let me tell you that I also have a B.Sc. in physics with highest distinction and honors. Your "lecture", with simple calculations to "explain" the results to me, which I can read anywhere or derive them myself, if I want, are an insult to the discussion I wanted to have here. Hence, as my Hakham taught us, "it is beneath the dignity of a son of Aaron to continue a discussion with someone who resorts to insults", I am ending this discussion with you. Shalom.