Great question Demetris,

One way to address it could be as follows: since there is an infinity of infinities (even though this statement is tautological and hence meaningless mathematically, it is quite understandable in everyday parlance), is the infinity with the strongest cardinality open or closed ?

If it is closed, then the complexity of external reality (rather than simply nature) is infinite but 'circumscribed' within an extant infinite set of possibilities.

If it is open (which I think can be demonstrated), then the complexity of external reality (rather than more narrowly 'nature') is both infinitely complex and open - i.e. uncircumscribed.

Great question Demetris,

One way to address it could be as follows: since there is an infinity of infinities (even though this statement is tautological and hence meaningless mathematically, it is quite understandable in everyday parlance), is the infinity with the strongest cardinality open or closed ?

If it is closed, then the complexity of external reality (rather than simply nature) is infinite but 'circumscribed' within an extant infinite set of possibilities.

If it is open (which I think can be demonstrated), then the complexity of external reality (rather than more narrowly 'nature') is both infinitely complex and open - i.e. uncircumscribed.

My friends, the steps to my personal induction were in chronological order the next ones:

Dear H Chris you gave a great answer and you was first in my next thought:

*My concern is not about the existence or not of universes, but about the cardinality of their infinity.

Your contribution on the open or closed is a next step.

I have to think about it, but, if I trust my mathematical instinct, I think the concept of openness is crucial for topological reasons....

I have to think about this.

Dear Demetris,

I agree with your post. In spite of all claims of mathematics as the language of nature, it is a tool developed by us in our finite wisdom to describe what we experience. As you no there is principle of under-determination, reduction and abstraction, when we model a complex system. Further, current thinking is that it is not always true that a set of causes results in a unique set of outcomes. There are many more probable sets which can be true. It is observer/interpreter as how one models and derive inferences from experience and experiments. As such, the question is reduced to objectivity. We can express in mathematics what we ourselves experience and what inference we draw from experiment out of other possibilities which others can go under experience or draw conclusion. Anyhow our knowledge is partial truth. It is also scientifically and philosophically proved our mind and science have limitations. As such, above claim to my mind is correct.

Dear @Demetris, I agree with you that nature is more complicated than any creation or of any one of its subsets, because we invent and build these creations and our thinking is bounded and influenced by the facts of nature that we can not surpass.

I think there are no two things: nature and the explanation of reality (humanity). The real story is that as knowledge progresses, new mathematics (very much as new physics, new languages, and so on) are being created that help understand better the new dimensions, scales and depths of nature.

The externalist point of view (God's eye, a view from nowhere, etc.) have been reckoned as highly limited and nowadays nearly nobody subscribes in spearhead science such a view.

The complexity of nature is revealed, invented or created (as you wish) accordingly to avery single step of the history of science; say, maths. In other words, there is no an a priori complexity of nature beforehand and then, later on, a description and explanation of it.

Dear Demetris,

Your reasoning sounds somewhat like the ontological proof of the existence of God by Thomas Aquinas...Anyway infinity has always been a source of problems for mathematicians, and openness needs some topological concepts, that is one needs to define a topology ont the niverse.

Actually there are several types of knowledge, because the question is about the knowledge of nature by human beings, The scientific knowledge is special in that it assumes the existence of assertions which are always true, and which can be checked. As a consequence the knowledge that humanity can acquire about the universe can always be challenged, and cannot be elevated to the quality of truth : this is just an efficient representation, valid until proven false. And because our experience is strongly limited, in space and time, our scientific knowledge is obviously small vs the universe.

However with non scientific knowledge one can enlarge the scope, and indeed this is what religions do. But here you need faith, meaning an act (supposedly conscient and free) of adhesion to shared assertions.

Complexity, is something that can be defined within some model. Complexity of nature seems to me not proper. Incomprehensible nature seems OK. On the other hand if the questions asks about cardinalities, then it is more complex. Goint from model to model cardinality is not preserved! May be I do not quiet understand  the question, but it seems to me that it is not well posed.

Right, dear Costas. Just to push the comment a bit farther, the tricky thing about the possible infinity of nature lies in that t goes against the most recent and solid arguments around quantum gravity - which clearly point out to the discrete nature of reality, i.e. the universe.

Let's put it in this way:

are the current and potential universes defined by math at time t

a subset of the current natural universe existing at time t'?

My answer:

if t = t' or t > t' YES

if t < t' NOT

(I assume some >, < or = relations for (trans)infinites (as Cantor proposed)).

sorry, correction:

if t = t' or t < t'  -YES

if t > t'               -NOT

I think the question also touches the problem of 'emergence': a denial of reductionism that occurs when the behavior or properties of a system cannot be predicted or explained as the sum of its simpler components. That problem is still contentious and is related with our question in the next way: if emerging behavior is true in mathematics then emerging complexity could take place out of simple elements, and surpass the universe's complexity even if we are not aware of the full universe's complexity whatsoever. The occurrence of emergence in mathematics has already been observed (Foote,2007. Science, 318, doi: 10.1126/science.1141754). Interestingly the paper use an example from group theory which is relevant by two reasons: 1) group theory is built upon a very simple axiomatic, and 2) group theory is quite a "pure" branch of mathematics which could be out of the reach of empirical sciences influence, i.e. not enriched by nature's complexity. In life sciences (I'm mathematician/biologist) examples abound showing exquisite levels of complexity "created" out of simpler systems due to a number of processes but noticeably by evolution.

Demetris, what do you mean by complex? Higher for number of degrees — then an algorithmical meaning of complexity — or greater by number of predicates?

Given Mathematics as a humanly intended science, it is certainly less complex than Nature in terms of number of predicates. Mathematics, for instance, could not "love", nor set preferences. As soon as mathematics should choose or decide, it would become Ethics.

On the other hand, in algorithmic terms the same mathematics is ontologically equally complex to Nature, because any natural algorithm is ontologically mathematable.

However, I agree that the thing is more interesting if considered in terms of space.

—g

Space? Well, please then take into account the very much tenable studies considering the atomic structure of space. Even if in some fields of math the claim for infinities is completely plausible, in physics the ongoing research goes quite in another direction.

The singular point lies in the strong coincidences between these findings in physics and the parallel research in biology - for instance, in the sense just mentioned by Edson, although with (quite) different lights.

Interesting opinions @Edson (the concept of emergence of complexity), @Giuseppe (as complexity we can define the number of the classes of the universes for example: expanding, flat, shrinking etc) and @Carlos(the different paths for infinities), @jean claude ('the knowledge that humanity can acquire about the universe can always be challenged,'- I disagree with homas Aquinas : example with First way 8. of http://web.mnstate.edu/gracyk/courses/web%20publishing/aquinasfiveways_argumentanalysis.htm ), @Costas (I do not define complexity as math object, rather I borrow the word in order to describe the whole set of potential existences. Let me expand the question: 'Since the Human creation of Mathematics have obtained such an enormous class of infinities, then it is not possible that the Nature has less infinities than one of its components. Thus we end up with the postulate that Nature has to be more 'infinite' than every creation of every subset of it')

@Juan, I liked your formalism and I think I will agree with you: Yes indeed as 'time' passes we have defined a broader and a broader class of universes. Take now the limit t-->oo and think about all civilizations (Humanity, Gumanity, Fumanity, ... artificial names for Humans, Gumans, Fumans ... say for all possible intelligent beings)

Isn't it a nice projection?

What if another, more advanced than us, 'Humanity', could share with us its findings about the infinity of universes...

I copy from:

http://www.ulc.org/wp-content/uploads/2012/09/King-James-Bible-KJV-Bible-PDF.pdf

"

The Epistle to the Hebrews

{1:2}

Hath in these last days spoken unto us by [his] Son, whom

he hath appointed heir of all things, by whom also he made

the worlds;

{11:3} Through faith we understand

that the worlds were framed by the word of God, so that

things which are seen were not made of things which do

appear."

In the Greek translation of Septuagint (70) there exist four mentions for the word "σύμπαντα=universes" at:

Psalms 38:6,Ψαλμοί, 38:6 ιδου παλαιστας εθου τας ημερας μου και η υποστασις μου ωσει ουθεν ενωπιον σου πλην τα συμπαντα ματαιοτης πας ανθρωπος ζων διαψαλμα

Psalms 103:28,Ψαλμοί, 103:28 δοντος σου αυτοις συλλεξουσιν ανοιξαντος δε σου την χειρα τα συμπαντα πλησθησονται χρηστοτητος

Psalms 118:91,Ψαλμοί, 118:91 τη διαταξει σου διαμενει η ημερα οτι τα συμπαντα δουλα σα

Ezekiel 7:14,Ιεζεκιήλ, 7:14 σαλπισατε εν σαλπιγγι και κρινατε τα συμπαντα

In other Religions?

Dear Arno,

The introduction of Aleph Numbers was really a revolution.

The problem is that, as far as we introduce mathematical objects, then we find that Nature is again one step in front of us and again we have not tools to describe it: It is something like the Radius of Knowledge: As this increases, then the Circumference of Ignorance also increases.

So, as intelligent entities, we will approach the explanation of Nature's complexity (or put it in another word) asymptotically and at a very low rate.

Dear All,

It would be interesting to report the references to the concept of universes or worlds -or whatever else similar word- in the main Religions of our planet.

I think that, sometimes, in Religion Texts there are hidden many scientific truths...

Dear Arno, is there any kind of a pre-publication of your work?

It would be interesting...

There is no definable sense of the word “complexity” outside of models– when we accept this, then we can still ask whether a state of the art-description of physical facts is more complex than a human invention (other than this description) can ever be. Apart from this, a biological description of the same fact can be more complex. Complexity on the side of Humanity can be arbitrarily raised by assuming iterations of “I believe that I believe…” and “I believe that you believe, that he believes…” etc. but I think this is not really interesting.

As to descriptions: once we fully understand a physical fact or process we could describe it at different coordinates or for more and more time points …- would that mean more complexity in an interesting sense of complexity?

As to infinity: there is an old rule saying “No difference without a difference!” It means that it should make a (possible) difference in our lives whether a sentence is true or not. An exception to this rule are sentences in mathematics, where proof procedures are involved. The proof procedures make a difference in our lives, of course.

I think that all of us should watch next video:

http://www.youtube.com/watch?v=rENyyRwxpHo

It is one more argument about the weakness of all human-centered cosmological approaches...

Dear Demetris,

thanks for the indication of the link. It's fantastic.

It is just proving that the general accepted cosmological postulates:

• homogeneity of space
• isotropy of space

simply do not exist...

Dear Demetris,

I think this is a misunderstanding: Homogenity and isotropy of space does not mean that agglomerations of matter are stereotyped. Instead it means that it should make no difference whether we look in one direction or another in relation to what we will have to expect. We do neither assume that the laws are different in other regions nor that the distribution (density) of matter has to be homogeneous.

That the matter is not homogeneously distributed can only be detected by assuming that the laws are not changed. We have to distinguish the scales.

So, looking towards the attractor located at Centaurus of the above video does not make a different conclusion than looking at the near empty space around Milky Way? I have some doubts about...

Yes, of course, extremely non-homogeneous, including curving. But that is an assertion about the distribution, not about the physical “frame” of laws in which the distribution is located. No postulates are violated.

Postulates:

E.g. a friend wrote a computer program that displays stars in a constellation resembling an easter egg in your telescope. Than you will say “Oh dear friend…. I got you.” No postulate was violated. It is a question of probability. But

E.g. The telescope shows a constellation within Laniakea that is congruent to another constellation within Laniakea then we will say that it is probably violating a postulate because curved space should not lead to the same picture within a range of 450 Mio light years. If it were a repeated image then a postulate about the geometry of light would be violated. It still would be possible that there are real “repeating” constellations.

My example: Let's suppose that we are in a region of a cluster where, locally, galaxies seem to come towards there from all observable directions: Wouldn't that fact had as a result that we face a contracting universe? (For the video: what do people in an Earth located at Centaurus will say: expansion or contraction, for their own cosmological model?)

The definition of “Local Group” is that the movements are statistically distributed. Some objects are approaching while others move away. The definition of “global” is that the red shift outweighs the statistically distributed movements in all directions (because of increasing red shift in larger distance). The two observations can coexist. There is no inconsistency between them.

And how are we sure that from that 'Earth of Centaurus' there exists an observation for objects that move away?

If I understand your question correctly we can be sure for the same reasons the graphic designer could make the film.

Yes, but if that film is accurate (or the relevant work more precisely), then at Centaurus "region" we have a strange situation, without any similarities to MilkyWay...

No, not at all. This is the point of the last 6 or 7 posts. There is no change in the physical laws. Gravitational forces, atom structure, quantum mechanics, speed of light etc. are not changed when we want our discussion to share common ground. Without this common ground the graphic designer could not make the film. And we would need a different definition of isotropy etc.

The graphic designer can also know how the Milky Way will look from outside (within certain limits). When A moves away from B, somebody living on A will say “The others are moving away with a speed of…” and this is the same for somebody living on B.

Yes, about physical laws, no about the evidences for cosmological conclusions. If I see everything to come towards me, then it is reasonable to conclude that my universe has a contractory nature.

Why should you see only objects that are approaching? 

Hi Demetris,

There is probably a misunderstanding about the word “attractor”. This word appears often in contexts of a phase space, not our normal space. In the video it is used in the meaning of a gravitational center, not in the sense of a “strange attractor” of chaos theory.

Anyway, we cannot answer my question, since we cannot travel far away to Centaurus attractor, where, according to video, see attachment figure, all galaxies seem to come inwards.

In the video Laniakea is only one out of many groups each of them having gravitational centers. And also within Laniakea there are galaxies that are not attracted by the “Great Attractor” (see the red and the blue elements in the video). This all happens within our local group (= “Laniakea”). But we are not limited to it. We can still see the background radiation, can’t we?

I have no idea whether organic life is possible in the near of the gravitational center. I guess it's not. But if it were and if there were a graphic designer then he should see also objects that are moving away of him

I am fascinated by the video, too, but at the same time I would say that we know already a lot. I think we would not learn much when travelling to Centaurus. And this is so for some logical reasons: We are assuming that the laws of physics are not changed there. If we take it that the structures of matter are decisively different over there, then we could not make the film that shows us Centaurus. Of course these are all assumptions on the basis of probability. But when you assume that everything could be different in other regions, then you will never be able to start your journey.

I think that our knowledege is a preliminary one, since we 'believe' more and prove less. Take for example the great literature about black holes: the only candidate observation is Cygnus X1-just one after so so many years. In Statistics 1=0! 

This work:

doi:10.1038/nature13674

http://www.nature.com/nature/journal/v513/n7516/full/nature13674.html

is a promising one, since it does not believe, but observes.

Don't forget to watch the submitted to Nature video:

http://www.nature.com/nature/journal/v513/n7516/full/nature13674.html#videos

http://bcove.me/9kbck303