Existing, non-existing and other universes

“According to modal realism, possible worlds really exist” writes Jennifer Fisher in his book On Philosophy of Logic (p. 91.). This approach is based on modal logic that interprets true and false statements in relation to possible worlds: E.g. necessity means that the statement “is true in all possible words” (Ibid, p. 75.) and our actual world is nothing more than a world which was chosen from the set of other, existing ones. I don’t accept modal realism’s logic (after all, possibility isn’t equal to existence), but it is interesting from our point of view that the logic of modal worlds is similar to the logic of classic multiverse hypothesis that assumes the existence infinitely many words and this similarity shall lead us to a strange type of imaginable universes.
Max Tegmark interprets the multiverse as the manifestation of every mathematically possible world. It is a form of mathematical Platonism, and the main thesis is that on the one hand, anything is possible mathematically exists in reality. On the other hand, everything is governed by the rules of mathematics. “Mathematical” means in this case that every combination of different sets of physical laws and constants, or even different equations exist. In other words: according to Tegmark, all words can be described by mathematics and every imaginable combination is manifested in a really existing world. The core of Tegmark’s concept is that mathematics is equal to physics in a certain sense, since it describes the world ruled by physical laws.
But it is not sure that even our universe can be described perfectly by mathematics and perhaps only our belief suggests that every natural phenomenon is controlled by either deterministic, or probability or evolutionary laws. Inter alia, it is possible that the Great Unified Theory (GUT) doesn’t exist, since there is no mathematics to describe every connection. It is perhaps only about our inability to give a coherent description about reality, since our tools (including our minds, mathematics and logic) aren’t appropriate for it.
Or, it is imaginable that there are universes that cannot be described by mathematics at all: after all, mathematics is based on the presumption of the conservation of some rules. Thus, it is not necessarily well-founded to state that every universe is mathematical in nature. So we can imagine whole universes (albeit they wouldn’t be biofil) without mathematically interpretable natural laws. In other words: although they exist, they cannot be described by one or other mathematical form of physical laws.
Traditionally, we distinguish existing and non-existing worlds and the main sin of modal realism is that it intermixes these two categories. Now we can introduce a third kind of universes which differ from both the “existing” and “non-existing” ones and since per definitionem it is impossible to give a scientific description about their features, they don’t belong to the realm of physics.

Mathematics as cellular automation

According to a popular belief, mathematics is nothing more than a big tautology, since it is a deductive system and a new result reachable through a process of steps specified by certain rules. The two sides of the equation means the same: To give an example: 2+2=4 (and it is held that the relations between the sets of axioms and the result of proof is the same).
Of course, the logic of mathematics makes possible to reach a result (e.g. a mathematical proof), but even if you accept the axioms as a given and unchangeable base, the possibility doesn’t means the necessity. Even a few elements can result “hyper astronomically” huge number of combinations (to borrow Quine’s term). Thus searching for an answer (e.g. examining whether a theorem is true) can be interpretable as (a random) walk in the phase space of mathematics. Perhaps we are convinced that there is a certain mountain pike in this virtual landscape, but we do not know the path to it; or we even don’t know whether the hill exists at all. Obviously, this image is more or less misleading, since there aren’t existing routes originally, and we have to build them–and in some cases this activity constructs the target itself. Furthermore, not only the phase space of a certain mathematics is enormously huge, but the phase space of mathematics based on different sets of axioms and considerations are similarly large as well.
It is an interesting question that to what extent overlaps different mathematics (=mathematics based different set of axioms and rules) each other. And since the “phase space” describes mathematics as an n-dimensional landscape where every point is defined by certain parameters, we can try to define them using another method. Ad analogiam: Remember Descartes’s idea to mathematize geometry.
Arithmetic is compressible: if you know the adding rules, then you can calculate the result of 2+2 directly, omitting the steps of 1+1=2; 2+1=3; 3+1=4. Simply speaking, there aren’t intermediate steps.
Opposite to it, in case of a mathematical theorem you cannot omit any part of the proofing jumping from the starting point directly to the end, thus it is history dependent and the first step is essential to reach the second one, etc. This uncompressible method is strongly resembles for the cellular automation’s operation mode.
Cellular automata (CA) is based on simple rules that determine the status of a certain cell (=a certain point of the landscape or cellular space, if you prefer) taking into consideration certain cells’ intermediate states. It is an uncompressible process: You cannot tell the result without executing the process itself.
It seems to be plausible that we can build a CA to generate point by point the path to any certain proof: After all, we can adapt the rules which guides the work of cellular automata to the objective to be achieved. What is more, probably we could construct a “universal mathematical CA” (UCA) to execute the whole mathematics. Or, we could build other UCAs to examine their ways of work. Perhaps it would result totally different mathematics.

Big Data as mathematics

Big data means that we process not only a small amount of data but all of them. And what is similarly important: we, at least partly, should stop the search for the reason–cause correlation (Victor Mayer-Schönberg and Kenneth Cukier: Dig Data, p. 14 – 15 (Hungarian Edition)) since the really big amount of data makes simply impossible to detect the causality. To give an example, the Google, examining the connection between the spread of flu and the changes in search words, tested a 450 million (!) algorithm to find the most effective version to predict the epidemic. (ibid, p. 10)
The big data approach can be applied to mathematics at least in two ways.
1. First, traditional mathematics is small data “science”: it manages only a small amount of data and tries to find more or less direct connections between certain features using a kind of deductive logics (which replaces causality in mathematics). E. g. we know the Euler theorem d^2=R(R – 2r) which describes the distance (d) between the circumcentre (R=circumradius) and incentre (r=inradius) in a triangle in geometry. Obviously, it is a proofed theorem, so we understand the cause of the correlation between these data. But why don’t try to adapt the big data approach and why we don’t try to analyze all the possible geometrical data to find new, although unproven, connections? Similarly, we could examine the distribution of prime numbers taking into consideration not only their places on the number line, but all the accessible data about numbers from HCNs (highly composite numbers) to triangle numbers to any other features to discover connections even we aren’t able to prove them.
2. There is another way to apply big data approach to a new level. Reverse mathematics is a program to examine which sets of axioms are necessary to build the foundations of mathematics. I.e. how should we choose a small amount of starting points to get a certain solution? It is, in accordance with its name, a reverse approach to the traditional mathematical way of thought which moves from a small set of axioms to theorems and which is a small data approach. But we can apply the big data “philosophy” more or less imitating Google’s solution examining different combinations of an enormously huge amount of possible axioms to create different data landscapes. Perhaps it would lead to a new kind of metamathematics.

Where is every time traveler?

Perhaps nowhere, since they simply don’t exist, as time travel is impossible–but Einstein’s twin paradox (time dilatation, if you prefer) is nonetheless real. Notice that the time dilatation is not a proof of time travel; and that time travel and time dilatation are different questions. The first one is about whether time is similar to space in a certain sense, the second one is about what happens at very high speeds.
Obviously, time is a problematic field of modern physics. As far as we know, “the second law is the only fundamental law of physics that distinguishes between past and future” [Melanie Mitchell: A Guided Tour to the Complexity, p. 43]. Obviously, it has a kind of physical background, as the rise of entropy wouldn’t be possible without an early phase of high order (=Big Bang) but entropy’s concept is purely mathematical. I.e. in the case of the law of gravity, the gravitational force is inversely proportional to the square of distance–and this proportion cannot be deduced solely from the mathematical equations. But to understand the second law is enough to know that there are more ways to make disorder than order. What is to say, in this case there is no an additional physical law to determine the results over the logic of mathematics (opposite to gravitational law where there is a second “layer” over the mathematical description to determine that the connection between the distance and force is not, say, linear. By the way: if Newton’s law is two-layered (physics over math), then it is interesting question whether exist laws with tree, four etc. layer).
The second law creates the “time of arrow”. So since there is no any physical effect to modify its mathematics, time can be regarded as a result of the mathematics which is the basis of our time description.
British historian Arnold J. Toynbee said that history was regarded to be nothing but "one damned thing after another.” According to the logic of our argumentation, it is defendable that although time, using a kind of mathematical abstraction, can be represented as a dimension, in reality it is nothing more than “one damned thing after another”. So the meaning of travel in time is simply uninterpretable: It has no meaning at all, but it doesn’t exclude the time dilatation where according to the observers of a different frame, events follow slower each other if your speed is close to the speed of light.
Of course, it can be argued that it is a natural law that there is no an additional natural law over the level of mathematics in connection with entropy, and it can be asked why.

A new kind of infinite machines

Since David Hilbert’s thought experiment, it is popular to demonstrate the strangeness of infinities describing a hotel with infinite rooms where new and new tourists/tourist groups arrives (even in a countably infinite number). The trick is that although all the rooms are full, the management always can find free ones – after rearranging the reservations. I.e. if only one tourist wants to check in, then the person occupying room 1 is can be moved to room 2; and the occupier of room 2 to room 3 etc. (and the occupier of room n moves to room n+1). If a countably infinite amount of new guest arrives, then the person from room 1 moves to room 2; the person from room 2 to room 4 (from room n to room 2n). After all, there as many odd as even numbers and the new visitors can occupy the odd-numbered rooms that are free now. This method works even if countably infinitely many buses arrives with countably infinitely many passengers on each.
The infinite hotel is misleading in a certain way, since it suggests that these algorithms are the simplest solutions for pairing the rooms and visitors. But there is a simpler method: at the time of the arriving of a new group with even countably infinitely visitors, we can ask every occupier to leave their room – the result is infinitely many free room with infinitely many persons (including the newly arrived ones) without room. Then we ask everybody to go into a still free places – and that’s all: we paired infinitely many persons with infinitely many rooms.
Keeping in mind the lesson of the infinite hotel, we can introduce a new kind of infinite machines with a new typology.
From our point of view, there are two fundamental parameters to determine these machines: the number of steps of the process to reach infinity and the needed time.
It’s obvious that there are impossible machines. You cannot build a machine that solve a problem in zero time even it infinitely fast; and similarly impossible that version that takes only finite number of steps in an infinitely long period – but not because it halts at a certain point in the process (i.e. since it is prescribed that it has to stop after a certain number of steps or reaching a number), but because – as a reversed Thomson lamp – its algorithm prescribes it.
So the simplest infinite machine is a Turing machine with an infinite tape – it can take infinitely many steps over an infinitely long period (and every step can be paired with the moment of the step).
Opposite to it, a Tomson's lamp takes infinitely many steps within a finite period of time. The solution is that 1+1/2+1/4…=2 so if we can press the Thomson lamp’s button two times faster at the n+1st step than at the nth step, then we can finish the process within 2 unit of time (i.e. within two seconds, if it took 1 second to press the button for the first time).
But it is possible a third type of infinite machine. Obviously, the last time we press the Thompson lamp’s button we have to do it infinitely fast, and the pressing process is infinitely short. It means on the one hand, that we handle (at least mathematically) an infinitely small amount. On the other hand: Why should we vary the pressing time to reach our aim? It is possible theoretically to press the button infinitely fast even for the first time; and even an infinitely small time is enough to do it infinitely many times. So this infinite machine finish its process not in infinite time (as a Turing machine) and not in a finite time (as a Thomson lamp), but in an infinitely short time.

machine type	number of steps	time
Impossible zero	infinite	zero
Impossible finite	finite	infinite
Turing	infinite	infinite
Tomson lamp	infinite	finite
Third type	infinite	infinitely small

How (not) to avoid the G-world in cosmology?

The first chapter’s title in Mary – Jane Rubinstein’s recently published book entitled Worlds Without End. The Many Lives of the Multiverse is: How to Avoid the G-word.
Namely, the God.
It is an interesting trend that not trying to avoid the G-word is an acceptable attitude for some modern cosmologist, although the Creator’s existence or non-existence hasn't play a role in modern optics or mechanics (what is more, it is not a question in the explanation
At the beginning of the 17th century Galileo argued that there were two books written by God. One of them was the the Bible and another was the book of nature. Although it was a common belief of that age that both of them used the same language, according to Galileo, the second one used the language of mathematics. [John Hedley Brooke: Science and Religion, p. 104] Supposing the existence of a Creator who created the World for humans, it is an open question why He had decided to use two different languages, and why an exotic and complicated system was introduced to describe the physical reality created by Him. In other words: why the physical reality is so complicated that it is impossible to describe it in everyday words? Was it a necessity to hide the structure of the world behind a complicated and non-evident formulism?
Obviously, Galileo’s concept based on a kind of Neoplatonic mathematical mysticism. According to it, the fundamental structure of our Universe is not only can be described by mathematics, but the nature itself is purely mathematical. But it is not an answer for the question why decided God to choose this language.
On the other hand, a group of physicists in the 17th century refused the Aristotelian concept which stated that an explanation cannot be complete without pointing out a final cause. Since Aristotle didn’t favor mathematics as a language of world description, Galileo’s approach offered a way to deny any argumentation based on final cause, and Descartes decided to focus on immediate causes of physical events. [Brooke, ibid, p. 72]
Since then it became a tradition in natural sciences to study those immediate causes and omitting either a final cause or the intervention of a Creator. But – to oversimplify the problem – when somebody studies either the “fine-tuned” nature of our Universe or the supposed existence of multiverses, there isn't an opportunity to study his or her subject from Descartes’ approach, because there is no an existing physical event to examine.
According to Bernard Carr, “If you don’t want God, you would better have a multiverse.” Namely: if you don’t want to refer to God, then you can replace Him with the idea of Multiverse – or vice versa. The “Multiverse replaces God with what is perhaps an equally baffling article of faith: the actual existence of an infinite number of worlds.” [Rubinstein, p. 29]
We have two different strategies to handle the causes. One can choose Descartes’ approach to always focus on immediate ones – and then he or she can ask another question about the solved problem’s immediate cause. We can move forward step by step – its mathematical analogy is the potential infinity where every piece of the result is finite and it can be achieved in a finite number of steps, but this process never has an endpoint.
The other approach is more problematic. If we are allowed to ask for a not immediate cause, then we could ask what caused that cause, etc. [Rubinstein, p. 21] and the result is an infinite regress without end.