31 March 2015

The origin of the multiverse

According to Aristotle our world which had a potentially infinite past and it’s spatially extension was finite, but a potentially infinite past is unacceptable today, since it necessarily ends in our present. So it is completed – and so it cannot be potential (but a potentially infinite future is still a possibility). Obviously, Aristotle’s finite space conception is similarly contradictory for us, as it is based on the belief in a finite, spherical shell of the stars that surrounds the Universe.
Newton believed in actually infinite space and finite time – namely, according to him the World was created some thousand years ago (of course, if we do not accept the existence of a final point in history, then it has a potentially infinite future).
The Big Bang Theory uses a finite spacetime model.
Last but not least, according to a common interpretation, the multiverse hypothesis presumes that both space and time is actually infinite and the Big Bang is only a bubble in the infinite sea of other bubbles. This resembles to the Steady State Theory, since it declares a spatiotemporal infinity without a beginning (and it is ironic that this obsolete model, which is forgotten in traditional cosmology, survives in this form). As far as I know, nobody examined the question of the possible origin of the multiverse as a scientific problem. Of course, if you believe that it has no origin (since it is spatiotemporally infinite), then it is a meaningless question.
But we can play either with the idea of a spatiotemporally finite multiverse that includes  only finitely many universes.The fact that we cannot able observe other worlds, does not mean that there are infinitely many of them.
Or there is a possible multiverse model where time is finite (or potentially infinite), but its space is actually infinite. In other words: it is a multiverse that was born a finite time ago but it is spatially infinite since the beginning, and it resembles to Newton’s model in a certain way. The appearance of infinity is always problematic in physics, so – at least from this point of view – this model with an instant spatially infinity is not more contradictory than the traditional multiverse interpretation with an infinite past.

25 March 2015

The Landauer number: in search of the biggest possible number

“One the most fundamental observations… about the natural numbers… is that there is no last or greatest,” writes Grahan Oppy in the introductory chapter of his book entitled Philosophical Perspectives of Infinity. It is widely accepted concept– but it raises some serious questions.
We usually distinguish two kinds of infinity: the potential and the actual. The potential one means that if you choose n, then I can choose n+1.Obviously, neither a biggest number nor an infinitely big number exist if we accept potential infinity.
In case of actual infinity means there is no a biggest number and the infinity actually exists with each of its strange qualities (see the infinite hotel or Thomson's lamp, for example). Since the actual infinite unreachable with counting (after all, even an unthinkably big number is finite), we have to presume its existence if we believe in the existence of actual infiniy. In other words: the presumption of actual infinity’s existence is the basis of the presumption of the existence of actual infinity (which is a tautology). But we can answer that actual infinity is a kind of  mathematical abstraction – similar to the complex numbers.
In cosmology it is controversial whether actual infinity exist or we can accept only potential infinity. This question is not surprising, since the realms of the pure mathematics and the real physics is not necessarily are the same. So we have to take into consideration the possible differences between them.
Rolf Landauer pointed out that the computation – opposite to a mathematics which can be imagined in a Platonist manner – has physical limits. I.e. it is impossible to compute every point of the number line (or only a line segment’s every point) unless we have unlimited (actually infinite) computing capacity.
In our Universe we have a limited computing capacity - unless our Universe is eternal and we suppose that opposite to entropy's law we will have enough energy for calculations in the far-far future). According to Paul Davies (Cosmic Jackpot), the real numbers that are served as a basis of the natural laws in the traditional physics, simply don’t exist.
What is more, it means that there is a biggest possible number in our Universe. To illuminate it, consider the following example: If you have only a minute to write down the biggest possible number, then the limited amount of time limits your possibilities. It is unquestionable that even a mere 60 second is enough for the construction of an enormously huge number – but it is unquestionable, as well, that if our Universe is not eternal, then we have only limited time to compute the biggest possible result.
It would be interesting to find the most efficient form/algorithm/solution to calculate the biggest number that can be constructed in one minute or 100 billion billion years, but it is more important from our point of view, that this aspect of our physical reality characterized neither actual nor potential infinity, but a very big, but finite number that can be called Landauer number.
Representatives of ultrafinitism in mathematics states that there is no either actually infinite sets of natural numbers or very big numbers (2^10000, for example), since they are inaccessibly by human minds. This argumentation seems to be flawed, as it presupposes that the existence of a mathematical object presupposes that it is imaginable by us. The introducing of the Landauer number doesn’t causes similar problems.
So we can imagine three kind of universes: They are determined by Landauer number or potential or actual infinity. But notice that it is only about the nature of the time of a given universe, and either of the space or the mass density can be potentially/actually infinite parallel to the existence of Landauer numbers.

17 March 2015

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

08 March 2015

The nature of the natural laws

Although both the existence and the nature of physical laws usually are taken for granted, it is worth examining these ideas’ origin, since it can influence even the approach of today’s science.
Aristotle developed a rather complicated idea about the causes (ending with a kind of “Formal Cause” – which “causes” the form). His “Final Cause” became the modern laws of nature and his “Efficient Cause” is close to modern cause (Barrow: World within World, p. 53). So following Aristotle’s logic we make a distinction between the laws and those phenomena they affect. But it is not surely a necessary distinction between causes and their subjects. The fundamental question is whether the laws exist in a certain sense or they are only practical descriptions of reality. According to Lee Smolin, supposing a kind of cosmic evolution and the existence of baby universes that are born with slightly modified natural laws than of their parent universe's, this “evolving laws seems to be a breakdown of the distinction between the state of a system and the law that evolves it.”
This hypothesis makes possible to imagine some different scenarios between the laws and those subjects that are affected by them.
Obviously, we can accept the traditional laws vs systems differentiation.
2. But it is more exciting to suppose that the formation of a baby universe means the formation (slightly) different laws. It can be happened three different ways.
  • The first one means that the law formation is restricted to the moment of creation (whatever it means). After the Big Bang we have a certain amount of matter, energy etc. and it won’t change. Similarly, the laws are “finished” as well and they won’t change. On the other hand, Smolin supposes that different universes can be determined by different laws.
  • But why to restrict the changes of natural laws for a very short period of time? A second solution proposes a longer, even continuous evolutionary process where the forming laws and the physical environment are in continuous interaction in the course of the universe’s history. According to this model, not only the initial laws of a new universe are different from its parent’s laws, but the changes can be influenced by some events in the history of the universe and slightly different initial conditions would result very different laws later.
  • To make the story even stranger, there is the problem of the saltation hypothesis. In evolutionary theory not accepted to suppose that biological evolution produces its effects via large and sudden changes, but cosmology isn’t about earthly ecosystems’ biology. The laws of our Universe seems to be fixed today, but it is imaginable that this idle period when our physical laws are static is only a transitional state, and then a sudden saltation would happen in the future and the nature of the laws will radically change.

02 March 2015

The tree of cosmological natural selection

According to Lee Smolin, it is not a well-founded idea to apply Newton’s paradigm about the nature of physical laws without an approximation to the whole Universe, since it is based on a local observation and it suggests the existence of timeless laws which can be applied to a timeless space.
What is more, the Newtonian interpretation is unable to explain why the observed laws exist instead of others. Similarly, both our Universe’s high homogeneity and that this Universe is so far from thermal equilibrium are in question. Boltzmann answered the second problem arguing that it was a result of the fact that our world was born in an incredibly low-entropy state (it is the “past hypothesis”). But he simply replaced the original problem with a new one, since it is not known why was so low the entropy originally.
Generalizing this problem, the Newtonian paradigm is “a theory [which] has infinite of solutions,” but we observe only one Universe.
To solve this problem, one can introduce the multiverse hypothesis without changing the Newtonian paradigm. According to this logic, our Universe is a part of a bigger ensemble and we live in a biofil cosmic environment, because all of the possible universe-variations are realized, and we are simply lucky enough to find ourselves in a world of life-friendly conditions. But there is no testable predictions to falsify the multiverse proposal.
Smolin’s answer is the “cosmological natural selection”. It states that “the laws of nature have evolved over time” and he “decided to copy the formal structure of population biology by which populations of genes or phenotypes evolve on so called fitness landscapes.” The reproduction happens via black holes and the result is the birth of new baby-universes which inherit the physical laws of their parent universes’ laws in a slightly changed form (thanks to small, random changes). This process selects the more successful (=more back hole reproductive) universes and we presumably live in a successful universe which laws offers favorable conditions for black hole creation. And it is not a surprise, since life seems to be improbable without long-lived stars and carbon – which are necessary to the formation of black holes.
But this coincidence seems to be strange. Why the preconditions of black holes and life are the same?
  • Perhaps it is only an observational bias and the physics of stars and the appearance of life are different phenomena without real connection, but there are other hidden, but cruical factors  – although we still didn’t realized them. Obviously, this explanation cannot be excluded, although it seems to be improbable.
  • Perhaps the presence of long-lived stars, back holes and carbon are necessary preconditions of life, and these and only these preconditions lead to life.
  • Or – and it seems to me the most probable – life can appear not only in carbon based universes, but our laws which leads to the creation of heavier elements allows its appearance. In other words: there are other, biofil universes without carbon, and it is another question whether there are another universes with different physics and different physical processes to reproduce themselves – perhaps even via more efficient solutions than the black holes. Namely: adapting Smolin's evolutionary approach, evolutionary processes are able to find only local maximums and the history of the descending baby-universes can be portrayed as an evolutionary tree. And it not sure that our branch is the most successful.