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.

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