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.

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.
1. 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.