17 June 2015
Big Data as mathematics
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.