From Observable-Based to Probabilistic Peasoning

In this work, using categorical techniques, I will give a mathematical definition of law of chance. I will also show that every proof in the multiplicative fragment of linear logic can be interpreted in a law of chance (validity). Laws of chance are defined as time and uncertainty invariants. I believe that they can give an interesting contribution to answer the following question: why is mathematics reliable? It is a common opinion that even a partial answer to this question could give some insight to the problem of the foundations of mathematics. There are many examples of the reliability of the mathematical method in different theories and fields: for instance the existence of the planet Pluto has been foreseen only on the basis of mathematical computations. Using the validity of the proof system the reliability of reasoning (and I believe also of computing due to the Curry-Howard isomorphism) is a consequence of the fact that these methods are based on the laws of chance. Such laws are satisfied by many possible outcomes that have not yet been observed. In fact proofs, in this semantics, define infinite sets of possible observables, while the available information is only finite. My claim is therefore that mathematics is reliable because it is able to grasp some of these invariants that remain stable also in the presence of the high variability of outcomes due to randomness.