Physics

There has always been an effort to formalize a physical theory from a small set of well defined physics principles or laws. Newtonian mechanics, special relativity and thermodynamics, for example, follow that pattern. Unfortunately, this practice has stopped in modern times. Lagrangian mechanics and Quantum mechanics, for example, simply posit the mathematical structure without a clear justification as to what is physically described and why those rules exist. Physics is recovered at a later stage, by "interpreting" the results of the mathematical theory.

In this project we want to revert back to that original spirit and put the physics at the center of our physical theories. We want to identify those physical starting points, our physical assumptions, and then derive the mathematical framework as a necessary consequence. By proceeding in this fashion, it is always clear under what conditions a particular theory can be assumed to be valid, it is clear what mathematical structure are physically meaningful and what are just artifacts introduced by mathematicians to make calculations or proofs easier to develop. It also allows us to think more deeply about our starting points, leading to new insights and possible new approaches.

Math

The foundations of mathematics settled during the beginning of the 20th century. Work by Cantor, Russell, Zermelo, Fraenkel, Godel and many others re-organized the whole discipline on top of logic and set theory. During that period physics was tackling new ideas in both theory and experiment, which lead to general relativity and quantum mechanics, and never really took that step at fully identifying the formal foundations of its discipline.

In this project we establish a general mathematical theory for experimental science, which aims to axiomatize the underpinning of all scientific disciplines in the same way that logic and axiomatic set theory establishes the underpinning of all modern math. The core concept is the idea of an experimentally verifiable statement: an assertion that, if true, can be verified to be so in a finite amount of time. We establish the logic of these objects and we see how they lead to topologies and sigma-algebras. A physical theory, then, is simply a collection of verifiable statement with a well defined semantic relationship between them.

Uncovering the assumptions of physics means understanding how a physical assumption is formally captured by the semantic relationship between statements, and how those relationships necessarily lead to a given mathematical structure.

Philosophy

Over the last century, there has been an increasing interest within philosophy of science on the foundations of the physical theories. Particularly in quantum mechanics, the effort is to better characterize the mathematical structure describing the theory in terms of "real" ontological entities. The main issue here is that the same mathematical equation can be used to describe different physical systems (e.g. linear systems are used in electronics, thermodynamics, biology, etc...) and the same physical system can be described by different mathematical structures (e.g. a massive point particle can be modeled by a point in a cotangent bundle or by a point in a tangent bundle).

In this project we develop the mathematical structures of science starting by clearly defining the physical objects we are studying, with particular attention to the philosophical aspects that arise with those definitions. As we do this from the very beginning, we are forced to clarify many aspects that are usually "swept under the rug". As everything else is built on top these definitions, the mathematical structures have one clear meaning and there are no issue of interpretation. This provides a better framework to understand what science is, what the laws of physics are and what they can or cannot be describing.