Introduction by field of study
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.
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.
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.
Overview of the framework
In this table we list the core ideas of our work and how they translate into the mathematical framework.
|Principle of scientific objectivity. Science is universal, non-contradictory and evidence based.||Our general mathematical theory of experimental science. A scientific theory is a collection of statements that can be experimentally tested, which ultimately determine the possible cases that can be experimentally distinguished. It must keep track of what can be verified (a topology), what can be predicted (a sigma-algebra) and a way to establish the precision of different statements (a measure).|
|I||Assumption of determinism and reversibility. Given the present state of the system under study, all future and past states are uniquely identified.||Dynamical system. The state of the system is the finest description that can be determined experimentally. Deterministic and reversible evolution preserves what can be tested experimentally (the topology), what can be predicted (the sigma algebra), the precision of statements (the measure) and the nature of the system (e.g. vector space, metric, ...). That is: it is a self-homeomorphism in the category (i.e. a continuous transformation for the topology, a measure preserving map for the measure, a linear transformation for a vector space, ...).|
|II.a||Classical assumption of infinitesimal reducibility. The system under study is made of a homogeneous material that can be decomposed into infinitesimal parts. Giving the state of the whole system is equivalent to giving the state of each infinitesimal part. The evolution of each part is deterministic and reversible.||Classical Hamiltonian particle mechanics. The state of the composite system is identified by a distribution of the material over the states of the infinitesimal parts. The state space of the infinitesimal part must allow to define the distribution independently from what coordinate system is used (a symplectic manifold). Deterministic and reversible evolution can only move the density from one state to another without changing it and must conserve the number of states as they evolve. That is: each infinitesimal part evolves according to Hamilton's equations (a symplectomorphism).|
|II.b||Quantum assumption of irreducibility. The system under study is made of a homogeneous material that can be decomposed into infinitesimal parts. Giving the state of the whole system does not provide any description for the infinitesimal parts. The evolution of the system as a whole is deterministic and reversible but not at the level of each infinitesimal part.||Quantum Hamiltonian particle mechanics. The state space of the homogeneous material is identified by a distribution within which each infinitesimal part moves chaotically. This random motion can be described in terms two independent stochastic variables (a complex distribution). While the overall system can still be assembled/disassembled into parts (a vector space), the correlation between the random variables will result into constructive and descructive interference. Deterministic and reversible evolution will preserve the total amount of material (unitary evolution) and therefore evolves according to the Schroedinger equation.|
|III||Kinematic equivalence. For the system under study, the kinematics (i.e. trajectories in physical space-time) and the dynamics (i.e. trajectories in state space) are equivalent. Giving a set of initial conditions, such as position and velocity, is equivalent to giving the initial state.||Lagrangian particle mechanics with scalar/vector potential forces. As we must be able to re-express the distributions defined over state variables as distributions over kinematic variables, ranges of initial states have to map linearly to ranges of initial conditions (symplectic form induces a metric). This constrains the possible relationships between velocities and momentum and limits the system to massive particles moving under forces described by scalar or vector potentials.|