Tuesday, 22 March 2016

Abstract Models and their Importance in Physics

Abstract models are of utmost importance to understand aspects of nature from first principles that would otherwise appear impossibly complex by observation alone. There is a clear basis in the theory of reasoning itself, as well as the history of science, for why abstract models are important for our understanding which is sometimes taken for granted.

The original basis of the forces of motion Galileo himself derived by setting up abstract and in many ways unnatural models such as describing perfect, frictionless inclines in a vacuum with perfect frictionless spheres rolling down on it to describe inertial forces and acceleration due to gravity.

Galileo described motion based on abstractions that would not have been an immediately obvious way to model motion. After all, one might ask the question that "We do not see frictionless inclines in vacuum on earth, so why start modelling the motion of objects this way?"

However by describing balls rolling down frictionless inclines he was able to evolve his model to the point where it described objects in free-fall.





In Galileo's model, steeper inclines give faster accelerations and that when the incline reaches the point of being absolutely vertical we have a maximum acceleration, g, which is acceleration due to gravity. Moreover all objects should move at this acceleration, neglecting air resistance.

This was a very strange idea at the time given the fact that most of his contemporaries were studying the natural forces of motion as they really appeared, i.e. rocks falling, leaves or apples falling which are in fact subject to chaos, i.e. small deviations in initial conditions which propagate outwards.

Furthermore, the basis of physics in Galileo's time had already been an accepted and quite restrictive dogma of sympathies and antipathies derived by the ancient Greeks, in particular Aristotle. The ancients had described the laws of motion based on direct empiricism, for example using the observation of steam rising to the sky or a weight falling to earth as being the same as the explanation for how it happens to move, i.e. "they are going to their natural place".

This dogma, and others like it were not only accepted as a fact for literally thousands of years but taken for granted to be true by educated individuals. It wasn't until people, such as Galileo, became puzzled (and most importantly allowed themselves to be puzzled, i.e. risking sounding uneducated) that the old dogma's of physics were thrown out. This was probably the single most important revolution in the history of thought.

Why is this so?

The thing about the natural world is that the data fed to us, either by direct observation or experiment, are almost always recalcitrant or, in other words, messy. All animals, including Humans, as well as the data recording machines we have invented to assist us in our understanding, have limits in defined boundary conditions which are much more limited than the scope of the environment or universe for that matter.



For this reason, our understanding and modelling of the world must be something which is based on defined limits - leading to the ideas of "naturalness" of the Ancient Greeks for example, or of occult, often anthropomorphic, religious models of science. It is not until we replace the messy, empirical and natural nature of reality with a more abstract mechanistic model that we have every made progress in terms of predictions and the relationship between causes and effects

Because  direct empiricism  was replaced with abstract models, starting with Galileo, the philosophers and scientists that followed were able to deduce the messy laws of physics from first principles and so Galileo’s abstractions were eventually accepted as being the only way to deduce physics.

Furthermore, Newton showed that mathematics, such as calculus, could only be done on such abstract models and generated verifiable predictions on objects operating within those abstract models of nature that contained certain constraints and boundary conditions. These all contributed to the mechanistic worldview, which was to be the main philosophical driving force in physics since Newton.

In quantum physics we have a different story but with a similar outcome, not leading to chaos but collapse. For example, in quantum physics we continue with Galileo’s modality and set up an abstract theoretical model, the wave function, based on the position and momentum variables of the hydrogen atom say. We can only ever hope to set up an equally abstract circumstance to match prediction with theory – (i.e. by isolating hydrogen atoms in magnetic traps in a uniform spin orientation.)

This is true in this case because the observations made of the subject we want to study, the atom, is so vulnerable to externalities that under the parameters of quantum mechanics the observation itself changes the state that we would want to observe. When we experiment on or even “observe” the atom (i.e. using Heisenberg’s gamma ray microscope on a free electron), physicists are influencing it and force the wavefunction of the atom, or any quanta for that matter, to collapse. The only way to describe any further what was going on was to create a yet more abstract model, the path-integral interpretation of quantum mechanics, to by-pass the use of wavefunctions which resulted in a more concise, fundamental model.

Constant empirical observation and modelling based on pure observations and experimentation alone can never fully reveal the fundamentals how natural systems work for the same reasons perhaps that pre-Galilean natural philosophers could not have deduced classical physics from first principles – in that it seems we need to think about doing modelling on first principles using abstract models which is not always an obvious task to undertake.