# The Philosophy of Science Part V: Mathematics Continued

This video is part of a continuing series exploring the underlying philosophy that governs the modern scientific method. In previous videos I have used the term the Metaphysics of Physics for more poetic effect.

In first four videos we explored what are the logical tests that are applied to a scientific theory to evaluate whether it is a valid explanation for a set of data we have obtained from making observations of the mysterious reality in which we all find ourselves.

If we considered how the ancient Greeks did this in respect to thinking about what is true, they demanded the statement, model or theory must satisfy two requirements: it must be logically coherent (in that it must not contain any logical contradictions) and it must be correspondent with reality (in that its predictions of what will happen in our reality must match what we observe).

However in order to have these two tests in place we also must make some further assumptions. We first assume that our reality exists independent of our own existence (my one word summary for this was coexistence) and that the operation of the universe itself was not constantly changing; in that it was possible to develop theories which were largely spacetime independent (we called this assumption or presupposition conservation).

By the end of our first and second videos we had 4 unpinning principles (all starting with the letter “C”) that power the scientific method and these were:

0 Coherence

1 Coexistence

2 Correspondence

3 Conservation

I purposely placed the laws of logic as a zeroth law to emphasize it is absolutely the bedrock of the scientific method. Now as we thought about our first 4 principles (another good word we can use here is axioms) we also considered what is meant by conservation and in doing so we realised we required our first scientific tool to help us optimise our scientific theories and this tool is known as Occam’s razor. Very roughly, in our terminology, Occam’s razor states that if two scientific theories are equally coherent and correspondent the simpler theory is to be considered the correct one. This formally makes the scientific method reductionistic (or perhaps more accurately, as Richard Dawkins once said, hierarchal reductionistic) and my one word term of Occam’s razor (keeping to words that begin with the letter “C”) is “Compaction” (The Metaphysics of Physics Part III).

However, there is one final giant tool of science which requires several videos to unpack and that is the wonderful tool of mathematics. Keeping to my words beginning with the letter “C” theme, we called this last tool “Calculation”. In my 4^{th} video (The Metaphysics of Physics Part IV: Calculation) on the underlying structure of the scientific method, we began our journey into mathematics starting at the very beginning with thinking about children learning to count.

In this video I stated that I am going to take my listeners on a very narrow path rapidly progressing from the most basic maths we learn as children through to the mathematics that is required to prove Einstein’s famous E=mcÂ˛ equation. In this video we complete our creating most of the syntax and mathematical structures that we need to develop Einstein’s equation by showing how the syntax of indices come about and how their meaning is generalised across all the values from the number system that the indices can take. If you have not watched my first video on seeing how maths is first born out of enumerating logical categories then you might want to look at this one first using the youtube link above.

Finally this video ends by starting to use our newly created mathematical system to describe mathematically some basic observations to do with motion that we experience in our day to day lives. In particular, we define the meanings of velocity and acceleration, we then look at basic observation that demonstrate that in order to change an objects motion, a force is required and this force equals the object’s mass multiplied by the acceleration of the object. Finally we discovered two useful conserved quantities by first multiplying our force over the time it acts to come up with the conserved quantity momentum (as Newton did) or multiply force over the distance the force acts to come up with another conserved quantity called kinetic energy as Leibniz did.