Introduction

Overview

For my first post, I decided to make use of some notes I comprised a few years ago while watching one of my favorite lectures given by Richard Feynman. This particular lecture is the second in a series given by Feynman at Cornell University in 1964. 

References

Richard Feynman - Wikipedia

The Character of Physical Law - Wikipedia

Resources

Feynman's Lectures on Physics - The Relation of Mathematics and Physics

My Scribbles

Concepts

Using a game of checkers to think about physical laws

• Checkerboard = "space" on which the rules of a checker game are played out
• Mathematics = "space" on which the rules of the fundamental laws are played out 

Mathematical relations in physics convey information

• Algebraic definitions of laws do not convey information about the laws underlying "machinery"
• More generally, algebraic relations tell us how the "machinery" (e.g., moon orbiting a planet) moves but not why
• E.g., the magnitude of the gravitational force between two objects is proportional to the product of their masses (scalar) and inversely proportional their squared distance (scalar that is also a contracted bivector) 

Example 1:  Kepler's Laws

• First Law: Every planetary orbit is elliptical with the Sun positioned at one of the two foci 
• Second Law (Inverse-Square Law): Orbits span a constant area per time
• Third Law (Harmonic Law): A planet's squared orbital period is proportional to the cubed distance of the orbit's semi-major axis 

Take-Home Message: In order to appreciate these laws, you need a sufficient understanding of certain set of mathematical concepts and their relations to one another

Newton made all his proofs geometrical in his book, Principia 

• Modern mathematics is usually based on calculus (which Newton played a large role in pioneering - apparently there is some controversy as to whether the original founder was Newton or  Wilhelm Leibniz)
• Analytical methods make use of geometrical relations to derive algebraic identities 

Two approaches towards mathematics:

1. Babylonian tradition = student learns by doing examples until catching on to a general rule

2. Greek tradition = student uses Euclidean methods only (based on sets of axioms)

• Around 400 BC Euclid discovered that all theorems of geometry can be "ordered" by a particularly simple set of axioms
• Q: What constitutes an axiom (e.g., is the Pythagorean theorem an axiom in itself? - Descartes said yes)
• It seems as if the best axioms are the most efficient way of relating different geometries 

Take-Home Message: Physics needs to employ the Babylonian method!

• This is because physics entails "wide principles that sweep across all the different laws" 
• The fact that physics is incomplete (we are trying to find new laws to reconcile current ones) is a consequence of the insufficiency of the Euclidean method

Examples : Three theories that are based off the same mathematics but that are physiologically unequivocal (i.e., they have yet to be reconciled in a way that can be tested by experiment) 

1. Newtons laws = three laws that describe classical motion 

2. Local field method = based on the principle of locality

3. Minimum principle = based on least action principle

• The fact that a minimum field exists suggests that at small scales, particles obey quantum mechanics 
• The current consensus is that the laws of physics must satisfy both the local field method and minimum principle (but we don't really know)

Quotes

• "Nobody knows the ultimate"
•  "(It) is impossible to (explain) in a way that a person can feel, the beauties of the laws of nature without their having some deep understanding understanding of mathematics. I'm sorry that seems to be the case."
• "Mathematics is not just a language. It is a language plus reasoning" 
• "Mathematics is a tool for reasoning"
•  "Mathematics is organized reasoning" 
• "Mathematics is a way to go from one statement to another"
• "As a matter of fact the total amount that a physicist knows is very little, he has only to remember the rules for getting from one place to another and her's able to do it then. In other words, all the various statements about equal times, the forces in the direction of the radius, and so on are all interconnected by reasoning.
• Now an interesting question comes up, is there some pattern to it? Is there a place to begin the fundamental principles and deduce the whole works? Or is there some particular pattern or order in nature that we can understand that the these are more fundamental statements and these are more consequential statements? "
• "And so I'm never quite sure of where I'm supposed to begin and where I'm supposed to end. I just remember enough all the time so that as the memory fades and the pieces fall out by repurchasing back together again" 
• "Laws often range beyond the range of their deductions"