Chapter 9: Patterns in Nature: Formulating laws of physics
“The main message in Newton’s Principia was not the specific laws of nature that he discovered and used, but the idea that such laws exist – together with evidence that the way to model nature’s laws mathematically is with differential equations.”
Types of differential equations
- Ordinary differential equations (ODE): unknown function y of a single variable.
- Partial differential equations (PDE): unknown function y of two or more variables. (Euler and d’Alembert)
D’Alembert (wave equation)
Euler (harmonics)
Gravitational Attraction: Colin Maclaurin, Clairaut, Legendre, Laplace, Poisson’s equation.
Heat and Temperature: Joseph Fourier (heat equation),
Fluid Dynamics: Euler, Claude Navier, Poisson (Navier-Strokes equations).
“It is fair to say that Newton’s invention of differential equations, fleshed out by his successors in the 18th and 19th centuries, is in many ways responsible for the society in which we now live. This only goes to show what is lurking just behind the scenes, if you care to look.” (page 180)
Types of differential equations
- Ordinary differential equations (ODE): unknown function y of a single variable.
- Partial differential equations (PDE): unknown function y of two or more variables. (Euler and d’Alembert)
D’Alembert (wave equation)
Euler (harmonics)
Gravitational Attraction: Colin Maclaurin, Clairaut, Legendre, Laplace, Poisson’s equation.
Heat and Temperature: Joseph Fourier (heat equation),
Fluid Dynamics: Euler, Claude Navier, Poisson (Navier-Strokes equations).
“It is fair to say that Newton’s invention of differential equations, fleshed out by his successors in the 18th and 19th centuries, is in many ways responsible for the society in which we now live. This only goes to show what is lurking just behind the scenes, if you care to look.” (page 180)
Chapter 10: Impossible Quatities: Can negative numbers have square roots?
“What really matters is not the individual numbers, but the system to which they belong – the company they keep.”
Number Systems: natural numbers, integers, rational numbers, real numbers.
“It took mathematicians a long time to appreciate that numbers are artificial inventions made by human beings; very effective inventions for capturing many aspects of nature, to be sure, but no more a part of nature than one of Euclid’s triangles or a formula in calculus. Historically, we first see mathematicians starting to struggle with this philosophical question when they began to realize that imaginary numbers were inevitable, useful and somehow on a par with the more familiar real ones.” (page 182)
It’s very interesting how we tend to forget that some things are concepts created by humans in order to understand the world around us. When we think of a number, we have the feeling that it is something concrete, when in reality it’s only an abstraction.
Important Figures: Ferro, Tartaglia, Cardano, Rafael Bombelli (all Renaissance algebraists), John Wallis (imaginary numbers as points in a plane), Johann Bernoulli, Leibniz (both on complex analysis), Augustin-Louis Cauchy (Cauchy theorem, true founder of complex analysis), Gauss.
“The most beautiful formula of all time”: eiπ = -1
“Today, complex numbers, and the calculus of complex functions, are routinely used as an indispensable technique in virtually all branches of science, engineering and mathematics.” (page 194)
Number Systems: natural numbers, integers, rational numbers, real numbers.
“It took mathematicians a long time to appreciate that numbers are artificial inventions made by human beings; very effective inventions for capturing many aspects of nature, to be sure, but no more a part of nature than one of Euclid’s triangles or a formula in calculus. Historically, we first see mathematicians starting to struggle with this philosophical question when they began to realize that imaginary numbers were inevitable, useful and somehow on a par with the more familiar real ones.” (page 182)
It’s very interesting how we tend to forget that some things are concepts created by humans in order to understand the world around us. When we think of a number, we have the feeling that it is something concrete, when in reality it’s only an abstraction.
Important Figures: Ferro, Tartaglia, Cardano, Rafael Bombelli (all Renaissance algebraists), John Wallis (imaginary numbers as points in a plane), Johann Bernoulli, Leibniz (both on complex analysis), Augustin-Louis Cauchy (Cauchy theorem, true founder of complex analysis), Gauss.
“The most beautiful formula of all time”: eiπ = -1
“Today, complex numbers, and the calculus of complex functions, are routinely used as an indispensable technique in virtually all branches of science, engineering and mathematics.” (page 194)