Chapter 19: Number Crunching: Calculating machines and computational mathematics
“Mathematicians have always dreamed of building machines to reduce the drudgery of routine calculations. The less time you spend calculating, the more time you can spend thinking.” (page 345)
This chapter goes into all the rise of the computers since the 17th century, going from Pascal, Napier, and Leibniz, all the way to Charles Babbage, Augusta Ada King, Willgodt T. Odhner, Newton, Boole, and an introduction to Alan Turing.
It talks about machines, new methods of calculations, computers, algorithms, and numerical analysis.
“These tools have opened up new areas, helped to solve long-standing problems, and freed up time for conceptual thinking. Mathematics has become much richer as a result, and it has also become applicable to many more practical problems.” (page 354)
This chapter goes into all the rise of the computers since the 17th century, going from Pascal, Napier, and Leibniz, all the way to Charles Babbage, Augusta Ada King, Willgodt T. Odhner, Newton, Boole, and an introduction to Alan Turing.
It talks about machines, new methods of calculations, computers, algorithms, and numerical analysis.
“These tools have opened up new areas, helped to solve long-standing problems, and freed up time for conceptual thinking. Mathematics has become much richer as a result, and it has also become applicable to many more practical problems.” (page 354)
Chapter 20: Chaos and Complexity: Irregularities have patterns too
“By the middle of the 20th century, mathematics was undergoing a rapid phase of growth, stimulated by its widespread applications and powerful new methods… One such topic, which achieved public prominence in the 1970s and 1980s, is chaos theory, the media’s name for nonlinear dynamics… Another is complex systems.” (page 356)
This chapter raises many philosophical questions about whether we are determined or have free will. It’s interesting to think of that question since chaos theory has showed us a possibility of thinking that what we think is complete uncontrolled actions, in fact, can be that are determined and act mechanically.
Important Figures: Douglas Adams (The Hitchhiker’s Guide to the Galaxy, supercomputer, answer to the universe… 42), Laplace’s determinism, Marcy Lucy Cartwright and John Littlewood.
Nonlinear dynamics: Stephen Smale, James Yorke and Tien-Yien Li (Chaos), Butterfly Effect, Strange Attractors (Edward Lorenz),
Theoretical Monsters: Benoit Mandelbrot (fractals), Lewis Fry Richardson (Does wind have a velocity?).
Chaos everywhere!: strange attractors, Moon’s tides, biological populations, ecosystems.
Complexity: George Cowan, Murray Gell-Mann; John von Neumann, Stanislaw Ulam, Konrad Zuse (Cellular automaton, complex systems), Hans Meinhardt, Stuart Kauffman, Hui-Hsien Chou and James Reggia (geology and biology, complex systems).
“Throughout its lengthy history, mathematics has taken its inspiration from these two sources – the real world and the world of human imagination. Which is most important? Neither. What matters is the combination. The historical method makes it plain that mathematics draws its power, and its beauty, from both… Mathematics has never been so active, it has never been so diverse, and it has never been so vital to our society.
Welcome to the Golden Age of mathematics.”
(page 373)
This chapter raises many philosophical questions about whether we are determined or have free will. It’s interesting to think of that question since chaos theory has showed us a possibility of thinking that what we think is complete uncontrolled actions, in fact, can be that are determined and act mechanically.
Important Figures: Douglas Adams (The Hitchhiker’s Guide to the Galaxy, supercomputer, answer to the universe… 42), Laplace’s determinism, Marcy Lucy Cartwright and John Littlewood.
Nonlinear dynamics: Stephen Smale, James Yorke and Tien-Yien Li (Chaos), Butterfly Effect, Strange Attractors (Edward Lorenz),
Theoretical Monsters: Benoit Mandelbrot (fractals), Lewis Fry Richardson (Does wind have a velocity?).
Chaos everywhere!: strange attractors, Moon’s tides, biological populations, ecosystems.
Complexity: George Cowan, Murray Gell-Mann; John von Neumann, Stanislaw Ulam, Konrad Zuse (Cellular automaton, complex systems), Hans Meinhardt, Stuart Kauffman, Hui-Hsien Chou and James Reggia (geology and biology, complex systems).
“Throughout its lengthy history, mathematics has taken its inspiration from these two sources – the real world and the world of human imagination. Which is most important? Neither. What matters is the combination. The historical method makes it plain that mathematics draws its power, and its beauty, from both… Mathematics has never been so active, it has never been so diverse, and it has never been so vital to our society.
Welcome to the Golden Age of mathematics.”
(page 373)