Chapter 17: The Shape of Logic: Putting mathematics on fairly firm foundations
“A series of foundational crises – in particular the controversies over the basic concepts of calculus and the general confusion about Fourier series – had made it clear that mathematical concepts must be defined very carefully and precisely to avoid logical pitfalls.” (page 308)
“The dreadful truth was that mathematicians had devoted so much effort to the discovery of deep properties of numbers that they had neglected to ask what numbers were.”
Important Figures: Dedekind (1858, gaps in the logical foundations of the system of real numbers, Dedekind cuts), Giuseppe Peano (1889, list of axioms for whole numbers), Gottlob Frege (1880s, sets and classes of whole numbers), Georg Cantor (Cantor’s theory of transfinite numbers – different sizes of infinity), David Hilbert (axiomatic view), Kurt Gödel (Gödel’s Incompleteness Theorem).
“Set theory led to major advances, including a sensible, though unorthodox, system of infinite numbers. It also revealed some fundamental paradoxes related to the notion of a set… it was a proof that mathematics has inherent limitations, and that some problems do not have solutions. The upshot was a profound change in the way we think about mathematical truth and certainty. It is better to be aware of our limitations than to live in a fool’s paradise.” (page 330)
“The dreadful truth was that mathematicians had devoted so much effort to the discovery of deep properties of numbers that they had neglected to ask what numbers were.”
Important Figures: Dedekind (1858, gaps in the logical foundations of the system of real numbers, Dedekind cuts), Giuseppe Peano (1889, list of axioms for whole numbers), Gottlob Frege (1880s, sets and classes of whole numbers), Georg Cantor (Cantor’s theory of transfinite numbers – different sizes of infinity), David Hilbert (axiomatic view), Kurt Gödel (Gödel’s Incompleteness Theorem).
“Set theory led to major advances, including a sensible, though unorthodox, system of infinite numbers. It also revealed some fundamental paradoxes related to the notion of a set… it was a proof that mathematics has inherent limitations, and that some problems do not have solutions. The upshot was a profound change in the way we think about mathematical truth and certainty. It is better to be aware of our limitations than to live in a fool’s paradise.” (page 330)
Chapter 18: How Likely is That?: The rational approach to chance
“The growth of mathematics in the 20th and early 21st centuries has been explosive. More new mathematics has been discovered in the last 100 years than in the whole of previous human history.” (page 332)
Probability theory is possibly more significant than any other single major branch of mathematics. It’s a measure. Its applied branch is statistics. It helps us with hypothesis testing.
Important Figures: Pascal, Jacob Bernoulli, Abraham De Moivre, Henri Lebesgue, Francis Galton.
“Probability is now one of the most widely used mathematical techniques, employed in science and medicine to ensure that any deductions made from observations are significant, rather than apparent patterns resulting from chance associations.” (page 344)
Probability theory is possibly more significant than any other single major branch of mathematics. It’s a measure. Its applied branch is statistics. It helps us with hypothesis testing.
Important Figures: Pascal, Jacob Bernoulli, Abraham De Moivre, Henri Lebesgue, Francis Galton.
“Probability is now one of the most widely used mathematical techniques, employed in science and medicine to ensure that any deductions made from observations are significant, rather than apparent patterns resulting from chance associations.” (page 344)