Chapter 11: Firm Foundations: Making calculus make sense
“The beauty and power of calculus had become undeniable. However, Bishop Berkeley’s criticisms of its logical basis remained unanswered, and as people began to tackle more sophisticated topics, the whole edifice started to look decidedly wobbly.”
Main Players: Fourier (heat flow, functions as infinite series of sines and cosines), Bernard Bolzano (continuous functions, polynomial), Augustin-Louis Cauchy, Niels Abel, Peter Dirichlet and, above all, Karl Weierstrass (power series).
“Taking criticisms like those made by Bishop Berkeley seriously did, in the long run, enrich mathematics and place it on a firm footing. The more complicated the theories became, the more important it was to make sure you were standing on firm ground.” (page 209)
Main Players: Fourier (heat flow, functions as infinite series of sines and cosines), Bernard Bolzano (continuous functions, polynomial), Augustin-Louis Cauchy, Niels Abel, Peter Dirichlet and, above all, Karl Weierstrass (power series).
“Taking criticisms like those made by Bishop Berkeley seriously did, in the long run, enrich mathematics and place it on a firm footing. The more complicated the theories became, the more important it was to make sure you were standing on firm ground.” (page 209)
Chapter 12: Impossible Triangles: Is Euclid’s geometry the only one?
“Essentially, it was assumed that there can be only one geometry, Euclid’s, and that this is an exact mathematical description of true geometry of physical space. People found it difficult even to conceive of alternatives. It couldn’t last.”
It all started with navigation…
Projective geometry entered in play from the early 17th century (it had a Euclidean frame).
The biggest issue was Euclid’s Fifth Postulate on parallel lines.
Geometry and Art: artists of the Renaissance, 3D, mathematics of perspective, Filippo Brunelleschi, Leone Battista Alberti, Piero della Francesca, Girar Desargues (arquitect/engineer).
Works on Euclid’s Fifth Postulate: John Playfair, Adrien-Marie Legendre (similar triangles), Gerolamo Saccheri, Johann Henrich Lambert (quadrilateral with 3 right angles, hyperbolic functions), Gauss (theorems on other geometries), Nikolai Ivanovich Lobanchevsky (non-Euclidean geometry, curvature), Elliptic Geometry (the ones before), Poincaré (hyperbolic geometry), Maurits Escher, Einstein (Theory of General Relativity).
“… the nature of physical space is a question for observation, not thought alone. Nowadays, we make a clear distinction between mathematical models of reality, and reality itself. For that matter, much of mathematics bears no obvious relation to reality at all – but is useful, all the same.” (page 227)
It all started with navigation…
Projective geometry entered in play from the early 17th century (it had a Euclidean frame).
The biggest issue was Euclid’s Fifth Postulate on parallel lines.
Geometry and Art: artists of the Renaissance, 3D, mathematics of perspective, Filippo Brunelleschi, Leone Battista Alberti, Piero della Francesca, Girar Desargues (arquitect/engineer).
Works on Euclid’s Fifth Postulate: John Playfair, Adrien-Marie Legendre (similar triangles), Gerolamo Saccheri, Johann Henrich Lambert (quadrilateral with 3 right angles, hyperbolic functions), Gauss (theorems on other geometries), Nikolai Ivanovich Lobanchevsky (non-Euclidean geometry, curvature), Elliptic Geometry (the ones before), Poincaré (hyperbolic geometry), Maurits Escher, Einstein (Theory of General Relativity).
“… the nature of physical space is a question for observation, not thought alone. Nowadays, we make a clear distinction between mathematical models of reality, and reality itself. For that matter, much of mathematics bears no obvious relation to reality at all – but is useful, all the same.” (page 227)