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On Hilbert's Sixth Problem

Part of the Synthese Library Book Series (volume 441)

DOI: 10.1007/978-3-030-83837-9

ISBN: 978-3-030-83836-2

Publisher: Springer International Publishing - Springer Journals

Aug 9, 2021

1 - 10 of 20 Chapters

LitStream

Chapter 1Preliminary

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 1 –15

[However things aren’t just as simple, for the specter of undecidability and incompleteness, which was unknown and, we may say, unsuspected in 1900, raises its head.]
LitStream

Classical Mechanics

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 19 –23

[We describe here what we mean by a “physical theory.” Actually we develop the main standard formalisms for it. There are actually three such formalisms]
LitStream

Variational Calculus

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 25 –27

[However we will handle mathematical objects the way physicists do. We are aware that there is a huge epistemological problem here: the way mathematics is handled by physicists looks different from the way mathematicians use their tools. Examples are everywhere, from the H-theorem to Dirac’s δ...
LitStream

Lagrangian Formulation

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 29 –33

[Let’s consider the (very general) example. Consider the motion of a system of N material points.]
LitStream

Hamilton’s Equations

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 35 –38

[Again we meet our notational convention for the indices of vector-like objects that sort of behave like tangent vectors to a curve of coordinates qi(t) with components.]
LitStream

Hamilton–Jacobi Theory

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 39 –41

[He discovered an exact solution for the Einstein equations which allows for closed timelike curves, that is, that allows for some kind of “natural” time machines. Lanczos also said that the Hamilton–Jacobi equation is a kind of sacred burning bush of classical mechanics.]
LitStream

Where the Action Is

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 43 –47

[We are back to our starting point; Hamilton’s principal function S is in fact (but for a constant) the Action Integral. Here we also reach a climatic point in our exposition.]
LitStream

From Classical to Quantum

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 49 –52

[We have sketched here how it is done: take a single particle in 3-space; its Hamilton–Jacobi equation is, from the general Hamilton–Jacobi equation.]
LitStream

Field Theory

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 53 –54

[The concept stems from a construction based on an infinite coupling of harmonic oscillators (and one later proves that quantized electromagnetism can be seen as a countable infinite collection of harmonic oscillators).]
LitStream

Electromagnetism

da Costa, Newton C. A.; Antonio Doria, Francisco

Pages: 55 –59

[We deal now with classical electromagnetic theory. We restrict our attention to the so-called vacuum fields. We start from another sacred spot, Maxwell’s equations which describe the electromagnetic field.]

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