Kindle Notes & Highlights
by
Lee Phillips
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May 8 - June 17, 2025
Her four closely connected results, collectively now called Noether’s theorem, supply the foundation of the present-day search for the holy grail of physics: a unified theory that will join quantum mechanics with gravity.
The theorem goes beyond providing the foundation of current physical theory and supplying the touchstone for the physics of the future. It provides the modern definition of the concept of energy and clarifies the importance of symmetry in nature.
According to Wilczek, Noether’s theorem is “the single most profound result in all of physics.”1
Renowned physicists Leon Lederman and Christopher Hill say that Noether’s theorem is “one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem” and that it “rules modern scientific methodology.”
Such a connection among concepts that had previously seemed unrelated, demonstrated by unambiguous mathematics, not only was unexpected but also seemed to have implications that reached beyond physics and toward philosophy.
Weyl wants us to understand his friend Emmy Noether’s attitude and state of mind: that the course of her life was fueled by her overwhelming passion for mathematics.
the atmosphere at Göttingen was still, at that time, rather formal and chilly, with academic ranks among the different levels of faculty and between faculty and students carefully observed.18 This was not Hilbert’s style.
Perhaps an even more telling hint of Hilbert’s unconventional attitudes is the story that when his son, Franz, began his schooling, he was asked what religion he belonged to, and the boy had no idea.28
was Hilbert’s friend Minkowski who planted the idea for the theme for the lectures.29 He suggested that, in planning a turn-of-the-century talk, Hilbert might consider a “look into the future” and construct a “listing of the problems on which mathematicians should try themselves during the coming century.
Noether’s career: “A greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.”32
First, why is Einstein’s theory called a theory of “relativity”? The theory deals with the question of how you would describe things from, or relative to, different points of view. In this case, the points of view are different reference frames.
The first explicit theory of relativity was laid out by Galileo, and we now call it Galilean relativity.
It was the theory of relativity that persisted for over three hundred years, that Einstein inherited, and that he showed could not be true if some other things that we had learned in the meantime were true.
supported by experiment, and ruthlessly applying simple logic to ingenious thought experiments led Einstein to the conclusions of his special theory of relativity.
The constancy of the speed of light was also implied by James Clerk Maxwell’s electromagnetic theory. In a sense, Maxwell’s theory was the first unified theory in physics: the great Scottish physicist used the criteria of mathematical beauty and symmetry to blend the existing theories of electricity and magnetism into a set of equations that showed that each was an aspect of the other. These equations showed that vibrating fields of electricity and magnetism progressed through space in the form of waves—of light, heat, or radio waves—with a velocity that was a constant of nature and was
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we owe our concept of four-dimensional space-time to Minkowski. It combines the three space dimensions with the time dimension, but not in the way that Galileo or Newton might have done, where time and space retain their wholly separate identities. In Minkowski’s space-time, the time and space coordinates are more intimate with each other: Minkowski’s rotation mixed time intervals and space intervals together.
By showing that Einstein’s space-time transformations were equivalent to a rotation, Minkowski had discovered a hidden symmetry in the equations of special relativity, one that Einstein hadn’t seen.
Minkowski’s trick neatly brings out a radical consequence to relativity: space and time are not eternally separate. All you have to do is step on a moving platform, and space and time, formerly distinct concepts, become blended.
“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
“Over the whole life of Göttingen shone the brilliant genius of David Hilbert, as if binding us all together.… Almost every word he said, about problems in our science and about things in general, seemed to us strangely fresh and enriching.”
Hilbert’s predilection for abstraction in a broader cultural context: “Early twentieth-century intellectuals—including mathematicians, artists, architects, musicians, dancers, writers, and physicists—were fascinated by the concept of abstraction.
David Hilbert, considered the greatest mathematician since Carl Friedrich Gauss, was using highly abstract methods at Göttingen. In Erlangen, Noether began applying his approach to algebra.”58
Peter Freund, in A Passion for Discovery, adopts a similar point of view. He notices the similarities between the increasing abstraction of early twentieth-century mathematics and physics, and the emergence of a higher degree of abstraction in the arts, which he feels is represented in the works of Wassily Kandinsky, Arnold Schoenberg, James Joyce, and others.59
Einstein’s puzzling over the apparent identity of the two forms of mass is related to what we now call the principle of equivalence. This principle can be formulated in several ways. One way is to insist that the two types of mass are identical because of a fundamental symmetry in nature—that the laws of physics must take the same form whether you are in a gravitational field or in a region of space with no gravity but, say, in a spaceship undergoing an equivalent acceleration.
