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Einstein to Fourier

Albert Einstein

We are accustomed to regarding as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perception.

'D' wing of R&D

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I believe that the first step in the setting of a "real external world" is the formation of the concept of bodily objects and of bodily objects of various kinds. Out of the multitude of our sense experiences we take, mentally and arbitrarily, certain repeatedly occurring complexes of sense impression (partly in conjunction with sense impressions which are interpreted as signs for sense experiences of others), and we attribute to them a meaning—the meaning of the bodily object. Considered logically this concept is not identical with the totality of sense impressions referred to; but it is an arbitrary creation of the human (or animal) mind. On the other hand, the concept owes its meaning and its justification exclusively to the totality of the sense impressions which we associate with it.

The overcoming of naive realism has been relatively simple. In his introduction to his volume, An Inquiry Into Meaning and Truth, Russell has characterized this process in a marvellously pregnant fashion:

We all start from 'naive realism,' i.e., the doctrine that things are what they seem. We think that grass is green, that stones are hard, and that snow is cold. But physics assures us that the greenness of grass, the hardness of stones, and the coldness of snow, are not the greenness, hardness, and coldness that we know in our own experience, but something very different. The observer, when he seems to himself to be observing a stone, is really, if physics is to be believed, observing the effects of the stone upon himself. Thus science seems to be at war with itself: when it means to be most objective, it finds itself plunged into subjectivity against its will.

Apart from their masterful formulation these lines say something which had never previously occurred to me.



Leonhard Euler

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

least action
least action

The calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Mathematically, this involves finding  of integrals of the form:

action formula

Furthermore, and now this is the point, this is the punch line, the symmetries determine the action.  This action, this form of the dynamics, is the only one consistent with these symmetries [...]  This, I think, is the first time that this has happened in a dynamical theory: that  the symmetries of the theory have completely determined the structure of the dynamics, i.e., have completely determined the quantity that produces the rate of change of the state vector with time.

Herbert Feigl

There's no question that Feigl recognized the difficulty of the problem. His problem of "sentience" is of course just the problem I'm concerned with. (David Chalmers)

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I can here only briefly indicate the lines along which I think the 'world knot' to use Schopenhauer's striking designation for the mind-body puzzles may be disentangled. The indispensable step consists in a critical reflection upon the meanings of the terms 'mental' and 'physical', and along with this a thorough clarification of such traditional philosophical terms as 'private' and 'public,' 'subjective' and 'objective,' 'psychological space(s)' and 'physical space,' 'intentionality,' 'purposiveness,' etc. The solution that appears most plausible to me, and that is consistent with a thoroughgoing naturalism, is an identity theory of the mental and the physical, as follows: Certain neurophysiological terms denote (refer to) the very same events that are also denoted (referred to) by certain phenomenal terms ... I take these referents to be the immediately experienced qualities, or their configurations in the various phenomenal fields.


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How do we model phenomenal fields?

There's no question that Feigl recognized the difficulty of the problem. His problem of "sentience" is of course just the problem I'm concerned with.


And perhaps, posterity will thank me for having shown it that the ancients did not know everything.

'D' wing of R&D

Among the more or less general laws, the discovery of which characterize the developmentof physical science during the last century, the principle of Least Action is at presentcertainly one which, by its form and comprehensiveness, may be said to have approachedmost closely to the ideal aim of theoretical inquiry. Its significance, properly understood,extends, not only to mechanical processes, but also to thermal and electrodynamicproblems. In all the branches of science to which it applies, it gives, not only an explanation of certain characteristics of phenomena at present encountered, but furnishes rules whereby their variations with time and space can be completely determined. It provides the answersto all questions relating to them, provided only that the necessary constants are known andthe underlying external conditions appropriately chosen.

It was a long time before it was clearly explained, and the principle of least action correctly understood. If the principle be said to have been discovered at this time, the honour should be given to Lagrange. This, however, would be an injustice to other men who had prepared the way for Lagrange to bring the work later to a satisfactory completion. Of these, the first was Leibniz; indeed, he was the chief, according to a letter dated 1707, the original of which has been lost. Then came Maupertuis and Euler. It was chiefly Moreau de Maupertuis (appointed president of the Prussian Academy of Sciences (1746-1759) by Frederick the Great) who not only recognized the existence and significance of the principle, but used his influence in the scientific world and elsewhere to procure its acceptance.

Maupertuis’s exposition of the principle of least action asserted no more than “that the action applied to bring about all the changes occurring in Nature is always a minimum.” Strictly, this formulation does not admit any conclusions to be drawn regarding the laws governing the changes, for as long as no statement of the conditions to be satisfied is made, no deductions can be made as to how the variations are balanced. Maupertuis had not the faculty of analytical criticism necessary to discern this want. The failure will be more easily understood when it is realized that Euler himself, a brilliant mathematician, did not succeed in producing a correct formulation of the principle, though he was assisted by many colleagues and friends.

Maupertuis’s real service consisted in his search for a principle that would be, above all, a minimum principle. That was the real object of his investigation. To this end he made use of Fermat’s principle of quickest arrival, although its bearing upon the principle of least action was very indirect and, at all events, unknown to the physics of his time.

Planck, The Principle of Least Action (PDF)


Richard Feynman

Nature has a great simplicity
and therefore a great beauty.

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If you take a physical state and do something to itlike rotating it, or like waiting for some time deltat you get a different state. We say, "performing an operation on a state produces a new state." We can express the same idea by an equation:

|phi> = A|psi>.

An operation on a state produces another state. The operator A stands for some particular operation. When this operation is performed on any state, say |psi>, it produces some other state |phi>.

color sphere

I would like to again impress you with the vast range of phenomena that the theory of quantum electrodynamics describes: It's easier to say it backwards: the theory describes all the phenomena of the physical world except the gravitational effect [...] and radioactive phenomena, which involve nuclei shifting in their energy levels. So if we leave out gravity and radioactivity (more properly, nuclear physics) what have we got left? Gasoline burning in automobiles, foam and bubbles, the hardness of salt or copper, the stiffness of steel. In fact, biologists are trying to interpret as much as they can about life in terms of chemistry, and as I already explained, the theory behind chemistry is quantum electrodynamics.



There is a very simple technique for analyzing any curve, no matter how complicated it may be, into its constituent simple harmonic curves. It is based on a mathematical theorem known as Fourier's theorem [...]

The theorem tells us that every curve, no matter what its nature may be, or in what way it was originally obtained, can be exactly reproduced by superposing a sufficient number of simple harmonic curves—in brief, every curve can be built up by piling up waves. ~Sir James Jeans 

'D' wing of R&D



The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound. A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency f and integer multiples of that frequency.

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.


The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other.The idea is that a general sound wave is a superposition of harmonics, the way a symphony is a superposition of the harmonics corresponding to the notes played by various instruments. Mathematically, this means expressing a given function as a superposition of the functions describing harmonics, such as the familiar functions sine and cosine. Automorphic functions are more sophisticated versions of these familiar harmonics.There are powerful analytic methods for doing calaculations with these automorphic functions. And Langlands' surprising insight was that we can use these functions to learn about much more difficult questions in number theory.