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Jacobi to Minkowski

Carl Jacobi

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'D' wing of R&D

Hamilton-Jacobi equation

Jacobi's name is probably best known to undergraduates from the Jacobian, an n×n determinant formed from a set of n functions in n unknowns. Jacobi was certainly not the first to use it; the "Jacobian" already appears in an 1815 paper of Cauchy. But Jacobi did write a long memoir about it in 1841, and proved that the Jacobian of n functions vanishes if and only if the functions are related (Cauchy had only proved the "if" part).

Jacobi also did important work on partial differential equations and their application to physics. Along with Hamilton, he developed an approach to mechanics based on generalized coordinates. In this approach, the total energy of a mechanical system is represented as a function of generalized coordinates and corresponding generalized momenta; for example, in a double pendulum the two generalized coordinates could be two angles. Hamilton-Jacobi theory is the technique of solving the system by transforming coordinates so that the transformed coordinates and momenta are constants.

William James
William James

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The ultimate of ultimate problems, of course, in the study of the relations of thought and brain, is to understand why and how such disparate things are connected at all […] We must find the minimal mental fact whose being reposes directly on a brain-fact; and we must similarly find the minimal brain event which will have a mental counterpart at all.

Hilbert space

The real color of a thing is that one color-sensation which it gives us when most favorably lighted for vision. Soon its real size, its real shape, etc. these are but optical sensations selected out of thousands of others, because they have aesthetic characteristics which appeal to our convenience or delight. But I will not repeat what I have already written about this matter, but pass on to our treatment of tactile and muscular sensations, as 'primary qualities,' more real than those 'secondary' qualities which eye and ear and nose reveal. Why do we thus so markedly select the tangible to be the real? Our motives are not far to seek. The tangible qualities are the least fluctuating. When we get them at all we get them the same. The other qualities fluctuate enormously as our relative position to the object changes.

color sphere

Now the immediate fact which psychology, the science of mind, has to study is also the most general fact. It is the fact that in each of us, when awake (and often when asleep), some kind of consciousness is always going on. There is a stream, a succession of states, or waves, or fields (or of whatever you please to call them), of knowledge, of feeling, of desire, of deliberation, etc., that constantly pass and repass, and that constitute our inner life. The existence of this stream is the primal fact, the nature and origin of it form the essential problem, of our science. So far as we class the states or fields of consciousness, write down their several natures, analyze their contents into elements, or trace their habits of succession, we are on the descriptive or analytic level. So far as we ask where they come from or why they are just what they are, we are on the explanatory level.

Wm James

When we're asked "What do 'red', 'blue', 'black', 'white' mean?" we can, of course, immediately point to things which have these colours,—but that's all we can do: our ability to explain their meaning goes no further.


Thus "this is red," "this is earlier than that," are atomic propositions.

Russell & Whitehead


All other psychological phenomena are derived from the combinations of these ultimate psychological elements, as the totality of words may be derived from the totality of letters. Completion of this task would provide the basis for a Characteristica universalis of the sort that had been conceived by Leibniz, and before him, by Descartes.


There is a branch of mathematics known as 'harmonic analysis' which deals with the converse problem of sorting out the resultant curve into its constituents. Superposing a number of curves is as simple as mixing chemicals in a test-tube; anyone can do it. But to take the final mixture and discover what ingredients have gone into its composition may require great skill.

Fortunately the problem is easier for the mathematician than for the analytical chemist. 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


In May 1926 Schrödinger published a proof that matrix and wave mechanics gave equivalent results: mathematically they were the same theory.

Theodor Kaluza

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Alongside the metric tensor of the 4-dimesnional manifold (interpreted as a tensor potential for gravity) a general relativistic description of world-phenomena requires also an electromagnetic four-potential Am.

The remaining dualism of gravity and and electricity, while not lessening the theory's enthralling beauty, nevertheless sets the challenge of overcoming it through a fully unified view.

A few years ago H. Weyl made a surprisingly bold thrust towards the solution of this problem, one of the great favorite ideas of the human spirit. Through a new radical revision of the geometric foundations, he obtains along with the tensor gmn a kind of fundamental metric vector which he then interprets as the electromagnetic potential Am. The complete world metric then becomes the common source of all natural events.


It is true that our previous physical experience contains hardly any hint of the existence of an extra dimension...


Now it may be asked why these hidden variables should have so long remained undetected.



The aspects of things that are most important for us are hidden because of their simplicity and familiarity.

Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them.


Felix Klein
Felix Klein

'D' wing of R&D

Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.

It became possible to affirm that projective geometry is indeed logically prior to Euclidean geometry and that the latter can be built up as a special case. Both Klein and Arthur Cayley showed that the basic non-Euclidean geometries developed by Lobachevsky and Bolyai and the elliptic non- Euclidean geometry created by Riemann can also be derived as special cases of projective geometry. No wonder that Cayley exclaimed, "Projective geometry is all geometry."



Oscar Klein
Oscar Klein

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In a former paper, the writer has shown that the differential equation underlying the new quantum mechanics of Schrödinger can be derived from a wave equation of a five-dimensional space, in which h does not appear originally, but is introduced in connection with a periodicity in x0. Although incomplete, this result, together with the considerations given here, suggest that the origin of Planck's quantum may be sought just in this periodicity in the fifth dimension.


Klein's adaptation of Kaluza's work had a major difference from the original in that the extra or fifth dimension was curled up into a ball that was on the order of the Planck length, 10-33 cm. It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature.
 Ian T Durham

The highlight of that conference, at least with the hindsight of history, was the remarkable paper by Oscar Klein in which he proposed a unified model of electromagnetism and the nuclear force based on Kaluza-Klein ideas. This paper stands out in its originality and its brilliance from the other contributions to the conference and it foreshadowed the later developments of non-Abelian gauge theories that are the foundation of our present theory of particle physics.

David J Gross

 Joseph-Louis Lagrange

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

action integral

If the total energy is conserved, then the work done on the particle must be converted to potential energy, conventionally denoted by V, which must be purely a function of the spatial coordinates x, y, z, or equivalently a function of the generalized configuration coordinates X, Y, and possibly the derivatives of these coordinates, but independent of the time t. (The independence of the Lagrangian with respect to the time coordinate for a process in which energy is conserved is an example of Noether's theorem, which asserts that any conserved quantity, such as energy, corresponds to a symmetry, i.e., the independence of a system with respect to a particular variable, such as time.)

Pierre-Simon Laplace

Such is the advantage of a well constructed language that its simplified notation often becomes the source of profound theories.

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The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation

Laplace equation

The wave equation is the important partial differential equation

wave equation

that describes propagation of waves with speed velocity. The form above gives the wave equation in three-dimensional space where Laplacian is the Laplacian...

An even more compact form is given by


where dAlembertian operator is the d'Alembertian, which subsumes the second time derivative and second space derivatives into a single operator.

A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined. 

Wolfram MathWorld

Gottfried Wilhelm von Leibniz

The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.

'D' wing of R&D

Mechanical causes

Besides, it must be confessed that Perception and its consequences are inexplicable by mechanical causes; that is to say, by figures and motions. If we imagine a machine so constructed as to produce thought, sensation, perception, we may conceive it magnified — to such an extent that one might enter it like a mill. This being supposed, we should find in it on inspection only pieces which impel each other, but nothing which can explain a perception. It is in the simple substance, therefore, — not in the compound, or in the machinery, — that we must look for that phenomenon [...]

Sphus Lie

Sophus Lie

Lie posed himself a momentous question. Is there a theory of differential equations analogous to Galois's theory of algebraic one? Is there a way to decide when adifferential equation can be solved by specified methods?

The key, once more, was symmetry. Lie now realized that some of his results in geometry could be reinterpreted in terms of differential equations. Given one solution of a particular differential equation, Lie could apply a transformation (from a particular group) and prove that the result was also a solution. From one solution you could get many, all connected by the group. In other words, the group consisted of symmetries of the differential equatiom.

It was a broad hint that something beautiful awaited discovery.

John Locke
John Locke

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These I call original or primary qualities of the body, which I think we may observe to produce simple ideas in us, viz., solidity, extension, figure, motion or rest, and number.

Secondly, such qualities which in truth are nothing in the objects themselves, but powers to produce various sensations in us by their primary qualities, i.e. by the bulk, figure, texture, and motion of their insensible parts, as colour, sounds, tastes, etc., these I call secondary qualities.


Earthly minds, like mud walls, resist the strongest batteries; and though, perhaps, somethimes the force of a clear argument may make some impression, yet they nevertheless stand firm, keep out the enemy, truth, that would captivate or disturb them.


Michael Lockwood

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Take some range of phenomenal qualities. Assume that these qualities can be arranged according to some abstract n-dimensional space, in a way that is faithful to their perceived similarities and degrees of similarity — just as, according to Land, it is possible to arrange the phenomenal colors in his three-dimensional color solid. Then my Russellian proposal is that there exists, within the brain, some physical system, the states of which can be arranged in some n-dimensional state space [...] And the two states are to be equated with each other: the phenomenal qualities are identical with the states of the corresponding physical system.


color sphere

We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form. Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R4 and an internal structure, or set of states, labeled by elements g of G.


Calabi-Yau space

A speck in the visual field, though it need not be red must have some color; it is, so to speak, surrounded by color-space. Notes must have some pitch, objects of the sense of touch some degree of hardness, and so on.


Ernst Mach
Ernst Mach

'D' wing of R&D

A color is a physical object as soon as we consider its dependence, for instance, upon its luminous source, upon temperatures, upon spaces, and so forth.


Without renouncing the support of physics, it is possible for the physiology of the senses, not only to pursue its own course of development, but also to afford to physical science itself powerful assistance.

James Clerk Maxwell
James Clerk Maxwell

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When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the color and luminance of the light. Now, though everyone experiences these sensations and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis, and can never become in themselves objects of thought. If we attempt to discover them, we must do so by artificial means and our reasonings on them must be guided by some theory.

Warren McCulloch

In 1943 Warren McCulloch and Walter Pitts proposed a general theory of information processing based on networks of binary switching or decision elements, which are somewhat euphemistically called "neurons," although they are far simpler than their real biological counterparts [...]

McCulloch and Pitts showed that such networks can, in principle, carry out any imaginable computation, similar to a programmable, digital computer or its mathematical abstraction, the Turing machine.

Muller, Reinhardt



The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. 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.


We will try to visualize the state of things by the graphic method. Let x, y, z be rectangular co-ordinates for space, and let t denote time. The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. [...] The multiplicity of all thinkable x, y, z, t systems of values we will christen the world.


[So] few and far between are the occasions for forming notions whose specializations make up a continuous
manifold, that the only simple notions whose specializations form a multiply extended manifold are the positions of perceived objects and colors.


The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colors, tones) may be specified by the giving of n quantities, the "co-ordinates," which are continuous functions within the manifold.


GE Moore

GE Moore
This mental occurrence, which I call 'seeing', is known to us in a much more simple and direct way, than are the complicated physiological processes which go on in our eyes and nerves and brains. A man cannot directly observe the minute processes which go on in his own eyes and nerves and brain when he sees; but all of us who are not blind can directly observe this mental occurrence, which we mean by seeing. And it is solely with seeing, in this sense — seeing, as an act of consciousness which we can all of us directly observe as happening in our own minds — that I am now concerned.



I hold up this envelope, then: I look at it, and I hope you all will look at it. And now I put it down again. Now what has happened? We should certainly say (if you have looked at it) that we all saw that envelope, that we all saw it, the same envelope: I saw it, and you all saw it. We all saw the same object. And by the it, which we all saw, we mean an object, which, at any one of the moments when we were looking at it, occupied just one of the many places that constitute the whole of space. Even during the short time in which we were looking at it, it may have moved — occupied successively several different places; for the earth, we believe, is constantly going round on its axis, and carrying with it all the objects on its surface, so that, even while we looked at the envelope, it probably moved and changed its position in space, though we did not see it move. But at any one moment, we should say, this it, the envelope, which we say we all saw, was at some one definite place in space.

But now, what happened to each of us, when we saw that envelope? I will begin by describing part of what happened to me. I saw a patch of a particular whitish color, having a certain size, and a certain shape, a shape with rather sharp angles or corners and bounded by fairly straight lines. These things: this patch of a whitish color, and its size and shape I did actually see. And I propose to call these things, the color and size and shape, sense-data things given or presented by the senses — given, in this case, by my sense of sight. Many philosophers have called these things which I call sense-data, sensations. They would say, for instance, that that particular patch of color was a censation. But it seems to me that this term 'sensation' is liable to be misleading. We should certainly say that I had a sensation, when I saw that color. But when we say that I had a sensation, what we mean is, I think, that I had the experience which consisted in my seeing the color.


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