study of nature

The scientist does nos study nature becaus it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.

H. Poincare

Elipse

Man is not a circle with a single centre; he is an ellipse with two foci. Facts are one, ideas are the other.

Victor Hugo, Les Miserables

Representation

A representation is a formal systems for making explicit certain entities or types of information, together with a specification of how the system does this.

Extracts from Symmetry by Hermann Weyl

A geometry, Klein said, is defined by a group of transformations, and investigates everything that is invariant under the transformations of this given group. Of symmetry one speaks with respect go a subgrup gamma of the total group.

Physical occurrences happen not only in space but in space and time; the world is spread out not as a three- but as a four-dimensional continuum. The symmetry, relativity, or homogeneity of this four-dimensional medium was firts correctlly described by Einstein.

What Einstein did was this: without bias he collected all the physical evidence we have about the real structure of the four-dimensional space-time continuum and thus derived its true group of automorphisms.

Galois´theory is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.

A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure endowed entitiy Sigma try to determine its group of automorphisms, i.e., the group of those element-wise transformations which leave all structural relations undisturbed.

applications of solitons

For more than 130 years solitary waves where considered rare curiosities. Not until the 1960's did it become clear that such waves could form in the ocean as seismic waves of enormous wavelengths generated by shocks in the ocean's floor. Moreover, soliton waves are common in a great variety of other physical systems. In some circumstances it is almost impossible ot prevent them from forming.
There are endless examples. When a double-helix molecule of DNA is put into solvent, there is an exchange of hydrogen atoms betwenn the solvent and the DNA. A break forms between the two helices and moves along the molecule as a stable wave. Similar energy pulses are solitons that traverse the helix of other protein molecules. Many solitons exhibit periodicity, fluctuating between extreme values of one or more properties. Such "second order" solitons turn up as pulses in laser light moving along a glass fiber. Energy travels along nerves in solitons pulses. Pressure waves can form solitons, such as the sound of an explosion or the movement of certain mechanical vibrations through solids.
Magnetic fields trapped in superconductors and superfluids form soliton vortices. It seems likely that Jupiter's famous red spot is a long-lasting soliton in the giant planet's turbulent atmposphere.
The most promising application of solitons is in fiber optics. Fibers have a natural tendency to keep light pulses from dispersing, and various techniques are being devised to keep the pulses even more uniform.
Recent years have seen a major trend towards constructing soliton models of elementary particles.

	Gardner, Martin, 1914-
Title	The ambidextrous universe : mirror asymmetry and time-reversed worlds / Martin Gardner ; illustrated by John Mackey.
Edition	2nd ed.
Published	Harmondsworth : Penguin, 1982, c1979.

solitons and particles

The superficial similarity between the properties of solitons and of elementary particles is striking. Solitons may propagate without change of form. A solitons may be regarded as a local confinement of the energy of the wave field. When two solitons collide, each may come away with the same character as it had before the collision. When a soliton meets and antisoliton, both may be anhihilated. Elementary particles share these properties. So, if an appropiate system of nonlinear field equations admits soliton solutions then theses solitons may represent elementary particles and have properties which may be confirmed by observations of particles.

(Drazin, Solitons: an introduction)

On progress

The progress of mathematics may be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example the theory of enumeration, appear boundless. Rules for the enumeration of sets subject to various conditions appear to obey an indefinite variety of recursions and seem to lead to a bounty of generating functions. We are naively led to conjecture that the class of enumerable objects is infinite and unclassifiable.
As cases pile upon cases, however, patterns begin to emerge. Freakish instances are quietly disregarded; impossible problems are recognized as such, and what is left gets organized along a few general criteria. We will do all we can to boil these criteria down to one, but we will probably have to be satisfied with a small finite number.

Gian-Carlo Rota

On teaching

A good teacher does not teach facts, he or she teaches enthusiasm, open-mindedness and values. Young people need encouragement. Left to themselves, they may not know hot to decide what is worthwhile. They may drop an original idea because they think someone else must have thought of it already. Students need to be taught to believe in themselves and not to give up.

Gian-Carlo Rota

Definition of math

Mathematics is the study of analogies between analogies. All science is. Scientists want to show that thins that don´t look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.

On solitons

The soliton fad is a catchy mixture of operator algebra and explicitlly solvable differential equations. How can anyone resist such temptation? So much for the good news. The bad news is that solitons seem unbudgingly one-dimensional, despite the insinuations of computer simulators. Chalk one up to one-dimensional physics.

Gian-Carlo Rota

Escapes

Of all escapes from reality, mathematics is the most succesful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same reality we are trying to evade. All other escapes -sex, drugs, hobbies, whatever- are ephemeral by comparison. The mathematician`s feeling of triumph, as he forces the world to obey the laws of his imagination has freely created, feeds on its own success. The world is permanently changed bye the workings of his mind and the certainty that his creations will edure renews his confidence as no other pursuit. The mathematician becomes totally commited, a monster, like Nabokov`s chess player who eventually sees all life as subordinate to the game of chess.

(Gian-Carlo Rota, Indiscrete Thoghts)