There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are

Part of the maturity of the sciences is an appreciation of which questions are best left to other disciplinary approaches.

But perhaps the rest of us could have separate classes in science appreciation, the wonder of science, scientific ways of thinking, and the history of scientific ideas, rather than laboratory experience.

It has always appeared to me that we sacrifice many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress: I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. ...[T]hese digressions may serve to relieve the dryness of a mere collection of Examples.

Natural abilities are like natural plants; they need pruning by study.

The amazing number of species; their curious forms, so infinitely varied, and yet so nearly and gradually approximating through an endless series of transitions from one species to another; the diversity of structure observable in those parts which afford generic characters, added to the wonderful changes in form which they undergo, with their surprising economy - are circumstances which contribute to render them objects of most curious speculation to the philosopher. And although the study of every class of animals is most indisputably attended with peculiar advantages, yet we shall venture to affirm, that is from a knowledge of the characters, metamorphoses, and various modes of life, this little animals are destined to pursue, that [the natural philosopher] will obtain a more intimate acquaintance with the great laws of nature, and veneration for the Great Creator of all, that can be derived from the contemplation of any other class in nature.

Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second?

For a large number of problems there will be some animal of choice or a few such animals on which it can be most conveniently studied.

We must also accept the possibility of Gestalt qualities comprehending complexes of elements of different categories.

We are planning a sequel... The Geometry of Low-Dimensional Groups and Lattices which will contain two earlier papers...

The classification of subject matter may be carried out for all sort of special purposes - to arrange books on shelves, to group inventions in patents, to classify the raw materials, intermediates and products of importance to a particular manufacturer, and so on. All such arrangements have their particular uses and their particular problems.

Applying this approach, systems belonging to different scientific disciplines are investigated in their natural forms. On the basis of experimental results, isomorphic relations between different systems are studied and, finally, some general principles applicable for all systems of a certain class are formulated.

…if we intend to pursue the history of our species in its further attainments, we may soon enter on subjects which will confine our observation to more narrow limits. The genius of political wisdom and civil arts appears to have chosen his seats in particular tracts of the earth, and to have selected his favourites in particular races of men.

Class field theory can be divided into two parts, local and global. In each part it is the study of all the abelian extensions of a certain base field. The underlying philosophy is to describe all abelian extensions in terms of objects residing within, or close to, the base field.