The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.
Gottlob FregeRelated topics
book concept content development euclid following last lie mathematics might number opinion perhaps prove should trying try view wayRelated quotes
The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication for the earlier concept; this earlier concept, or sometimes the term used for it, is called the explicandum; and the new concept, or its term, is called an explicatum of the old one.
Rudolf Carnap
The Object of this Treatise is-(1) To point out to the student of Mathematics, who has not the advantage of a tutor, the course of study which it is most advisable that he should follow, the extent to which he should pursue one part of the science before he commences another, and to direct him as to the sort of applications which he should make. (2) To treat fully of the various points which involve difficulties and which are apt to be misunderstood by beginners, and to describe at length the nature without going into the routine of the operations.
Augustus De Morgan
It is conceivable that animal life might have the attribute of using the heat of surrounding matter, at its natural temperature, as a source of energy for mechanical effect.... The influence of animal or vegetable life on matter is infinitely beyond the range of any scientific enquiry hitherto entered on. Its power of directing the motions of moving particles, in the demonstrated daily miracle of our human free-will, and in the growth of generation after generation of plants from a single seed, are infinitely different from any possible result of the fortuitous concurrence of atoms.
William Thomson
Individuals who have been wronged by unlawful racial discrimination should be made whole; but under our Constitution there can be no such thing as either a creditor or a debtor race. That concept is alien to the Constitution's focus upon the individual. ...To pursue the concept of racial entitlement - even for the most admirable and benign of purposes - is to reinforce and preserve for future mischief the way of thinking that produced race slavery, race privilege and race hatred. In the eyes of government, we are just one race here. It is American.
Antonin Scalia
Modern man, seeking a middle position in the evaluation of sense impression and thought, can, following Plato, interpret the process of understanding nature as a correspondence, that is, a coming into congruence of pre-existing images of the human psyche with external objects and their behaviour. Modern man, of course, unlike Plato, looks on the pre-existent original images also as not invariable, but as relative to the development of a conscious point of view, so that the word "dialectic" which Plato is fond of using may be applied to the process of development of human knowledge.
Wolfgang Pauli