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5. lappuse - The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.
384. lappuse - LXXV-LXXVI are intended to revise and give further illustrations of ideas about the nature of functions of a single variable which have already been acquired in Part I. The essentials of the ideas connoted by the terms " indeterminate value " and " singular points " find their place here. In Ex. LXXVI an inquiry into the properties of a few functions of two variables is made the occasion for extending the method of rectangular coordinates to the analysis and description of curved surfaces. The investigation...
412. lappuse - To wit, when we seek to subject them to numeration ... we find that they flee away perpetually, so that not one of them can be apprehended precisely in itself. . . . Now that cannot be called a true number which is of such a nature that it lacks precision. . . . Therefore, just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity.
79. lappuse - It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
46. lappuse - ... is less compact and less easily reproduced. Its message is frequently inarticulate and obscure. For these and similar reasons it should be regarded as a subsidiary algebraic instrument which fulfils its best office when it either leads up to a formula by which it may itself be superseded, or serves to unfold more fully the implications of a formula whose properties have been only partially explored.
20. lappuse - ... in the algebra course and exclude sin x or tan x. All alike are pieces of symbolism invented for the description and interpretation of facts of the external world. Each represents a typical kind of function. To each corresponds a specific form of curve which may be regarded as the graphic symbol of the function. Both algebra and trigonometry would gain by fusion the former through an added variety and richness in the illustration of its main themes, the latter by the removal of the excessive...
16. lappuse - Mathematical truths always have two sides or aspects. With the one they face and have contact with the world of outer realities lying in time and space. With the other they face and have relations with one another.
17. lappuse - One never has existed and probably never will exist apart from the other. The view that they represent wholly distinct forms of- intellectual activity is partial, unhistorical, and unphilosophical. A more serious charge against it is that it has produced an infinite amount of harm in the teaching of mathematics. Our purpose in teaching mathematics in school should be to enable the pupil to realize, at least in an elementary way, this two-fold significance of mathematical progress. A person, to be...
474. lappuse - We have in fact a double algebra as the instrument for the complete treatment of all higher analysis, except that in which one of higher multiplicity is used. The field of Quantics has been brilliantly cultivated by Cayley, Sylvester and others.
19. lappuse - ... perfecting the technique of his subject, finds it natural as well as most effective to take a special group of allied methods or allied problems and to develop them as far as he can without concerning himself too greatly about the practical value of his work. Now this systematic exploration of special arts of mathematics is, no doubt, of vital importance for the continued growth of the science.