## Mysticism in Modern MathematicsH. Frowde, 1910 - 264 páginas |

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abstrac abstract admit algebraic quantity algebraic symbolism analogy analytical geometry angles argument arithmetical assumption calculus called Cayley Cayley's circle conceive conception conclusion conic connexion convention defined derived direction distinction doctrine elementary algebra equal equation Euclid Euclidean Euclidean geometry experience explanation expression fact geometrical axioms geometrical entities given idea identity imaginary magnitude imaginary points imaginary quantity interpretation invisible points involution involved judgement kinds of space language length linear shape Lobatschewsky logical manifold mathematical mathematicians Max Müller measure of curvature mental merely metageometers metaphor metrical mind mystical nature negative quantity non-Euclidean non-Euclidean geometry notion of space object ordinary paradoxical particular plane positive premisses process of reasoning process of thought projective geometry properties proposition purely quadric question relation Riemann's root rule of signs self-evident sense straight line substitutive signs suppose surface systems of geometry term theory things tion words

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Página 191 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...

Página 27 - process of tunnelling, of tunnelling through a sand-bank. " In this operation it is impossible to succeed, unless " every foot, nay almost every inch in our progress, be " secured by an arch of masonry, before we attempt the " excavation of another. Now, language is to the mind " precisely what the arch is to the tunnel.

Página 73 - I would myself say that the purely imaginary objects are the only realities, the ovTuf ovra, in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object. If there is no conception of straightness, then it Is meaningless to deny the existence of a perfectly straight line.

Página 78 - Second, that the formal processes of solution or demonstration be conducted throughout in obedience to all the laws determined as above, without regard to the question of the interpretability of the particular results obtained...

Página 224 - ... the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space — the most important system for our present purpose...

Página 27 - A sign is necessary, to give stability to our intellectual progress, — to establish each step in our advance as a new starting-point for our advance to another beyond. A country may be overrun by an armed host, but it is only conquered by the establishment of fortresses. Words are the fortresses of thought.

Página 69 - There are therefore two points of intersection — viz. a straight line and a circle intersect always in two points, real or imaginary. It is in this way that we are led analytically to the notion of imaginary points in geometry. The conclusion as to the two points of intersection cannot be contradicted by experience: take a sheet of paper and draw on it the straight line and circle, and try. But you might say, or at least be strongly tempted to say, that it is meaningless. The question of course...

Página 14 - A definition is nothing else but an explication of the meaning of a word, by words whose meaning is already known. Hence it is evident, that every word cannot be defined ; for the definition must consist of words ; and there could be no definition, if there were not words previously understood without definition.

Página 143 - In the case of two given curves, there are thus two equations satisfied by the coordinates (x, y) of the several points of intersection, and these give rise to an equation of a certain order for the coordinate x or y of a point of intersection. In the case of a straight line and a circle, this is a quadric equation; it has two roots, real or imaginary. There are thus two values, say of x, and to each of these corresponds a single value of y. There are therefore two points of intersection — viz....

Página 41 - ... he had only to lay the accent on truly, and he would have understood what I meant — namely, that in the true sense of these words, as defined by myself, no one thinks who does not directly or indirectly speak, and that no one can be said to speak who does not at the same time think. We cannot be too charitable in the interpretation of language, and I often feel that I must claim that charity more than most writers in English. Still, I am always glad if such opponents as Mr.