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algebra analysis appears applied arithmetic arrangements assert become branch calculus called century character collection combinations conclusions consider consists constructions continuous course created creation define definite dependent determination differential division elements enables entities equations example existence experience expression fact field figures follows functions geometry give given human hypernumbers idea ideal imaginary importance individuals infinite infinity instance integers intuition invariants invention irrational kind laws leads logic logistic mathe mathematician mathematics matics means merely method mind nature notion numbers objects operations particular philosophy physics plane possible present problems processes properties propositions purely question range rational reality reason reduce REFERENCES relations roots sense simple solutions solve space square statement structure successive symbols theorems theory things thought tion transformations true truth unit universe variables various whole
Seite 41 - 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...
Seite 2 - Conterminous with space and coeval with time is the kingdom of mathematics ; within this range her dominion is supreme ; otherwise than according to her order nothing can exist ; in contradiction to her laws nothing takes place. On her mysterious scroll is to be found, written for those who can read it, that which has been, that which is, and that which is to come.
Seite 34 - Geometry is not an experimental science; experience forms merely the occasion for our reflecting upon the geometrical ideas which pre-exist in us. But the occasion is necessary, if it did not exist we should not reflect, and if our experiences were different, doubtless our reflections would also be different. Space is not a form of...
Seite 123 - Mathematics is not the discoverer of laws, for it is not induction ; neither is it the framer of theories, for it is not hypothesis ; but it is the judge over both, and it is the arbiter to which each must refer its claims ; and neither law can rule nor theory explain without the sanction of mathematics.
Seite 186 - ... undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp. This it is which gives body, a structure so to speak, to the iridescent appearances which flatter our senses, and without this support the beauty of these fugitive dreams would be only imperfect, because it would be vague and always fleeting. On the contrary, intellectual beauty is sufficient unto...
Seite 61 - The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age ; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
Seite 162 - Then a page or so later: the analogy with red and white roses seems, in the end, to express the matter as nearly as possible. What is truth and what falsehood, we must merely apprehend, for both seem incapable of analysis. And as for the preference which most people so long as they are not annoyed by instances - feel in favour of true propositions, this must be based, apparently, upon an ultimate ethical proposition: 'It is good to believe true propositions and bad to believe false ones.
Seite 169 - Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of...
Seite 54 - The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features.
Seite 186 - The scientist does not study nature because 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.