The Foundations of MathematicsIn this brief treatise, Carus traces the roots of his belief in the philosophical basis for mathematics and analyzes that basis after a historical overview of Euclid and his successors. He then examines his base argument and proceeds to a study of different geometrical systems, all pulled together in his epilogue, which examines matter, mathematics, and, ultimately, the nature of God. |
Common terms and phrases
absolute abstract anyness apriority arithmetic Bolyai boundary called causation Common Notions conception concrete congruent COSIMO curvature definite determined direction domain element eternal etry Euclid Euclidean geometry Euclidean systems existence experience feature figures formal sciences four-dimensional fourth dimension G. B. Halsted Gauss geom Grassmann halves idea ideal construction infinite Kant Kant's laws of form Lobatchevsky logic manifold mathe mathematical space mathematicians matics matter and energy measurement Mephistopheles metageometry method mind motility nature non-Euclidean non-Euclidean geometry norm parallel axiom parallel lines philosophical philosophy of mathematics plane geometry plane P₁ position possible posteriori postulate priori priori constructions problem produce pseudosphere pure form pure reason pure space purely formal real space relations result Riemann right angles scope of motion sense-impressions space-conception sphere straight line straightest line surface theorem of parallel thinking subject three co-ordinates three planes tion tive trace law transcendental triangle tridimensional truth two-dimensional
Popular passages
Page 4 - If a straight line meets two straight lines, so as to make "the two interior angles on the same side of it taken " together less than two right angles...
Page 4 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.