An Essay on the Foundations of GeometryThe Foundations of Geometry was first published in 1897, and is based on Russell's Cambridge dissertation as well as lectures given during a journey through the USA. This is the first reprint, complete with a new introduction by John Slater. It provides both an insight into the foundations of Russell's philosophical thinking and an introduction to the philosophy of mathematics and logic. As such it will be an invaluable resource not only for students of philosophy, but also for those interested in Russell's philosophical development. Foundations of Geometry consists of four chapters which explore the various concepts of geometry and their philosophical implications, including a historical overview of the development of geometrical theory. |
Contents
PREFACE BY BERTRAND RUSSELL | 1 |
And necessarily leads to distance when quantity is applied | 2 |
This may be the essential postulate of our science or | 7 |
And Bolyai | 14 |
Hence the axiom of distance also is à priori in a double sense 170 | 18 |
The second the measure of curvature of a manifold grew | 20 |
The first period was inaugurated by Gauss | 25 |
But these three are necessary to the direct measurement | 29 |
He attacks nonEuclidean spaces on the mistaken ground that | 112 |
Section A THE AXIOMS OF PROJECTIVE GEOMETRY | 119 |
And are required for qualitative spatial comparison | 124 |
Lotzes objections fall under four heads | 125 |
Two pairs of points on one straight line or two pairs | 131 |
The conception of a form of externality | 135 |
CHAPTER IV | 138 |
And to the systematic unity of the world | 141 |
Beltrami gave Lobatchewskys planimetry a Euclidean | 34 |
Projective coordinates have been regarded as dependent | 35 |
Metrical Geometry has three indispensable axioms | 47 |
Kant contends that since Geometry is apodeictic space must | 53 |
Among the successors of Kant Herbart alone advanced | 59 |
Though mathematically invaluable his view of space as | 65 |
Both sets of axioms are necessitated not by facts but | 69 |
Erdmann accepted the conclusions of Riemann | 74 |
Is wholly false if it means that the axiom of Congruence | 80 |
And rejects it owing to a mathematical misunderstanding | 87 |
Two philosophical questions remain for a final chapter | 96 |
All homogeneous spaces are à priori possible and | 101 |
Section B THE AXIOMS OF METRICAL GEOMETRY | 147 |
Some objections remain to be answered concerning | 150 |
What is the relation to experience of a form of externality | 151 |
Which however is logically and philosophically untenable | 153 |
Free Mobility includes Helmholtzs Monodromy | 158 |
This form is the classconception containing every possible | 162 |
Since two points must have some relation and the passivity | 164 |
What relation does this view bear to Kants? | 173 |
And that knowledge requires the This to be neither simple | 181 |
What are we to do with the contradictions in space? | 185 |
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Common terms and phrases
à priori absolute analytical anharmonic ratio apodeictic certainty apriority argument axiom of Free axiom of parallels Bolyai Chap Chapter circle comparison Congruence construction contradictions coordinates criticism deduced defined definition determine dimensions discussion distance distinguish element elliptic elliptic Geometry empirical empty space endeavour Erdmann Euclid Euclidean Geometry Euclidean space experience finite follows form of externality Free Mobility géométrie given Hence homogeneity of space imaginary important impossible independent infinitely divisible infinitesimal intuition involves judgment Kant Kant's Kantian Klein knowledge logical Lotze Lotze's manifold mathematical matter means measure of curvature Metageometry method metrical Geometry Monodromy motion non-Euclidean Geometry non-Euclidean spaces object philosophical plane possible presupposes principle priori projective Geometry projective transformation properties proposition prove purely quantity question regarded relativity of position Riemann and Helmholtz rigid bodies sense space-constant spatial figures spatial magnitudes spatial relations Sphereland spherical spherical Geometry straight line surface theory things unique wholly