Space, Time, Matter Quotes

“By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry.”
― Hermann Weyl, Space, Time, Matter

“Or, we may use Cartesian co-ordinate systems from the outset:”
― Hermann Weyl, Space, Time, Matter

“Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.”
― Hermann Weyl, Space, Time, Matter

“when placed in two positions successively, realises this idea of the equality of two portions of space ; by a rigid body we mean one which, however it be moved or treated, can always be made to appear the same to us as before, if we take up the appropriate position with respect to it. I shall”
― Hermann Weyl, Space, Time, Matter

“We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry.”
― Hermann Weyl, Space, Time, Matter

“the problem is then to develop a theory of invariance with respect to arbitrary linear transformations, in which, however, in contra-distinction to the case of affine geometry, we have a definite invariant quadratic form, viz. the metrical groundform once and for all as an absolute datum.”
― Hermann Weyl, Space, Time, Matter

“Two possibilities present themselves for the analytical treatment of metrical geometry.”
― Hermann Weyl, Space, Time, Matter