necessary to make clear to ourselves the definition of geometry as it will appear if we accept the standpoint of the above investigations and the definition of space and time arising from them: geometry will no longer be a “science of the properties of space” as it is usually defined, but a “science of the properties of ordered aggregates, or continua”. These continua will possess different (metrical) properties depending upon the way they are ordered and upon the way we define “interval” in them—upon the postulates we lay down in the foundations of our geometry.
One of such continua is our perceptual Experience: it is the investigation of its properties which is the task of physics. Between physics and geometry there is only this difference: that in geometry we can give orders up to a certain limit, i.e., as far as we are not offending against internal consistency, whereas in physics we must obey orders.
We say that geometry is a priori, deductive, physics a posteriori, inductive, founded on experimental knowledge; the truth seems to be that both sciences are to a certain extent inductive, based on experience, from which they generalise, abstract, and to a certain extent deductive, building on concepts and postulates obtained by generalisation. The difference between geometry and physics (I speak of course only of theoretical physics) is therefore not, as generally assumed, so great as regards their material: both deal with concepts, i.e., abstractions, generalisations from experience:[1] the difference between them being, that geometry can do with its concepts whatever it pleases, as long as it remains consistent with itself, while physics must strive to arrive at conclusions which agree with physical experience, to make its concepts correspond to certain physical experience.
Finally, geometry once more approaches physics when, to facilitate its work, it borrows from physics its aids: the perceptual representation of its concepts; but here must be kept in mind that this representation is but an approximation to the ideal concepts of geometry, just as (if I may be allowed to present the fact of physical simplification of phenomena in this reverse order) actual empirical physical experience is but a rough approximation to the ideal Experience of physical theory.
- ↑ With due apologies to those who define geometry as a purely formal branch of logic, having nothing to do with the so-called geometrical “intuition” of lines, points, etc., but with pure hypothetical judgments: “If we assume A, then B follows”. If we admit this definition of geometry, why differentiate geometry from other branches of pure mathematics? For our present purpose we shall prefer to think of geometry as the science of points, lines, angles, etc.
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