Sidelights on Relativity quotes

… yet this theory evoked a lively sense of discomfort among Newton’s contemporaries, because it seemed to be in conflict with the principle springing from the rest of experience, that there can be reciprocal action only through contact, and not through immediate action at a distance.

It is only with reluctance that man’s desire for knowledge endures a dualism of this kind.

How was unity to be preserved in his comprehension of the forces of nature?

But neither Maxwell nor his followers succeeded in elaborating a mechanical model for the ether which might furnish a satisfactory mechanical interpretation of Maxwell’s laws of the electro-magnetic field.

The laws were clear and simple, the mechanical interpretations clumsy and contradictory.

Almost imperceptibly the theoretical physicists adapted themselves to a situation which, from the standpoint of their mechanical programme, was very depressing. They were particularly influenced by the electro-dynamical investigations of Heinrich Hertz.

Now comes the anxious question:—Why must I in the theory distinguish the K system above all K′ systems, which are physically equivalent to it in all respects, by assuming that the ether is at rest relatively to the K system?

For the theoretician such an asymmetry in the theoretical structure, with no corresponding asymmetry in the system of experience, is intolerable.

If we assume the ether to be at rest relatively to K, but in motion relatively to K′, the physical equivalence of K and K′ seems to me from the logical standpoint, not indeed downright incorrect, but nevertheless inacceptable.

But on the other hand there is a weighty argument to be adduced in favour of the ether hypothesis.

To deny the ether is ultimately to assume that empty space has no physical qualities whatever.

The fundamental facts of mechanics do not harmonize with this view.

There can be no space nor any part of space without gravitational potentials; for these confer upon space its metrical qualities, without which it cannot be imagined at all.

The existence of the gravitational field is inseparably bound up with the existence of space.

On the other hand a part of space may very well be imagined without an electromagnetic field;

At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things.

In my opinion the answer to this question is, briefly, this:—As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

At this point an enigma presents itself which in all ages has agitated inquiring minds.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things.

Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things.

In my opinion the answer to this question is, briefly, this:—As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or “Axiomatics.”

as void of all content of intuition or experience.

These axioms are free creations of the human mind.

All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only). The matter of which geometry treats is first defined by the axioms.

as void of all content of intuition or experience. These axioms are free creations of the human mind. All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only).

The matter of which geometry treats is first defined by the axioms.

Schlick in his book on epistemology has therefore characterised axioms very aptly as “implicit definitions.”

But a presentation of its principles thus clarified makes it also evident that mathematics as such cannot predicate anything about perceptual objects or real objects.

In axiomatic geometry the words “point,” “straight line,” etc., stand only for empty conceptual schemata.

That which gives them substance is not relevant to mathematics.

Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the relations of real things to one another.

The very word geometry, which, of course, means earth-measuring, proves this.

For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc.

To accomplish this, we need only add the proposition:—Solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations of practically-rigid bodies.

Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics.

Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry “practical geometry,” and shall distinguish it in what follows from “purely axiomatic geometry.”

From such a deviation it could be proved indirectly that the universe is finite. It would even be possible to estimate its spatial magnitude.

Can we picture to ourselves a three-dimensional universe which is finite, yet unbounded?

The usual answer to this question is “No,” but that is not the right answer. The purpose of the following remarks is to show that the answer should be “Yes.”

First of all, an observation of epistemological nature.

A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts.

But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind.

We may without difficulty impart more depth and vigour to these ideas by carrying out special imaginary constructions. Nor would it be difficult to represent the case of what is called elliptical geometry in an analogous manner.

My only aim to-day has been to show that the human faculty of visualisation is by no means bound to capitulate to non-Euclidean geometry.

• You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg works calculated using the method you already use to calculate your applicable taxes.