Physical Meaning Of Geometrical Propositions

by Einstein (simplified)

I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann.

We were taught that Euclid’s geometry is true. But are we really sure?

Geometry has concepts like “plane,” “point,” and “straight line.” These lead to axioms which we are inclined to accept as “true.”

From this, all the remaining propositions are shown to follow from those axioms, i.e. they are proven.

A proposition is then correct(“true”) when it has been derived in the recognised manner from the axioms.

The question of the “truth” of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. But his is actually unanswerable by the methods of geometry. It is also in itself entirely without meaning.

We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called “straight lines.” These lines have the property of being uniquely determined by two points situated on it.

The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object.

Our Perceptions are Relative and Dynamic and Not Absolute and Static

Geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.

This is why we feel constrained to call the propositions of geometry “true.”

Geometrical ideas correspond to objects in nature. This correspondence is the exclusive cause of the genesis of those ideas.

Geometry should refrain from such a course, in order to give to its structure the largest possible logical unity.

For example:

• we have the deep habit of seeing two marked positions on a rigid body
• we are used to regard three points on a straight line when seen with one eye.

The propositions of Euclidean geometry become relative to the position of rigid bodies if we decree that the two points on that rigid body always correspond to the same line-interval-distance, independently of any changes in the position of that body,

For example, three points A, B and C on a rigid body lie in a straight line when:

• A and C are given
• B is chosen such that the sum of the distances AB and BC is as short as possible

This incomplete suggestion will suffice for our present purpose.

This kind of geometry is then to be treated as a branch of physics. The geometry is true if it matches those real things that we associate with those geometries.