# Einstein's Relativity book Chapter 1

In chapter 1 of Relativity by Einstein, while talking about Geometrical Propositions, he said that the concept of "true" doesn't tally with the assertion of pure geometry.

I didn't understand how the assertions of pure geometry don't tally with the "true" concept. Also in the follow-up lines, he said that we are in habit of designating the word "true" with "real" objects.

The following is the extract from that passage:

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," to each of which is ascribed 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; 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.

What does he actually mean by the words: "true", "real" objects, here? Please elaborate by giving an example of such a case from geometry.

Einstein is pointing out the difference between Mathematics and Physics. In Mathematics axioms are assumed and then theorems are derived from this axioms. For example Euclidean geometry is derived from Euclid's five axioms. Assuming different axioms results in different geometries like hyperbolic geometry and elliptic geometry. The important point here is that the axioms are not true or false - they are just assumptions. A mathematician is free to choose any axioms they want (as long as no contradiction arises).

By contrast Physics is concerned with describing the world around us. In physics something is true if it matches experimental observations and false if it does not. Unlike a mathematician we are not free to make any assumptions we want.

The point Einstein is making is that Euclidean geometry is neither true nor false because it is a mathematical construction. By contrast, if a physicist made the statement "the geometry of the universe is Euclidean" then this statement can be true or false because this is a statement about the universe that can be tested by experiment.

• A mathematician is free to choose any axioms they want as long as no contradiction arises. One could argue this condition makes mathematics search for all possible consistent and inequivalent theories, thus mathematician is not free to choose any axioms he wants, but his job is to search for axioms that make sense, which is independent of the mathematician. Jul 9 at 6:18
• @Umaxo a fair point. Jul 9 at 6:20
• @John Rennie Thanks a lot! Your answer also clears a lot of things.I was struggling at that passage since a week. Jul 9 at 7:53

He must be talking about how the ideas of geometry need not correspond to real world objects. Geometry was initially conceived as the study of real world shapes that we see around us. Stuff like "The sum of angles of a triangle is 180 degrees", or "parallel lines never meet each other" were seen as self-evident truths. Euclid's axioms were inspired by real world objects.

This was the case with most of early mathematics. Real world inspired us to create numbers, calculus, geometry, etc. But later, we realised that math is fundamentally about the inter-relationships between abstract ideas. The relationship of those ideas with the real world comes second. One can define numbers without giving any real world meaning to the abstract symbols 1,2,3,4, etc. One can define abstract relationships between these symbols like 1+1=2. This abstract world of numbers happens to have a correspondence with real world objects. But all mathematical systems need not correspond to real world entities (One could say that these abstract ideas still reside in the human brain, which is a real-world entity. Some people called Platonists say that mathematical ideas exist independently of the human mind, in a "universe" of their own)

Geometry, too, need not be about the real world shapes that we see around us. One can just postulate some meaningless words like: 'Point', 'Line', 'Curvature', etc. One can then define the inter-relationships between these words like "What does a point have to do with lines?". We're giving these words meanings only with respect to each other, not with respect to the real world.

Long before Einstein discovered General Relativity, mathematicians had developed abstract geometries like Hyperbolic and Spherical geometries. At that time, it was thought that Euclidean geometry was the one that corresponded to the real world. The "curved" geometries were thought of as purely abstract mathematics. General relativity showed us otherwise.