Others say...

"Still The Best"
After reading books by Hawking, Kaku, Greene, and others on the topic, none are more lucid as Einstein himself in describing relativity to the non-physicist.

"Worthwhile Read"
This is a very short book and is quick reading. I have read other books with better explainations of relativity. I didn't like the writing style -- it was translated by an Englishman in the early 20th century so the sentence structure doesn't flow like normal reading.

"Readable, for the Science-Minded"
This is a good, concise exposition of general and special relativity from the man who wrote the book. However, it will be above the heads of some... If you're interested in reading this book, I'd suggest looking for it in the library to see if it's really the book for you. Or even browse the text in a bookstore before you buy. If you think you'll need a course in Calculus, find a review of the theories that you can follow.

"Not as good as the real thing"
Having owned a 1968 paperback reprint of the genuine, 1954, Methuen, London, 15th edition publication, for many years, I was extremely disappointed to receive this Penguin out-of-copyright-rip-off replacement for my original (now falling apart) paperback copy. The vitally important (1956) fifth appendix is missing, as is the earlier 4th appendix. The reason for this is obvious. Penguin have merely re-published that part of the real thing which is now already more than 70 years old, and, therefore, out of copyright.

Even the quality of the employed paper is at, or below, newspaper standard. I now know why the Amazon price seemed cheap. As Einstein's definitive final word on this subject, this particular excuse for Einstein's popular exposition is, IMO, not even worth the paper it is written on.

"The "
This objection comes, not from challenging any of Einstein's assumptions, but rather, from the history of mathematics, which has undergone a renaissance during the past twenty years.

Among the most interesting studies are those which have examined the history of Cantorian set theory and the response to it. The response is what is relevant here.

"Natural" mathematics developed in response to the supposed "paradoxes" of set theory. Regardless of whether those are paradoxes or not (see A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES'), "natural" mathematics claimed that, in response, a new doctrine had to be developed. The new doctrine (best explained in P. Maddy, NATURALISM IN MATHEMATICS) said that mathematics inherently generated paradoxes, which could only be avoided or solved by the insertion into arguments the idea that mathematics is inherent in human beings, even though the idea does not play any internally consistent role in the arguments into which it is inserted.

Poincare was an ardent advocate of "natural" mathematics (along with Hilbert and Russell) and he set it out in SCIENCE AND HYPOTHESIS. Einstein was an ardent advocate of this position, based on his reading of the Poincare book (see Stachel and Howard, EINSTEIN THE FORMATIVE YEARS).

Since it does not play an internally consistent role in any argument in which it is inserted, if "natural" mathematics is found in relativity, it will constitute a contradiction in the argument.

Here is the "train" experiment as given by Einstein in RELATIVITY:

Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the embankment. But the events A and B also correspond to positions A and B on the train. Let M1 be the mid-point of the distance AB on the traveling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves...with the velocity...of the train.

The objection is "naturally coincides" (fallt zwar...zusammen). This is a pure expression of "natural" mathematics. Poincare said that "the mind has a direct intuition of this power ["proof by recurrence" or "mathematical induction"], and experiment can only be for [the mind] an opportunity of using it, and thereby of becoming conscious of it." In geometry "we are brought to [the concept of space] solely by studying the laws by which...[muscular] sensations succeed one another." "Natural" coincidence "allows" one point to "succeed" another, in conformity with the demands of "natural" mathematics.

Einstein never justifies the term. Neither can it be dispensed with. If it is eliminated, the geometrical contradiction emerges automatically. If two Cartesian parallel coordinate systems coincide at one point, they coincide at all points and are one such system. This contradicts the assumption of two such systems.

This is the objection--from WITHIN the relativity of simultaneity and NOT by challenging or changing any of Einstein's assumptions--which brings relativity to an end. This is because we cannot get to special or general relativity or QED if there is a flaw in the relativity of simultaneity.

This is not the first occasion on which Einstein incorporated "natural" mathematics in his arguments, and in which "natural" mathematics caused problems for the argument. He used it in his 1905 paper on Brownian motion. As S. Sarkar of the University of Texas says in the Stachel and Howard book: "Einstein begins with an assumption whose status is still problematic and troubled his contemporaries: that there exists `a time interval ô, which shall be very small compared with observable time intervals but still so large that all motions performed by a particle during two consecutive time intervals ô may be considered as mutually independent events....This is essentially a very strong Markov postulate. Einstein makes no attempt to justify it....[W]here mathematics ends and physics begins is far from clear...."

It is, I think, sobering to realize that this objection existed for 100 years without ever being noticed--sobering because it shows how pervasive a prejudice can be. It is doubly interesting that even critics of relativity shared the prejudice--they too bought into "natural" mathematics. But I didn't.

That's the end of relativity. Where to next:

1. "Natural" mathematics provided very facile arguments. It was immediately taken up in all the disciplines. It was used by Godel, Sraffa, Kimura--everywhere. Where do the flaws appear in arguments in other disciplines?

2. It is likely that "natural" mathematics was only given its name in the twentieth century, when there was an intense interest in paradoxes. However, supposed "paradoxes," such as Zeno's, have been around for a long time. It is likely that "natural" mathematics is the immemorial effort to "avoid" or "solve" these paradoxes. For example, where does Darwin use "natural" mathematics?

3. The status of the Pythagorean theorem is in question again. Under general relativity it is invalid. However, "natural" coincidence does not permit us to reach general relativity. So the question arises once again: is the Pythagorean theorem internally consistent? If not, why not?

So research goes beyond and before relativity. But relativity itself is over.

The discussion in this post was first published as follows:

Ryskamp, John Henry, "Paradox, Natural Mathematics and Twentieth-Century Ideas" (April 14, 2006).[...]