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I am reading Big Bang by Simon Singh. Here is the short story (as I understood it):

Einstein gets a contradiction by assuming that

1- The speed of light $c$ is constant with respect the ether.

2- Galilean relativity is true.

because if you were traveling at the speed of light, you wouldn't be able to see yourself in a mirror, since you would be moving with respect the ether at the same speed at the light leaving your face. So you would be able to tell that you are moving, contradicting galilean relativity.

From here seems logical to deduce (assuming galilean relativity) that you will see your face on the mirror, therefore the speed of light can't be constant with respect the ether. And here comes where I am stuck, to me the most obvious way of solving this problem is thinking that light will act as any other thing we know, if you are moving at the speed of light and you throw a ball in the same direction of the movement at a speed $v$, you will see the ball leaving you at $v$ and someone that doesn't move with respect to you will see it at $c+v$. But according to the book, Einstein's first thought was to say that light has a constant speed for every observer.

So finally here comes my question: Did Einstein (or anyone before him) consider the first argument I gave here and, if so, why did he reject it?

P.D. In case it is useful to know something about my background, I am a mathematician with very little knowledge of physics, just the things I read in these kind of divulgative books. That said, sorry about incongruences or inaccuracys, I would very much apreciate corrections.

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    $\begingroup$ I suggest you read Subtle is the Lord by Abraham Pais, who knew Einstein personally. $\endgroup$ – Jan Peter Schäfermeyer Mar 25 '17 at 22:40
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    $\begingroup$ It was "Einstein's first thought" only after Fitzgerald, Lorentz and Larmor formulated a theory of ether, where observer's speed with respect to it was absolutely undetectable because it is invariant under Lorentz transformations, see Einstein and time dilation. To Einstein this made it, and ether itself, unphysical. Since Lorentz transformations transform frames into each other and $c$ is unchanged by them it should be the same in every frame. $\endgroup$ – Conifold Mar 28 '17 at 1:04
  • $\begingroup$ Ob-xkcd: xkcd.com/1584 $\endgroup$ – IanF1 Apr 1 '17 at 15:01
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Einstein was reading Maxwell.

Axiom a) follows from Maxwell's electromagnetic theory of light. Axiom b) is stated imprecisely in your message. The actual axiom is that you cannot detect the absolute motion of the observer (or that the laws of nature must be independent of the motion of the observer (rectilinear and with constant speed)). This is the generalization of Galileo's relativity principle which Einstein stated.

In more precise mathematical language, Maxwell's equations which describe light are not invariant under Galilean transformation. This is why "Galilean relativity" has to be replaced by "special relativity". This was understood by several people at approximately the same time (Einstein, Lorentz and Poincaré). They found the correct transformations which preserve Maxwell's equations. The group of these transformations is called the Poincaré group and the subgroup consisting of homogeneous transformations is called the Lorentz group.

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Einstein himself wrote, in a letter of 14 December 1915 to Moritz Schlick, that he was primarily inspired by David Hume and Ernst Mach. According to the paper How Hume and Mach Helped Einstein Find Special Relativity, "It was more Hume than Mach."

Quoting from Einstein's letter to Schlick:

Your exposition is also quite right that positivism suggested rel. theory, without requiring it. Also you have correctly seen that this line of thought was of great influence on my efforts and indeed E. Mach and still much more Hume, whose treatise on understanding I studied with eagerness and admiration shortly before finding relativity theory.

It is not clear if Einstein is referring to Hume's A Treatise of Human Nature or his An Enquiry Concerning Human Understanding.

Einstein goes on to say:

It is very possible that without these philosophical studies I can not say that the solution would have come.

In Einstein's Autobiographical Notes of 1949 he writes:

Today everyone knows, of course, that all attempts to clarify this paradox [of light that leads to special relativity] satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution of the problem. The type of critical reasoning required for the discovery of this central point was decisively furthered, in my case, especially by the reading of David Hume’s and Ernst Mach’s philosophical writings.

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