Scientific American is focusing on Einstein in September, it being 100 years since he came up with the general theory of relativity. It briefly describes how he came up with the theory, which makes it a lot easier to understand and is interesting to me because it contains a logical fallacy.

Einstein claimed that the happiest thought in his life was: “If a person falls freely, he will not feel his own weight.” To be honest, this sounds like his personal troubles were causing depressive episodes. However, he noted that if a man was in an enclosed chamber in free fall, he would feel weightless. He would not be able to tell if he was in free fall or if he was floating in zero gravity. Likewise, if the chamber was in zero gravity and a constant force was pulling the chamber up at an accelerated rate he would feel his feet pressed to the floor, and would have no way of telling if he were in a stationary chamber under gravity or was being accelerated in zero gravity.

Einstein dubbed this “the equivalence principle.” The local effects of gravity and of acceleration are equivalent. Therefore, they must be manifestations of the same phenomenon, some cosmic field that accounts for both acceleration and gravity.

This seems to be an invalid argument. It takes the form:

If A, then C.

If B, then C.

Therefore, A = B.

Just because two things have the same effect doesn’t mean they are the same thing. For example, saying “if an elephant sat on me I would die, and if a rhinoceros sat on me I would die, therefore an elephant is a rhinoceros” is clearly incorrect.

Of course, this insight of Einstein’s allowed a theory that is very good at explaining the universe that we observe. That could be because it got translated into mathematics, and in mathematical logic it is actually true. If A=C and B=C, then A=B must be true.

This might indicate something very profound about mathematics and its relation to the universe, but I’m not sure what.