The argument runs like this: suppose we say that a variable is ‘close to the edge’ when it is within 10% of its possible extreme values on either side. If there were only one variable that determined our universe, 20% of its possible values would be close to the edge. If we observed such a variable as being very close to the edge, we might suspect that something fishy was going on or that our universe was designed. But for two variables, 1 − 0.8^2 = 32% of values will be close to the edge. And in general, for n variables, 1 − 0.8^n of the values will be close to the edge.
This is simply mathematically incorrect. What would be correct is to say that there is a probability of 1 − 0.8^n that at least one of the variables is “close to the edge” in the sense described. This is not the same thing as saying the expected proportion of the variables is 1 − 0.8^n; that quantity remains constant no matter how many variables you have. (And it is this latter quantity that is relevant for the fine-tuning argument: it is not surprising that the universe contains a single variable that looks to be optimized for our existence, but that it contains a whole host of variables so optimized.)
I don’t know whether this is a mistake Deutsch himself made, or whether he had a better argument that you (the reviewer) simply summarized sloppily. Either way, however, it doesn’t speak well to the quality of the book’s content. (And, moreover, I found the summaries of most of the book’s other theses rather sloppy and unconvincing as well, though at least those contained no basic mathematical errors.)
This is simply mathematically incorrect. What would be correct is to say that there is a probability of 1 − 0.8^n that at least one of the variables is “close to the edge” in the sense described. This is not the same thing as saying the expected proportion of the variables is 1 − 0.8^n; that quantity remains constant no matter how many variables you have. (And it is this latter quantity that is relevant for the fine-tuning argument: it is not surprising that the universe contains a single variable that looks to be optimized for our existence, but that it contains a whole host of variables so optimized.)
I don’t know whether this is a mistake Deutsch himself made, or whether he had a better argument that you (the reviewer) simply summarized sloppily. Either way, however, it doesn’t speak well to the quality of the book’s content. (And, moreover, I found the summaries of most of the book’s other theses rather sloppy and unconvincing as well, though at least those contained no basic mathematical errors.)