Turing develops the idea of machine intelligence in a series of lectures and papers between 1947 and 1952. In some of them he addresses the mathematical objection (his term) whose gist is the claim that humans can assert some mathematical truths that exceed the abilities of computing machines. We first ask why Turing took so seriously the mathematical objection. After all, even if some humans surpass machines in their mathematical abilities, this by itself does not undermine the project of machine intelligence. Our answer is that the mathematical objection raises a dilemma with respect to Turing’s core claims about machine intelligence and forces him to relinquish at least one of them. We then clarify Turing’s reply to the mathematical objection. Based on the textual evidence, we argue that, according to Turing, the machine that plays against the human in the Turing test is not a static machine but an enhanced machine.