"Mathematician's thinking" refers to the "decisive problem" put forward by Hilbert in 1928: to find a general method to judge whether any proposition in mathematical logic can be proved.

"Failure" refers to Turing's paper "On Computable Numbers and Their Applications in Deterministic Problems" published in 1936, which points out that "Deterministic problems are unsolvable".

The by-product of this paper is Turing's "Turing Machine" invented to prove that the deterministic problem is insoluble. Turing machine is a mathematical model, and later people invented its physical realization "computer" based on Turing machine.

This historical story is vividly summarized as "computer is the product of mathematicians' thinking failure."