Think of it, any concept that can be logically described--any algorithm--can be carried out by a machine, a computer. So, some philosophers argue that what goes on in your brain has some kind of internal logic (i.e., cause and effect relationships) and thus can be exactly mimicked by a computer.

Put it another way, how can you tell the difference between what a brain does and what a computer does? And if you can't, what's the difference between a brain and a computer anyway?

Thus is born the field of machine-based intelligence, commonly called Artificial Intelligence usually referred to as AI. (See here and/or here and/or here for initial definitions of this term.)

Turing himself addressed the relationship of computing machine and brain in a paper called "Computing Machinery and intelligence," (Mind, vol. 59, no. 236.1950). What he said is well described in Rheingold's Tools for Thought chapter, so I'll just quote that part here (with apologies for a bit too much text on one page--but it's worth it):

"The 1950 article is worth reading by anyone interested in the issue of artificial intelligence. The very first sentence still sounds as direct and provocative as Turing undoubtedly intended it to be: "I propose to consider the question 'Can machines think?' "

"Turing . . . began . . by describing a game! He called this one "The Imitation Game," but history knows it as the "Turing Test." Let us begin, he wrote, by putting aside the question of machine intelligence and consider a game played by three people--a man, a woman, and an interrogator of either gender, who is located in a room apart from the other two. The object of the game is to ask questions of the people in the other room, and to eventually identify which one is the man and which is the woman--on the basis of the answers alone. In order to disguise the appearance, voice, and other sensory clues from the players, the interrogation takes place over a teletype.

"Turing then asks us to substitute a machine for one of the unknown players and make a new object for the game: This time, the interrogator is to guess, on the basis of the teletyped conversation, which inhabitant of the other room is a human being and which one is a machine. In describing how such a conversation might go, Turing quoted a brief "specimen" of such a dialog:

• Q: Please write me a sonnet on the subject of the Forth Bridge.
• A: Count me out on this one. I could never write poetry.
• Q: Add 44957 to 70764.
• A: (pause about 30 seconds and then give as answer)105621.
• Q: Do you play chess?
• A: Yes.
• Q: I have K at my K1, and no other pieces. You only have K at K6 and R at R1. It is your move. What do you play?
• A: (After a pause of 15 seconds) R-R8 mate.

"Note that if this dialog is with a machine, it is able to do faulty arithmetic (39457 + 7064 does not equal 105621) and play decent chess at the same time.

"Having established his imitation game as the criterion [of] determining whether or not a machine is intelligent . . . Turing explained his own beliefs in the matter:

". . . I believe that in about fifty years' time it will be possible to program computers, with a storage capacity of about [ten billion bits] to make them play the imitation game so well that an average interrogator will not have more than 70 percent chance of making the right identification after five minutes of questioning. The original question, "Can machines think?" I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and educated opinion will have altered so much that one will be able to speak of machines thinking without expecting it to be contradicted."