[Some philosophers] think that if something belongs to the realm of a priori knowledge, it couldn't possibly be known empirically. This is just a mistake. ... To give a really common example: anyone who has worked with a computing machine knows that the computing machine that the computing machine may give an answer as to whether such and such a number is prime. No one has calculated or proved that the number is prime. We, then, if we believe that the number is prime, believe it on the basis of our knowledge of the laws of physics, the construction of the machine, and so on.
If this already seems completely ridiculous to you, you already have the basic idea of my argument. Computers do not do anything unless they are instructed to by a programmer. Computers are an extension of our own computational power. How do we know the processes on the computer tell us whether a number is prime or not? The knowledge that the algorithm on the computer is correct is a priori. We have merely taken an a priori knowledge and loaded it onto another computational machine than our own brains. It is the same process our brains would use to determine whether a number is prime or not. At the very least, this supposedly simple example is anything but that.
Why is Kripke so intent on treating the computer differently from our brain? If I taught someone how to do the prime detection algorithm in their head and asked them instead of the computer, would this not be the exact same situation? The human brain is not one piece of computational machinery, but a massively parallel device. The process of outsourcing computation from one part of the brain to another is normal and essential to us. There's no reason why the outsourcing of computation to my laptop should be treated differently.
This misunderstanding of what computation is deep, and probably the source of my strong and general disagreement with his arguments.