I was reading Introduction to the Theory of Computation by Michael Sipser and I found the following paragraph quite interesting:
During the first half of the twentieth century, mathematicians such as Kurt Godel, Alan Turing, and Alonzo Church discovered that certain basic problems ¨ cannot be solved by computers. One example of this phenomenon is the problem of determining whether a mathematical statement is true or false. This task is the bread and butter of mathematicians. It seems like a natural for solution by computer because it lies strictly within the realm of mathematics. But no computer algorithm can perform this task. Among the consequences of this profound result was the development of ideas concerning theoretical models of computers that eventually would help lead to the construction of actual computers
As a CS student, completely new to the theory of computation, this is hard to believe for me. It said that a computer can't solve a basic task such as determining whether a mathematical statement is true or false. Can't it really!? I have programmed lots of codes that determine if a mathematical statement is true or not. For example, a simple code such as
return 6 == 2*3 will return true if this statement is true, so why does the text says that a computer can't perform this task?
I'm sure I'm missing something here. Perhaps I'm mistaken by the definition of "Mathematical statement". But I'm quite sure "Is 6 equal to 2 * 3?" is a mathematical statement and can be validated by a computer. So what did the text mean by that? I'm confused!
PS: Sorry if the Complexity theory tag is misplaced here. As I said I'm new to the field and on the same page the author of the book stated that the theories of computability and complexity are closely related.