When we talk about mathematics, we talk about logic, conciseness, completeness and abstraction. So that means, everything in this world, can be described in terms of mathematics? Even religion? Atleast thats a common belief of a lot of atheists that if something cant be proven through logic or lacks empirical evidence, it shouldnt be believed. Have they tried proving mathematics to be correct? Hmmm…so thats where the problem starts because of Gödel’s incompleteness theorems.
In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. He proved it impossible to establish the internal logical consistency of a very large class of deductive systems – elementary arithmetic, for example – unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves.
The proof of Gödel’s Incompleteness Theorem is very simple, a lil confusing but surely makes it clear that even logic can be illogical. His basic procedure is as follows:
- For instance there is a machine, UTM, that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
- Lets call the program P(UTM) for Program of the Universal Truth Machine.
- Gödel writes out the following sentence: “The machine constructed on the basis of the program P(UTM) will never say that this sentence is true.” Call this sentence G. Note that G is equivalent to: “UTM will never say G is true.”
- Now the UTM is asked whether G is true or not.
- If UTM says G is true, then “UTM will never say G is true” is false. If “UTM will never say G is true” is false, then G is false (remember G = “UTM will never say G is true”, point 3). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
- We have established that UTM will never say G is true. So “UTM will never say G is true” is in fact a true statement. So G is true (since G = “UTM will never say G is true”).
- So we finally we know something that although G can be true, it cannot be universally true!
G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics!