Computers are getting faster and faster… and the software is getting more and more complex. When will computers be smarter than humans? Couldn’t the brain just be a giant computer which slowly evolved though millions of years of evolution? It makes perfect sense doesn’t it? If you are anything like me, you would think this is absurd. There is no way that the concepts used in Windows XP are the same concepts that my brain uses. Computers are not self-aware…. They can’t be creative! But why can’t a computer ever obtain these things? To find this answer, we will have to jump into the world of mathematics.
In 1936, Alan Turing submitted a paper called “On Computable Numbers, with an Application to the Entscheidungsproblem”. In this paper he defined what are now called “Turing Machines”. A Turing Machine is basically a mathematical model of a computer. However, unlike modern computers, these “Turing Machines” had infinite memory and processing speed. This means that a computer, no matter how fast it was, would always be inferior to a Turing Machine. After a computer was rigorously defined, mathematicians and computer scientists could now figure the limitations of computers. One of the more notable limitations was the “Turing Halting Problem”. The problem is that a Turing machine will never know if a program will finish or run forever when given a finite input.
Next, I will jump to another mathematician name Kurt Gödel. In 1931, Kurt Gödel came up with Gödel’s incompleteness theorem which states:
“For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.”
Basically, what the theory is saying, no matter how many axioms (or mathematical rules) you have, there will be another mathematical statement that can’t be derived from the original axioms. This means that it would take an infinite amount of axioms to define all of mathematics.
We can now use Gödel’s Incompleteness Theorem and Turing Halting Problem to show that the human mind can’t be reduced to a Turing Machine. First we will create a program which determines if a mathematical statement is true or false (in general). In order to show that a mathematical statement is false, the Turing machine would have to compare the statement with every possible mathematical axiom (in general). Since there is an infinite amount of them, it would take an infinite amount of time to figure it out. Because of the “Turing Halting Problem”, the Turing Machine has no way of knowing if the program will stop or run forever. This means there is no way of knowing if the mathematical statement is true or false. If we assume that, Human beings operate according to the laws of physics and that the behavior of all physical systems can be predicted using algorithmic calculations. From the behavior of a human mathematician we can then extract reliably correct theorems about non-terminating programs .
Although my argument isn’t mathematically rigorous and contains some gaps in logic, it contains the essence of the proof. For a more detailed proof, please read “Shadows of the Mind” by Roger Penrose. It is an excellent book, but would be a challenge to understand without a mathematical background.
There has been a lot of opposition to Penrose’s assertions by the scientific community. The biggest challenge is whether human mathematician can extract reliably correct theorems about non-terminating programs. So the argument shouldn’t be interpreted as completely mathematically sound. However, it is the strongest argument I have read regarding the nature of human conscience.