This article follows on from an earlier article. In that article, I explained how evolution taken to its limit implies the existence of God. This article explains how this is not so, as the end product of evolution will be in some definite way something that is strictly less than God. First I will examine the arguments of Lucas and Penrose and show why they are not valid. I will use the insight generated from studying their arguments to shore up the end result.
This section is a paraphrase of an article written by myself under the supervision of Jack Copeland for the course PHIL607 Formal Logic at the University of Canterbury. In the 1960s the philosopher John Lucas wrote an article "proving" that our conscious minds are more than just finite machines. More recently the philosopher/mathematician Roger Penrose in his book The Emperor's New Mind used a very similar argument to show a very similar result. I will now attempt to explain why the arguments of Lucas and Penrose are invalid.
The arguments of Lucas and Penrose use the standard logical technique of reductio ad absurdum to arrive at their result. The following is a summary of their arguments: Assume that we are (in principle simulatible by) finite machines. Let M be one such finite machine. Then M is consistent (we assume!) and contains basic arithmetic. Then Kurt Gödel's Incompleteness Theorem shows that there is a Gödel Statement G(M) that is true but unprovable by M. However by reasoning about its own algorithm, M can prove G(M), which leaves us at a contradiction.
To see why this contradiction occurs, suppose we have an artificial intelligence M_{2} that is consistent and contains basic arithmetic. Then by Gödel's theorem there exists a statement G(M_{2}) that is true by unprovable by M. If M_{2} can prove G(M_{2}) by reasoning about itself we arrive at a contradiction M_{2} knows G(M_{2}) and M_{2} doesn't know M_{2}. Therefore we can conclude that a machine reason about itself to know its own algorithm.
The argument of Lucas and Penrose is of the form A and B ⇒ C and not(C), where A is the statement: "We are finite machines", B is the statement: "We can know our own algorithm" and C is the statement: "M can prove G(M)". From the contradiction C and not(C) we can deduce by elementary logic: not(A) or not(B). The conclusion is not the negation of A as Lucas and Penrose would have us believe, but rather the negation of B: a machine cannot know its own algorithm. Giving a machine its own algorithm is like building a better machine, it gives the original machine in a definite way more power than the original machine had.
Consider the totality of life on earth as a single theoremproving machine M. Assuming that living beings are finite machines that could, in principle, be simulated on a computer then M is also. Assuming that the set of theorems generated by M is consistent and contains basic arithmetic then the Gödel statement G(M) of M is something that is true but unprovable by the system M. One of the properties of God is that he/she is allknowing (omniscient). Therefore we have a discrepancy between M and God and it is true that in this definite way, M is strictly less than God. In the previous section I explained how a machine cannot know its own algorithm, which shores us up against the possibility that M can "Gödelise" itself, that is to come to learn G(M) by reasoning about itself.
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