Roger Penrose, who I consider to be a very important mathematical physicist and great expositor, thinks that quantum mechanics is necessary to explain consciousness. I disagree. Very roughly Penrose argues that human consciousness can comprehend Gödel’s proof and no computer can. He advances this argument in “The Emperor’s New Mind”. I maintain that humans merely jump to conclusions as a practical means of surviving and have thus stumbled upon a real pattern that we call logic. These conclusions are encoded in our heads isomorphically with logical propositions. I claim that we came upon logic because logic, as expressed in the first order predicate calculus is good at expressing useful patterns such as “if the sun is setting then the lions are not at the watering hole.”. That sentence is useful even though it is sometimes false. Mathematics is now often seen as a huge elaboration of logic and why such logic should be so vital to our modern society is indeed a mystery which need not be explained for my argument. For this is an explanation of something that has worried me since I was 20: “How did we evolve on the savanna to be smart enough to do Math?”. It reduces, however, to a non-biological mystery of the unreasonable usefulness of mathematics. Perhaps that mystery has to do with a universe complex enough to support life.
We jump to conclusions and sometimes encounter contradictions such as Russell’s paradox and abandon some of our guesses. Indeed one of Gödel’s results is that we can do no better than this. When computers compete with us doing mathematics they will do no better; they too will jump to conclusions and try to find arguments to convince other mathematicians (computers and people). We aspire to rigor and that aspiration improves our math, but still we must jump and back off sometimes.
Here I make note of a sloppiness in the language that I and others use to describe software constructs. That we are not confused seems remarkable to me but also suggests that we use inaccurate language successfully (usually) and are hardly aware of it. When we program computers at least up to the last decade or so, we teach them to be precise in their reasoning. I specialize in that sort of programming because I enjoy it. I am disconcerted sometimes when Google (or other search engine) leaps to a conclusion that I might be interested in something else that Google somehow associated with my query. I am disconcerted but sometime thankful. I do not suggest that Google is using non-deterministic computers, but that deterministic computers can jump to conclusions and be wrong. And it may not after all be a bug.