204 Are the Androids Dreaming Yet? Every correctly formed theorem number has another number, which demonstrates the proof of that number. If this is universally true there should be no contradiction. Unfortunately if you apply the theorem to itself you get something similar to the liar’s paradox. “This proof number is not a proof of the truth of this theorem number.” The proof number proves the theorem number is true, but the truth of the statement is that it can’t be a proof of the statement... Paradox. The only way to resolve the paradox is to go back one step and realize that not every correctly formed theorem number has a proof number using only the rules of that system. Concisely, Goédel’s theorem says, “Within any formal system of mathematics there can be statements that are true but are not provable using only the rules of that system” When Hilbert heard of Gédel’s proof, his first reaction was anger. After all, he had spent 30 years of his life trying to prove mathematics was tidy and complete. Gédel had just shown it was not. Hilbert never worked on formalism again, but the rest of the mathematical establishment largely ignored the result. Godel’s proof did not stop mathematicians proving new theorems nor doing useful mathematics. They went on much as before, using a mixture of intuition and analysis. The only difference was someone had told them analysis alone would not succeed. The repercussions of Gédel’s theory have more to do with understanding our place in the Universe and the nature of knowledge discovery. These are ‘big’ philosophical questions, which don’t greatly affect the day-to- day ability of a mathematician to do their job. However, it is important to understand that knowledge discovery is not simply analysis. Knowing this helps us understand human creativity. Inconsistency In the proof above, I said the only way to resolve the paradox is by saying there cannot be a proof number for every mathematical statement and therefore mathemat