Many people seem to be misunderstanding the nature of mathematical proof in discussions of AI and software correctness, security, and safety. In this post, I’ll describe some of the background and context for this.
Almost every aspect of logic and mathematical proof has its origins in human language which emerged about 100,000 years ago. 2500 years ago, Aristotle and Euclid began the process of making the natural language rules precise. Modern logic began in 1677 when Leibniz tried to create a “calculus ratiocinator” to mechanically check precise arguments. The job was finished by Frege, Cantor, Zermelo, and Fraenkel in 1922 when they created a precise logical system capable of representing every mathematical argument and which stands as the foundation for mathematics today. Church and Turing extended this system to computation in 1936. Every precise argument in every computational, engineering, economic, scientific, and social discipline can be precisely represented in this formalism and efficiently checked on computer.
Some had hoped that beyond checking arguments, every statement might also be proven true or false in a mechanical way. These hopes were dashed by Goedel in 1931 when he published his incompleteness theorem showing that any logical system rich enough to represent the natural numbers must have statements which can neither be proven true nor false. In 1936 Turing found a simple computational variant now called “the halting problem” which showed there are some properties of some programs which cannot be proven or disproven.
But the Goedel statements and the uncomputable program properties are abstruse constructions that we never want to use in engineering! Engineering is about building devices we are confident will behave as we intend! Any decent programmer will have an argument as to why his program will work as intended. If he doesn’t have such an argument, he should be fired! If his argument is correct, it can be precisely represented in mathematical logic and checked by computer. The fact that this is not current standard practice is not due to limitations of logic or understanding but to sloppiness in the discipline and poor educational training. If engineers built bridges the way that we write programs, no one would dare drive over them. The abysmal state of today’s level of software correctness and security will likely be looked at with wonder and disgust by future generations.
Many countries are now building autonomous vehicles, autonomous drones, robot soldiers, autonomous trading systems, autonomous data gathering systems, etc. and the stakes are suddenly getting much bigger. If we want to be confident that these systems will not cause great harm, we need precise arguments to that effect. If you want to learn more about some of the extremely expensive and life harming consequences that have already happened due to ridiculous sloppiness in the design of our technology check out the beginning of a talk I gave at Stanford in 2007: https://www.youtube.com/watch?v=omsuTsOmvsc