This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 1 What is Inference? I think the most interesting question in the world is how we think . This is one of the basic questions of life – but how well do we understand it? Our difficulty is not a shortage of ideas, but rather that different fields give discordant answers. For example, mathematics and science provide two very different outlooks on how we think. On the one hand, mathematical logic is all about reasoning under absolute certainty – through mathematical proof. On the other hand, science always operates under uncertainty , both in its hypotheses and its empirical tests. To see how these differ, let’s imagine a scientist, Sonya, discussing this question with a mathematician, Matt. He says, “Mathematical logic is a rigorous theory of truth that is the definition of correct thinking. So the problem is already solved: thinking just means deriving a conclusion from some starting information according to the principles of mathematical proof.” Sonya says, “That’s an interesting theory. How could we test it? Does it make any predictions?” Matt chuckles. “Actually, my argument is that this is true by definition . After all, a thought is correct only if it can be proved mathematically. So correct thinking must follow the process of mathematical proof. Any departure from that process will be incorrect, and therefore lies outside our definition. Surely there is no need to dissect a hamster brain, or do some such experiment, to test a statement that is more a definition than a theorem.” “I see what you mean. Still, your idea seems to make specific predictions that are testable. For example, computers can perform billions of logical operations per second, far faster and more exactly than any human being. If thinking is just applying the rules of mathematical logic, we could program a computer to use mathematical logic to solve math problems, and it ought to be able to outthink you mathematicians.” “Oh no! People have tried that. You can’t just give a computer the basic rules of logic and some axioms, and ask it to prove a desired statement. To get a program to generate even a simple proof, like the irrationality of √ 2 , you have to input endless lists of hints and rules. I think the problem is that there’s a combinatorially infinite number of ways to try to prove a given statement, and the computer doesn’t know which one will work. So it has to try each possible path until it hits a deadend, and then try the next path...” “But wait. Doesn’t a human mathematician face the same problem?” “Sure. The only way to be logically certain that a given path will lead to a proof, would be if you already had the complete proof. That is an allornothing proposition. But a mathematician doesn’t just mindlessly try every possible path, like a computer. Instead he thinks about what strategy is likely to work, and uses his intuition.” “You seem to be saying that whereas the computer only has the allornothing method of logical proof, a good...
View
Full
Document
 Spring '10
 CHRISLEE
 Conditional Probability, Probability, Bayes Law

Click to edit the document details