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
mathematician is guided by a feeling for the probabilities, and can estimate the likelihood that a strategy will work.”
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 Spring '10
 CHRISLEE
 Conditional Probability, Probability, Bayes Law

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