Re: [Haskell-cafe] Re: Induction (help!)

On Tue, May 6, 2008 at 4:53 AM, Achim Schneider wrote:

PR Stanley wrote:

...

Hi I don't know what it is that I'm not getting where mathematical induction is concerned. This is relevant to Haskell so I wonder if any of you gents could explain in unambiguous terms the concept please. The wikipedia article offers perhaps the least obfuscated definition I've found so far but I would still like more clarity. The idea is to move onto inductive proof in Haskell. First, however, I need to understand the general mathematical concept.

Top marks for clarity and explanation of technical terms. Thanks Paul

Induction -> from the small picture, extrapolate the big Deduction -> from the big picture, extrapolate the small

Induction has two meanings in mathematics, and I don't believe this is the type of induction the OP was asking about. See Daniel Fischer's response for the type you are asking about, and try not to be confused by the irrelevant discussion about inductive logic. Luke

Thus, in traditional logic, if you induce "all apples are red", simple observation of a single non-red apple quickly reduces your result to "at least one apple is not red on one side, all others may be red", i.e, you can't deduce "all apples are red" with your samples anymore.

As used in mathematical induction, deductionaly sound:

1) Let "apple" be defined as being of continuous colour. 2) All "apples" are of the same colour 3) One observed "apple" is red Ergo: All "apples" are red

Q.E.D.

The question that is left is what the heck an "apple" is, and how it differs from these things you see at a supermarket. It could, from the above proof, be a symbol for "red rubberband". The samples are defined by the logic.

Proposition 2 should be of course inferable from looking at one single apple, or you're going to look quite silly. It only works in mathematics, where you can have exact, either complete or part-wise, "copies" of something. If you can think of a real-world example where this works, please speak up.

That's it. Aristotlean logic sucks.

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