[Haskell-cafe] Re: Induction (help!)
Achim Schneider
barsoap at web.de
Tue May 6 06:53:46 EDT 2008
PR Stanley <prstanley at ntlworld.com> 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
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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