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

[Luke Palmer]

On Tue, May 6, 2008 at 4:53 AM, Achim Schneider <barsoap at web.de> wrote:
> 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

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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