PR Stanley
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.
Paul: surely, you wouldn't come up with an incorrect premise like "all apples are red" in the first place. Sorry, still none the wiser Cheers, Paul
well, "all 'apples' are 'red'" is the proof you want to make. And it can be correctly induced, if and only if you can prove that a) all 'apples' share the same behaviour regarding the property 'colour' b) one 'apple''s colour property is 'red' which leads you directly to c) every 'apple' (that is, potentially infinite many) 'apples' are 'red'. if b) concludes that one 'apples''s colour property is not 'red', of course, you can't say "all 'apples' are 'red'" and still write QED beneath your proof without having to admit that that wasn't what you wanted to demonstrate. As an example, you might construct a proof that every faculty of any number is positive: fac 0 = 1 fac n | n > 0 = n * (fac (n-1)) fac _ = undefined You don't have to go on to proving an infinite number of possible inputs, you only have to prove 1) that 1 is positive (by definition) 2) that any number greater than 0 is positive (by definition) 3) that multiplication of two positive numbers results in a positive number (by definition) 4) that only positive numbers get multiplicated (by analysing haskell's semantics, read: data flow). So, as soon as you have proved the properties of the fixed point, the whole, infinite set is proven. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited.