This post follows on (loosely) from a previous discussion on maths and computing and asks what it really means to ‘prove’ something in each discipline.
An apocryphal story has an Oxbridge maths don lecturing to a group of undergraduates … After some time completely filling a huge blackboard with heavy calculus – with accompanying commentary, he turns to the class and casually notes, “So then, it’s clear that …” (the exact claim isn’t important). As he turns to resume his chalk-work, a particularly bold student enquires, “Excuse me, Professor; but is that really ‘clear’?” The don steps back and surveys his work; studying the entire board from top-left to bottom-right, with numerous head and eye movements to-and-fro – even some pointing – to cross-check various parts with each other. After a full five minutes of silent contemplation, he turns back to the students, smiles, announces, “Yes!”, and carries on as before.
So who’s defining ‘clear’ here?
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When the Four Colour Theorem (FCT) was finally ‘proved’ in 1976, it upset a lot of mathematicians. It was the first significant mathematical concept to be proved with a good deal of help from a computer and, for many, that didn’t make it a real proof. Although we’re largely (maybe not entirely) OK with it now, the objections at the time weren’t just theorists’ snobbery. At the heart of it all were some fundamental questions about the role a computer could or should play in formal logic.
Essentially, the FCT says that the maximum number of different colours needed to colour a map, so that no bordering countries are the same colour, is four. (Colours can touch at a point but not at an edge.) It’s easy to show that five will always do the trick and, in fact, most normal maps only need three. However, certain types of map certainly seemed to need four so was four always enough? Continue reading
1 Comment | tags: Computer proof, Formal methods, Formal proof, Four color theorem, Math, Mathematics, Maths, Problem solving, Program correctness, Program verification, Sunday Times | posted in Algorithms, Computer Science, Computing, Mathematics, Philosophy, Programming, Software