The limits of rigour?

It is possible to show that a proof must be a least a certain number of steps in logical systems. This sparked a thought.

(Don’t worry – no mathematics is needed to understand this article)

Huh? Ethics ain’t math!

For one moment let’s put aside one crucial difficulty: it is very hard to define suitable notation and axioms for questions phrased in English. If I asked you the following (hard) question…

Let G = D6. What is n2, the number of Sylow 2-Subgroups?

You would look blankly at me. However, that is because I just threw at you a whole load of notation which you didn’t understand – sorry! [And if you were taught some relevant notation and did some practice, you wouldn’t find it so hard.] However, this question at the very least has a set of well defined definitions and notation. Further, the method of proof and axioms are well known and defined. Moreover, it is much clearer in Group Theory (don’t ask) what the burden of proof is (i.e. how much do you need to do to show something is true.

It is reasonable to think that, if these questions can be answered, there exists some axioms we can use and a method of proof to prove them! [Regardless is we know what they are yet]

Some questions are hard

Even if you found the previous question hard/incomprehensible, these questions have a relatively short nummber of steps to solve. I heard from a mathematician once that someone had spent years of their life trying to solve a hard problem. They were trying to create a set of constructions on how to build a very many sided polygon (maybe 600 or so?). After they completed their work, and had compiled their great tome, an Oxford maths professor did some calculations and found the minimum number of steps needed to construct the polygon was far higher than his book. Thus, he was wrong. The unfortunate man may even have been attempting a problem with a number of steps so long, solving it was impossible.

But others are impossible

Philosophers might disagree so much on key questions because their answers are simply too complicated – require too many steps – to be solved (at least by humans). However, an agreed set of axioms, method of proof and burden of proof would be helpful as well! Without these, it is no wonder they cannot agree on answers – they can’t even agree on the question.

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