Im increasingly of the opinion that people only use the word axiom/axiomatic as a cop-out to excuse lazy thinking. Sort of: 'This is difficult to understand, so I wont try. Ill just decide what I want to be the case and say that the baseless assumptions I make to ge there are "axioms". Thatll make my stupid position seem well thought out.'
People reify things all the time, it allows them to make sense of complex, inexplicable events and concepts. Its always been the case, and I doubt it will change anytime soon.
Yeah, but there's the baseless assumption that the world is run by evil sorcerers, and the baseless assumption that 1 + 1 = 2. The former is in principle provable, or at least could be argued for, whereas the latter is merely stateable. Hence the latter is an axiom whereas the former shouldn't be.
This is what axioms are [i]for[/i]. Suppose there is some untapped area of mathematics which depends crucially on some key conjecture which you hope will turn out to be a theorem. You could spend years trying to prove it, but that would be a bit wasteful. Instead, go ahead and investigate what happens if your conjecture is true by making it an axiom. If the resulting area of mathematics turns out to be fruitful then you know it's worth going back and proving your result.
Why this turns out to be important is because the axiom may actually not be provable at all. The most famous example of this is in Euclidean geometry. For many, many years nobody could properly prove that Euclid's Fifth Postulate was true. But they got on with doing geometry (or, as we now call it, Euclidean Geometry) anyway. It was some use for stuff like, say, architecture and navigation.
Later it turned out that the Fifth Postulate wasn't actually true !
Disaster ? No. Because it's not actually false eitherOf course, stuff like this doesn't really arise
( ... )
there is nothing specifically evil about evil acts except that the force of evil is involved
That neatly sums up the circularity of the argument... its conclusions are essentially the same as its axioms, thereby proving absolutely nothing other than its adherents' wish to adhere. Depressing!
Oh well, reading this and posting about it has allowed you to benefit from bateleur's wise instruction, so it wasn't time completely wasted ;-)
This is a paper that has a "body and soul" section that is all about different sorts of makeup/shampoo... Not that there are any better alternatives out there...
(late entry but can't resist an opportunity to bash the Grauniad)
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People reify things all the time, it allows them to make sense of complex, inexplicable events and concepts. Its always been the case, and I doubt it will change anytime soon.
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Umm... that's exactly what an axiom is ! If you can prove something to be true, it's a theorem, not an axiom.
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This is what axioms are [i]for[/i]. Suppose there is some untapped area of mathematics which depends crucially on some key conjecture which you hope will turn out to be a theorem. You could spend years trying to prove it, but that would be a bit wasteful. Instead, go ahead and investigate what happens if your conjecture is true by making it an axiom. If the resulting area of mathematics turns out to be fruitful then you know it's worth going back and proving your result.
Why this turns out to be important is because the axiom may actually not be provable at all. The most famous example of this is in Euclidean geometry. For many, many years nobody could properly prove that Euclid's Fifth Postulate was true. But they got on with doing geometry (or, as we now call it, Euclidean Geometry) anyway. It was some use for stuff like, say, architecture and navigation.
Later it turned out that the Fifth Postulate wasn't actually true !
Disaster ? No. Because it's not actually false eitherOf course, stuff like this doesn't really arise ( ... )
Reply
That neatly sums up the circularity of the argument... its conclusions are essentially the same as its axioms, thereby proving absolutely nothing other than its adherents' wish to adhere. Depressing!
Oh well, reading this and posting about it has allowed you to benefit from bateleur's wise instruction, so it wasn't time completely wasted ;-)
Reply
(late entry but can't resist an opportunity to bash the Grauniad)
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