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#1 2005-12-31 12:40:16
- mikau
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obsessed with proofs
I few weeks ago I learned about the critical number (closed interval) theorem. It stated that if f(x) is a function continious on an interval [a,b] the maximum and minimum values on the interval are either at f(a), f(x) or the critical numbers of the function f. I would not have realized this, but its quite obvious once its presented to you. But my mathbook said "the proof of this concept is beyond the scope of this book".
To me it seems obvious. Why would we have to prove it?
Last edited by mikau (2005-12-31 12:45:59)
A logarithm is just a misspelled algorithm.
#2 2005-12-31 15:37:03
- ryos
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Re: obsessed with proofs
I'm not a stickler for proofs, but I believe it's to avoid basing any work on unfounded assertions, obvious though they may seem.
Perhaps the proof is not difficult, but, because it's a concept that's simply and intuitively grasped, it is not necessary to the text, and therefore beyond its scope.
El que pega primero pega dos veces.
#3 2005-12-31 15:42:17
- mikau
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Re: obsessed with proofs
perhaps. But can you prove 2 + 2 will always equal 4? We obvserve it happens every time we try but technically can't prove it. Does that make it less true?
I agree proofs should be done most of the time, but sometimes proving an obvious concept feels redundant.
A logarithm is just a misspelled algorithm.
#4 2005-12-31 17:48:50
- ryos
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Re: obsessed with proofs
Proofs use the rules and definitions of math. I don't think you can prove the rules; you must define them before you can prove things (chicken and egg, sort of thing). Right?
2+2 = 4 is an identity. It is also a proof in and of itself, because it is evident from the rules of addition that 2+2 = 4; IOW, no further steps are required to show that it is true.
OK, Ok, ok, I'll do it:
Assert: 2+2 = 4.
By the rules of arithmetic, 2+2 = 4.
Substitute: 4 = 4
Therefore, 2+2 = 4.
Woohoo!
El que pega primero pega dos veces.
#5 2005-12-31 17:59:49
- mikau
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Re: obsessed with proofs
lol! Couldn't resist huh? Don't make me pull out that y = x formula to prove 1 = 2 :-P
But I think I've demonstraighted that some things cannot be proven technically, but are so obvious they don't need to be. In my oppinion, the critical number theorem is about as obvious as 2 + 2 = 4.
Ok not quite as obvious, but think about it, if f(a) and f(b) are not the maximums and/or minimums then there is to be a point somewhere between them thats higher or lower then they are. If its higher then it has to turn around to come back down, the turning points of a function are the critical numbers. Nothing to prove in my mind.
A logarithm is just a misspelled algorithm.
#6 2006-01-01 00:18:48
- krassi_holmz
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Re: obsessed with proofs
The things that cannot be proved are the axioms.
They are true without proving.
But something very very interesting:
Guedel proof that there doesn't exist a full set of axioms, e.a. there exist a statement that cannot be proved using finite number of axioms.
IPBLE: Increasing Performance By Lowering Expectations.
#7 2006-01-11 01:40:03
- darthradius
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Re: obsessed with proofs
Hey Mikau...if it's true, then we should be able to prove it, right? You sound surprised that we need to be that rigorous...There are many theorems that seems to be entirely common sense, but we still need to prove them...that is the essence of mathematics...observation, conjecture, and proof...That is the whole point of axiomatic proof...we can only build on what we know when we are certain that we know it...
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
-Bertrand Russell