Rolle Rant

One must accept certain criteria to be able to prove that 1+1=2; in a general mathematical sense, it does not - in fact, neither number (or digit) may even exist!

You forgot that it also brought us.......MicroSoft!

Um, isn't that the definition of '+'? ...or I suppose '2', if you want to go that way.

This depends on the axioms. But then you could prove 2+2=4, which needs

Well, no, the conditions ARE important. The function 1/x, between X=3D1 and X=3D-1 doesn't fit the condition (and the local slope doesn't ever match the 'mean' slope in that range). Between X=3D1 and X=3D2, though, the function IS defined at all points, smooth and continuous, and the theorem correctly predicts a behavior of the derivative.

But the theorem does not apply to 1/x *at all* - it only applies to functions which have the same value at two points, between those two points.

You mean that seeing something that is totally obvious is not a "proof" until written down?

That may well be true. After all, pencil and paper really are not essential to doing maths. Some people can do it in their heads.

And, of course, smooth and continuous.

Well, if they perfect VDMs and their functionality through real hardware hooks, who care?

Are you trying to read between the lines or write or append between the lines? :-)

I'm just surprised nobody mentioned the typo

You were just on such a great roll(e) with your rant that nobody wanted to interrupt it with "x marks the spot" or similar. :-)

Ed

But would the theorem still work if it was done in pictures instead of Roman/Greek letters? Answer that if you're so smart!

Well, no, the conditions ARE important. The function 1/x, between X=1 and X=-1 doesn't fit the condition (and the local slope doesn't ever match the 'mean' slope in that range). Between X=1 and X=2, though, the function IS defined at all points, smooth and continuous, and the theorem correctly predicts a behavior of the derivative.

-------Are you saying 1/x is defined at all points between -1 and 1?

Trolle's Theorem: people will rant about the most ridiculous things.

Yes, but this is more intellectual than most. It would be wasted in most NGs

I thought those were fly specks ;-) ...Jim Thompson

Russell & Whitehead's "Principia Mathematica" goes through more than 500 very dense pages before getting to the point of proving that 1+1=2.

So it's obviously very plausible, but isn't certain till it's proven. Rolle's theorem is part of the scaffolding for the theory of continuous functions (and therefore of calculus), so it's pretty important. There have been some pretty counterintuitive theorems proven in that field, e.g. Fourier's theorem!

Cheers

Phil Hobbs

The problem with intuition is that it doesn't get you very far once you move away from familiar territory (integers, reals) and, worse, can lead you astray. E.g. applying the rules of real arithmetic to complex numbers, assuming Euclidean geometry, etc.