Rolle Rant

[Dirk Bruere at NeoPax]

Rolle's Theorem.
http://en.wikipedia.org/wiki/Rolle's_theorem

Something I first came across at university 40 years ago.
It really pissed me off, and still does.
Isn't it so totally, utterly obvious that it does not need proving?
A bit like claiming that given two numbers, x and x where x<y any number
greater than x and less than y will be between x and y?

 

[Rich Grise]

Rolle's Theorem.
http://en.wikipedia.org/wiki/Rolle's_theorem

Something I first came across at university 40 years ago.
It really pissed me off, and still does.
Isn't it so totally, utterly obvious that it does not need proving?
No.

A bit like claiming that given two numbers, x and x where x<y any number
greater than x and less than y will be between x and y?

That doesn't prove that there will be a point with slope == 0.

Good Luck!
Rich

 

[Dirk Bruere at NeoPax]

That doesn't prove that there will be a point with slope == 0.

It still seems totally obvious.
And for Rolle's next trick - proving 1+1=2

 

[John Devereux]

It still seems totally obvious.

Agreed. It either starts off directly towards the other point - in which
case slope is 0. Or else it starts out at some angle from the straight
line between the two points. In which case it has to turn around at some
point, which means the slope must go through through 0 (if it is
differentiable, as stated).

 

[Nobody]

Rolle's Theorem.
http://en.wikipedia.org/wiki/Rolle's_theorem

Something I first came across at university 40 years ago.
It really pissed me off, and still does.
Isn't it so totally, utterly obvious that it does not need proving?

I take it you didn't study maths?

Nothing, and I mean *nothing*, is considered so obvious that it doesn't
need proving.

And for Rolle's next trick - proving 1+1=2

Principia Mathematica didn't get around to such advanced topics until
nearly 400 pages in; Wikipedia says:

Quotations:

"From this proposition it will follow, when arithmetical addition has
been defined, that 1+1=2." – Volume I, 1st edition, page 379 (page 362
in 2nd edition; page 360 in abridged version).

 

[Dirk Bruere at NeoPax]

Only pot smokers rant over such things

It was a big impediment to my starting maths at uni.
I could not understand why such a thing needed proving.

 

[Dirk Bruere at NeoPax]

I take it you didn't study maths?

Nothing, and I mean *nothing*, is considered so obvious that it doesn't
need proving.

Well, Rolle's Th is.
And it is provably so

Principia Mathematica didn't get around to such advanced topics until
nearly 400 pages in; Wikipedia says:

Quotations:

"From this proposition it will follow, when arithmetical addition has
been defined, that 1+1=2." – Volume I, 1st edition, page 379 (page 362
in 2nd edition; page 360 in abridged version).

I am familiar with that, but maths courses at uni do not *start* with
such things.

 

[Dirk Bruere at NeoPax]

The first rule of mathematics is "assume nothing".

The motto of engineers everywhere is "without assumptions, no progress
will be made". This has brought us airplanes (and Langly's Aerodrome),
steamships (and the Titanic), the Golden Gate Bridge (and the Tacoma
Narrows Bridge), the PC (and the PC), etc.

I did not say I "assumed" it, but that it was totally obvious.

 

[Dirk Bruere at NeoPax]

Ah, but then you must *prove* that it is _totally_ obvious,
as opposed to say, somewhat obvious, or even mostly obvious...

The proof is too trivial to bother with IMHO

 

[Dirk Bruere at NeoPax]

I guess that's the difference between mathematics and
engineering. In engineering we have "rules of thumb"
and various accepted approximations, perhaps based
soley on experience or tradition. In mathematics we
strive to have every step and possible loophole
explicitly proven and filled in all the way back to
the founding axioms, otherwise it's just a guess.

Godel showed that it's just a guess, except in a rather restricted set
of mathematics eg Peano arithmetic

 

[[email protected]]

It was a big impediment to my starting maths at uni.
I could not understand why such a thing needed proving.

We could also impose Sharia, but it wouldn't be my first choice.

 

[DarkMatter]

It still seems totally obvious.
And for Rolle's next trick - proving 1+1=2

Coin toss.

100 times and there is nothing from keeping all 100 from being either
heads or tails as each toss has no bearing on subsequent tosses, nor is
it affected by any count of previous toss outcomes in the set.

Then, there is chaos, of course.

 

[Dirk Bruere at NeoPax]

Where have _you_ worked? Everywhere that I've ever worked, "That's
totally obvious" is the boss's synonym for "I'm making an assumption,
and if you question it you're fired".

But strangely, after viewing the proof - it's totally obvious.

 

[Tim Williams]

But strangely, after viewing the proof - it's totally obvious.

Doesn't matter for math. Everything is either a:
1. Posulate, or
2. Theorem derived from the postulates, no matter how trivial.

I am reminded of a Feynman story. In his university days, he would
sometimes tease the math students. Feynmen felt that all mathematical
theorems were obvious and intuitive from the prepositions, without having
to work through the proof itself.

So one day, he challenged a student to present any theorem, and he would
decide on the spot if it were true or false. The student chose,
http://en.wikipedia.org/wiki/Banach–Tarski_paradox
a classic case of unintuitive results. The student explained the
prepositions in terms a lowly physicist would understand: you take an
orange, and cut it up into finitely many very small, very complex pieces,
rearrange them, put them back together and you get two oranges.
"Aha!" Feynman said, for oranges are *not* infinitely divisible -- they
are made of atoms, not infinitely divisible spheres. So the student
failed to explain the theorem with an appropriate analogy.

http://www.multitran.ru/c/m.exe?a=DisplayParaSent&fname=Richard Feynman\Chapter12

Tim

 

[[email protected]]

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!

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

 

[Frank Buss]

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
25,933 steps, if you are really pedantic:

http://us.metamath.org/mpegif/mmset.html#trivia

 

[John Devereux]

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.

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.

 

[Dirk Bruere at NeoPax]

Doesn't matter for math. Everything is either a:
1. Posulate, or
2. Theorem derived from the postulates, no matter how trivial.

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

I am reminded of a Feynman story. In his university days, he would
sometimes tease the math students. Feynmen felt that all mathematical
theorems were obvious and intuitive from the prepositions, without having
to work through the proof itself.

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.

 

[Dirk Bruere at NeoPax]

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.

And, of course, smooth and continuous.

 

[TheGlimmerMan]

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

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