The red herring principle

[ajb]

We have a metatheorem (due to Urs Schreiber?)

 

Metatheorem A “red herring” need not, in general, be either red or a herring.

 

So, so simple examples.

 

1) A manifold with a boundary is not a manifold.

2) A supermanifold is again, not a manifold.

3) Noncommutative geometry considers "spaces" that may or may not be commutative.

4) A Grassmann number is not really a number (as a mathematical object used in counting and measuring).

 

and so on.

 

Anyone suggest some other "nice examples"?

[liarliarpof]

Dimensions. A point is 'dimensionless', or Zero. The line is constructed by aligning infinite #s of Zero-dimensional points. So how does the line acquire a 'dimensionality' of One? And so on toward the 'plane' & upwards.

[ajb]

Dimensions. A point is 'dimensionless', or Zero. The line is constructed by aligning infinite #s of Zero-dimensional points. So how does the line acquire a 'dimensionality' of One? And so on toward the 'plane' & upwards.

 

Any sub space of another space is going to consist of a collection of points.

 

To parametrise a point one needs no parameters. It is just specified as a selected point.

 

A line requires you to use one parameter. Let us work on [math]\mathbb{R}^{n}[/math] and use local coordinates [math]\{x^{a} \}[/math].

 

A point we can think of as a map

 

[math]pt \rightarrow \mathbb{R}^{n}[/math].

 

In local coordinates this is simply given by [math]x^{a}_{p}[/math]. You specify a particular coordinate.

 

 

A line or really what we call a curve is a map

 

[math] \gamma : I \subset \mathbb{R} \rightarrow \mathbb{R}^{n}[/math].

 

If [math]t [/math] is a coordinate on [math]\mathbb{R}^{n}[/math] then in local coordinates we have

 

[math]\gamma^{*}x^{a} = x^{a}(t)[/math].

 

So you specify a one parameter family of points.

 

A plane or really a sheet requires two parameters, say [math]\{t , s\}[/math] and we have in local coordinates

 

[math]x^{a}(t,s)[/math] a two parameter family of points etc...

 

So I am not sure this really qualifies for the red herring principle.

[D H]

Anyone suggest some other "nice examples"?

A multi-valued function (e.g. asin(x)) is not a function. Neither is the Dirac delta function.

 

Cartesian tensors are not tensors. There are Cartesian tensors that are not tensors and tensors that are not Cartesian tensors.

[ajb]

DH, these are good examples. Just the sort of thing I had in mind.

 

As an aside the lack of a distinction between covariant and contravariant Cartesian tensors courses mind blocks when passing to more general manifolds or indeed from more general manifolds to Euclidean spaces. (I always have trouble anyway!)

[alan2here]

If SIN is not a function what is it, an algorithm? If I were coding it from scratch I would use a function, in a programming languege, which is really an algorythm.

 

In this sence sum, product and factorial are not functions either. How about to the power of?

[the tree]

sin is a function, asin isn't (nor is it an algorithm) since it fails the well definedness property. The dirac delta isn't a function because it doesn't really work with the sets it's supposedly defined on.

[D H]

If SIN is not a function what is it, an algorithm?

I never said sin(x) is not a function. It most certainly is. A function is something that has a well-defined output for all values in the domain of the function.

 

It is asin(x) that is not a function. asin(x) is short for arcsin(x), in other words the inverse of the sine function.

 

The problem arises from the fact that [math]sin((2n+1)\pi-x)=sin(2n\pi+x)=sin(x)\,\forall\, n\in\mathbb I[/math]. There is not a single value that can be assigned to asin(x).

 

asin(x) is what is called a multi-valued function. Using "multi-valued" as a qualifier of the word "function" makes it sound like multi-valued functions are a special kind of function. This is not the case. In fact, it is the other way around. The set of all functions is a proper subset of the set of all multi-valued functions.

 

So, in a sense, the term "multi-valued function" is a bit of a mis-nomenclature, and that is exactly what ajb is looking for in this thread.

[Amr Morsi]

May be the below is following this criteria, ajb:

"Minimum surface area, generated by rotating a curve between 2 fixed points around a fixed axis, doesn't imply that the curve is a straight line, although the shortest distance between two fixed points is a straight line."