## Dissection

Began dissecting at the weekend. Clowns to the left of me, jokers to the right is the story so far. But there aren’t half still some jokers to the right.

### 2 Responses to “Dissection”

1. sigfpe Says:

Conor,

Just responding to your comment here: http://sigfpe.blogspot.com/2006/06/taylor-series-for-types.html I had actually already found and read this dissection document. Interesting. It’s nice to see a rigorous derivation of something like a Taylor expansion for types.

If you don’t mind recklessly throwing all caution to the wind here’s another way to ‘derive’ the Taylor series, different to my previous ‘derivation’:

If F[X] is an F-container of X’s
dF[X] is an F-container with a hole
d^2F[X} is an F-container with an ordered pair of holes
(1+d^2/2)F[X} is an F-container with no holes or an unordered pair of holes, etc.
G[d]F[X] is an F-container with a G-container of holes in it(!!)
A.dF[X] is an F-container with an A-labelled hole
G[A.d]F[X] is an F-container with a G-container of A-labelled holes in it
exp(A.d)F[X] is an F-container with a set of A-labelled holes in it
But that’s just an F[X+A].
Expanding the left hand side gives the Taylor series.

But yes, I’m fully aware this is a long way from rigour. But it’s fun to play with, and a container of holes sounds like a useful thing to me. I think it’d be nice to make it rigorous somehow.

2. sigfpe Says:

Write the dissection operator as \|/.

Then the operator D defined by D F X = \|/ F X 1 is Fox’s free derivative.