anderson@kogs11.informatik.uni-hamburg.de (James Anderson) writes:
> But I would like to understand this argument by analogy (a) with
> matrices.
> Suppose that a pop array with no bounds is analogous to a
> 0 x 0 matrix (whatever that might be). Then a 0 x 0 matrix
> contains an element.
> Now my question. Persuing this analogy, what does a
> 0 x N matrix contain?
An array can be regarded as a representation of (a) a simple kind of
memoised function and (b) a 2-D array can be regarded as a representation
of a matrix.
But an array with an empty bounds list should not be confused with an array
whose bounds are empty, which corresponds to a 0x0 matrix.
vars A = newarray([1 0 1 0],identfn);
Now it is quite clear what a 0x0 matrix is, if we recall the correspondence
between matrices and linear transformations. A 0x0 matrix with
entries drawn from a field F represents (in the mathematical sense) a
linear transformation from a 0-dimensional vector space over F to a
0-dimensional vector space over F. This transformation is unique (or as
they say in Congress, very unique). So there is no need for an entry.
Likewise n*0 and 0*n matrices represent linear transformations to and from
(or from and to depending on your pre/post multiplication conventions) zero
dimensional vector spaces. These are likewise unique. And the array A above
does not have any entries and so is a correct representation of a 0x0
matrix.
But an array with a null boundslist corresponds to a nullary function,
which in the lambda calculus is just a -constant-. So it should have an
entry.
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