RE: The nature of number
July 28, 2012 at 12:52 pm
(This post was last modified: July 28, 2012 at 12:52 pm by Categories+Sheaves.)
Yes, these 'x0 +n' transformations are, in a sense, part of the same "field". But much in the same way that multiplication by zero isn't the same as multiplying by 3 (I can 'undo' the latter by multiplying with 1/3, but I can't undo the former) these functions aren't carrying a line to a line, but collapsing a line to a point. Here, for example: You have a 'field' of n-by-n matrices, some of which have "rank n"/"nullity 0" and map an n-dimensional euclidean space onto an n-dimensional euclidean space/map only one point in the domain to zero, some of which have "rank n-1"/"nullity 1" and map an n-dimensional euclidean space onto an (n-1)-dimensional euclidean space/map a line in the domain (or 1-dimensional euclidean space) to zero, some of which have "rank n-2"/"nullity 2" and map an n-dimensional euclidean space onto an (n-2)-dimensional euclidean space/map a plane in the domain (or 2-dimensional euclidean space) to zero...
A lot of things have caveats that pop up in the "zero"/"trivial" case
A lot of things have caveats that pop up in the "zero"/"trivial" case