Hello,
Some have defended conventionalism in mathematics. I shown that hard to sustain
in recursion/computability theory, and thus arithmetic. Here something which
shows that it is hard to maintain conventionalism in the study of finite
symmetries.
Groups (mainly set of symmetries) can be decomposed into some composition of
“prime groups” (called simple group).
Who is the guy who decided that a all finite simple groups belong to either 18
infinite families of groups, except for 26 exceptional one, the sporadic
groups, which does not, and who decided conventionally that the biggest one is
Monstruously big, the Monster, which has
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
elements.
All groups can be represented by a group of matrices, with the coefficients
belonging to some field (usually the complex numbers), with the usual product
of matrice. A field is itself a special sort of double group. (Not to confuse
with quantum field, of filed of forces).
The minimal dimension needed for that representation is the dimension of the
group. It is dimension of the space in which the element of the group represent
the symmetries.
The Monster group has dimension 196,883 with the matrix coefficient taken in
the field of complex numbers, but it has dimension 196.882 on the field z_2
with two elements {0, 1}.
Who decided that the dimension of the monster group is 196.882. Divine
convention? Could a God makes this in another way?
It might play some role in physics, notably conformal fields, strings,… (cf
Munshine).
A rather nice video on the Monster group is:
https://www.youtube.com/watch?v=mH0oCDa74tE
<https://www.youtube.com/watch?v=mH0oCDa74tE>
As I have explained, the non algorithmic distribution of the codes of the total
computable function is enough kicking back for me to be realist on arithmetic,
but the Monster group presents, I think, some difficulties for the
conventionalist too.
Bruno
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/everything-list/C232267E-B98D-4177-AE23-CCACBBD41BE9%40ulb.ac.be.
