Wacek, (01)
The question of how to or whether to represent a null value of
some kind is a context-dependent issue about how to regularize
the operators of some mathematical system. (02)
vQ> If you and me are just you and me, then nothing is nothing,
> no entity at all, and not the empty set. You can well
> interpret 'nothing' as a sheet of paper on which there is
> no drawing, though there is the sheet -- how do such
> interpretations help? (03)
The number 0, for example, simplifies the statements of many
arithmetic principles. Similarly, the empty set simplifies
many of the axioms of set theory. In lattices, the bottom
symbol simplifies many axioms. In a Boolean lattice, the
bottom corresponds to a proposition that is always false;
such a proposition doesn't say anything useful, but it makes
it possible to formulate the axioms more systematically. (04)
For some mathematical structures, a null value has no useful
role. In most versions of mereology, for example, there is
no empty part. An atom in mereology is defined to be something
that has no part other than itself. In such systems, the word
'nothing' is just a way of saying 'no thing'. Unlike the empty
set, which is assumed to exist in set theory, the word 'nothing'
(or a formal symbol that represents it) would be a way of saying
"It is false that there exists an x such that..." (05)
In short, the concept of 'nothing' or a 'null value' depends
on the operations needed to regularize some system. (06)
John (07)
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