A constructivist's view of numbers and sets
- From: "John Jones" <jonescardiff@xxxxxxx>
- Date: 8 Jun 2006 13:16:38 -0700
Particular numbers are interesting. And taken as a whole, numbers are
interesting. But can we take numbers 'as a whole'? Let's review some of
the points we have established from a constructivist perspective:
1) Numbers cannot be counted.
2) Numbers cannot be collected, grouped, arranged, or placed in
families.
3) Numbers have no intrinsic properties.
4) Numbers occur only in the application in which they are found.
2) is a consequence of 4). But we still feel, or need to be able to
suggest, or demand, that numbers can occur in groups or collections.
Indeed, it is necessary to the modern idea of number. So let us return
to intuitive roots here, in the hope that a new principle, as a
solution, can arise. We might, for example, want to be able to compare
outcomes by comparing numbers from different applications. Yet this
brings us into conflict with 4). So we need to be able to refer to
'numbers' without, at the same time, commiting ourselves to a view from
a new application. We can formulate our search in this way 'what allows
us to speak of 'numbers from different applications'? :
a) We cannot use 'groups' to refer to numbers from different
applications. The elements of a group confer their properties on each
other and the outcome is the group: this conflicts with 3).
b) We cannot use 'collections' to refer to numbers from different
applications. The term is often confusingly synonomous with 'groups'
and 'sets'. In spite of these difficulties, it is still not clear what
it is that can bring numbers together in a collection. For if it cannot
be the properties of numbers themselves (from 3), then it seems that we
must appeal to an organizing paradigm or framework within which numbers
can be brought together. The nature of this hidden, but assumed
paradigm is never revealed. It is often confused with the objects in
the collection themselves - e.g. the set of sets.
c) We may be able to use the term 'sets' to refer to numbers from
different applications, but it is clear that we must distinguish a set
from a collection or find ourselves in difficulties, briefly indicated
above (b)). Our solution is this. It is clear, from 4) that we must not
represent 'relationship' when referring to numbers from different
applications for that indicates the construction of a new application,
and we want to generalize applications, not refer all applications to a
meta-application. Sets offer us the option of 'mapping'. Here, no
relationship is implied, and we can speak of numbers from different
applications. However, this tightening up of definitions, although far
from offering a prescriptive and unfamiliar view of sets, does require
a re-working of set theory to remove the flaws introduced when sets are
confused with collections.
(Please quote this source if the above material is used elsewhere.)
.
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