INTRODUCTION

The term 'set' is used in a 'technical' sense in mathematics. What does
this mean? For our purposes, it means that there is a set that is used
in mathematics that is not empirically encountered. Immediately this
throws out any attempt to translate the meaning of the mathematical term
'set' into a universal epistemology, simply because the field of
operation of a technical meaning is restricted to its own domain.
Nevertheless, despite this absence of a translative link between
mathematical sets and empirical sets, set theory is still empirically
useful if it is 'mapped' to empirical set events. This is because a
mapping is not configured through a relation such as a translation. This
means that a pragmatic set theory does not require the services of
translation to be empirically useful.

But my concern here is the nature of the set that mathematics employs.
How can we begin to answer this question - which requires the powers of
translation between a mathematical and empirical domain - if the
mathematical (technical) set is restricted to its own domain or known
only on its own terms?

DISCUSSION

First, we can note that empirical sets such as herds, bouquets, canteens
of cutlery, etc., cannot be integrated. There is no set union, for
example, of a herd and a canteen. Any attempt in this direction will
compromise the nature of the collective, or set. We don't speak of a
herd of cutlery.

Yet, oddly, as it might seem at first, set theory allows the integration
of sets and the formation of subsets. This can mean only one thing.
Mathematical sets are all of one type. An empirical analogy can be made:
empirical sets are multitudinous, yet we can isolate any one set and
make variations on it. Take, for example, the set 'bouquet'. We can have
a set or bouquet of carnations, and a bouquet of roses. It is legitimate
to speak of a mixed set in this case - a set of mixed flowers
(carnations and roses).

So now the business of determining the nature of the mathematical set is
made simpler for us because mathematical sets are of one type. We can
easily determine this type, but we encounter a difficulty, for which I
must make a short diversion: the difficulty arises from the fact that,
as empirical sets show us, sets are emergent properties. That is, sets
are known not from their individual elements or 'members' but arise
independently of them. A bouquet, for example, is not constituted of its
members but arises from its members. It would not be legitimate to speak
of the set 'a bouquet of carnations and a herd of cows' for no set
arises naturally in repect of it.

In other words, we are familiar with a set, not through its members, but
as a form that arises as a possiblity of its members.

There is, then, one mathematical set and, like any set, its properties
are independent of any elements it may or may not contain, or have as
its members. The difficulty I alluded to, above, now comes in. The
notion or form of a set is intentional, or innate as it were. That is, a
set is not determined by its elements, but is a subjectively familiar
form that arises from its elements. A bouquet, for example, is not an
object above and beyond its elements but neither is it determined by them.

Familiarity, then, is a necessary property of a set. Bouquets, herds and
cultlery are all sets we are familiar with. Unless we make further
specifications to, for example - "a set of flowers" - we do not describe
a familiar set, and so do not decribe a set. And, we can make no appeal
to its individual elements or objects (flowers in this case) to provide
us with our (familiar) set.

CONCLUSION

The question we must ask if we want to determine the nature of the
mathemtical set is to ask what familiar property this set has. And, it
is true, there is no immediately, empirically, familiar property. We
would not expect anything else if the mathematical 'set' is announced
as being 'technicall' determined. All that mathematics gives us for this
'set' are bracketed elements. We are not familiar, at least empirically,
with the idea of any particular mark or sign as itself announcing a set.
Yet, to reiterate, it is the brackets, and not the elements, that
constitute the properties of the mathematical set. The brackets announce
a unique set, just like a bouquet, or herd announce unique sets. The
brackets tell us that mathematics itself is the nature of the set
deployed by mathematics, and that its objects are all of one type.
Doubts we may have against the proposal that bracketing announces an
immediately discernible, familiar, intentional set must cast a shadow
over the intelligibility of the set theory enterprise.

(Quote this source if used please. This material is part of a proposed
dissertation)