Re: Mathematical objects and Discernment


John Jones wrote:
LauLuna wrote:
It appears to me that John Jones' idea of interpreting a set as a
collection of objects upon which our attention is directed might be
useful for the treatment of paradoxes. For instance Russell's paradox:
when I'm trying to consider all sets that are not elements of
themselves to build with them a new set, my attention cannot be
directed upon the very set I'm trying to construct.

Perhaps this is why we always fall short when we try to build up the
set of all sets: the set we are building cannot be one of those we had
in mind. This is related to Cantor's paradox and set theoretical
reflection principles.

I suggest to resort to Husserl's phenomenological terms in order to get
a more technical vocabulary. Instead of "attention" we could speak of
"intentionality!, i. e. the fact that every mental (intentional) act
its directed upon an (intentinal) object: when we think there is always
something thought, or wanted when we want, or hated, and so on.

In order to propose a solution for paradoxes and circularities we could
establish as an "eidetic" feature of intentional act that no
intentional act can be equal to or contained in its own intentional
object: So, Liar-like acts of thinking like "what I'm thinking in this
very act of thinking is wrong" or "...is right" are phenomenologically
impossible.

A logical or mathematical object is always introduced by means of a
linguistical expression that in turn has to be the objectivation of an
act of thought. If we ban all objects that are no objectivation of any
possible act of thinking, we would surely ban exactly the paradoxical
and circular ones.

The outcome is that the phenomenology of our acts of thinking could be
relevant for mathematics and logic, rather surprisingly!

Best regards.


A number of related points to your interesting reply:
The establishment of a 'collection' presents us with some difficulties.
Clearly, a set cannot be a collection unless we pre-suppose a framework
within which elements or objects are differentiated. This framework
remains implicit (unrecognised) in set theory. If the framework is
considered at all, I would suggest that it may be mistaken for a set,
as in 'the "set" of all sets'. Whatever the analyses of that idea may
turn up, we must, at least, first consider whether the framework is
ontologically prior to, or necessary to the set and its 'collection'.

We can explore the consequences of removing the framework that supports
the notion of 'collection' (as the term is employed by sets). By
removing the framework that establishes differences we remove the
identity of the elements of the set; we also remove numeration. Do we
still have the makings of a set? I would suggest that we do.

Let us re-call the plea, injunction, slogan, what have you, that 'there
can be a set of anything'. There are limitations to this which I won't
explore here, and which you also have discovered. But for the moment
let us say that we can have a set of anything we like. We then can have
a set of objects from disparate paradigms or frameworks. Now I suggest
that in this case, where there is no common paradigm, then the elements
of the set are non-numerable and identityless. This is not fatal to the
idea of sets. All that is left for us to do is re-define the set to
remove the concept of 'the common framework' - A set is defined by its
name and not by its elements. A consequence is that sets are 'composed'
of non-numerable empty mark(s).

I suggest that this re-definition is not radically prescriptive, and
ties up with what you have said. We converge. I suspect that the above
definition is capable of supporting much of what goes for set theory
today. Much is also lost - a 'set' of sets not elements of themselves
(which you have also pointed out), the numeration and transferability
of the elements of sets, and sub-sets. For example, concerning
sub-sets, the 'sub-set' must be installed in the name of the set,
thereby removing the obscure framework that divides the two.

From these arguments I can address your point about intentionality. If
we take an OLP approach we may find that the term 'attention' might
suffice, and I for one have always been a little confused by the the
term 'intentionality'. Nevertheless, 'intentionality' is the
'technical' or installed term within philosophy today, so I will use it
here. If an act of intention defines the elements of a set and the
elements of a set are identityless and non-numerable, as I claim, then
it might seem that we have a problem. But isn't what happens, this?:
when I make a set I pick out objects by an intentional act. I then
place them in a framework so that I can count them and identify them.
In the set I have no framework, so instead I install the objects in the
name of the set, and what falls under the name of the set is a
non-numerable identity-less collection of indistinguishable marks or
acts. An argument against this is that I can have five cows in a set. I
would counter that the name of the set must be more closely addressed
and in this case the name of the set should be the set 'the set of five
cows' - and so we retain the integrity of the new definition of sets.

As an afterthought, Frege used concept and function as a foundation for
his arithmetic. I suggest that act and mapping might have been equally
if not more succesful if, as I understand it, the latter removes the
problem of the unrecognised framework that has plagued set theory. Act
and mapping also emphasises the phenomenology of mathematics, whereas
'concept' flits back and forth uneasily between sense and reference.

For 'concept' you can read 'thought' or 'postulate' all considerations
of the mind.

'Collect' can mean 'place them mentally, or in thought'

.

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