Re: Infinitesimals

On Dec 10, 6:37�am, herbzet <herb...@xxxxxxxxx> wrote:
John Jones wrote:
herbzet wrote:

'The real numbers in their natural order' is a perfectly ordinary
and common remark which means nothing profound, it just means that
we usually regard the reals in their size ordering.

It makes it sound as if they (number or 'x') have a range of
properties. How would you write that? something like x(p)?

So for example, if I list the emotions alphabetically: aggression,
amouressness, avarice, beastliness, etc., then I have only listed
their signs in a memorable array. The emotions themselves are not
ordered alphatically, but merely have an order imposed on their
signs.


Of course, if we order the animals in a zoo by, say, their weight,
is it only the signs for the animals that are ordered, or the animals
themselves?

This brings it to a head. Do I 'order' the signs of an element or the
element itself? The thing to note is that 'order' exhibits
relationship. I therefore need to distinguish between a relationship
held between things that 1) is a result of a property of the thing
itself and 2) a relationship that arises independently of the
properties of a thing.

1) is an example of a group, e.g. a herd of cows (a herd is a property
of a cow in a group of cows).
2) is an example of a set, e.g. a set of cows.
(So a set of cows is not a herd.)

In 1) The animals are 'ordered' by their own properties. It is not
merely their signs that are ordered. The order of the group is not
that of the order of singleton cows.
In 2) The animals do not order themselves, but their signs are
ordered. The order of the set is that of the order of singleton cows.

So if you weigh animals in a zoo then we can make a set whose name is
the set 'the set of weighed zoo-animals'. But their weights are placed
in an order, and that order is a group because it is an emergent or
group property of weights that they are sized.

So the answer, I think, is that weighed animals in a zoo are a group
of weights in the set of weighed zoo animals. It is the sign of the
animals that are ordered but it is weights themselves and not merely
the sign of a weight that is ordered. How's that?



This is what I meant about mathematicians transferring orders or
memorable arrays between incommensurable domains (between numbers and
infinitesimals, or between emotions and the alphabet, etc.)

Sounds utterly plausible, offhand. �However, if two domains of objects
can be similarly ordered, are they not thus, somehow, commensurable?
Are domains that cannot be similarly ordered necessarily incommensurable?

(...by all accounts a set makes everything commensurable. But then can
a set distinguish between singletons and groups?...)
To continue: Lets say that a mapping constitutes your 'similar
ordering'.
I have said in other posts that a mapping does not necessarilly
constitute a relationship between the things ordered. In other words a
mapping can be made between incommensurables. Let's see: I can map the
object of one idea with the object of another idea. There are some
proviso's. The fundamental factors that need to be in place for the
success of a mapping (is there a better way of putting that?) is that
the objects that are mapped are both countable and re-identifiable,
otherwise I cannot map them.

So all I have to do to show that two domains of objects can be
similarly ordered (mapped), and yet not commensurable, is to
demonstrate a mapping between an element that is countable or
reidentifiable, and an element that is not countable or
reidentifiable. I can do this if I map Liebnizean monads to Newtonian
objects, or if I map numerals to numbers (numerals are not countable),
or concepts to their particulars or extensions. Of course, we are not
speaking here of merely mapping the signs.

The reals when ordered by size are a dense set (for any two reals
r1 < r2 there is a third real r1 < r3 < r2). �The rationals when
ordered by size are also a dense set. �The reals can't be ordered
sequentially (since they are uncountable) but the rationals can be
(since they are countable). �Are the real numbers and the rational
numbers incommensurable?

It's coming up to bedtime. A quick answer to that last point is that
by my lights, they are incommensurable. That means they have no
relationships and cannot be derived from a common source. Can they?
However, by 'uncountable' you are referring to the totality and not to
the particulars. That may make a difference.
.

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