Re: Infinitesimals



John Jones wrote:
herbzet 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?

I think the distinction between (1) and (2) is workable, but your
working through of this particular example is mind-boggling to
an 'umble logic dilettante such as meself.

I would just say that there are various "memorable arrays" of the
animals, and that ordering by weight is one of them. I'm not
worried about whether the relation is intrinsic or extrinsic.

(If we ordered them by name, lexicographically, that would be
obviously ordering by an extrinsic property. Ordering by
weight seems more like an intrinsic property of the animals
to me, offhand. But both orders are "memorable arrays" and
are happy. I don't see why I need to care about intrinsic vs.
extrinsic properties to recognize a "memorable array".)

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.

OK.

More later, maybe.

[...]

--
hz
.

Relevant Pages

  • Re: The Future of Agriculture and the Importance of Developing Our Skills and Knowledge Base
    ... Read that first and then write about raising animals ... Power Steer ... cows are far more common here than developments catering to people. ... solar-powered system for producing food on land too ...
    (rec.gardens)
  • Re: Yep, this RFID stuff is really gonna be great!
    ... The article isn't talking about medical viruses, ... It would take an engineer with a background in RFID to ... these animals will be "read" a minimum of 19 times from origin, ... standard should apply to BSE in cows. ...
    (soc.retirement)
  • Re: The really floating rebuilt ark
    ... that is half the length of the real Ark, ... me to be average) and estimate the cage space needed to hold 2 animals ... I did weight calculations (the major problem - if you have more available ... at c21,000 tonnes, and take the hull component of that ...
    (talk.origins)
  • Re: Yep, this RFID stuff is really gonna be great!
    ... It would take an engineer with a background in RFID to ... these animals will be "read" a minimum of 19 times from origin, ... standard should apply to BSE in cows. ...
    (soc.retirement)
  • Re: Infinitesimals
    ... Of course, if we order the animals in a zoo by, say, their weight, ... of a cow in a group of cows). ... group property of weights that they are sized. ... I have said in other posts that a mapping does not necessarilly ...
    (sci.logic)