Cardinality or Nominality?

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kirby urner

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Mar 4, 2011, 11:01:33 AM3/4/11
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I've been checking with the statistics curriculum and notice that
"nominal" values are closer to what I and some others have 
been referring to as "cardinal" values, meaning they don't 
have much use for > or < (ranking in terms of greater or lesser).

The important thing is to make the distinction:  using names 
to tell things apart (distinguishing), independently of any 
ranking or ordering.  One phases over into the other (e.g.
book titles may be alphabetized -- an ordering).

Nomenclature issues...

Kirby

in Philadelphia (@ AFSC corporation meeting (Quakers))

John Bibby

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Mar 5, 2011, 5:31:37 AM3/5/11
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The key pedagogic point is 'Does it matters?' If dealt with properly, it is rather profound, and not as simple as usually indicated.

The canonical confusion goes back to S,S. Stevens' book which has paralysed statistics teaching for decades.

We wrestled with this at the Open University and adopted the following categorisation. The important thing is to distinguish between two distinctions viz: continuous/discrete and ordered/unordered.

Thus we may have data which is
1. discrete + ordered
2. discrete + unordered
3. continuous + ordered.

Then there's a further distinction that the ordering scale may be A. interval, B. ratio, or C. none (noting that B includes A)

This leads to the following possibilities:
1A: nu
1B: number of siblings
1C: certain psychological attributes (I forget - this is rare)
2: names of siblings
3A: temperature
3B: height
3C: certain psychological attributes (I forget - this is rare).

A final important point is that the decision regarding the status of any variable is model-determined (some would say 'ideological') e.g. social class may be regarded as ordered or unordered. Also, some would wish to put interval or ratio measurements on their measures of social class.

Stevens had the following categories which are subsumed by the others as follows:
  • 'nominal': 
  • 'ordinal'
  • ratio'
  • 'interval'
I hope this helps. but realise it may not!

JOHN BIBBY


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Maria Droujkova

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Mar 5, 2011, 5:53:50 AM3/5/11
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John,

One of the principles of mathematics education that I hold dear is to bring the multitude of topics back to a few global "roots" - fundamental ideas that organize everything. From the point of view of this principle, I very much appreciate the fundamental ideas you just named. I think it helps students to orient themselves in their daily work if they are aware of a taxonomic mind map, so to speak, with few big overarching categories of concepts.

Cheers,
Maria Droujkova

Make math your own, to make your own math.

kirby urner

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Mar 5, 2011, 8:59:57 AM3/5/11
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On Sat, Mar 5, 2011 at 4:31 AM, John Bibby <johnbibby...@gmail.com> wrote:

<< snip >>
 
Stevens had the following categories which are subsumed by the others as follows:
  • 'nominal': 
  • 'ordinal'
  • ratio'
  • 'interval'
I hope this helps. but realise it may not!

JOHN BIBBY


Yes sir, helpful.  Statistics is where the rubber meets the road, as the say, 
in terms of sucking in data from the "empirical world" (as if there were others),
as I'm starting to re-appreciate.

DIGITAL MATH

As a worker on the digital mathematics railroad (formerly "discrete math"
but morphing), I'm keen to abet numeric content, so familiar from arithmetic,
with lexical content, as executable math notations (aka "programming 
languages") tend to work with words, characters, semantic content 
other than "just numbers".

SAME VS DIFFERENT

By focusing on the nominal (naming) functions of language, irrespective 
of sorting and ordering, I get back to basic taxonomies such as mammals,
fungi, plants, micro-organisms... a biological beginning where the first 
step is to tell things apart and to recognize when we have another one
of the "same thing".  Simple counting (none, 1, 2, 3...) depends on 
recognizing "this is another one of those" (sheep, apples....).

MATH OBJECTS

In the world of "math objects", we have a similar need to distinguish
different types of "thing" (object) without worrying about ranking them.
A "vector" is a mathematical "creature", as is a "polyhedron", a "set",
a "rectangular table" (of data) and so forth.  The mathematical vista 
is populated with these "animals" we distinguish nominally (by name).

NAMESPACES

Then comes the concept of "namespace", which sounds too advanced
in some contexts, but it simply means the names we're using depends
on the language in play.  The concept of fluid. We might be talking
about Russian vs. English.  We might be talking about a mathematical
language which invents new meanings for familiar words (happens 
all the time).

NOMINAL VS ORDINAL

In my own experience, I've had a couple library jobs, both involved
with sorting books, putting them where they belong on the shelves,
in an order corresponding to cards in a card catalog.  Sometimes 
the titles were in Arabic.  When being trained, I was exposed to 
extended alphabetical rules that deal with the interleaving of 
multiple languages and character sets (so-called collations, a
concept in unicode).  

HOW IT WORKS

Mathematics classes don't typically deal with such as the Dewey 
decimal system (that's library science), but the new digital math 
track might go there (physical libraries are still a reality).  I also 
want to dissect the concept of URL in the context of the Web, 
to tell stories about "how it works", about protocols, tcp/ip...  all 
"advanced topics" when studied in detail, but also useful and 
relevant from the proverbial "10,000 feet".  URLs, DNS, IP numbers...
-- lots of "nominalism" there.

Kirby

kirby urner

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Mar 5, 2011, 9:25:59 AM3/5/11
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NAMESPACES

Then comes the concept of "namespace", which sounds too advanced
in some contexts, but it simply means the names we're using depends
on the language in play.  The concept of fluid. We might be talking
about Russian vs. English.  We might be talking about a mathematical
language which invents new meanings for familiar words (happens 
all the time).

"The concept is fluid" I meant to say.  

The digital math track I'm working on introduces computer languages.

The Python language in particular has a (deserved) reputation for 
being friendly to newcomers.  The concept of "namespace" is well
defined therein.

Likewise, the concept of "creature" is somewhat reinforced by the
look of the language "on paper" (its aesthetics):

class Snake:
    def __rib__(self): 
    def __rib__(self): 
    def __rib__(self): 
    def __rib__(self): 
    def __rib__(self): 
    etc.

.... looks like a head with a long tail (like a snake, like a python).

The so-called __ribs__ stand for a variety of "special names" 
which give behavior to an object, which is reminiscent of a 
spinal cord supporting neural reflexes.  Blueprints for math
objects actually look like creatures in some way (a good segue
from all this talk about animals).

For example, a Vector object would have __add__ and __mul__ 
behaviors (addition and scalar multiplication).  

A Polyhedron object would have methods to scale, rotate, translate... 
(we'd stray from the double-underline aesthetic here, now that the 
"look and feel" is established).

Polyhedrons are a great place to start because the generic idea
of a polyhedron (something with an inside and outside, a membrane
in between) is the generic idea of a container or "thing".  Snakes,
people, dogs... puppets:  all polyhedrons (with behaviors).

This is getting into the nitty-gritty of a particular implementation.

I'm not suggesting all digital math tracks need to invest in the
same computer languages.  It'll be up to the various faculties
to import what they need, and to customize for local use.

Kirby

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