On 07 Mar 2012, at 18:36, Pzomby wrote:



On Mar 7, 5:29 am, Bruno Marchal <marc...@ulb.ac.be> wrote:



OK.
But it is not valid to infer from this, that mathematics is *about*
description.
On the contrary, mathematicians reason on "models" (realities,
structures), and they use description like all scientists.
mathematical logic is the science which study precisely the difference
between description (theories) and their interpretations (in from of
mathematical structure).
As you mention the notion of cardinal, a discovery here made by
logicians is that the notion of cardinal is relative. A set can have a
high cardinality in one model, and yet admit a bijection with N in
another model.


Yes, but even the symbols =, +, x, *, are notations that are
substitutes for words. Eg. Equals, addition or union, multiplication.
The operational notations are words used to describe the formulation
of the model.

Hmm... OK.
In logic they are symbol associated with axioms and rules, and they have (standard) semantics, for exemple the mathematical "meaning" of + is given by the set {(0,0,0) (0, 1, 1), (1,0, 1) (1,1,2) .... (6,7, 13), ..., (1, 23, 24), ....}.






“In common usage, an ordinal number is an adjective which describes
the numerical position of an object, e.g., first, second, third,
etc.”  http://mathworld.wolfram.com/OrdinalNumber.html

Are the “ordinal” numbers actually adjectives describing the
relational position in a sequence (first, second,…one-ness, two-ness
etc.)?

They can be used for that. But they can be much more than that.


Yes. Then it is Ok to use it for that.  eg. 1stness, 2ndness, 3rdness
in sport races gives a quality of feeling to the participants,
observers/bettors.

OK. But I would say the "quality" of being the first is more in the mind of the machine winning the competition, or in the mind of the machines members of the jury, than in the ordering relation itself.





Are numbers (ordinal) necessarily qualitative descriptions?

Perhaps. In the comp frame, I prefer to ascribe the qualities of
numbers, by the possible computational relation that they have with
respect to their most probable universal environment. This is more
akin with the human conception of quality as being a lived experience.
But what you say might make sense in some other contexts.


It is the “lived experience” that is reality as I understand.

OK. That is the reality of subjective experience, but we can bet there is something independent of that reality, and which might be responsible for that reality.



The condition of the universal environment is influenced by an event
at a point in time of the evolutionary process.  eg. Certain
qualitative conditions existed in Oct. 1066 in Britain. Also,
9/11/2001.  In nature: January in central Europe exudes certain
environmental qualitative conditions.

Once universal numbers are in relation with other one, many qualitative conditions can happen, assuming digital mechanism.





Numerals symbolize number position (as in particular instants in the
sequence of the continuum of time).

OK. But that's quantitative for me, or at least a "3p" type of notion.
Quality is more 1p, and can be handled at the meta-level by modal
logic, or by (often non standard) logics.

Bruno


Duration of time is quantitative.  Existing conditions in the duration
are qualitative.

I doubt this. I would bet that if time can be quantitative, and objectively measured by different observers, the duration notion is more qualitative, and subjective.



You state: “Quality is more 1p” but it is not exclusive to 1p.  Humans
observe and have  empathy for others qualitative conditions and
states.

I agree.

Bruno


http://iridia.ulb.ac.be/~marchal/


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