Thursday, April 23, 2015

Reflections on Number (3)

We will show here how each of the four meanings of a number can be represented in mathematical terms.

What is crucial here - in what properly represents a dynamic interactive form of understanding - is to express every number with respect to a dimension.

So in the general case ab, a represents the base and b the dimensional number respectively.

Thus now using once again the number 3 to illustrate we will go through the 4 distinct meanings

1) This again is the standard quantitative interpretation of 3 as representing a cardinal number.

This can be written as 31. So here the emphasis is explicitly on 3 as the base quantity.
Because of complementarity this means that the dimensional number 1 is merely implicit enabling 3 to be uniquely identified (from all possible members on number line).

So 3 as base is quantitative (in explicit terms); 1 as dimension is qualitative (in implicit terms).

2) This corresponds to our second definition in yesterday's entry where 3 as base number now explicitly takes on a holistic qualitative meaning as the notion of "threeness" which enables the collective identification of any group containing 3 members.

This can be written as 31. So here the emphasis is explicitly on 1 as the dimensional quantity (i.e. applying to all members on the number line).
Then the emphasis on 3 is now implicit where 3 has a unique qualitative meaning that is potentially infinite.

Notice how in the case of 1) 3 represents a specific number quantity; however by contrast in the case of 2), 3 now represents a holistic number quality (applicable to all possible groups of 3 members).

3) We now switch to ordinal notions

3 now takes on the meaning of a distinct group of 3 members that is explicitly defined in terms of its 1st, 2nd and 3rd members. This thereby represents a qualitative meaning of 3 that is actually finite.

This can be represented as 13.  So the emphasis here is explicitly on 3 as the dimensional number in qualitative terms which implies that implicitly the base number of 1 is understood in a quantitative manner. What this implies is that before we can rank members of a group ordinally (i.e. in qualitative terms) we must implicitly recognise each as a separate unit (in a quantitative manner).

4) We finally have the notion of ordinal identity that can be applies collectively to any number of groups (with 3 members).

This is written as 13. Here each group of 3 is identified explicitly as separate unit (which then is implicitly recognised as containing members that are arranged in an ordinal fashion). This in fact represents a quantitative meaning of 1 that is potentially infinite.

Therefore what happens in the dynamics of experience is that the number 3 here keeps switching as between its cardinal and ordinal meanings in both an actual finite and potentially infinite manner.
Alternatively it keeps switching as between quantitative and qualitative meanings in both an analytic and holistic fashion collectively.

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