## Monday, April 13, 2015

### Reflections on Number (1)

Once again we return to the crucially important notion of number to discover that it has laden with a great deal of hidden subtlety which needs to be carefully deciphered before coherent mathematical interpretation is possible.

Let us illustrate for example with respect to the number "2".

1) Now 2 has an accepted quantitative meaning in specific analytic terms.

So for example I identify 2 cars in my driveway, I am using number in this conventional sense.

Thus here 2 = 1 + 1 where the individual units are literally understood as homogeneous, without qualitative distinction (i.e. no unique relationship to each other).
Put another way, this represents the interpretation of the number as an independent entity (integer) in an impersonal individual manner.

2) However 2 equally has a quantitative meaning in general holistic terms. This equates with the dimensional (rather than the base notion of 2). So in the dimensional expression ab, a is the base and b the dimensional number respectively!

So if I for example identify classes of objects with respect to the stipulation that each contains 2 members (e.g. 2 cars, 2 chairs, 2 names etc,  then I am using 2 in this collective holistic sense where it can apply to any number of object groups (of 2).

Thus the crucial distinction here is that 2 now serves a collective - rather than individual - role in identifying a number property (i.e. 2) that is common to all classes defined in an impersonal collective manner.

In the two examples so far we have defined the number 2 in a quantitative manner (with respect to both its specific (analytic) and collective (holistic) properties.

However we can now equally define 2 in qualitative terms with respect to both aspects.

3) So 2 now is a number with a qualitative meaning in specific analytic terms. We could refer to this quality of 2 as "twoness" which thereby gives the number a unique personal identity.
Now whereas the quantitative counterpart notion of 2 is defined in a cardinal manner so that 2 = 1 + 1, this corresponding qualitative notion is defined by contrast in an ordinal manner.

Thus it is understood here that 2 = 1st + 2nd members (that are  qualitatively distinct).

Therefore whereas the quantitative notion of 2 (as an independent integer) entails no unique relationship between units) the qualitative notion by contrast implies a relationship of interdependence as between units (where each is uniquely distinct).

4) Finally 2 equally has a qualitative meaning in general holistic terms.

So we are now referring to the number 2 once again in a dimensional sense, but where it now is identified in ordinal terms as a number identifier with respect to a collection of groups.

In other words according to agreed criteria we could identify a number of different groups with respect to unique 1st and 2nd members respectively. So in this sense all the groups share the same qualitative identity of "twoness".

Put more simply, numbers representing both base and dimensional aspects respectively, repeatedly switch as between cardinal and ordinal meaning (in a quantitative and qualitative manner).

So again 2 as the base aspect has a cardinal interpretation (in quantitative terms) with a specific application to an independent individual entity.

However 2 also  representing a dimension (power or exponent) has a cardinal interpretation (in quantitative terms) with a holistic application (as applying in common to all instances of 2).

Then 2, again as base aspect, has an ordinal interpretation (in qualitative terms) with application to the two distinct members of an individual group (as 1st and 2nd respectively).

Finally 2 now representing a dimension has an ordinal interpretation in qualitative terms with a holistic aspect (as applying to all distinct instances of two unique members.

Conventional Mathematics however is riddled throughout with a gross form of reductionism, whereby the qualitative aspect of appreciation is continually interpreted in a quantitative manner (indicating a corresponding failure to properly distinguish finite and infinite notions).

Likewise the holistic aspect of appreciation (where number carries a collective sense) likewise is reduced in a merely analytic type manner (with a merely individual interpretation).

Thus instead of the number 2, as in my example, being given at least 4 distinctive meanings (that dynamically interact in experience), in conventional mathematical terms  it is given but a grossly reduced interpretation (i.e.where the qualitative aspect is reduced to the quantitative and the holistic aspect to the analytic).

We will develop these insights further in the next entry.