I have mentioned that number recognition entails - relatively - both cognitive and affective modes of understanding.
Through the former aspect, one comes to appreciation of the quantitative
(impersonal) nature of number in cardinal terms; through the latter one
comes to corresponding appreciation of its qualitative (personal) nature
in an ordinal manner.
So number is thereby inherently dynamic, entailing both
collective whole and individual part aspects.
So again the collective whole aspect of a natural number (as cardinal)
entails the homogeneous similarity of its independent part units (which thereby
lack a qualitative identity); however the individual part aspects of the number
(as ordinal) entail a unique distinction with respect to its unit parts (with
the collective sum thereby lacking a quantitative identity).
Therefore, when properly understood, in a dynamic interactive manner, these
two aspects of number are revealed as fully complementary with each other!
However in conventional mathematical terms due to its rigid absolute
framework, the qualitative
aspect of number is thereby reduced to mere quantitative interpretation
Now, I am aware that I have stated these points repeatedly. However I
believe it is necessary so as to fully convince you that the
present accepted mathematical framework - which is rarely ever questioned
- is simply not fit for purpose.
Therefore an enormous revolution in understanding now awaits, which promises
to be the greatest yet to occur in our intellectual history. This
will intimately affect every possible notion in mathematics and the sciences
with dramatic consequences for the future evolution of our world.
So far in the present discussion, I have concentrated on the analytic understanding
Once again, the analytic has two aspects relating properly
to cognitive and affective type appreciation respectively.
It is interesting how in accepted understanding, the highest form of
reason requires the ability to abstract from more limited concrete
information provided by the senses.
Therefore in earlier childhood, one only can come to a knowledge of number
with reference to simple concrete type examples (where counting is still
associated with the concrete objects of counting).
So both cognitive and affective aspects are here naturally involved in number
experience in a somewhat immature manner.
Then later one becomes able to continually abstract from mere concrete
understanding to obtain a purer mathematical appreciation of number (based on
However the reverse also is the case. Therefore one develops the pure
affective appreciation of number through the corresponding ability to detach the senses from reason.
In this way, one becomes able as it were to properly distinguish the
intuitive aspect of sense understanding from reason.
As we have seen however, because this specialised ability is not even
recognised in conventional mathematical terms, genuine intuition, where it
arises, is simply reduced in rational terms.
So from a dynamic interactive perspective, both cognitive and affective dimensions
are necessarily involved in number experience, which are - initially - associated
with reason and intuition respectively.
Thus in the past few entries I have been attempting to clarify the proper
role of both the cognitive aspect (refined reason) and the affective
aspect ( refined intuition) in the analytical understanding of number.
Thus it requires a very developed form of intuition (where the affective can
be properly differentiated from the cognitive aspect) to recognise that the
ordinal recognition of number is not strictly provided by reason. Once again
when one attempts such recognition through reason, ordinal notions lose
their qualitative uniqueness and become thereby reduced in a mere quantitative
However there is also the holistic recognition of number (entailing the
simultaneous juxtaposition of complementary opposite reference frames of
And this also has both cognitive and affective aspects, which I will return
to in the next entry.