I have mentioned before how a future golden age of Mathematics will contain at least three distinctive ways of interpreting mathematical symbols.
1) The conventional rational approach based on the quantitative
interpretation of mathematical symbols in a conscious manner.
2) The - largely - unrecognised intuitive approach based on the qualitative interpretation
of mathematical symbols in an unconscious manner. Though this approach is
indeed directly based on refined intuitive type recognition that cannot be
successfully reduced in standard rational terms, indirectly it can be intellectually
translated in a (circular) rational manner (entailing paradox from a dualistic perspective).
3) the comprehensive radial approach based on the mutual interpenetration,
in a coherent integrated fashion, of mathematical symbols in both a conscious
and unconscious manner.
However the great surprise that awaits entails the additional recognition
that all mathematical symbols have both cognitive rational and affective sense interpretations.
So comprehensive mathematical appreciation of symbols entails the emotional
as well as rational domain!
Indeed ultimately it entails also the volitional domain as the very means
for successfully harmonising - relatively - both conscious and unconscious
aspects with respect to cognitive and affective aspects is through the
volitional aspect (i.e. will).
Now in this context it would be helpful to carefully distinguish both these
two aspects of the psychological recognition of mathematical symbols through a
Imagine one is looking at 3 cars (say parked in a driveway)!
The cognitive recognition here relates to the common collective identity of
the cars (as belonging to the same class).
This directly concurs with the cardinal notion of number (where each unit of
the number in question enjoys an impersonal homogeneous identity in quantitative
So 3 = 1 + 1 + 1.
However affective (sense) recognition is quite distinct in relating to the
unique individual identity of each car (arising from their ordinal relationship
with each other.
.This then directly concurs with the ordinal notion of number where each unit
now enjoys a distinct personal identity in a qualitative manner.
So from this perspective 3 = 1st + 2nd + 3rd!
In conventional mathematical terms, the latter each interpretation is
simply reduced in a cardinal manner. So the personal unique identity of each item
- corresponding initially with sense recognition of an affective kind - is
However properly understood - when we recognise the true complementary
nature of both aspects of number recognition - cognitive and affective aspects
are necessarily involved in the dynamics of all number recognition.
Indeed these dynamics relate directly to the true relation of whole and part
(and part and whole).
Once more each prime, from the conventional mathematical perspective is
considered in a quantitative whole manner (where all units are considered as
homogenous and thereby lacking any qualitative distinction).
So again to illustrate, 3 (as a prime) = 1 + 1 + 1 (where the quantitative
units lack any qualitative identity).
This concurs with the standard rational (i.e. cognitive) interpretation of
number where each prime is considered as a "building block" of the cardinal
natural number system.
However, from the complementary (unrecognised) perspective 3 (as a
prime) is uniquely defined by its ordinal members in natural number
So 3 = 1st + 2nd + 3rd.
Now here, in reverse, 3 (as the unique combination of individual ordinal
units ) strictly lacks a quantitative identity. So 3, in this context,
properly relates to "threeness" (as the qualitative nature of 3).
However this latter qualitative recognition i.e. that number units bear a
necessary relationship with each other, pertains directly to sense recognition
(of an affective kind).
So when one fails to recognise the necessary interaction of both cognitive and
affective recognition with respect to each prime, a fundamentally distorted interpretation
of the relationship of whole and parts results.
So in conventional mathematical terms - reflecting the dominance of the
merely cognitive (rational) aspect of understanding - the number system is interpreted
in a merely reduced quantitative manner (where primes are unambiguously viewed
as the "building blocks" of the natural numbers).
However when one properly allows for the corresponding affective (sense)
aspect of understanding, the number system is likewise seen in a true qualitative
manner (where each prime is defined by its natural number members in an ordinal
So from the customary analytic perspective, the relationship between the
primes and natural numbers (and natural numbers and primes) is considered in a
one-way unambiguous manner.
So in standard (Type 1) terms each prime serves as a quantitative "building
block" of the natural number system a cardinal manner.
Then in corresponding (Type 2) terms each prime is already uniquely defined by its natural number mebers in an ordinal manner.
Then the simultaneous recognition of both Type 1 and Type 2 aspects (i.e. Type 3) requires
true holistic understanding in the inherent understanding of the number system in
a dynamic interactive manner where both aspects - that appear unambiguous from
within each reference frame considered in isolation - now appear as deeply
paradoxical (when indirectly conveyed in a circular rational manner).