Early Questioning

An important early moment occurred at the age of 10, when during arithmetic class one day in Ireland, the topic related to areas of rectangular fields.

I remember thinking to myself that the area of a field of 80 by 60 yards would closely approximate 1 acre (4840 sq. yards). This was the era in Ireland before metric measurements became widely established! 

Thus in multiplying these two numbers, a change in the dimensional nature of the units thereby takes place from linear (1-dimensional) to square (2-dimensional) format. 

However it then occurred to me that in the customary treatment of multiplication, no account is taken of this dimensional change, with the resulting product represented on the real (1-dimensional) number line.


So for example, the primes are customarily represented as the “building blocks” of the natural number system. 

From this perspective 2 * 3 = 6 (with the three numbers interpreted in similar 1-dimensional terms).

However properly understood, with the multiplication of two or more numbers, a dimensional change in the nature of the units necessarily occurs.

Expressed another way, both a quantitative and a qualitative transformation in units is thereby involved.

So 2 * 3 = 6 (in quantitative terms).

However 2 * 3 equally entails a qualitative change in the dimensional nature of units (i.e. from 1-dimensional to 2-dimensional).

Therefore in the customary treatment of multiplication a limited - and ultimately distorted - interpretation is given, whereby the qualitative aspect is reduced to the quantitative.

In this way all composite natural numbers are assumed to lie on the same number line (as the primes from which they are derived).


Now this early unease with the conventional interpretation of multiplication, proved far from a passing phase.

By the time I got to college, it had developed into a full-blown revolt against the conventional mistreatment (as I saw it) of the infinite notion in mathematics.

I was beginning to realise now that all numbers possess two distinct interpretations (again directly akin to wave/particle duality in physics) which keep switching with each other in the dynamics of experience. 

For example when we use “1” to represent a dimensional power (or exponent), in a certain sense, its meaning is clearly finite (in an actual manner)


So, from this perspective with 2¹ signifies an actual part measurement of 2 units (with respect to the 1-dimensional real line).

However equally, “1” represents the whole line (to which the finite measurement of 2 relates), which is potentially infinite.

Thus the dynamics of understanding, in this dimensional context, entail that the meaning of “1” keeps switching as between twin notions, which are finite (actual) and infinite (potential) with respect to each other.


When looked at from a psychological perspective, the understanding of “1” switches as between rational and intuitive notions, which - though necessarily related - are very distinctive in nature. 

However, with respect to conventional mathematical interpretation, a continual reduction of the (potential) infinite notion takes place in an (actual) finite manner; this equally entails from the psychological perspective, a similar reduction of intuitive type appreciation in merely rational terms.

This could also be expressed by saying that conventional mathematical interpretation - certainly from a formal perspective - necessarily entails a reduction of true qualitative type understanding in quantitative terms, or alternatively, in the manner that I customarily employ, a reduction of holistic type meaning in an analytic fashion.

Note: It is important to appreciate that the term “analytic” here simply refers to the conventional manner of numerical interpretation i.e. in merely quantitative terms.
Therefore though it necessarily includes the conventional mathematical definition of analytic e.g. an analytic function, in the manner that I employ the term, it has a much wider usage in applying to all accepted mathematical interpretation.