And this proved far from a passing concern. In fact I spend some considerable time thereafter in the attempt to "capture" the qualitative dimension.

Thus to focus clearly on this missing aspect, I considered a square with side of 1 unit (i.e. expressed in 1-dimensional terms). The area of the square is then given as 1 square unit (i.e. now expressed in a 2-dimensional manner). Therefore, though the quantitative result has remained unchanged as 1, clearly a qualitative dimensional change has taken place.

1

^{2}thereby represents the quantity of 1 (expressed in 2-dimensional units).
Then as all
real number quantities are ultimately expressed with respect to linear
(1-dimensional) units, I considered the possibility that the “dimensional”
could be thereby “converted” in a quantitative manner, by obtaining the two
roots of 1

^{2}.
This idea
of “conversion” in fact occurred quite naturally to me as much earlier, before
encountering them in maths class, I had - using 2 as a base - discovered the
principle behind logs, whereby multiplication could be “converted” to addition.

And the two
roots of 1 are + 1 and – 1 respectively. So, something
very strange seemed to be happening here, whereby two answers resulted that
were the direct opposite of each other.

Though I
was not able to properly understand this conundrum, it laid the firm foundation
for an eventual solution while attending University.

However
first year at University studying Mathematics proved a most disillusioning
experience whereby my former misgivings, regarding its reduced nature, were to
return in a much deeper fashion.

The course
involved extensive exposure to the notion of limits. Though apparently great
refinement had been brought to the modern discussion of the limiting notion, I
quickly realised that at bottom it was in fact a sham so as to enable treatment of
the infinite in reduced terms.

In fact
this represented just another important example of how a fundamental qualitative
notion is again reduced in a merely quantitative manner.

And the
relationship as between finite and infinite is completely unavoidable, as it
underlies every possible mathematical relationship.

For example
the concept of number potentially applies to every number
(and in this sense is infinite). However all actual numbers are necessarily of
a finite nature. Thus there is a dual sense in which the very notion of number
is used (entailing both finite and infinite aspects).

However the
infinite notion is then reduced in formal interpretation. So instead of the
realisation that this notion correctly applies in a holistic potential fashion,
the infinite is misleadingly identified in a finite actual manner.

From this
perspective, one therefore approaches the infinite through linear extension,
whereby it is considered as greater than any finite number.

However,
strictly speaking, this is just nonsense! What is greater than any finite
number is always another finite number! So we cannot meaningfully approach the
infinite in this manner!

Thus
infinity is not greater than any pre-assigned finite number (however large).
Also the infinitesimal notion is not less than any finite number (however
small).

This simply
represents the reduction of holistic (qualitative) notions in a reduced
quantitative (analytic) manner.

Then I
realised that this has important implications for what we know as mathematical
proof.

We might
maintain that a proof applies universally to all cases within its class.

So, the
Pythagorean Theorem for example (that the square on the hypotenuse of a right
angled triangle equals the sum of squares on the other two sides) thereby
universally applies to all triangles in this class.

However,
properly speaking there is an important distinction as between potential and
actual meaning.

So if we
say that the theorem potentially applies (infinitely) in all cases, this
strictly this does not entail any (finite) cases (in an actual manner).

Then when
we try to maintain the reduced conclusion that it applies in all actual situations,
we are left with an inevitable problem as the very notion of “all” does not
have a determinate meaning in finite terms.

Therefore
the very truth of a proposition (such as the Pythagorean Theorem) with respect to
a limited set of finite cases necessarily entails an unlimited set of other finite
cases which can never be determined.

Now this
not of course mean that there is no value in mathematical “proof” (as currently
pursued) but rather that such truth is of a relative - rather than
absolute - nature.

So an
uncertainty principle strictly attaches to the proof of every proposition
(which correctly must be understood in a dynamic interactive manner).

Therefore,
we have the inevitable interaction of two aspects, which are - relatively - analytic
(quantitative) and holistic (qualitative) with respect to each other.

Thus if we
attempt to concentrate solely on the analytic aspect - which represents the
present mathematical approach – we thereby reduce the infinite notion in a
finite manner.

If on the
other hand, we concentrate solely on the holistic aspect (as what is
potentially infinite), the finite then is elevated to the infinite notion, so that
we can no longer apply mathematical propositions in actual terms.

This
inevitable dynamic interaction as between analytic and holistic aspects of
understanding corresponds in psychological terms with the interaction of
conscious and unconscious aspects through reason and intuition respectively.

So reason
by its very nature tends to be analytic, whereas intuition is of a
holistic

nature.

So the
reduction of qualitative to quantitative type understanding in present
Mathematics directly complements the corresponding formal reduction of
intuitive to rational type interpretation.

Therefore,
while at University for me the bottom completely fell out of the mathematical
world (as conventionally understood).

I realised
then that a completely new approach was required, which - even if remaining in a minority of one for the rest of my life - I must pursue.

And this
was to have profound implications for the integration of mathematical with
spiritual type reality, for in the correct appreciation of the infinite notion,
Mathematics intrinsically possesses an inescapable spiritual dimension.

## No comments:

## Post a Comment