Thursday, April 28, 2016

Approaching the Infinite

I mentioned that around the age of 10, I formed serious misgivings regarding the nature of multiplication. To my way of thinking, even at that young age, a comprehensive explanation required a qualitative - as well as quantitative - aspect. However - quite literally - the qualitative dimension was effectively removed from conventional interpretation.

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. 

12 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 12.

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 does 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 

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, the bottom completely fell out of my 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.

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