Though with some misgivings - as I was already at odds with its fundamental rationale - I opted to pursue a degree in Mathematics at University (together with Economics).

Very much in the context of the times (mid to late 60's) in Ireland it was a broadly based Arts degree, which also included as subjects English Literature and Latin. Looking back, I consider that this in fact proved very healthy for my overall intellectual development and would consider that present degree programmes have in the main become much too specialised in an increasingly narrow manner.

However at the time I did not enjoy the experience at all (especially the 1st year) as my growing conflict with standard approaches came to a head not only in Mathematics but equally with respect to Economics.

With the benefit of hindsight it is now much easier for me to understand what was really happening at this time.

I would put it like this! Just as electromagnetic energy can be viewed as a spectrum with many distinctive bands, properly understood intellectual understanding should be seen in the same light.

However standard intellectual discourse especially with respect to Mathematics and the Sciences is presently heavily concentrated on just one narrow band of the overall spectrum.

This is what I refer to as linear (1-dimensional) understanding. In any context for discussion, this approaches truth in terms of dualistic notions based on just one polar reference frame. So typically for example in Mathematics the external (objective) is abstracted from its internal (subjective) pole; likewise the independent (quantitative) is likewise abstracted from the interdependent (qualitative) pole.

However just as natural light forms just one small band on the overall electromagnetic spectrum, properly appreciated, such dualistic linear understanding likewise forms just one small band on the overall spectrum of possible mathematical and scientific understanding.

Now, in the various spiritual contemplative traditions, it has long been recognised that many further stages of potential development exist with respect to intuitive understanding of an increasingly refined nature. Indeed detailed accounts testifying to the universal features of such states (both East and West) have been made available by the spiritual pioneers of the various religions..

However what has not yet been properly realised is that these advanced intuitive states of an increasingly holistic nondual nature, have not only relevance in a contemplative spiritual context, but equally have enormous implications for the advanced appreciation of all mathematical and scientific relationships.

So a major breakdown with respect to the rigid dualistic notions that especially inform our existing understanding of Mathematics (and related Sciences) was taking place with respect to my personal development during this period.

Thus I was already moving towards this distinctive holistic approach to Mathematics (and also Economics) for which no formal recognition whatsoever existed in the culture (especially at University level).

I remember for me it proved an intensely lonely and disillusioning experience with a marked decline in ability and motivation for what was conventionally acceptable, together with a deeper emerging holistic appreciation (in which no one displayed the slightest interest).

In fact I found my time at University pursuing Mathematics such an alienating experience that it led me to drop the subject altogether at the end of the year.

However, a much deeper philosophical questioning regarding the standard assumptions of Mathematics had taken place which especially centred around the treatment of the infinite notion.

I could readily see that that this key notion was being treated in a reduced rational manner, which was strictly nonsensical.

For example it was customary to define an infinite number as greater than any finite number. But to me this lacked any coherence! If one designates any finite number in actual terms (regardless of how large) another finite number necessarily can be found that is greater than it. Thus we cannot meaningfully approach the infinite notion from an actual finite perspective. Therefore a key qualitative distinction separates the finite from the infinite notion with both necessarily occurring in a dynamic interactive framework.

So, for example, the infinite notion (with respect to number) strictly applies in a merely potential manner to all numbers, whereas the finite notion always necessarily applies in a specific actual - and thereby limited -context.

Crucially therefore, both notions strictly can only meaningfully apply to number in a dynamic relative (i.e. approximate) context.

Indeed there were strong parallels here with the existential crisis I was going through at the the time with all the comforting belief systems in my life seemingly breaking down. In such a situation one searches for a deeper resolution to the problem through developing an authentic belief system, intimately based on personal conviction. So I now was led to adopt the same approach with respect to Mathematics with a willingness to challenge all conventional notions that did not stand up to serious scrutiny.

In fact this experience, though chastening at the time, was to quickly lead on to a truly dynamic interactive appreciation of the nature of the number system.

Initially this understanding was based heavily on philosophical type understanding. However I was later enabled to interpret it all coherently in a new holistic mathematical manner.

So the dynamic interaction of finite and infinite notions is inseparable from our very understanding of number.

The number concept is infinite in the sense that it applies potentially to all numbers. However any specific number identified (as a corresponding perception) is then necessarily of a finite nature.

Thus in dynamic terms, the very ability to posit (or determine) specific numbers (as perceptions) always implies another set of finite numbers (that always - by definition - must remain undetermined). In this sense our knowledge of number necessarily takes place against a background of uncertainty.

From this dynamic perspective, it makes no sense to attempt to approach the infinite notion from a finite perspective. So once again they are qualitative and quantitative with respect to each other.

In terms of experiential understanding, the very appreciation of the infinite is directly associated with intuition, whereas finite notions are directly interpreted in a rational manner.

Thus the dynamic interaction of both finite and infinite in number terms is replicated in psycho spiritual terms through the dynamic interaction of both reason and intuition.

However because of the reduced nature of Conventional Mathematics, in formal terms this relationship is defined in a merely static rational manner. Thus the inevitable interaction of reason and intuition in experience is reduced in merely rational terms. This of course equally implies that the true - merely potential - nature of the infinite notion is likewise reduced in a merely finite actual manner. This then leads directly to the confusion of the (potential) infinite with (actual) finite notions.

Therefore to put it briefly, I quickly realised that Conventional Mathematics - despite its appearance of great rigour with its modern treatment of limiting notions - could only operate by effectively reducing the infinite notion (in every context) in a finite rational manner. This much sought for rigour was in fact at bottom just a sham, diverting attention away from the crucial limitations imposed by its reductionist approach.

This opened up for me the much greater issue of how finite and infinite (or alternatively quantitative and qualitative) notions can be consistently related with each other within a mathematical context..

Indeed properly understood, this is the very issue to which the Riemann Hypothesis points.

However, this crucial overriding need to establish dynamic consistency as between quantitative and qualitative, is not even recognised within Conventional Mathematics (as its axioms are already based on the reduction of qualitative to quantitative meaning).

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ReplyDeleteThank you for the kind comment, which I appreciate very much!

ReplyDeletePeter