I do not generally refer much to childhood experience.
Partly this is due to the fact that I do not remember it as an especially happy time. Perhaps because of this, memories do not readily come to mind.
However recently one such memory has indeed returned in a vivid fashion suggesting to me an event that played a crucially important role with respect to subsequent childhood development.
Like so many young boys growing up in Ireland in the 1950's , my earliest interests centred strongly around sport. Indeed I have maintained this interest over the years with opinions to offer - to anyone who cares to listen - on a wide variety of different sports.
In my earlier years I also actively participated at home and school level. For years the back garden of our house was a popular venue where neighbourhood kids congregated to play football.
In early 1958 - at the age of 9 - I was playing Gaelic football regularly in the school leagues. Things were obviously going well, as I came to the attention of the main teacher in charge of football who picked me for the school team as a potential young "star of the future". I was then one or two years younger than most of my team mates.
However my first outing with the school team proved a disastrous experience. I remember on the morning of the match feeling deeply self conscious as if submerged at the bottom of a swimming pool. Needless to say it that state of mind I did not perform well. In fact the play on the field completely passed me by without my making a touch. Then after about 20 minutes I was mercifully substituted (by that same teacher who earlier had sung my praises). Sitting on the sidelines, I was oblivious to what was happening in the match, just wishing that the ground would open to swallow me up.
Later that afternoon, I remember standing at the front gate of my house looking up the street.
I realised then clearly that there would be no future for me in football, with the customary camaraderie of team mates. Something had fundamentally changed that I could not properly understood, where I stood entirely alone. And at that moment I felt far from any comfort and far from home.
Looking back now, I realise that what I was in fact experiencing was an existential crisis that was already defining the kind of future I could lead.
Now normally one does not associate such crises with young children who have not reached puberty. However - for me anyway - such a viewpoint makes little sense.
Just as it is accepted that it is possible in childhood to have a peak mystical experience (of temporary duration) likewise it is perhaps even more likely that a young person can have an intense peak - or perhaps more accurately - intense valley experience of a profound - though temporary - existential kind. And this is especially the case if one is destined to undergo such experiences on a more permanent basis later in life.
So looking back now, that day was perhaps my first introduction into - what I would in future recognise as - "the dark night of the soul" And there were many other similar occurrences (sometimes of much longer duration) before adult development seriously commenced.
However there was a compensating flip side to that event, the significance of which I only have recently come to properly recognise.
In Ireland in those days, students sat a state examination, called the Primary Certificate at the end of preparatory school. One of the subjects in that examination was Mental Arithmetic, which entailed the ability to make a variety of calculations without external assistance. The advent of pocket calculators was still some time away!
A few weeks after that match we were introduced in class to a number of exotic field measurements (most of which subsequently I have never heard mentioned). For example there was the furlong (which admittedly is important in horse racing). Then there were the more obscure terms of perches and roods and then the widely used measurement of acres. So 40 perches = 1 rood and 4 roods = 1 acre.
And most important of all, 1 acre = 4,840 square yards.
So therefore if we had a rectangular field for example with length 80 yards and width 60 yards the area of the field would thereby be nearly 1 acre (i.e. 4,800 square yards).
And then I had that big insight that has been central to my life ever since!
To illustrate, let us make it even simpler, by considering a small rectangular table with length 3 feet and width 2 foot. Now the area of that table is then 6 square feet.
However if we now consider the multiplication of numbers 3 and 2 the answer is given simply as 6!
In other words though we clearly can see in the case of the table that a dimensional change in the nature of the units takes place, (i.e. from 1-dimensional to 2-dimensional) when 3 feet is multiplied by 2 feet , in the case of the multiplication of the two numbers (i.e. 3 * 2) this dimensional change is simply ignored. Thus quite simply, when we multiply numbers a qualitative (i.e. dimensional) - as well as quantitative - change is necessarily involved in the nature of the units.
However, Conventional Mathematics overlooks this qualitative aspect of transformation.
So, I clearly realised in that eureka moment that the conventional mathematical understanding of the nature of multiplication is wrong (or as I would now say in more refined fashion, is of a greatly reduced nature).
And strange as it might seem, I resolved in that same moment that I would eventually get to the bottom of this fundamental issue.
Thus as I walked out of that classroom, I had already sown in my mind the seed for a radical reinterpretation of Mathematics (a task with which I am still intimately involved nearly 60 years later).
I now realise that these two defining moments in my life were in fact closely connected as death in one instance and rebirth in the other.
So the death of my sporting ambitions and the social comfort of being an accepted member of a group was to lead to a new solitary ambition (where I was confident that I could resolve a fundamental problem that others still refused to recognise).
I have never wavered in that belief. It would not concern me if the entire mathematical profession was to unite against what they might see as "a delusional obsession" as I have experienced that moment of revelation, where I could see clearly the gross reductionism of accepted mathematical thinking. One day others will confirm this when they too see the light.
Original discoveries in essence are very simple. However to make them it requires the capacity to stand outside the group (and its accepted conventional wisdom) to see reality in a new way that initially receives the confirmation of no one. And very few people are sufficiently willing to go against accepted opinion to possess this capacity.
Thus we are inclined to attribute original discovery with respect to ideas to inherent genius and great intellectual ability. However I would see the key requirement as this existential capacity to stand alone (whereby one can confidently affirm one's personally discovered truth against the world).
Then maybe on reflection such a capacity in fact does constitute one important aspect of genius!
Partly this is due to the fact that I do not remember it as an especially happy time. Perhaps because of this, memories do not readily come to mind.
However recently one such memory has indeed returned in a vivid fashion suggesting to me an event that played a crucially important role with respect to subsequent childhood development.
Like so many young boys growing up in Ireland in the 1950's , my earliest interests centred strongly around sport. Indeed I have maintained this interest over the years with opinions to offer - to anyone who cares to listen - on a wide variety of different sports.
In my earlier years I also actively participated at home and school level. For years the back garden of our house was a popular venue where neighbourhood kids congregated to play football.
In early 1958 - at the age of 9 - I was playing Gaelic football regularly in the school leagues. Things were obviously going well, as I came to the attention of the main teacher in charge of football who picked me for the school team as a potential young "star of the future". I was then one or two years younger than most of my team mates.
However my first outing with the school team proved a disastrous experience. I remember on the morning of the match feeling deeply self conscious as if submerged at the bottom of a swimming pool. Needless to say it that state of mind I did not perform well. In fact the play on the field completely passed me by without my making a touch. Then after about 20 minutes I was mercifully substituted (by that same teacher who earlier had sung my praises). Sitting on the sidelines, I was oblivious to what was happening in the match, just wishing that the ground would open to swallow me up.
Later that afternoon, I remember standing at the front gate of my house looking up the street.
I realised then clearly that there would be no future for me in football, with the customary camaraderie of team mates. Something had fundamentally changed that I could not properly understood, where I stood entirely alone. And at that moment I felt far from any comfort and far from home.
Looking back now, I realise that what I was in fact experiencing was an existential crisis that was already defining the kind of future I could lead.
Now normally one does not associate such crises with young children who have not reached puberty. However - for me anyway - such a viewpoint makes little sense.
Just as it is accepted that it is possible in childhood to have a peak mystical experience (of temporary duration) likewise it is perhaps even more likely that a young person can have an intense peak - or perhaps more accurately - intense valley experience of a profound - though temporary - existential kind. And this is especially the case if one is destined to undergo such experiences on a more permanent basis later in life.
So looking back now, that day was perhaps my first introduction into - what I would in future recognise as - "the dark night of the soul" And there were many other similar occurrences (sometimes of much longer duration) before adult development seriously commenced.
However there was a compensating flip side to that event, the significance of which I only have recently come to properly recognise.
In Ireland in those days, students sat a state examination, called the Primary Certificate at the end of preparatory school. One of the subjects in that examination was Mental Arithmetic, which entailed the ability to make a variety of calculations without external assistance. The advent of pocket calculators was still some time away!
A few weeks after that match we were introduced in class to a number of exotic field measurements (most of which subsequently I have never heard mentioned). For example there was the furlong (which admittedly is important in horse racing). Then there were the more obscure terms of perches and roods and then the widely used measurement of acres. So 40 perches = 1 rood and 4 roods = 1 acre.
And most important of all, 1 acre = 4,840 square yards.
So therefore if we had a rectangular field for example with length 80 yards and width 60 yards the area of the field would thereby be nearly 1 acre (i.e. 4,800 square yards).
And then I had that big insight that has been central to my life ever since!
To illustrate, let us make it even simpler, by considering a small rectangular table with length 3 feet and width 2 foot. Now the area of that table is then 6 square feet.
However if we now consider the multiplication of numbers 3 and 2 the answer is given simply as 6!
In other words though we clearly can see in the case of the table that a dimensional change in the nature of the units takes place, (i.e. from 1-dimensional to 2-dimensional) when 3 feet is multiplied by 2 feet , in the case of the multiplication of the two numbers (i.e. 3 * 2) this dimensional change is simply ignored. Thus quite simply, when we multiply numbers a qualitative (i.e. dimensional) - as well as quantitative - change is necessarily involved in the nature of the units.
However, Conventional Mathematics overlooks this qualitative aspect of transformation.
So, I clearly realised in that eureka moment that the conventional mathematical understanding of the nature of multiplication is wrong (or as I would now say in more refined fashion, is of a greatly reduced nature).
And strange as it might seem, I resolved in that same moment that I would eventually get to the bottom of this fundamental issue.
Thus as I walked out of that classroom, I had already sown in my mind the seed for a radical reinterpretation of Mathematics (a task with which I am still intimately involved nearly 60 years later).
I now realise that these two defining moments in my life were in fact closely connected as death in one instance and rebirth in the other.
So the death of my sporting ambitions and the social comfort of being an accepted member of a group was to lead to a new solitary ambition (where I was confident that I could resolve a fundamental problem that others still refused to recognise).
I have never wavered in that belief. It would not concern me if the entire mathematical profession was to unite against what they might see as "a delusional obsession" as I have experienced that moment of revelation, where I could see clearly the gross reductionism of accepted mathematical thinking. One day others will confirm this when they too see the light.
Original discoveries in essence are very simple. However to make them it requires the capacity to stand outside the group (and its accepted conventional wisdom) to see reality in a new way that initially receives the confirmation of no one. And very few people are sufficiently willing to go against accepted opinion to possess this capacity.
Thus we are inclined to attribute original discovery with respect to ideas to inherent genius and great intellectual ability. However I would see the key requirement as this existential capacity to stand alone (whereby one can confidently affirm one's personally discovered truth against the world).
Then maybe on reflection such a capacity in fact does constitute one important aspect of genius!
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