End of an Era
Sequel to ‘The Square Root of –1’
Assessing mathematics on two fronts, application and education, in the former, where calculators and computers apply it to machinery, it underpins civilization on this planet and reaches beyond to the stars. In teaching it is far behind, caught in the legacy of a dead and dictatorial past, its ultimate nature not understood.
We could even assume that the coals of the Flat Earth controversy are still hot.
In mid-16th century Copernicus waited until he was on his deathbed before ordering the publication of his reasons for believing in a global earth. His work was condemned as a heresy and banned on pain of death until the Inquisition closed its doors in 1820. The attack upon his findings did not cease however until the Flat Earth Society also shut down just before the outbreak of the First World War.
Almost three hundred years after Copernicus, and a few years after the Inquisition closed, Lobachevsky was dismissed from his professorial position in the Moscow University and his work suppressed for publishing a non-Euclidean geometry. Bolyai in Hungary, who published a similar paper recanted under pressure. He distanced himself from his findings and escaped with no more than a reprimand. Then came Riemann’s work on circular geometry. It was too scholarly for the troublemakers to understand so it caused no waves, but it provided the underpinning for the derivative mathematics that led Einstein to his epochal findings that set science and the world on a new course.
Something however is still missing, and in the trickledown it is the conception of a sense of perfect circularity in counterpoise to Euclid’s worldly straight. With the flat earth dismissed, and along with it, its mathematical justification, Euclid’s Fifth Postulate, the battle, it might be thought, could stop right there, but there is a tag end. The straight and the circular are relative, and Euclid’s postulate, although incorrect in the form of its original expression, is not to be dismissed altogether.
In 1979 I published my circular mathematical system as a mental arithmetic. Capably assisted by my son Robert we took it along to the heads of the Education Departments in Wellington and Auckland, and Teacher’s Training Colleges thinking it would be uniformly accepted. It met with interest and we were invited to lecture to the Colleges, Mathematical Association and schools. However there were two currents; welcome at the grass roots level, reservations caveats and stipulations from the top. Finally these hardened into prohibition to publish or teach in the school system. Eventually we realized that it was par for the course.
In support of their position the school Math Advisers held that in a universe containing gravitating bodies the path of light is curved. There is therefore no such thing as a straight line to infinity, nor any criterion for determining it. Without an anchor in the world of reality, Euclid's Fifth Postulate is void. We can forget it.
This however was going too far. There is ideality, and within it two extremes, the perfect straight and perfect circularity. Our thinking proceeds in the grip of opposites, and we are looking at a foundational pair in the domain of mathematics. Each is built into our thinking, where they oppose, reflect into and support each other. Euclid’s straight to infinity is ideal and expresses this ideality, and the ideal is real.
If the ideal straight did not exist in our mind how would we know of the existence of any curvature at all if we had no criterion, no inherent sense of the straight against which to measure it? The ideal straight is the criterion of the ideal curve, and each serves this purpose to the other.
Ideality, and with it, soul mind and consciousness are real. We, as humans, think, and this power of thinking emanates from its inherent self-retirement into invisibility, its power of obligatory self-effacement. It has its ground however, and its mechanism. It is there. Dismiss the ideal and we dismiss the very platform on which we stand.
As a criterion the ideal straight is perfect, for imperfection would cancel its very nature. It needs however, a counterpoise, and this equivalent but opposite idea is that of perfect circularity. Infinite circularity is the template of thinking in the mind. The perfect or ideal circle becomes the perfect and ideal straight, when, in the process of generating a world in appearance the mind subtends the straight as that world’s foundation. The circular then founds the mind; the straight the world. Inverting everything, the mind, as object to itself, becomes the object of our knowing, wherein we say that our sense of reality exists.
Returning to the historical development, after Lobachevsky and Bolyai mathematicians quickly presented new work in support of their findings. However it was high-tech and incomplete, and this left teachers facing a gap in mathematics they could neither fill nor conjure away. The Flat Earth was gone, and Euclid's straight with it, but through all this cleansing a coercive attitude lingered like the smile of a Cheshire cat. The presentation of mathematics continued to be rote and ritualistic.
Mind is the anchor of the world, which it projects tour-de-force as a psychical and metabolic accomplishment.[i] The world in turn anchors our sense of existence as thinking beings. We therefore put it first. In this boomerang sequence, the straight is the anchor of the curved insofar as we can be cognitively aware of it, and the infinite straight is the anchor of infinite curvature. The fact that the absolute and infinite straight is ideal does not, as said, expunge it from reality. It is, at bottom the foundation of our sense of reality.
Euclid's Fifth belongs per naturam to the logical foundation of reasoning mind. The key point in resolution is to see that ideality, in the sense of subjectivity, is as essential as objectivity in the foundation of reality. The one stands or falls with the other. All that was needed was a re-consignment, setting out the role and contribution of both sides, not condemning one or the other to a rubbish bin death. Euclid’s straight will live on as long as our understanding, but it only becomes sensible when partnered with its soul mate, the perfect circle. The straight, to use poetic language is the King of the world; the circular is the Queen of the mind.
* * *
Standing in line for first aid is ‘i’, the square root of -1, but this has already been covered in “The Square root of –1”. The following diagram from that work says it all (well, not quite:)
Figure 1
–1 is not a number but a direction to travel anticlockwise 1 place. In 10-circle we can see that it comes to 9, whose square root is 3, and also 7 in that circle. This statement, as the parent article declares, “is the touch of death for the mystery sequestered in the square root of –1.” i, the ‘imaginary number’ will never be the same again. It will retain its importance, but as a fulcrum between subjectivity and objectivity, being in itself neither imaginary nor a number. “Even for those who are far from grasping the pattern of the whole, the illusion of i, the square root of –1 will begin to fade and will continue to melt away until it vanishes.”
The Sign Laws
School math stands foursquare on its sign laws, but the building is askew. Hammered into place over centuries it is a pariah bereft of logical explanation, anathema to children and teachers alike. Perched on top like a penthouse, higher math teeters one-legged on a mystery, the ‘square root of –1’, like a modern Tower of Babel about to fall over.
The sign rules, ±x * ±x = ±y, are sound. Here they are, applied to the most simple multiplication:
(a) +1 times +1 = +1 –1 times –1 = +1
(b) +1 times –1 = –1 –1 times +1 = –1
The four shown exhaust the field. In (a) the factors are identical. In (b) they are different, because +1 ≠ –1.
Now, a square root is a number that, multiplied by itself, gives a given number. The given ‘number’ here is –1. We are looking for the square root of –1, an identity that, when multiplied by itself will give –1.
In line (a) above we find, +1*+1 and –1*–1. This meets the first condition; something multiplied by itself, but in each example the result is +1. The desired –1 result does not appear.
It does appear in line (b), but in each case, +1*–1 and –1*+1, the first condition, ‘multiplied by itself’ is lost.
It follows that ‘–1’ is not a square in the context of mathematics as currently taught. This makes ‘i’, the √–1, not only imaginary, but impossible and illogical. The failure applies to the square root of any and every negative amount. It attaches, in other words, to the subject in its very presentation.
Against this, mathematicians across the ages have discovered that if the square root of –1 is allowed a new discipline arises, though it does not fit into our conception of reality, for although as a mental discipline mathematics arises in the mind, like the mind its products must apply to something tangible in the world. We must re-examine the question then; to what, in the mind/ world relation does ‘i’, the imaginary number apply?
Does anything in our mundane reality correspond to the formulation that consigns mathematics to related but discrete sides, one side readily known as natural, existing; the other, as an epiphenomenon, ‘higher’, and between them an enigmatic relation?
The answer is Yes! The mind itself takes this form. We have in consciousness a world of objects, things that we can count and treat mathematically. Sense, understanding and its content are within consciousness in a totality we call mind, which allows us to know and interact with the objects that constitute objectivity; an existent world of objects in motion and relation, and knowledge of that world, so knower and known.
As set out in ‘The Mathematical Foundation’, mathematics reflects not only upon objectivity but also upon the mind in consciousness. In the process of establishing the details wherein the mind and world interact within the framework of understanding it establishes a mathematical foothold within the structure of mind itself.[ii]
Appendix
Number means, ‘more than 1’. 0 and 1 double as numbers, but logically they represent the mind-world link. 0 is the symbol for understanding and for mind itself,[iii] as in subjectivity. 1 is the same for the world in objectivity. Together they constitute the dynamo of sense and understanding. 0 anchors number in the world and takes the whole back to the mind. 1 is the mind's comprehension of unity or functional integrity in the world and its every object. In circular array each number has a clockwise and anticlockwise value. Thus +1 in 10-circle is –9.
Figure 2
The compact, opened out (fig. 3), expresses the mind-world relation, mathematically joined, corresponding and distinct.
Figure 3
The circular presentation overcomes the confusion that attaches in school math to the teaching of negative numbers, a confusion that has only advanced in the passing years since Fibonacci introduced them in the 12th century to explain debt, and their introduction into calculation in the 15th and 16th centuries.
Conventional math stacks what it calls positive and negative numbers (as if they were different species) like sheep in a pen to the right and left of 0, as counting lines to infinity, one a line of positive numbers, matching another of negatives running the other way, unable to see how they could come together in a circular completion.
Numbers, it says, are ‘concepts of quantity’, and negative numbers are concepts of quantities less than zero. Linguistically this makes sense, but not logically, other than within the confines of the mathematical orthodoxy, which is subject to the charge of an inability to fit positive and negative numbers into a rational framework anyway.
The remedy is to see that mathematics, in truth, is an expression of the physiology of mind, whose touchstone is circularity and understanding. The world reflects the mind in systematic inversion, and in the world the straight is king. Mechanically, mathematics reaches for the stars. In terms of comprehension and teaching it limps along on 16th century crutches.
Stephen W. Taylor MbChB © 2004 July 17