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Where
Zero Is Hero; And Vice
Versa
The
Weird Long-March of Wonderful Mathematics
Does
the road wind uphill all the way?
Yes, to the very end
Shall I find comfort, travel-sore and weak?
Of labour you shall find the sum
--
Christina Rossetti, 1830-94
It
was there to usher in Civilisation, with which it has
remained ever entwined, through many a twist and turn; not
free from human behaviour. It has been inspired by innate
illumination, imagination and ingenuity, alongside elevated
learning and scholarship, in spite of ‘unspeakable’
monstrosities (e.g. curves without tangents), frailties,
foibles and scandals, intellectual and otherwise. Indeed,
from murky beginnings through mazy trails, it has become
majestic and ever-mighty. A language, yes; but not everyone
agrees. Yet, a distinguished mathematician Karl Weirstrass
(1815-97) averred that a ‘mathematician who is not also
something of a poet will never be a complete
mathematician’. Jacob Bronowski (1908-74) wrote about
the poetry of mathematics and science in his classic text
and film The Ascent of Men. Others see beauty and
freedom in mathematics. And the latter has a long
multi-faceted history (see Annexe online: Mathematical
Medley). Yet all along, one haunting philosophical
thought has lingered: does mathematics exist solely in the
human mind, pur et dur, or it is free-living in
Nature?
Something
Special
The
origin and functioning of society crucially depend on the
need for the latter to collect, produce, distribute,
protect, and store food. In turn, those have necessitated
language -- humanity’s, most distinctive and important
trait. The latest studies thereon appear to indicate the
following. First, human language probably arose from our
upright posture, freeing hands, and leading to a bigger,
more complex brain over the last 40,000 to 100,000 years.
Second, the distinction between human and non-human
communication is more blurred than often thought; third, the
difference between ‘language’ and ‘non-language’ may
be more in terms of degree than of kind; and fourth,
multi-stranded language brings together the ‘expressive’
and the ‘non-expressive’, the ‘verbal’ and the
‘non-verbal’.
Thus
any ordinary language, essentially, vague, does evoke
emotions, and results in :
--
‘small’, readily understood words, used by everyone
conversant with that language e.g. tree, forest, water,
waterfall, as well as
--
‘big’ words thus, phenomenalism, transcendalism, which
necessitate the use of a dictionary, necessarily circular
(since the latter attempts to define every word, when that
very word includes those used in the latter’s own
definition).
To
remove ambiguity and tone down emotion considerably, we make
use of a ‘special’ language e.g. logic or mathematics.
Logic relates to ‘rules of thought’, of ‘right
reasoning’, and of ‘valid argumentation’ concerning
the truth of statements, based completely upon the meaning
of the terms they contain. Logic was invented by Aristotle
(384-322 BCE) whose views held sway for almost two thousand
years, until the 19th century, despite René
Descartes (1596-1650, I think, therefore I am) and
Wolfgang Liebititz 1646-1716 (the ‘one-man academy’),
among others. Change came – inevitably, with George Boole
(1815-64) and his ‘An Investigation of the Laws of
Thought; Augustus de Morgan (1866-71) concerning
relationships, rendered axiomatic, not least by
philosopher-mathematicians Bertrand Russell (1872-1970) and
Alfred North Whitehead (1861-1949). By that time, logic and
mathematics had become inseparable.
Numbering,
Shaping, Relating
Unlike
the earlier-civilised Babylonians and Egyptians, the ancient
Greeks, in their quest for truth, tried to avoid building on
the crude and the empirical. They searched for ‘unchanging
entities’ and ‘permanent relations’. Thus, from
stretched strong, boundaries of fields or even light rays,
they discovered the mathematical straight line, without
thickness, colour, molecular structure or tension. Their
mathematics became highly abstract; it was the latter, they
felt, that constituted the ‘reality’, while the
empirical was but a crude approximation of that reality.
Since
they had to start somewhere, they inevitably had recourse to
the ‘undefined’ such as ‘point’, ‘line’, and
‘number. From those notions, they derived more complex
ones, such as ‘triangle’, ‘square’ and ‘circle’.
The Greeks insisted that axioms (self-evident postulates) be
laid down to be followed by rigorous deductive reasoning
(proof) towards conclusions. Unfortunately, they also found
that common whole numbers (1,2,3 … at their simplest) were
insufficient, and in the process, discovered irrational
numbers, seemingly without ends, implying infinity. They had
discovered the ‘Unspeakable’ (Arrhetos), and some
of them even threatened to kill any one would reveal such a
secret. They knew from practical (empirical) experience that
a right handed triangle of side 1 would have a longest side
(hypotenuse) of √2, an irrational number
(1.414……); yet one that could be measured accurately.
Number had turned suspect. Shape was seemingly more
reliable.
Around
2500 years ago, Hippocrates of Chios composed his Element
of Geometry, a century prior to
that of Euclid (330? 275? BCE). But, it was the
latter who was to achieve universal and lasting fame, with
his publication, in several variants, destined to become the
most published mathematical textbook the world has known so
far. Despite certain shortcomings, the overall impact of
Euclidean geometry on intellectual thought proved immense,
neatly encapsulated in the saying ‘Euclid is the Truth,
and the Truth is Euclid.’
And
yet, Euclid had grave misgivings about his Parallel Axiom
(fifth Postulate). He knew that whatever it was, it was not
self-evident. His own version of his fifth axiom ran thus: ‘If
a straight line falling on two straight lines makes the
interior angles on the same side less than two right angles
then the straight lines will meet on the side of the
straight line on which the angles are less than two right
angles.’ No mention of infinity; but it was there, all
right, that scourge of the ancient Greeks. And keeping all
the other assumptions intact, but only altering the Parallel
Axiom was to lead to other equally valid geometries, as
found by C.F. Gauss, 1777-1858; J. Bolyai, 1802-60; and N.
Lobatchevsky, 1785-1856, on the one hand (on
‘saddleback’ surfaces, with the internal angles of a
triangle being less than 1800 and on the other by
B. Reimann, (1936-66, on global surfaces with the sum of the
internal angles being more than 1800). A single
Euclidean geometry (internal angles of plane triangle equal
to 1800) might have shown the way to the Truth,
but with more than one available, the foundations of
Mathematics came to be again perceived as insecure, even
when based on geometry i.e. on space notions.
Mathematics
had seemingly, reached a kind of impasse. Should the
emphasis shift back to number? Yes, if the latter could be
placed on a more rigorous basis; with the concepts of
‘real number’, ‘continuity’, ‘limit’, and
‘infinity’ made more precise and consistently usable --
with logic sorting out various problems by the end of the 18th
century, including those of relations (see annexe online:
‘All is Number’ – dream Pythagorean).
In
the process, Euclidean geometry also was to be placed on a
thoroughly rigorous basis e.g. by David Hilbert (1862-1943).
Or so it seemed. Then, came Kurt Godel (1906-78), probably
the greatest logician in the 20th century. He it
was who showed that logical principles implicit in existing
mathematical systems could not prove the consistency, or the
completeness, of the latter. Confusion had, indeed, become
worse confounded; logic had turned out to be its worst
enemy.
Initially,
logic had emerged as a result of the probings of the very
basic empirical practices of mathematics. Twenty-three
decades later, it was being used to delve into the
theoretical foundations of mathematics, to place the latter
on the sort of rigorous basis that the very epitome of
reasoning warranted. All this culminated in Mathematical
Logic of the kind of Principia Mathematica, 1910, by
Bertrand Russell and Alfred North Whitehead: ‘Logic is
the youth of mathematics and mathematics in the manhood of
Logic’. Altogether then, logic, had become more
mathematical, while mathematics had become more logical --
it was no longer possible to draw a line between the two –
in fact, the two had become one. And defects
notwithstanding, the redoubtable duo forged ahead with
renewed vigour.
At
the end of the 20th century, Algebra, Analysis
and Topology were generally deemed to be the main groups
of prevailing mathematics, with any mathematical expert
being unable to master more than one-tenth of the overall
subject, and to discuss intricacies of advanced work with
more than three dozen fellow experts, in the same subfield,
or closely allied ones. While in the late 19th
century, 12 sub-divisions of Mathematics were recognised,
the corresponding number at the turn into the 21st
century, was 3000, in most of which new mathematical
knowledge was being increasingly created, with an annual
production of more than 200,000 hand-crafted theorems.
Scandal!
Scandal!
Twixt
Politics and Religion
Pythagoras
(e 585-500 BCE) and his followers, blended mathematics,
music, philosophy and religion into a ‘whole’ on the
general theme of ‘All is Number’. But they had to flee
for the lives, not primarily because of ‘unspeakable’
irrationals and their threats to kill anyone disclosing the
latter, but more prosaically, because they meddled into
politics – not for the first time probably and not for the
last, almost certainly. Much later, Omar Khayyam
(c1040-c1125) austere philosopher-mathematician, ran into
trouble with religious authorities whom he accused of
‘having lost contact with reality’ – again, something
not unfamiliar. Of course, religious authorities were often
adamant, especially in the west. Thus St Augustine
(340-430), celebrated Bishop of Hippo, near Carthage in
North Africa, having reconciled the tenets of the teachings
of Plato, (427-349, BCE, the ‘maker of mathematicians’)
with the values and principles of Christianity, nonetheless
felt it necessary to warn that mathematicians might have
worked out ‘a convenant with the devil to darken the
spirit and to confine man to the bonds of Hell’.
Elsewhere
and earlier in China, the Thang Annals was recording
that ‘mathematicians, surveyors, physicians and
magicians were charlatans. The sages did not regard them as
educated!’ No wonder St Thomas Aquinas (c 1244-74)
chose the views of Aristotle (384-322 BCE) more prone to
science, in order to blend with Catholicism to formulate
scholasticism, initially beneficial to the intellect but
eventually unduly constraining.
Oh
Calculus!
Lietnilz
and Newton, shared deep antagonisms, beyond rival claims for
the paternity of the Calculus, with the former accusing
Newton of contributing to the decline of religion, in
England. In parenthesis, Liebnitz was eventually to render a
remarkable tribute to Newton, when he said that of all the
mathematics up to the time of Newton, the most important
half was due to the latter. Incidentally, John Maynerd
Keynes the celebrated economist, after examining the record
of Newton in 1947 in alchemy, religious texts and chronology
of Ancient Kingdoms described the latter as ‘the last
of the magicians’.
Those
who had been advocating a ‘completely rational religion’
(Deism) saw therein reason as God, the Principia of
Newton as the Bible and Voltaire as the Prophet. By then,
George Berkeley Bishop of Cloynes (1685-1753), had become
gravely concerned with the emphasis in society of material
aims, threatening both God and Soul. He chose to attack
through the Calculus and the process of differentiation (to
obtain the slope or gradient of a curve, or more generally
‘derivatives’ – ghosts of departed quantities, quipped
Berkeley) felt by many to be akin to dividing zero by zero
– not acceptable in Mathematics. However, the Calculus was
working successfully and expediency became the order of the
day. For Berkeley, such derivatives would not be allowed in
Theology; those using them in calculus should therefore, ‘not
be squeamish about any point in divinity’,
mathematicians ‘so delicate in religious points’
should be equally so in Mathematics – in the latter do
they ‘not submit to authority, take things on trust,
believe points in conceivable, have their mysteries,
repugnances and contradictions?’ It was not until late
in the 19th century that the Calculus was placed
on a rigorous basis; until then, it was used, perforce,
because it was, somehow, most successful.
For
that formidable counter-attack for God and Religion, George
Berkeley no doubt deserved a sainthood.
The
Future: Patents Pending
A
‘patent’ confers a ‘right‘ or ‘title’ to make
use or sell some invention; to be ‘patentable’, the
latter must be non-obvious, novel and industrially
applicable. The laws of nature, Newton’s Gravitation
or Einstein’s E = mc2, or a new plant
discovered in the wild should not be patentable,
regarded as free to all people. To the bitter controversies
on living organisms, new ones have emerged concerning
patents well beyond ‘products and processes’ of
yesteryear to include for example, whole chunks of
mathematics, especially those closely linked with computer
and information technologies or with finance and insurance
generally. The mind simply boggles at the prospects and
possibilities available, when one considers that even angles
in solar-powered cookers are patentable.
There
is however a current world shortage of top class
mathematicians, a problem being actively addressed and
pursued in many parts of the world, not least in
Scandinavia, the economic Tigers of South East Asia, other
Asian giants and in Eastern Europe, among others.
In
Dodo-land, there is also need to remedy mathematical
deficiencies in Education at all levels and in research
notably in the Environment, Food, Health and Economic
Development interfaces.
Tremulous
Eyes: Paradise Unlimited?
Clearly,
Mathematics is not simply a compendium of procedures and
operations to be mastered (or else!), a sort of
‘intellectual cookery book’ or its equivalent. It is
more, far more. And the more one appreciates the deeper and
widest aspects of the subject, the better it will be for all
of us. Perhaps we should reconsider new methods by borrowing
from the ancient world. Thus Baskara in 12th
century Ujjain would pose mathematical problems starting
‘pretty girl with tremulous eyes, if thou knowest the
correct method of inversion…’ And he was the most
original and influential scholar of his time, in India. So
why not?
Why
does Mathematics work? In part, it is because of its origin
in empirical experience. And, yet, why should long chains of
pure reasoning produce such remarkably applicable
conclusions? Is it because Nature is inherently mathematical
in design? Or does ‘reality’ simply reflect our own
thoughts à la Immanuel Kant? Or do we simply
‘modify physical laws to make the mathematics fit’?
And
yes! Is it solely a product of the mind or does it exist
freely in nature? The jury is still out on that question,
and is likely to be out for a long time yet, possibly
unravelling the following quatrain from Omar Khayyam in The
Rubaiyat (c1040-c1125) as decrypted by Edward Fitgerald
(1809-83):
Into
this Universe, and Why not knowing
nor
Whence, like Water willy-villy flowing;
And
out of it, as Wind along the waste,
I
know not whither, willy-willy blowing
Prof
J. Manrakhan
26 August 2008
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