Milchar
JuneJuly 2003 issue

The Natural IceLingam of Shri Amarnathji
(Photo
from ‘Kashmir’ by Francis Brunel)




Acedemics
… Tribhuwan N. Bhan
Revolution in Mathematics
Newton's
monumental work 'Calculus' opened new avenues for mathematics in 1966.
Though mathematicians remained busy exploring new fields of mathematics,
no one took pains, or perhaps they had no time to organise this
fastexpanding intellectual discipline. Due to the expansion in the
content of Mathematics, the variety of problems that it could deal with,
also expanded. This variety made largescale unification of its branches
rather difficult to attain.
Before the middle of the 19th Century, mathematics were
exploring new ideas. They made very little effort at organising the
subjectmatter or unifying the various branches of Mathematics. Euclid's
'Elements' however represented a major synthesis and Descartes'
'Analytical Geometry' was indeed a great unification of Algebra and
Geometry. That is all that can be said about the effort at the
'unification' before the middle of the 19th Century.
Then came the middle of the 19th Century. It heralded a
reaction, a change, a reformation and a reorganisation in Mathematics. All
this marked a beginning of a new epoch for Mathematics. By this time it
had become so vast and complicated that the link between its various parts
was beginning to get snapped, and Mathematics was breaking up into
unrelated compartments; which was about to put mathematics into trouble.
Mathematicians realised that to save the situation, some reformation
coupled with an examination of the fundamental concepts was the need of
the day. It was at this time that George Cantor (18451918) came on the
scene. Cantor has truly been called the Father of Modern Mathematics. To
think that someone else deserves this title is inconceivable. He said,
"The essence of Mathematics is its freedom." This slogan changed
the very approach to Mathematics.
Cantor introduced the 'Theory of Sets'. This
mathematical theory provided the answer to the much needed unification of
the vast subject of Mathematics. Cantor's 'Theory of Sets' created a stir
in the circles of Mathematics; and all other advances of that time fade
into insignificance before this revolutionary concept. In 1847, when
Cantor first published his paper on the 'Theory of Sets', a violent storm
of protest was led by Kroncker and Poincare. As they were Mathematicians
of no mean repute, their criticism discouraged many mathematicians from
even trying to understand the novel concepts of Cantor. He, however, got
enough support from Dedekind, MittagLeffler and others. Later on, in
early 20th Century, academic honors were showered on Cantor by many
countries. This late recognition could not stem the nervous breakdown
which Cantor first had in 1884 as a result of the barrage of criticism to
which he was subjected. This trouble recurred from time to time to the end
of his life. Cantor died in 1918 in a psychiatric clinic at Halle.
The 'Theory of Sets' went along two clearly different
lines of approaches. One was the Mathematical Theory of Sets, and the
other, the study of Mathematical System (Mathematical logic). The point
set topology was evolved from the first approach, because of its concept
of sets of points on a line, in a plane or in other dimensions of
Euclidean spaces. The latter approach mixed with logic, since little
regard was given to the nature of sets. Though the development of Set
Theory bifurcated in two distinct ways, both were logically well mixed in
Cantor's concept of Sets. By using very simple methods, Cantor arrived at
some amazing results. Due to the results he arrived at, it was possible
for Mathematicians to treat the concept of infinity along absolute logical
lines.
Not only Cantor, but logicians like Boole, DeMorgan
and Peone, constructed Mathematical systems which are responsible for the
present edifice of the Set Theory.
No doubt, the Set Theory holds the pride of place in
the world of Modern Mathematics, but it is 'Group Theory' which goes to
the very foundation of what happens when a particular mathematical
operation is applied to various elements or when different operations,
following a sequence, are applied to just one element of a set. It is the
Group Theory which has been used and applied in sophisticated electronic
systems. The Theory of Groups was introduced by a fiery French teenager
Evariste Galois. He wrote most of his theory in an unintelligible writing
covering about 30 pages in a single night, little did he know that the
next day he would be killed in a foolish duel over a girl of illfame whom
he did not even know. This tragic prodigy repeatedly proved unsuccessful
at the examinations, fought with his parents and elders, disobeyed his
teachers, was rejected by his family, was considered an outcast by society
and was imprisoned for threatening the King's life. At the time of his
death, he was hardly twenty years old, yet he is considered to be one of
the most creative and original mathematicians of all times. What made
Evariste Galois write out his theory that particular night? Being a
genius, could he foresee how close his death was. Could he have heard the
knock of death at his door and therefore resolved to complete his allotted
work hurriedly (the fact is obvious from the unintelligible writing of his
manuscript) before his end? Or was there some unseen power from above that
incited him to fulfill his destined duty towards the world of Mathematics,
just a few hours before his death and thus make a mark in this field.
Whatever be the answers to these questions is immaterial. Normal death due
to sickness or old age would not have been a fitting finale to the
controversial life of this tragic genius. It would have been an
anticlimax. Every aspect of Galois' life was an enigma and his death
provides a sort of poetic justice to the life he led. Nevertheless no one
can deny the fact, had Evariste Galois lived for just ten years more,
Mathematics would have advanced manifold.
With the introduction of logic in Mathematics, logical
senses grew more refined and subtle. People in general and mathematicians
in particular did not believe or trust anything which was not backed by
proof. About Euclid, people would say, "Euclid is Truth and Truth is
Euclid". Educated people would swear by Euclid and not by God. But
even Euclid was subjected to a thorough, critical and logical analysis.
Euclid had constructed a magnificent edifice by compiling the entire
available geometric data and putting these in the form of his monumental
work 'Elements', which is the basis of traditional Geometry. When his work
was put to a logical test, fissures appeared in his otherwise impressive
edifice. Logicians were shocked to find that Euclid had completely omitted
the concept of a straight line with infinite length. He only used line
segments. He also omitted the idea of 'betweenness' or 'lying between two
points', from his entire work. Anything that he found difficult to prove,
he and his followers took for granted as selfevident truths. His method
of using axioms to derive proofs was not without fault. Many of Euclid's
arguments are based on the theorem that a point D on a line AB lies
between points A and B. The familiar proof that a triangle, in which
AB=BC, then /A=/C, needs bisection /B, this bisector
intersects AC at D, but to complete the proof, one needs the fact that D
is between A and C. To know this, one must have a preknowledge of 'betweenness'
and must know the condition under which a point will be between the other
two points. This, however, was not done by Euclid. To make these points
and many other doubts clear, nonEuclidean geometries were created by
Lobachevasky of Russia, Janos Bolyai of Hungary and Bernhard Riemann of
Germany.
Nevertheless, the first person to conceive the idea of
nonEuclidean Geometry was Guass. He believed that new kinds of Geometry
could be developed from an unusual new axioms, that through a point that
does not lie on a given line, more than one line can be drawn parallel to
that line. Such an idea was contradictory to common sense and Euclid, who
believed that through a point that is not in a line, 'one and only one
line' can be drawn parallel to that line. The three men whose names are
mentioned earlier, carried out a revolution in Geometry which was foreseen
by Guass. Riemann, Guass's distinguished pupil, created a strange Geometry
in 1854, by saying that 'lines cannot be parallel'  i.e. they must meet
at both ends like meridians on the Earth. Using this concept, he created
perfectly consistent Geometry. It was this concept which became the
mathematical language for describing the curved space of 'relativity'.
Einstein used to some extent, this concept of Riemann as a mathematical
tool for derivation of the famous equation E=MC2. It is this equation
which shook the world by demonstrating the immense energy of the atom.
For over one and a half centuries up to 1950,
mathematicians and educationists had been trying to introduce drastic
revisions in the instruction of Mathematics, but its teaching had not
changed much. It was in the 1950's with the dawn of the satellite age that
people realised that the world rests on Science, and Mathematics forms the
backbone of all sciences. New programmes were introduced which lay stress
on fundamental concepts, structure and logic  not just 'how' to tackle a
mathematical problem, but 'why' to approach a problem in a particular
manner. Some decades ago, the unification of Mathematics and logic
appeared the most remote mathematical discipline. Suddenly it has turned
out to be the most practical and useful, and the knowledge of which is
most essential for using computers, and understanding the fundamental
concepts underlying Modern Mathematics.
Lately Modern Mathematics has become a subject of controversy and its
utility questionable. All this is due to the propaganda carried out
against it by the very people (not all of them) who are supposed to work
for the advancement of this discipline. They are the people all over the
world whose duty is to give instructions in this subject to the new
generation. They are either not willing to learn the new concepts or the
fear of the unknown is making them carry out a sabotage of 'Modern
Mathematics'. The success of the new programmes will depend on the sincere
effort and hard labour, mathematics teachers all over the world will put
in to master the new concepts and then willing to part with their
knowledge to their pupils. Of course, the cooperation of the parents of
the students learning 'Modern Mathematics' will go a long way in
contributing towards the successful implementation of the new syllabi.
 

