Revolution in Mathematics
By Tribhuwan N. Bhan
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.
Source: Milchar
