¡ßNon-Euclidean Geometry
Parallel postulate, Euclidean the 5th postulate, (¡¸Through a
given point not on a given line can be drawn 'just one' line parallel to the given line.¡¹). This situation is equivalent, respectively, to the fact that the sum of three
internal angles of a triangle is 180 degrees.
This parallel postu late is so complex that it seems like theorem.
Many mathematicians, therefore, made on effort to prove the postulate but satisfactory
results were not made.
So mathematlcians used another way, indirect proof, to deny the
postulate and find out contradictions.
But they only failed to prove the postulate further more they
got new theorems through the way
Finally, they found these new theorems called non-Euclidean geomerry.
¡Ý Saccheri, Girolamo (1667~1733) : These three possibilities are
refered to by Saccheri as the 'hypothesis of the acute angle,'
the 'hypothesis of the right angle', and the 'hypothesis of the
obtuse angle'. He tried to prove these three possibilities
were contradictions
,but rather had admitted his inability to find one, Saccheri
would today unquestionably be credited with the discovery
of non-Euclidean geometry.
¡Ý Lobachevsky : Lobachevsky and Bolyai asserts that parallel postulate is
axiom not tneorem.
He said "Through a given point not on a given line can be drawn
'more than one' line parallel to the given line".
He discovered new geometry.
¡Ý Riemann : He asserted that "Through a given point not on a given line can
be drawn 'no' line parallel to the given line." He discovered consistent geometry.
Ridigity and generalization of abstraction are the characteristics
of Riemann's mathematics.
He widened the geometry to the variety of space.
¡Ý Klein
He classified Euclidean geometry and non-Euclidean geometry
in 1871.
Non-Euclidean geometry means the possibility of another geometry's existence
except Euclidean geometry.
It, therefore, is clear that mathematicians don't have to cling
to practical physical space.
¡ßThe Abstraction of Algebra
¡Ý Hamilton : As Euclidean geometry was regarded as an only geometry, so arithmetic
algebra was regarded as an only algebra until the early 17th century.
But new algebraic system in which commutative law for multiplication
is broken is observed.
Hamilton's quaternions(ordered real number guadruples)_the commutative and assocrative law for addition and the
assocrative and distributive for multiplication is broken.
¡Ý Cayley : One more noncommutative
algebra the matric algebra devised by the English mathematician Arthur Cayley (1821-1895) in 1857.
¡Ý Boole : Boole maintained that the essential character of
mathematics lies in its form rather than in its content; mathematics is not (as
some dictionaries today still assert) merely "the science of measurement and
number," but, more broadly, any study consisting of symbols along with precise rules of operation upon those symbols, the rules being subject only to the
requirement of inner consistency.
He established both
formal logic and a new algebra_the algebra of ests known today as Boolean
algebra. In more recent times, Boolean algebra has found a number of applica
tions, such as to the theory of electric switching circuits.
¡ßThe Arithmetization of Analysis
The demand for an even deeper understanding of the foundations of analysis
was strikingly brought out in 1874 with publicizing of an example, due to
the German mathematician Karl Weierstrass, of a continuous function having
no derivative, or, what is the same thing, a continuous curve possessing no
tangent at any of its points.
It became clear that the theory of
limits, continuity, and differentiability depend upon more recondite properties
of the real number system than had been supposed. Accordingly, Weierstrass
advocated a program wherein the real number system itself should first be
rigorzed, then all the basic concepts of analysis should be derived from this
number system. This remarkable program, known as the arithmetization of
analysis, proved to be difficult and intricate, but was ultimately realized by
Weierstrass and his followers, so that today all of analysis can be logically
derived from a postulate set characterizing the real number system.
¡ßMathematicians ranges from the 19th to the 20th century
This section will be devoted to a brief consideration of Georg Cantor and Henri
Poincare, two mathematicians with life spans astride the nineteenth and twentieth
centuries, and who exerted a considerable influence on much of the mathematics
of present times. It is also natural to insert a few words about Leopold
Kronecker, the harsh and relentless critic of Cantor's of Cantor's mathematics of the infinite.
¡Ý Cantor : He commenced in 1874 his revolutionary work on set
theory and the theory of the infinite. With this latter work, Cantor created a
whole new field of mathemaitcal research.
He classified infinite set according to power.
He tought if one - to - one correspondence is possible between two sets, the two
sets have the same numbers of element
countable set: N~Z~Q, non-conutable set:(0,1)~R
Today, Cantor's set theory has penetrated into almost every branch of mathematics,
and it has proved to be of particular importance in topology and in the
foundations of real function theory.
¡Ý Kronecker : As a finitist, he condemned the work of Cantor,
regarding it as theology and not as mathematics. Believing that all of mathemat
ics must be based by finite methods upon the whole numbers, he was a nine
teenth-century Pythagorean. "God made the whole numbers,
all the rest is the work of man"
¡Ý Poincare : Poincare has been described as the last of the universalists in the field of
mathematics. It is certainly true that he commanded and enriched and astodnishing
range of subjects.
He was also one of the ablest popularizer's of mathematics and scirence.
¡ßThe Three Greatest Mathematicians of Womens
¡Ý Hypatia : Theon's (Theon lived in the turbulent colsing period of the fourth
century A.D.)daughter. Hypatia, was distinguished in mathematics, medicine,
and philosophy, and wrote commentaries on Diophantus Arithmetica and
Apollonius, Conic Sections. She is the first woman mathematician to be men
tioned in the history of mathematics. Her life and barbarous murder by a mob
of fanatical Christians in March, 415, are reconstructed in Charles Kingsley's novel.
¡Ý Kovalevsky : She had earlier studied under Karl
Weierstrass, contributed to the field of partial
differential equations.
¡Ý Noether : Amalie Emmy Noether, one of the most outstanding mathematicians.
Although Noether was a poor lecturer and lacked pedagogical skill, she
managed to inspire a surprising number of students who also left marks in the
field of abstract algebra.
