This document discusses the history of set theory and problems in the foundations of mathematics. It begins by covering the birth of set theory with George Cantor in the late 1800s. It then discusses paradoxes that arose in set theory, such as Russell's paradox, and early attempts to address these issues through axiomatization by mathematicians like Zermelo. The document also covers the foundational crisis in mathematics around this time and different viewpoints on the issue, such as formalism and intuitionism. It discusses Hilbert's program and Godel's incompleteness theorems. Finally, the document briefly touches on non-Euclidean geometry and number theory.

1. BIRTH OF SET THEORY AND PROBLEMS IN THE
FOUNDATION OF MATHEMATICS
HISTORY OF SET
George Cantor
•He was a German mathematician who developed the set
of theory.
•Born on March 3, 1845 at Saint Petersburg, Russian
Empire and died on January 6, 1918 at Halle, Province of
Saxony, German Empire.
•He first encountered sets while working on “problem on
trigonometric series”.
•Cantor published a six-part treatise on set theory from the
years 1879-1884. This work appears in Mathematische
Annalen and it was a brave move by the editor to publish
the work despite a growing opposition to Cantor’s ideas.
•The next wave of excitement in set theory came around
1900, when it was discovered that Cantorian set theory
give rise to several contradictions.
Bertrand Russell and Ernst Zermelo independently
found the simplest and best known paradox, now called
Russell’s Paradox: consider “ the sets of all sets that are
not members of themselves”.
•The ‘ultimate paradox’ was found by Russell in 1902. It
simplify defined a set A={x|x is not a member of x}.
•Zermelo in 1908 was the first to attempt an
axiomatisation of set theory.
PROBLEMS IN THE FOUNDATION OF MATHEMATICS
•“Formalism" and "intuitionism" and about the GODELS
incompleteness.
• The foundation crisis is a celebrated affair among
mathematicians and has also reached a large
nonmathematical audience. A well trained mathematician
is supposed to know something about the viewpoints
above.
•A professional aims to seek and be opinionated about the
topic, either dismissing the foundational discussion as
irrelevant.
• The Early Foundation of Mathematics historically
speaking logicism was a neat intellectual to the rise of
Mathematics, and particularly to the set, theoretical
approaches and methods.
• Dedekind approached (1871) to ALGEBRAIC NUMBER
THEORY - his set theoretical definition of number fields
and ideas and the methods by which he proved results
such as the fundamental theorem of unique composition.
• Around 1900Cantor's CONTINUUM PROBLEM.
Questions in set theory and with the problem of whether
every set can be well ordered. His second problem
amounted to establishing the consistency of the notion of
the set R of real number.
2.1 PARADOXES, CONSISTENCY Comprehension
principle, is the set of all real numbers, corresponding to
the property of being an ordinal there is the set of all
ordinals. The principle was thought of as basic logical law.
Zermelo - Russell paradox shows that the comprehension
principle is contradictory and it does so by formulating a
property that seems to be as basic and purely logical as
possible.
2.2 PREDICAVILITY For instance Richard paradox which
is one of the linguistic or semantic paradoxes.
2.3 CHOICES a paradox Logically self-contradictory
statement or a statement that runs contrary to one's
expectations. As important as the paradoxes were, their
impact on the foundational debate has often been
overstated. The crisis in the strict sense
•In 1921, the mathematische Zeitschrift published a paper
by Weyl in which the famous mathematician, who was a
disciple of Hilbert, openly espoused intuitionism and
diagnosed a "crisis in the foundation" of mathematics. The
crisis pointed towards a "dissolution" of the old state of
analysis, by Brouwer "revolution“.
•INTUITIONISM-idea that mathematics is a creation of the
mind. Brouwer was not worried, foras he would say in
1933, " the spheres of truth are less transparent than
those of illusion", but Weyl, although convinced that
Brouwer delineated the domain of mathematical intuition
in a completely satisfactory way.
• HILBERT'S PROGRAM. The 20th century was a
proposed solution to the foundational crisis of
mathematics, when early attempts to clarify the foundation
of mathematics were found to suffer from paradoxes and
inconsistencies.
• Godel and the Aftermath extremely ingenious
development of metamathematics-allowed Godel to prove
that systems like axiomatic set theory or Dedekind-Peano
arithmetic are incomplete. That is, there exist propositions
P formulated strictly in the language of the system such
that neither P nor -P is provable in the system.
•In the end, the debate made it clear that mathematics
and its modern methods are still surrounded by important
philosophical problems. But when it comes to laying out
the bare beginnings, philosophical issues cannot be
avoided.
NON-EUCLEDIAN GEOMETRY
Greek mathematician Euclid employed a type of
geometry, which studies the plane and solid figure of
geometry with the help of theorems and axioms, also
known as Euclidean geometry. Non-Euclidean geometry
is a part of non Euclid mathematics. It is the opposite of
Euclidean geometry. It discusses the hyperbolic and
spherical figures. It is also known as hyperbolic geometry.
The figures of non-Euclidean geometry do not satisfy
Euclid's parallel postulate.
INVENTION TO NON-EUCLIDIAN GEOMETRY
Greek mathematician Euclid presented the concept of
Euclidean geometry. At that time, people used to think
that there is only one type of geometry called Euclidean. A
wrong idea was present that all the geometrical figures
satisfy Euclid's parallel postulate. Here comes the concept
of non Euclidean geometry. The great mathematician Carl
Friedrich Gauss realized that all the geometrical figures
could not satisfy Euclid's parallel postulate. Gauss
described those figures figures that don't satisfy Euclid's
parallel postulate as non-Euclidean. Thus the concept of
non-Euclidean space arrived in geometry.
TWO MAIN FIGURES OF NON-EUCLIDEAN
GEOMETRY
-SPHERE
-HYPERBOLA
TYPES OF NON-EUCLIDEAN GEOMETRY
 HYPERBOLIC GEOMETRY- Hyperbolic
geometry is a branch of non Euclidean
geometry. It is not valid for the fifth parallel
postulate of Euclid. The fifth postulate statesthat
one given line is parallel with only one other line
through a point, not a line. There are at least
two lines in hyperbolic geometry that are
parallel with a given line through a point, not a
line. The properties of a triangle are different
from the Euclidean geometry. The sum of
angles in Euclidean geometry is 180. The sum
of angles of a triangle is less than 180 degrees
in this branch. The area and surface formulas of
hyperbolic geometry are different from the
Euclidean geometry.
 ELLIPTICAL GEOMETRY- It is the study of the
figures created on the surface of an ellipse. It
studies three-dimensional figures, unlike
Euclidean geometry. It is used in linear algebra,
arithmetic geometry, and complex analysis. One
possible representation of elliptical geometry is
ON a sphere. This is because the closest
analogue to lines on a sphere is a great circle,
and any two great circles on a sphere must
intersect For accurate calculation of area, angle,
distance on the earth, elliptical geometry is
used.
Non Euclidean geometry has a considerable
application in the scientific world. The concept
of non- Euclidean geometry is used in
cosmology to study the structure, origin,
constitution, and evolution of the universe.
Furthermore, it is used to state the theory of
relativity, where the space is curved. The
measurement of the distances, areas, angles of
different parts of the earth is done with the help
of it. Lastly, non-Euclidean geometry is applied
in celestial mechanics.
• Any geometry that is different from Euclidean
geometry
• Consistent system of definitions, assumptions, and
proofs that describe points, lines, and
planes
• Most common types of non-Euclidean geometries
are spherical and hyperbolic geometry
• Opened up a new realm of possibilities for
mathematicians such as Gauss and Bolyai
• Non-Euclidean geometry is sometimes called
Lobachevsky-Bolyai-Gauss Geometry
• Was not widely accepted as legitimate until the 19th
century
• Debate began almost as soon the Euclid’s
Elements was written
Euclid’s five postulates
The basis of Euclidean geometry is these five
postulates
1. Two points determine a line
2. A straight line can be extended with no limitation
3. Given a point and a distance a circle can be drawn
with the point as center and the distance
as radius
4. All right angles are equal
5. Given a point p and a line l, there is exactly one
line through p that is parallel to l
Euclidean & non-Euclidean
• Euclidean geometry marks the beginning of
axiomatic approach in studying
mathematical theories
• Non-Euclidean geometry holds true with the rest of
Euclid’s postulates other than the fifth
Fifth postulate
• The fifth postulate is very different from the first four
and Euclid was not even completely
satisfied with it
• Being that it was so different it led people to
wonder if it is possible to prove the fifth
postulate using the first four
• Many mathematicians worked with the fifth
postulate and it actually stumped many
these mathematicians for centuries
• It was resolved by Gauss, Lobachevsky, and
Riemann
Playfair’s Axiom
• Proclus wrote a commentary on the Elements and
created a false proof of the fifth postulate,
but did create a postulate equivalent to the
fifth postulate
• Became known as Playfair’s Axiom, even though
developed by Proclus because Playfair
suggested replacing the fifth postulate with
it
• Playfair’s Axiom: Given a line and a point not on the
line, it is possible to draw exactly one line
through the given point parallel to the line
Girolamo Saccheri

2. • Saccheri assumed that the fifth postulate was false
and then attempted to develop a
contradiction
• He also studied the hypothesis of the acute angle
and derived many theorems of non-
Euclidean geometry without even realizing
it.
Lambert
• Studied a similar idea to Saccheri
• He noticed that in this geometry, the angle sum of a
triangle increased as the area of a triangle
decreased
Legendre
• Spent 40 years working on the fifth postulate and
his work appears in a successful geometry
book, Elements de Geometrie
• Proved Euclid’s fifth postulate is equivalent to: the
sum of the angles of a triangle is equal to
two right angles and cannot be greater
than two right angles
Gauss
• Was the first to really realize and understand the
problem of the parallels
• He actually began working on the fifth postulate
when only 15 years old
• He tried to prove the parallels postulate from the
other four
• In 1817 he believed that the fifth postulate was
independent of the other four postulates
Gauss continued…
• Looked into the consequences of a geometry where
more than one line can be drawn through a
given point parallel to a given line
• Never published his work, but kept it a secret
Farkas & Janos Bolyai
• However, Gauss did discuss the theory with
FarkasBolyai
• FarkasBolyai: Created several false proofs of the
parallel postulate
• Taught his son Janos Bolyai math but told him not
to waste time on the fifth postulate
• Janos Bolyai: In 1823 he wrote to his father saying
“I have discovered things so wonderful that
I was astounded, out of nothing I have
created a strange new world.”
Janos Bolyai
• It took him 2 years to write down everything and
publish it as a 24 page appendix in his
father’s book
• The appendix was published before the book
though
• Gauss read this appendix and wrote a letter to his
friend, FarkasBolyai, he said “I regard this
young geometer Bolyai as a genius of the
first order.”
• Gauss did not tell Bolyai that he had actually
discovered all this earlier but never
published
Lobachevsky
• Also published his own work on non-Euclidean
geometry in 1829
• Published in the Russian Kazan Messenger ,a local
university publication
• Gauss and Bolyai did not know about Lobacevsky
or his work
• He did not receive any more public recognition than
Bolyai
• He published again in 1837 and 1840
• The 1837 publication was introduced to a wider
audiences to the mathematical community
did not necessarily accept it
Lobachevsky continued…
• Replaced the fifth postulate with Lobachevsky’s
Parallel Postulate: there exists two lines
parallel to a given line through a point not
on the line
• Developed other trigonometric identities for
triangles which were also satisfied in this
same geometry
• All straight lines which in a plane go out from a
point can, with reference to a given straight
line in the same plane, be divided into two
classes - into cutting and non-cutting. The
boundary lines of the one and the other
class of those lines will be called parallel to
the given line. Published in 1840
Lobachevsky's diagram
Beltrami
• Bolyai’s and Lobachevsky’s thoughts had not been
proven consistent
• Beltrami was the one that made Bolyai’s and
Lobachevsky’s ideas of geometry at the
same level as Euclidean
• In 1868 he wrote a paper Essay on the
interpretation of non-Euclidean geometry •
Described a 2-dimensional non-Euclidean
geometry within a 3-dimensional geometry
• Model was incomplete but showed that Euclid’s fifth
postulate did not hold
Riemann
• Wrote doctoral dissertation under Gauss’
supervision
• Gave inaugural lecture on June 10, 1854, which he
reformulated the whole concept of
geometry
• Published in 1868, two years after his death
• Briefly discussed a “spherical geometry” in which
every line through p not on a line AB meets
the line AB
Klein
• In 1871, Klein finished Beltrami’s work on the Bolyai
and Lobachevsky’s non-Euclidean
geometry
• Also gave models for Riemann’s spherical
geometry
• Showed that there are 3 different types of geometry
3 types
• Bolyai-Lobachevsky type
• Riemann
• Euclidean
Euclidean & Non-Euclidean Geometry
• Euclidean: the lines remain at a constant distance
from each other and are parallels.
• Euclidean: the sum of the angles of any triangle is
always equal to 180°
• Hyperbolic: the lines “curve away” from each other
and increase in distance as one moves
further from the points of intersection but
with a common perpendicular and are
ultraparallels
• Hyperbolic: the sum of the angles of any triangle
is always less than 180°
• Elliptic: the lines “curve toward” each other and
eventually intersect with each other
• Elliptic: the sum of the angles of any triangle is
always greater than 180°; geometry in a
sphere with great circles
NUMBER THEORY
• the study of the properties of numbers, where
by “numbers” we mean integers and, more
specifically, positive integers
• As early as 3500 BC, Sumerians kept a
calendar which means that they know some
form of arithmetic. By 2500 BC, they developed
base 60 number systems which was passed
to Babylonians.
• Babylonians elaborate it in clay tablets.
• Some people began to speculate about the
nature and properties of numbers that lead to
mysticism or numerology that even today
3,7,11, and 13 are considered omens of good or
bad luck.
• The first scientific approach to the study of
integers is attributed to Greeks.
Around 600 BC, Pythagoras and his disciples
classified the integers into even, odd,
prime and composites. They also linked
number to geometry such as the idea of
polygonal numbers and Pythagorean
triangles. The corresponding triple of
numbers (x,y,z) representing the lengths of
the sides is called a Pythagorean triple.
A Babylonian tablet, Plimpton 322 shows list of
Pythagorean triples.
Later on, Plato found a method for determining all
the triples which are in modern notation it
is: x=4n, y=4n2
-1 and z=4n2
+1.
EUCLID
• Around 300 BC, Euclid’s Elements appeared
consisting or 13 books where three of it (Books
VII, IX and X) are devoted to theory of numbers.
• Book IX- there are infinitely many primes
• Book X- method for obtaining all Pythagorean
triples
• Fundamental Theorem of Arithmetic —prime
numbers are the building blocks of all integers.
In other words, integers ≥ 2 are made up of
primes; that is, every integer ≥ 2 can be
decomposed into primes.
Euclid’s theorem:
• There are infinitely many primes.
• Every integer N = 2n−1
(2n
−1), where 2n
−1 is a
prime, is a perfect number.
Euclidean Algorithm
• The Euclidean algorithm for finding (a, b) is a
successive application of the division algorithm
and is based on the following result, where a ≥
b:
• Let r = a mod b. Then (a, b) = (b,r).
• The algorithm provides a systematic
method for expressing (a, b) as a linear
combination of a and b.
• The number of divisions needed to
compute (a, b) by the euclidean algorithm
is no more than five times the number of
decimal digits in b, where a ≥ b ≥ 2.
(Lamé’s theorem)
Contributed in finding all perfect numbers (such
as 6=1+2+3) that the Pythagoreans posed. In
his formula, he found all even perfect
numbers:
2p-1
(2p
-1)
Today, (2p
-1) is called Mersenne Numbers, in
honor to Mersenne who studied it in 1644.
*No odd perfect numbers are known.
DIOPHANTUS OF ALEXANDRIA
• Published 13 books which is the first Greek
work to use systematic algebraic symbols.
• Many of the problems in his book originated
from number theory and seeks for integer
solutions.
Diophantine Equations
— equations to be solved with integer values of
the unknowns
— integral solutions of equations with integral
coefficients
Diophantine analysis — the study of such
equations
The Pythagorean triples, x2
+y2
=z2
is an example
of Diophantine equation
PIERRE DE FERMAT (1601-1665)
• Father of modern number theory
• not a mathematician by profession; he was
trained in law
• Derived his inspiration from the work of
Diophantus and the only sources on number
theory were Diophantus’ Arithmetic and Books
VII-IX of The Elements
• Fermat wrote to several mathematicians
through the correspondence of Mersenne.
• They relayed information from the most
prominent mathematicians.

3. • Introduced Fermat numbers
• Fermat corresponded with men including the
following: Bernard Frénicle de Bessy, (a fellow
“number lover”), Descartes, Étienne Pascal,
Blaise Pascal, Gilles Personne de Roberval,
and Wallis
• Among them, only Fermat started and studied
number theory.
• He sent letters to Wallis but rejected Wallis
solutions that reinforced Wallis view of the
unimportance of number theory.
He posed the problem: “To find a pythagorean
triangle in which the hypotenuse and the
sum of the arms are squares”. Fermat
wrote to Mersenne and claimed that he
had found the smallest such pythagorean
triangle. The triangle that Fermat found
was triangle (4565486027761,
1061652293520,4687298610289).
•On October 18, 1640, Fermat wrote a letter to
Frenicle. In his letter, Fermat
communicated the following result: If p is a
prime and pa, then pl a-1. Fermat did not
provide a proof but enclosed a note that he
would send along a proof. This result is
known as FERMAT'S LITTLE THEOREM.
•Incidentally, the special case of Fermat's little
theorem for a = 2 was known to the
Chinese as early as 500 B.C.
•The first proof of Fermat's little theorem was given
by Euler in 1736.
•Leibniz had a proof in an unpublished work, but
once again Leibniz did not receive his
share of credit.
Pythagorean triples (a2
+b2
=c2
) relate to Fermat’s
Last Theorem. Fermat shows that, for this
to be true, then a and b would be squares.
He uses a proof by infinite descent but
ends with “the margin is too small to
enable me to give the proof completely and
with all detail”. The preservation of
Fermat’s work sure had an on-going battle
with margins.
In 1770, Euler provided the first proof for n = 3, but
contained a few gaps.
It was later perfected by Legendre.
Fermat himself gave a proof for n = 4, using the
method of infinite descent.
Dirichlet and Legendre, confirmed the conjecture
for n = 5.
Lamé established the conjecture for n = 7.
Andrew Wiles of Princeton University finally proved
Fermat’s Last Theorem in 1994.
Toward the end of his life, Fermat wrote to
Huygens about “handing on the torch.” No
one picked up “the torch” until the 18th
century, when the “rebirth” of number
theory took place. This rebirth came
through the works of number theorist,
Leonhard Euler.
Has no formal publication. Samuel de Fermat, son
of Pierre de Fermat, published his father’s
works.
Only 1 proof in number theory was published, the
Observation 45 on Diophantus.
LEONHARD EULER (1707–1783)
• is fascinated with Fermat’s work. He solved n=3
and n=4 on Fermat’s last theorem.
• Presented his method, Euler’s method to solve
Linear Diophantine Equations (LDEs).
Euler’s theorem- extends Fermat’s little theorem to
arbitrary moduli.
- useful for finding remainders of numbers
involving large exponents even if the divisor is
composite, provided the divisor is relatively
prime to the base.
- Every even perfect number is of the form
2n−1(2n − 1), where 2n − 1 is a prime.
- Let m be a positive integer and a any integer
with (a,m) = 1. Then a ϕ(m)
≡ 1 (mod m).
Euler’s phi function, one of the most important
number-theoretic functions (also known as
arithmetic functions). Arithmetic functions
are defined for all positive integers. Euler’s
phi function belongs to a large class of
arithmetic functions called multiplicative
functions.
Euler discovered the eighth perfect number in
1750.
JOSEPH LOUIS LAGRANGE (1736–1813)
• developed a delightful proof of Wilson’s
theorem as an application of Fermat’s little
theorem and Euler’s formula
• Also worked on quadratic reciprocity
• Lagrange developed the fundamental
properties of periodic continued fractions
• Continued fractions-multi-layered fraction and
a term coined by the English mathematician
John Wallis
ADRIEN-MARIE LEGENDRE(1752-1833)
• In 1823 he provided a beautiful demonstration
of Fermat's last theorem for the case n = 5.
• He published the first textbook in number theory
in 1798.
KARL FRIEDRICH GAUSS (1777–1855)
• three years after Legendre published a book, he
published Disquisitiones Arithmeticae
— laid the foundation for modern number theory
— where he presented the theory of
congruences, a beautiful arm of divisibility
theory, published in 1801 when he was only 24
— it is no accident that the congruence symbol ≡,
invented by Gauss around 1800
“Mathematics is the queen of the sciences and
arithmetic the queen of mathematics.”
• he called the law on quadratic reciprocity,
which Legendre had published a couple of
years earlier, the theorem aureum, or the gem
of arithmetic and gave the first complete,
rigorous proof of the law; he was only 18 years
old then
• He wants to extend the meaning of the word
integer to include the so-called Gaussian
integers, that is, numbers of the form a + bi,
where a and b are integers.
• At the age of 19, proved that the 17-sided
regular polygon is constructible.
SOPHIE GERMAIN (1776-1831)
• female mathematician that has interest in
number theory.
• obtain the lecture notes of Lagrange and other
scholars. She sent Lagrange a paper on
analysis, under the pseudonym M. Leblanc. She
discovered a result of Fermat's Last Theorem,
which bears her name. Sophie Germain's
Theorem relates to solutions to cases of
Fermat's Last Theorem and different divisibility
properties of these solutions.
Sophie Germain Primes
• Primes of the form 2p + 1.
• A chain of such primes is a Cunningham chain.
• It has been conjectured that there are infinitely
many Sophie Germain primes
GUSTAV PETER LEJEUNE DIRICHLET (1805–
1859)
• he was inspired by Gauss’ masterpiece,
Disquisitiones Arithmeticae (1801). He
established Fermat’s Last Theorem for n = 14.
• laid the foundations of the new branch of math
called analytic number theory
• Dirichlet’s Theorem: If a and b are relatively
prime, then the arithmetic sequence a, a + b, a
+ 2b, a + 3b,... contains infinitely many primes.
• The Pigeonhole Principle is also known as the
Dirichlet box principle.
• The Pigeonhole Principle: If m pigeons are
assigned to n pigeonholes, where m > n, then at
least two pigeons must occupy the same
pigeonhole.
• *pigeonhole=remainders. What is explained is
just the divisibility relation.
The Modern Period of Mathematics(19th
Century)
• in the sense the term is used by working
mathematicians these days—took shape in the
period from 1890 to 1930, mainly in Germany
and France.
• Modern mathematics approaches things
differently. It primarily studies structures whose
interactions have certain patterns. For instance,
it turns out the geometric properties needed to
build calculus can be boiled down to: (a) a
metric and (b) a space with certain properties.
On reflection, this makes sense.
Carl Friedrich Gauss (1777–1855)
• Epitomizes this trend. He did revolutionary work
on functions of complex variables, in geometry,
and on the convergence of series, leaving aside
his many contributions to science.
• He also gave the first satisfactory proofs of
the fundamental theorem of algebra and of
the quadratic reciprocity law.
• This century saw the development of the two
forms of non-Euclidean geometry, where the
parallel postulate of Euclidean geometry no
longer holds.
Nikolai Ivanovich Lobachevsky( Russian
Mathematician) and his rival, János Bolyai,
(Hungarian Mathematician)
• independently defined and studied hyperbolic
geometry, where uniqueness of parallels no
longer holds.
Bernhard Riemann ( German Mathematician)
• In this geometry the sum of angles in a triangle
add up to less than 180°. Elliptic geometry was
developed.
• here no parallel can be found and the angles in
a triangle add up to more than 180°.
• Riemann also developed Riemannian geometry,
which unifies and vastly generalizes the three
types of geometry, and he defined the concept
of a manifold, which generalizes the ideas of
curves and surfaces.
Hermann Grassmann
• Germany gave a first version of vector spaces,
• William Rowan Hamilton
• Ireland developed non-commutative algebra.
• The British mathematician
George Boole
• a British Mathematician
• devised an algebra that soon evolved into what
is now called Boolean algebra, in which the only
numbers were 0 and 1. Boolean algebra is the
starting point of mathematical logic and has
important applications in electrical engineering
and computer science.

4. Augustin-Louis Cauchy, Bernhard Riemann, and
Karl Weierstrass
• reformulated the calculus in a more rigorous
fashion.
Niels Henrik Abel, (Norwegian) and Évariste
Galois, (Frenchman)
• proved that there is no general algebraic
method for solving polynomial equations of
degree greater than four (Abel–Ruffini theorem).
• - utilized this in their proofs that straightedge
and compass alone are not sufficient to trisect
an arbitrary angle, to construct the side of a
cube twice the volume of a given cube, nor to
construct a square equal in area to a given
circle.
Georg Cantor
• established the first foundations of set theory,
which enabled the rigorous treatment of the
notion of infinity and has become the common
language of nearly all mathematics. Cantor's set
theory, and the rise of mathematical logic in the
hands of Peano, L.E.J. Brouwer, David Hilbert,
Bertrand Russell, and A.N. Whitehead, initiated
a long running debate on the foundations of
mathematics.
(20th
CENTURY)
Wolfgam Haken and Kenneth Appel (1976)
• proved the four color theorem, controversial at
the time for the use of a computer to do so.
• Four Color Theorem, or the four color map
theorem, states that no more than four colors
are required to color the regions of any map so
that no two adjacent regions have the same
color.
Andrew Wiles
• building on the work of others, proved Fermat's
Last Theorem in 1995.
• Fermat's Last Theorem (sometimes
called Fermat's conjecture, especially in older
texts) states that no three positive integers a, b,
and c satisfy the equation an
+ bn
= cn
for any
integer value of n greater than 2. The cases n =
1 and n = 2 have been known since antiquity to
have infinitely many solutions.
Paul Cohen and Kurt Gödel
• proved that the continuum
hypothesis is independent of (could neither be
proved nor disproved from) the standard axioms
of set theory.
Thomas Callister Hales
• proved the Kepler conjecture in 1998.
A group of French mathematicians, including
Jean Dieudonné and André Weil, publishing
under the pseudonym "Nicolas Bourbaki
• tempted to exposit all of known mathematics as
a coherent rigorous whole. The resulting several
dozen volumes has had a controversial
influence on mathematical education.
Albert Einstein
• used it in general relativity. Entirely new areas
of mathematics such as mathematical logic,
topology, and John von Neumann's game
theory changed the kinds of questions that
could be answered by mathematical methods.
All kinds of structures were abstracted using
axioms and given names like metric spaces,
topological spaces etc. As mathematicians do,
the concept of an abstract structure was itself
abstracted and led to category theory.
Grothendieck and Serre
• recast algebraic geometry using sheaf theory.
Large advances were made in the qualitative
study of dynamical systems that Poincaré had
begun in the 1890s. Measure theory was
developed in the late 19th and early 20th
centuries. Applications of measures include the
Lebesgue integral, Kolmogorov's axiomatisation
of probability theory, and ergodic theory. Knot
theory greatly expanded. Quantum mechanics
led to the development of functional analysis.
Other new areas include Laurent Schwartz's
distribution theory, fixed point theory, singularity
theory and René Thom's catastrophe theory,
model theory, and Mandelbrot's fractals. Lie
theory with its Lie groups and Lie algebras
became one of the major areas of study.
Abraham Robinson
• he introduced the non standard analysis
• rehabilitated the infinitesimal approach to
calculus, which had fallen into disrepute in
favour of the theory of limits, by extending the
field of real numbers to the Hyperreal numbers
which include infinitesimal and infinite
quantities. An even larger number system, the
surreal numbers were discovered by John
Horton Conway in connection with combinatorial
games.
Derrick Henry Lehmer's
• he use of ENIAC to further number theory and
the Lucas-Lehmer test; Rózsa Péter's recursive
function theory.
• Claude Shannon's information theory; signal
processing; data analysis; optimization and
other areas of operations research.
Kurt Gödel
• found that this was not the case for the natural
numbers plus both addition and multiplication;
this system, known as Peano arithmetic, was in
fact incompletable. (Peano arithmetic is
adequate for a good deal of number theory,
including the notion of prime number.)
• A consequence of Gödel's two incompleteness
theorems is that in any mathematical system
that includes Peano arithmetic (including all of
analysis and geometry), truth necessarily
outruns proof, i.e. there are true statements that
cannot be proved within the system.
Paul Erdős
• published more papers than any other
mathematician in history, working with hundreds
of collaborators. Mathematicians have a game
equivalent to the Kevin Bacon Game, which
leads to the Erdős number of a mathematician.
This describes the "collaborative distance"
between a person and Erdős, as measured by
joint authorship of mathematical papers.
Emmy Noether
• has been described by many as the most
important woman in the history of
mathematics.[183] She studied the theories of
rings, fields, and algebras.
• In 2000, the Clay Mathematics Institute
announced the seven Millennium Prize
Problems
• 2003 the Poincaré conjecture was solved by
Grigori Perelman (who declined to accept an
award, as he was critical of the mathematics
establishment).
Modern Algebra is the set of advanced topics of
algebra that deals with abstract algebraic structures
rather than the usual number systems.
-The most important of these structures are groups,
rings, and fields..
History of the Development of Modern Algebra
Nicolas Bourbaki identifies three main streams
leading to the development of modern Algebra:
(1) The theory of algebraic numbers, developed by
Gauss, Dedekind, Kronecker, and Hilbert.
(2) The theory of groups of permutations (and, later,
groups of geometric transformations), where the
work of Galois and Abel was fundamental.
(3) The development of linear algebra and hyper
complex systems
Early Group Theory
Leonhard Euler- algebraic operations on numbers -
generalization of Fermat's little theorem
Friedric Gauss- cyclic &general abelian groups. In
1870, Leopold Kroneckerabelian group particularly,
permutation groups.
Heinrich M. Weber gave a similar definition that
involved the cancellation property.
Lagrange resolvants by Lagrange.
The remarkable Mathematicians are..Kronecker
Vandermonde, Galois Augustin Cauchy. Cayley-
1854-...Group may consists of Matrices.
The end of the 19th and the beginning of the 20th
century saw a tremendous shift in the methodology
of mathematics.
- Abstract algebra emerged around the start of the
20th century, under the name modern algebra.
- Its study was part of the drive for more Intellectual
rigor in mathematics.
- Initially, the assumptions in classical algebra, on
which the whole of mathematics (and major parts of
the natural sciences) depend, took the form of
axiomatic systems.
The developments of the last quarter of the 19th
century and the first quarter of 20th
century were
systematically exposed in "Bartel van der Waerden's
Moderne algebra."
The two-volume monograph published in 1930
1931 that forever changed for the
mathematical world the meaning of the word...
algebra" from the theory of equations to the
theory of algebraic structures'.
The Fundamental Theorem
The "fundamental theorem of algebra" refers to this
statement:
(1.1) The number of roots of a nonzero polynomial
over the field C (multiplicities counted), is equal to
the degree of the polynomial.
The works leading to this theorem rest heavily on Girard
(1595-1632). Girard's theorem on the existence of
imaginary roots was certainly necessary in proving the
fundamental theorem. One should note that this version of
Girard's theorem is Gauss's interpretation which he
created to make it applicable to the fundamental theorem
(James 58)
Joseph-Louis Lagrange Italian Mathematician
Lagrange's work was applied to group theory after
the times of Abel and Galois. In the late 1800s,
algebra was growing into the subject we now call

5. pure mathematics. In the late 1800s, algebra was
growing into the subject we now call pure
mathematics. By now, the advances made in solving
the polynomials, and all that was gained through
Cardano and Ferrari (whose methods for solving
third and fourth degree polynomials are in the
appendix), were now commonly know. Also, the facts
obtained from the work of De Moivre on the roots of
complex numbers gave mathematicians such as
Lagrange and his contemporaries great insight into
the theory of equations. We have already discussed
the contribution made by Lagrange known as the
Fundamental Theorem of Algebra. The following
section looks at some of the work Lagrange did
which is directly related to the theory of groups.
Girard's Theorem is, essentially, the piece that
Lagrange left out of his proof of the fundamental
theorem. Lagrange (1736-1813) had progressed, in
1772, in proving the fundamental theorem of algebra.
Unfortunately Lagrange neglected to show that n
imaginary roots of any equation of degree n indeed
existed. What Lagrange did prove, however, was that
the form of these imaginary roots is a+B-1 with a, R.
It wasn't until 1815 that Gauss filled this gap by
interpreting Girard's Theorem (Shenitzer and Stillwell
99).
Johann Carl Friedrich Gauss German
Mathematician
Disquisitiones Arithmeticae is the heart of Gauss's
work. It contains research important to the theory of
numbers and the theory of equations such as the
Fundamental Theorem of Gauss and the Law of
Quadratic Reciprocity of Legendre.
According to Dunnington, Disquisitiones would have
placed Gauss among the top ranking mathematicians
of his day. In March 1775, by induction, he
discovered that for odd primes p, -1 is a quadratic
residue of prime numbers of the form 4n+1 and is a
quadratic nonresidue of primes of the form 4n+3, that
is, 1= x² (mod p) for some x Z if and only if p is of the
form 4/+1. (38) He wrote the first proof of it at the age
if nineteen and published it the following year. Now
restless about this class of research, he published six
different principles concerning residues and
nonresidues of prime numbers.
Niels Henrik Abel (1802–1829)
Norwegian Mathematician
Abel was born and raised in Norway. As a boy he
already was intrigued by the work of Lagrange and
Cauchy's permutation work. Once he reached
college, he progressed rapidly and in 1821 began
writing original papers (Hollingdale 298). Abel
eventually grew frustrated with one problem in
particular for which no mathematician had yet offered
a proof. It was this very problem that led Abel to
discovering what are now known as Abelian
equations that is, equations that are solvable if
g(h(x)) = h(g(x)). Abel ignored Gauss's opinion that
the quintic was unsolvable, and relentlessly searched
for the general solution to it. According to Pesic, it
was not until Abel submitted samples of his work to
his teachers that he knew this solution was much
more complicated than he had thought. (90) From
there he could have spent a lifetime trying to prove
its solvability. Instead he intuitively turned completely
around and proved its unsolvability by contradiction.
That is, he assumed that the general solution
existed, and showed that it led to an absurdity
(impossibility).
Évariste Galois (1811- 1832)
French mathematician
Everiste Galois (1811-1832), whose work agreed
entirely with Abel's, also proved the unsolvability of
the quintic in 1830. Unaware of Abel's work, Galois
gave a complete account of the solvability of
algebraic equations that was more thorough than
Abel's. He did this before the age of 20. Many minds
today are intrigued at the notion that two
mathematicians so young could independently prove
something so brilliant and then die such mysterious
deaths. Galois is responsible for the abstraction of
algebraic systems. He understood solvability better
than any mathematician before him (and probably
after), even Abel. What Galois did, that Abel didn't,
was discuss the permutations of the roots, not case
by case, but by an abstraction of the algebraic
system in which they are elements. He called these
systems groups!