"Symbolic logic
has been disowned by many logicians on the plea that its interest is mathematical
& by many mathematicians on the plea that its interest it logical."
Alfred North Whitehead
"One of the greatest philosophers in Western culture, was born in Stagira,
a small town in northern Greece. His father was the personal physician of the
king of Macedonia. Orphaned young, Aristotle was raised by a guardian.
At the age of 18, Aristotle entered Plato's academy in Athens. He was the
'brightest and most learned student' at the Academy which he left when Plato died
in 347 BC.
About 342 BC, the king of Macedonia invited him to supervise the
education of his young son, Alexander, who later became Alexander the Great.
Aristotle taught him until 336 BC, when the youth became ruler following the
assassination of his father.
Around 334 BC, Aristotle return to Athens and
founded a school called the Lyceum. His philosophy and followers were called
peripatetic, a Greek word meaning 'walking around,' since Aristotle taught his
students while walking with them. The Athenians, perhaps resenting his
relationship with Alexander the Great, who had conquered them accused him of impiety soon
after the Emperor's death in 323 BC. Aristotle, knowing the fate of Socrates,
who had been condemned to death on a similar charge, fled to Chalcis, so the
Athenians would not 'sin twice against philosophy.' He died there the following year."
(Koshy, p. 2)
"A member of the most distinguished
family of mathematicans, was born in
Basel, Switzerland. His grandfather, a pharmacist in Amsterdam, had
become a Swiss through marriage, and his father was a town councilor
and a magistrate.
Bernoulli received his M.A. in philosophy in 1671 and a theological
degree 5 years later. During this time, he studied mathematics and astron-
omy against his father's will. He spent the next 2 years tutoring in Geneva.
In 1687 he became professor of mathematics at the University of Basel,
remaining there until his death. His brother Johann succeeded him at
Basel.
In May 1690 he used the term integral in the calculus sense known
today. Bernoulli's most famous work, Ars Conjectandi, was published posthumously in 1713. It contains
significant contributions to probability theory, the theory of series, and gravitational theory."
(Koshy, p. 212)
"The son of a cobbler whose main interests were mathematics & the making
of optical instruments, was born in Lincoln, England. Beyond attending a local
elementary school and briefly a commercial school, Boole was self-taught in
mathematics and the classics. When his father's business failed, he started
working to support the family. At 16, he began his teaching career, opening
a school of his own four years later in Lincoln.
In his leisure time, Boole read mathematical journals at the Mechanics
Institute. There he grappled with the works of English physicist and mathematician
Sir Isaac Newton and French mathematicians Pierre-Simon Laplace and Joseph-Louis
Lagrange.
In 1839, Boole begin contributing original papers on differential equations
to The Cambridge Mathematics Journal & on analysis to the Royal Society. In
1844, he was awarded a Royal Medal by the Society for his contributions to
analysis; he was elected a fellow of the Society in 1857.
Developing novel ideas in logic and symbolic reasoning, he published
his first contribution to symbolic logic, The Mathematical Analysis of Logic,
in 1847. His publications played a key role in his appointment as professor
of mathematics at Queens College, Cork, Ireland, in 1849, although he lacked
the university education.
In 1854, he published his most important work,
An Investigation to the Laws of Thought, and he presented the algebra
of logic now known as boolean algebra. The next year he married Mary Everest,
the niece of Sir George Everest, for whom the mountain is named.
In addition to writing about 50 papers, Boole published two textbooks,
Treatise on Differential Equations (1859) &
Treatise on the Calculus of Finite Differences; both were used as texts
in the United Kingdom for many years.
A conscientious & devoted teacher, Boole died of pneumonia in Cork."
(Koshy, p. 4)
"Was born in St Petersburg Russia where his father was a successful merchant
and broker. Cantor showed great interest in mathematics from early childhood.
In 1856 the family moved to Germany. Six years later, he entered the
University of Zurich, but in the following year he moved to the University
of Halle to study mathematics, physics, and philosophy. There he was greatly
influenced by the eminent mathematician Karl Weierstrass (1815-1897). Although
his father wanted him to become an engineer, Cantor relentlessly pursued his
interest in mathematics and received his doctorate of philosophy at 22 from
the University of Berlin for his Works in number theory.
In 1869, Cantor began his professional career as an unsalaried
lecturer at the University of Halle. Five years later, he published his
revolutionary work on set theory. Cantor developed an arithmetic of
transfinite numbers analogous to that of finite numbers, thus creating
another area of mathematical study. He proved that the set of real numbers
is uncountable & he also established the existence of infinitely many
ordinal numbers by ingenious methods. He also made significant contributions
to indent equations in trigonometric series. Deeply religious, Cantor was
also interested in art, music, and philosophy.
Being unhappy with his low salary at the university, Cantor
tried to secure a better paid position at the University of Berlin, but was
sabotaged by Leopold Kronecker (1823-1891), an eminent mathematician at the
University, who severely criticized Cantor views on sets.
Relentless attacks by contemporary mathematicians intensified
the manic depression he suffered from. Cantor died in a mental hospital in
Halle in 1918.
Cantor was 'one of the greatest intellects of the 19th century,'
according to Bertrand Russell. He 'was an imaginative geniuses work has
inspired [every aspects of] mathematical thought,' Hazel Perfect of the
University of Sheffield wrote in 1994."
(Koshy, p. 68)
"Was born in Richmond, England. At 14 he entered King College, London. His
teachers, recognising his superb mathematical talents, encouraged him to
be a mathematician
At 17, Cayley entered Trinity College, Cambridge, where he was
rated to be in a class by himself, "above the first." By age 25, he had
published 25 papers, the first one by age 20.
In 1846, he left his position at Cambridge to study law and
became a successful lawyer. Feeling unfulfilled, he left the law after 14
years, although during this period he had published more than 200 papers.
In 1863 Cayley joined the faculty at Cambridge University. He pursued
his mathematical interests, until his death." (Koshy, p. 164)
"Born in Mandurai, Tamil Nadu, India where his father was a colonel in the
Indian army. When the young De Morgan was 7 months old, the family moved to
England. He attended a private schools, where he mastered Latin, Greek and
Hebrew and developed a strong interest in mathematics. After graduating in
1827 from Trinity College, Cambridge, he pondered a career either in medicine
or law, but pursued mathematics. His professional career began in 1828 at
University College, London. Three years later, however, when the college
dismissed a colleague in anatomy without explanation, De Morgan resigned
on principle. He returned to Trinity in 1836 when his successor died and
remained there until a second resignation in 1866.
A fellow of the Astronomical Society and a founder of the London
Mathematics Society, De Morgan greatly influenced the development of mathematics
in the 19th century. He exuded his passion for the subject in his teaching,
stressing principles over techniques.
An incredible prolific writer, De Morgan authored more than
1,000 articles in more than 15 journals, as well as a number of textbooks,
all characterized by clarity, logical presentation and meticulous detail.
De Morgan's original contributions to mathematics were mainly
in analysis and logic. In 1838, he had been he coined the term "mathematical
induction" and gave a clear justification to this proof method, although it
had been in use. His the Differential and Integral Calculus (1848)
gives the first precise definition of a limit and some tests for convergence
of infinite series.
De Morgan was also interested in the history of mathematics.
He wrote biographies of Sir Isaac Newton and Edmund Halley. His wife wrote
De Morgan's biography in 1882.
His researches into all branches of knowledge and his prolific
writing left him little time for social or family life, but he was well-known
for his sense of humor."
(Koshy, p. 23)
"Was born near Tours, France. At eight, he entered the Jesuit school at
La Fleche, where because of poor health he developed the habit of lying in
bed thinking until late in the morning; he considered those times the most
productive. He left the school in 1612 & moved to Paris, where he studied
mathematics for a brief period.
After a short military career & travel through Europe for
about 5 years, he returned to Paris & studied mathematics and philosophy.
He then moved to Holland, where he lived for twenty years writing several
books. In 1637 he wrote Discours, which contains his contributions
to analytic geometry.
In 1649 Descartes moved to Sweden at the invitation of Queen
Christina. There he contracted pneumonia & died."
(Koshy, p. 87)
"Was born in Duren, Germany. The son of a postmaster, he first attended a
public school & then a private school that emphasized Latin. After attending
the Gymnasium in Bonn for 2 years, Dirichlet entered a Jesuit college in Cologne
where he received a strong background in theoretical physics under the
physicist Georg Simon Ohm. In May 1822, he moved to the University of
Paris.
In 1826, Dirichlet returned to Germany & taught at the University
of Breslau. Three years later, he moved to the University of Berlin where
he spent the next 27 years.
Dirichlet's primary interest in mathematics was number theory,
inspired by Gauss' masterpiece, Disquisitiones Arithmeticae (1801). He
established Fermat's Last Theorem for n = 14. Among the many results
he discovered include the proof of a theorem presented to the Paris Academy
of Sciences on algebraic number theory in 1837: The sequence {an
+ b} contains infinitely many primes, where a & b are
relatively prime.
In 1855, when Gauss died, Dirichlet moved to the University
of Göttingen. Three years later, he went to Montreaux, Switzerland, to
deliver a speech in honor of Gauss. While there, he suffered a heart attack
& was barely able to return home. During his illness his wife succumbed to
a stroke, & Dirichlet died."
(Koshy, p. 145)
"Was born in Budapest, Hungary. Except for about three years in schools,
Erdös (pronounced air-dosh) was taught at home, mostly by his father, who
had returned from a Siberian prison after 6 years.
A child prodigy, Erdös, at age 3, discovered negative numbers
for himself. In 1930 Erdös entered the Peter Pazmany University in Budapest.
Three years later, he discovered a beautiful proof of the celebrated
Chebyshev theorem that there is a prime between any positive integer n
and 2n. In 1934 he received his Ph.D. from the university.
An author of about 1500 articles and coauthor of about 500, Erdös was
one of the most prolific writers in mathematics. A tribute in 1983 described
him as "the prince of problem-solvers & the absolute monarch of problem-posers."
As "the Euler of our time," he contributed extensively to number theory,
combinatorics, function theory, complex analysis, set theory, group theory,
& probability, the first two areas being closest to his heart.
"Always searching for mathematical truths,” he deemed worldly
possessions a nuisance, so he never had a home, a car, checks, or even an
address. Always traveling from meeting to meeting, carrying a half-empty
suitcase, he would stay with mathematicians wherever he went & donate the
honoraria he earned as prizes to students.
A recipient of many honors, Erdös died of a heart attack while
attending a mathematics meeting in Warsaw."
(Koshy, p. 147)
"Little is known about Euclid's life. He taught at the University of Alexandria
and founded the Alexandrian School of Mathematics. When the Egyptian ruler
King Ptolemy asked Euclid if there were an easier way to learn geometry than
by studying The Elements, he replied, 'There is no royal road to geometry.'
Euclid is called the father of geometry.
'No work, except for the Bible, has been more widely read,
studied, or edited,' according to J. E. Lightner of Western Maryland College,
Westminister, Maryland. 'More than 2000 editions of the work have appeared
since the first printed one in 1482; however, no extant copy of The Elements
dates from Euclid's own time.'"
(Koshy, p. 192)
"Was born near Toulouse as the son of a leather merchant. A lawyer by
profession, he devoted his leisure time to mathematics. Although he published
almost none of his discoveries, he did correspond with contemporary mathematicians.
Fermat contributed to several branches of mathematics, but he
is known for his work in number theory. Many of his results appear in margins
of his copy of the works of the Greek mathematician Diophantus (250 A.D.).
He wrote The following about his famous conjecture: 'I have discovered a truly
wonderful proof but the margin is too small to contain it.'"
(Koshy, p. 5)
"Was born in Konigsberg, Prussia. He studied medicine and mathematics at
the University of Konigsberg and became professor of mathematics at the
Imperial Academy of Sciences in St Petersburg in 1725. In 1728, he moved
to Moscow to tutor Tsarevich Peter II and his cousin Anna of Courland.
From 1729 to 1763, he corresponded with Euler on number theory. He
returned to the Imperial Academy in 1732, when Peter's successor Anna
moved the Imperial Court to St Petersburg.
In 1742, Goldbach joined the Russian Ministry of Foreign
Affairs, and later became privy councilor and established guidelines for
the education of royal children.
Noted for his conjectures and number theory and work in analysis, gold buck died in moscow.
"
(Koshy, p. 5)
"Was born in Chicago, graduated
from the University of Chicago in 1937, and received an M.S. from the Uni-
versity of Nebraska 2 years later. After receiving his Ph.D. in mathematics in
1942 from the University of Illinois, he began his teaching career at the univer-
sity and moved to the university of Louisville until 1945. After a year working
on the Manhattan project at Los Alamos Science Laboratory, he joined the tech-
nical staff at Bell Telephone Labs in 1946; he headed the numerical methods
research department from 1964 to 1967, and then the computer science research
department until 1977. He left Bell in 1977 and became an adjunct professor in
computer science at the Naval Postgraduate School, Monterey, California.
Recipient of numerous awards, Hamming made significant contributions to
algebraic coding theory, numerical methods, statistics, and digital filters."
(Koshy, p. #)
"Was born in educated in Konigsberg, Germany. He made significant contributions to algebra, analysis, geometry and mathematical physics. He described the importance of set theory in the development of mathematics: 'No one shall expel us from the paradise which Cantor has created for us.'" (Koshy, p. 74)
"[He] ranks with Leonhard Euler . . . as one of the greatest mathematicians
of the 18th century. The eldest of 17 children in a wealthy family in Turin,
Italy, Lagrange was forced to pursue a profession after his father, an
influential cabinet official, lost all his wealth by engaging in unsuccessful
financial speculations. While studying the classics at the College of Turin,
the 17-year old Lagrange found his interest in mathematics kindled by an
essay by the astronomer Edmund Haley on the superiority of the analytical
methods of calculus over geometry in the solution of of optical problems.
In 1754, he began corresponding with several outstanding
mathematicians in Europe. The following year, was appointed professor of
mathematics at the Royal Artillery School in Turin. Three years later, he
helped to found a society that later became the Turin Academy of Sciences.
While at Turin, Lagrange developed revolutionary results in the calculus of
variations, mechanics, sound, and probability, winning the prestigious Grand
Prix of the Paris Academy of Sciences in 1764 and 1766.
In 1766, when Euler left the Berlin Academy of Sciences, Frederick
the Great wrote to Lagrange that "the greatest king in Europe" would like
to have "the greatest mathematician of Europe" of his court. Accepting the
invitation, Lagrange moved to Berlin to head the Academy and remained there
for 20 years. When Frederick died in 1786, Lagrange moved to Paris at the
invitation of Louis XVI. He was appointed professor at the Ecole Normale
and then at the Ecole Polytechnique, where he taught until 1799. He died
in Paris.
Lagrange made significant contributions to analysis, analytic
mechanics, calculus, probability, and number theory, as well as helping to
establish the French metric system." (Koshy, p. 159)
"An outstanding German mathematician, philosopher, physicist, diplomat
and linguist, was born into a Lutheran family. The son of a professor of
philosophy, he 'grew up to be a genius with encyclopedic knowledge.'
He had taught himself latin, Greek and philosophy before entering
University at Leipzig at age 15 as a law student. There he read The Works
of great scientists and philosophers such as Galileo, Francis Bacon, and
Rene Descartes. Because of his youth, Leipzig refused to award him the
degree of doctor of laws, so he left his native City forever.
During 1663 to 1666, he attended the universities of Jena and Altdorf,
and receiving his doctorate from the latter in 1666, he began legal services
for the electorate of Mainz. After the elector's death, Liebniz pursued
scientific studies. In 1672, he built the calculating machine that could
multiply and divide and presented it to the Royal Society in London the
following year.
In late 1675, Leibniz laid the foundations of calculus, an honor he
shares with Sir Isaac newton. He discovered the fundamental theorem of calculus,
and invented the popular notations d/dx for differentiation and ∫ for
integration. He also introduced such modern notations such as dot for
multiplication, the decimal point, the equal sign, and the colon for ratio.
From 1676, until his death, Leibniz worked for the Duke of Brunswick
at Hanover and his estate after the Duke's death in 1680. He played a key
role in the founding of the Berlin Academy of Sciences in 1700.
Twelve years later, Leibniz was appointed councilor of the Russian
Empire and was given the title baron by Peter the Great.
Suffering greatly from gout, Leibniz died in Hanover. He was never married.
His Works influenced such diverse disciplines as theology, philosophy,
mathematics, the natural sciences, history and technology."
(Koshy, p. 3)
"A British philosopher and mathematician, was born into a prominent,
aristocratic and progressive-minded family near Trelleck, Wales. His mother
died in 1874 and his father 2 years later; so the young Russell was brought
up by his father's parents.
Russell was home educated by tutors. In 1890 he
entered Trinity College, cambridge, where he excelled in both mathematics
and moral sciences. In 1895, he was awarded a fellowship for his original
dissertation on the foundations of geometry, published in 1897. After
graduation, he worked briefly in the British Embassy in Paris and then he
went to Germany, where he wrote his first book, German Social Democracy
(1896). In 1910, Trinity appointed him a lecturer and logic and the
philosophy of mathematics.
Russell's outspokenness and liberal views often landed him
in controversies. Around 1907, Russell fought hard for women's suffrage in
the United Kingdom. During World War I, he was dismissed by Trinity for
his protests in pacifist views. In 1918, he was in prison for 6 months for
an article that was branded sedicious. While in prison, he wrote
Introduction to Mathematical Philosophy. When he was about 90 years
old, he was imprisoned again for campaigning for nuclear disarmament.
& emsp; In 1925, Trinity, realizing that the 1916 dismissal was
excessively harsh, invited Russell back. He served there as a fellow from
1944 until his death.
Russell wrote more than 40 books on diverse subjects,
including philosophy and physics; his greatest work is the three volume
Principia Mathematica (1910-1913), which he co-authored with the
Cambridge philosopher Alfred North Whitehead (1861-1947). It describes the
logical constructions of the foundations of mathematics from the set of
primitive axioms.
Russell won the 1950 Nobel Prize for literature "'as a
defender of humanity and freedom of thought.'"
(Koshy, p. 42)
"Attended Cambridge University, which for several years denied him the degrees
he earned, because he was Jewish.
At 24, he became professor of natural philosophy at the
University of London. Three years later, he taught at the University of
Virginia for a year and then returned to England to become an actuary while
continuing his mathematical investigations.
Sylvester was professor of mathematics at Johns Hopkins
University from 1870 to 1883. In 1878 he founded The American Journal of
Mathematics."
(Koshy, p. 164)
"Was born into a philanthropic family and Hull, England. After attending
the high schools at Highgate and Islington, in 1853 he entered Gonville &
Caius College Cambridge, and graduated in mathematics 3 years later. He was
elected a fellow of the College, position he held until his death.
In the 1859 then was ordained in the Church of England, but
after a brief period of church work, he returned to Cambridge as a lecturer
on moral sciences. In 1883 he gave up his priesthood. The same year, he
received D. Sc. from Cambridge and was elected a fellow of the Royal Society
of London.
Venn was greatly influenced by Boole's work in symbolic logic.
Venn's masterpiece, Symbolic Logic (1881), clarifies the inconsistencies
& ambiguities in Boole's ideas & notations. He employed geometric diagrams
to represent logical arguments, a technique originated by Liebniz & developed
further by Euler. Venn added a rectangle to represent the universe of discourse.
Venn published two additional books, The Logic of Chance
(1866) and The Principles of Empirical Logic (1889).
"
(Koshy, p. 72)
"Was born in Baku, Azerbaijan. An alumnus of the University of Tehran (1942)
& the Massachusetts Institute of Technology (1946), he received his PhD from
Columbia University in 1949 for his dissertation on frequency analysis of
time-varying networks. He began his professional career in the Department of
Electrical Engineering at Columbia. In 1959, he joined the Department of
Electrical Engineering & Computer Science at the University of California,
Berkeley, serving as its chair during the years 1963-1968. Currently he is
a professor at Berkeley & Director of Berkeley Initiative in Soft Computing.
Zadeh's earlier "work was centered on systems analysis, decision
analysis & information systems. Since then his current research has shifted
to the theory of fuzzy sets & its application to artificial intelligence.
His research interest now is focused on fuzzy logic, soft computing, computing
with words & the newly developed computational theory of perceptions &
precisiated natural language," according to the University of California
website.
A truly gifted mind & an expert on ai, Zadeh has authored about
200 journal articles on a wide variety of subjects relating to the conception,
design and analysis of information/intelligence systems. He serves on the
editorial boards of more than 50 journals and on the advisory boards of a
number of institutions related to AI."
(Koshy, p. 92)
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