2. THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA
(ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA
(PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.
2
3. Introduction
Mathematics and Philosophy have had historically close ties: many thinkers appear
both in books than in Mathematics in Philosophy and often the subjects of
investigation have boundless from one to another discipline.
This study aims to highlight just a few of these links, we studied in these years of high
school.
The origins: Greeks philosophers
Greek Mathematics is mainly Geometry and Greek Philosophy often uses
mathematical tools to affirm or refute theories. Logic and demonstrations as we know
them today have their bases in the Hellenic peninsula, Southern Italy and Sicily.
Greek philosophers and mathematicians had great fame in their society and the
subtlety of their ideas still has influence on our modern way of thinking.
Above the entrance to the Academy was inscribed the phrase "Let None But Geometers Enter
Here.": this means they considered essential knowing Geometry and mainly the way of
thinking students get from learning deductions and arguing in a precise way. This is
something Maths and Philosophy have still in common today.
Pythagoras
Often described as the first pure mathematician, he is a very important
figure in the development of Mathematics yet we know relatively little
about his mathematical achievements as we have none of Pythagoras's
writings: the society which he led, half religious and half scientific,
followed a code of secrecy which certainly means that today
Pythagoras is a mysterious figure.
Fortunately we do have details of Pythagoras's life from early
biographies which use important original sources: there is a general
agreement on the main events of his life but most of the dates are
disputed.
Pythagoras he was born and spent his early years in Samos but traveled
a lot with his father: it seems that he also visited Italy. Certainly he was well educated,
learning to play the lyre, learning poetry and to
recite Homer, for sure he heard about Thales and
his pupil Anaximander who both lived on Miletus
and probably he met them there when he was
between 18 and 20 years old: by this
time Thales was an old man and, although he
created a strong impression on Pythagoras and
contributed to his interest in Mathematics and
Astronomy advising him to travel to Egypt to learn
more of these subjects. The tradition says he
attended some lectures from Anaximander and he
got interest into Geometry and Cosmology.
A few years after the tyrant Polycrates seized control of the city of Samos, in about 535 BC,
Pythagoras went to Egypt, on behalf of the tyrant himself, who was his friend.
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4. Many of Pythagoras's beliefs can be related to Egyptian customs: for instance the secrecy of
the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from
animal skins, and their striving for purity were all customs that Pythagoras would later adopt.
In 525 BC the king of Persia Cambyses II invaded Egypt: Polycrates abandoned his alliance
with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses
won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis,
Egyptian resistance collapsed and Pythagoras was taken prisoner and taken to Babylon. In
about 520 BC Pythagoras left Babylon and could return to Samos as both Polycrates and
Cambyses died in 522 BC. We don't know how he obtained his freedom. After his return to
Samos, Pythagoras made a journey to Crete where he studied the system of laws there. Back
in Samos he founded a school which was called the semicircle, but the Samians were not very
keen on this method and treated him in a rude and improper manner. So he left Samos and
went to Croton, where he founded a philosophical and
religious school that had many followers: Pythagoras was
the head of the society with an inner circle of followers
known as mathematikoi. who lived permanently with the
Society, had no personal possessions and were vegetarians
and both men and women were permitted to become
members of the Society. The outer circle of the Society
were known as the akousmatics and they lived in their own
houses, only coming to the Society during the day. They
were allowed their own possessions and were not required
to be vegetarians.
Pythagoras's Society at Croton, despite his desire to stay out of politics, was affected by
political events. Pythagoras went to Delos in 513 b.C. to nurse his old teacher Pherekydes
who was dying. He remained there for a few months until the death of his friend and teacher
and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris
and it seems probable that Pythagoras was involved in the dispute. Then in around 508 b.C.
the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself.
Pythagoras escaped to Metapontium and probably he died there, perhaps committing suicide
because of the attack on his Society. The Pythagorean Society expanded rapidly after 500
b.C., became political in nature and also split into a number of factions.
The beliefs that Pythagoras held were:
(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all brothers of the order should observe strict loyalty and
secrecy.
We don't know anything about Pythagoras's actual work: his school
practiced secrecy and communalism making it hard to distinguish
between the work of Pythagoras and that of his followers. Certainly his
school made outstanding contributions to mathematics, and it is possible
to be fairly certain about some of Pythagoras's mathematical
contributions. They were not acting as a mathematics research group
does in a modern university, there were no 'open problems' for them to
solve, and they were not in any sense interested in trying to formulate or solve mathematical
problems.
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5. Rather Pythagoras was interested in the principles of Mathematics, the concept of number, the
concept of a triangle or other mathematical figure and the abstract idea of a proof. Pythagoras
believed that all relations could be reduced to number relations and studied properties of
numbers which would be familiar to mathematicians today, such as even, odd, triangular and
perfect numbers. He noticed that vibrating strings produce harmonious tones when the ratios
of the lengths of the strings are whole numbers, and that these ratios could be extended to
other instruments: he was a fine musician, playing the lyre, and he used music as a means to
help those who were ill.
Today we particularly remember Pythagoras for his famous geometry theorem. Although the
theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years
earlier he may have been the first to prove it.
Here is a list of theorems attributed to Pythagoras, or rather more generally to the
Pythagoreans.
(i) The sum of the angles of a triangle is equal to two right angles: the Pythagoreans knew the
generalization which states that a polygon with n sides has sum of
interior angles 2n
(ii) The theorem of Pythagoras: to Pythagoras the square on the
hypotenuse would certainly not be thought of as a number
multiplied by itself, but rather as a geometrical square constructed
on the side and to say that the sum of two squares is equal to a
third square meant that the two squares could be cut up and
reassembled to form a square identical to the third square
(iii) Constructing figures of a given area and geometrical algebra: for
example they geometrically solved equations such as a (a ─ x)
= x2
(iv) The discovery of irrational numbers: it seems unlikely to have been due to Pythagoras
himself, as it was against his philosophy the all things are numbers, since by a number he
meant the ratio of two whole numbers. However, because of his belief that all things are
numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right
angled triangle had a length corresponding to a number. He also knew about Golden
Number and many properties of the regular pentagon (it was a sacred figure as the
number 5 was sacred itself)
(v) The five regular solids (Plato learned about them and now they are known and Platonic
solids)
(vi) In astronomy Pythagoras taught that the Earth was a sphere at the center of the Universe.
He also recognized that the orbit of the Moon was inclined to the equator of the Earth and
he was one of the first to realize that Venus as an evening star was the same planet as
Venus as a morning star.
Primarily, however, Pythagoras was a philosopher and he had many ideas sometimes original,
sometimes learned during his staying in Egypt:
• the dependence of the dynamics of world structure on the interaction of contraries, or
pairs of opposites
5
6. • the viewing of the soul as a self-moving number experiencing a form of
metempsychosis, or successive reincarnation in different species until its eventual
purification
• the understanding that all existing objects were fundamentally composed of form and
not of material substance.
Zeno
Zeno from Elea is famous for his paradoxes that stumped
mathematicians for millennia and provided enough aggravation to
lead to numerous discoveries in the attempt to solve them. He was
born in the Greek colony of Elea in southern Italy around 495 b.C.
and we know very little about him: he was a student of the
philosopher Parmenides and accompanied his teacher on a trip to
Athens in 449 b.C. where he met a young Socrates. On his return to
Elea he became active in politics and was arrested and tortured for
taking part in a plot against Nearchus, the city's tyrant.
He was a philosopher and logician, not a mathematician, even if
many mathematicians studied his ideas: he probably invented
dialectic, a form of debate in which one arguer supports a premise
while another one attempts to reduce the idea to nonsense, that is at the basis of the process
of reductio ad absurdum, often used in Maths demonstrations. Parmenides believed that
reality was one, immutable and unchanging: motion, change, time and
plurality were all mere illusions.
The four most famous paradoxes are the Dichotomy, the Achilles,
the Arrow, and the Stadium. For that time, they were not considered very
important, but they were studied by XX century's mathematicians as
Bertrand Russell and Lewis Carroll. Today, knowing the converging
series and the theories on infinite sets, these paradoxes can be explained.
However, even today the debate continues on the validity of both the
paradoxes and the rationalizations.
The Dichotomy: Motion can't exist as before that the moving object can reach
its destination, it must reach the midpoint of its course, but before it can reach the middle, it
must reach the quarterpoint, but before it reaches the
quarterpoint, it first must reach the eigthpoint, etc. Hence,
motion can never start.
The Achilles: Achilles while is running can never reach the
tortoise ahead of him because he must first reach the point where
the tortoise started, but when he arrives there, the tortoise has
moved ahead, and Achilles must now run to the new position,
which by the time he reaches the tortoise has moved ahead, etc.
Hence the tortoise will always be ahead.
The fundamental error in the theory is now obvious: Achilles and
the tortoise are moving independently through space-time. Achilles'
position is not dependent upon that of the tortoise: he overtakes the
tortoise when his trajectory crosses that of the tortoise. The only
problem was in calculating the precise point in time when that
happened. The wrong Math was being used to calculate it so it
could not give the correct answer. In fact, things were more
complicated than Xeno was allowing for, and this shows the danger of relying on intuition to
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7. assess reality. The logic seemed sound because it was intuitive. Maths tells us intuition was
wrong. The main flaw of Zeno’s paradox is that he uses the concept of “eternity”: if we record
the story mathematically, the time taken for Achilles to run the footrace is (if it took him 10
seconds to run 100m): 10 + 1 + 0.1 + 0.01 + 0.001… = 11.111… So, the tortoise is only ahead
of Achilles for less than 11.2 seconds (rounded). After 11.2 seconds pass, the time passed
exceeds the sum of the infinite series and the paradox no longer applies. What Zeno didn't
know was the infinite concept, that was finally systematized around the XVII century.
The Arrow: According to Zeno, time is made up of instants (“nows”), which are the smallest
measure and indivisible. Consider an arrow, apparently in motion, at any instant. First, Zeno
assumes that it travels no distance during that instant —‘it occupies an equal space’ for the
whole instant. But the entire period of its motion contains only instants, all of which contain
an arrow at rest, and so, Zeno concludes, the arrow cannot be moving. Zeno was so clever in
formulating his arrow paradox that it has resisted convincing resolution into our days.
The commonly accepted "resolution" is the "at-at" theory formulated by Bertrand Russell: he
says that the motion of an arrow appearing at different positions at different times can be
described by a continuous function where is the position at time and with
a non-constant continuous function of the real variable, the arrow is thus
changing position with increasing time and thus is moving.
The Stadium: Half the time is equal to twice the time. The three rows
here start at the first position. Row A stays stationary while rows B & C
move at equal speeds in opposite directions. When they
have reached the second position, each B has passed twice
as many C's as A's. Thus it takes row B twice as long to
pass row A as it does to pass row C. However, the time for
rows B & C to reach the position of row A is the same. So
half the time is equal to twice the time: if there is a
smallest instant of time and if the farthest that a block can move in that instant is the length of
one block, then if we move the set B to the right that length in the smallest instant and the set
C to the left in that instant, then the net shift of the sets B and C is two blocks. Thus there
must be a smaller instant of time when the relative shift is just one block.
Plato
Plato (427-347 B.C.) is considered to be one of the
philosophers who contributed much in shaping
western Philosophy.
He was born from an aristocratic family and he had
early interests were in poetry and politics: he
learned philosophy from Socrates. As a consequence
of the political execution of his teacher, he
abandoned temporarily his political interests and
left Athens. In his travels, he got in touch with the
Pythagoreans from whom he gained the interest in
Mathematics: when he finally returned to Athens
about 385 B.C., he founded his Academy, that was
considered one of the main centers of intellectual
life at the time, where he was lecturing for the rest
of his life, attracting many talented people in Greece. The Academy flourished until 529
when it was closed down by the Christian Emperor Justinian, who claimed it was a pagan
establishment. The topics of study in the Academy included Philosophy, Mathematics
and Law. One of his student, Eudoxus from Cindus, became one of the most able
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8. mathematicians at the time, gained fame from his work on the theory of proportions
and developing the idea of the method of exhaustion.
Plato’s contributions to Mathematics were focused on its
foundations: he discussed the importance of examining the
hypotheses of Maths, underlying the importance of making
mathematical definitions clear and precise as these definitions
are fundamental entities in this discipline. Another indirect
contribution of Plato was the important role he played in
encouraging and inspiring people to study mathematics: he
proposed many problems and encouraged the students of the
Academy to investigate.
Why did Plato stress on the study of Maths? We can find the
answer in the seventh book of his masterpiece, The Republic, where he stated some of
his views on its importance: to him, the idea of good is the ultimate objective of
Philosophy as “in the world of knowledge the idea of good appears last of all, and is
seen only with an effort; and, when seen, is also inferred to be the universal author of
all things beautiful and right.”. Arithmetic and Geometry have two important
characteristics that make them valuable in comprehending the
idea of good:
• these subjects lead the mind to reflect and hence
enabling the mind to reach truth;
• the advanced parts of Arithmetic and Geometry have
the power to draw the soul from becoming to beings as
“arithmetic has a very great and elevating effect,
compelling the soul to reason about abstract number
and repelling against the introduction of visible or
tangible objects into the argument.”. Studying
arithmetic as an amateur or like a merchant with a view
to buying or selling will not help the soul to make this
transition from becoming to being.
He also thought that people who are good in Mathematics will
do well in any other field of knowledge and payed specifically
attention to the irrational numbers, as for him Science is incomplete without them
and that a comprehensive study of irrationals is necessary to build “a coherent and
universal philosophy free of the difficulties that wrecked the Phythagorean system.”.
He concentrated on the idea of "proof" and insisted on accurate definitions and clear
hypotheses: this laid the foundations for Euclid's
systematic approach to Maths.
He didn't deny the important applications of
Mathematics in people’s daily life, but, to
him, the philosophical importance of this
subject is more important and more
rewarding as it may affect one’s
understanding of his being, even if he didn't
made any important discovery in this filed: he
supplied many Mathematical problems in his
writing, especially in the Meno, where Socrates
asks to a slave boy to double the area of a square
or asks to the audience if a given triangle can be
inscribed in a given circle.
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9. In Mathematics Plato's name is linked to the “Platonic solids”, that he
learned from Pythagorean school: in the Timaeus, there is a mathematical
construction of the elements earth, fire, air, and water being represented by
the cube, tetrahedron, octahedron, and icosahedron respectively. The fifth
Platonic solid, the dodecahedron, is Plato's model for the whole universe:
this was the first attempt Kepler used when he had to fit Brahe's data
collection studying the Solar System structure.
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10. Aristotle
Aristotle (Stagirus 384 a.C. - Euboea 322
a.C.), another giant of the Western culture
whose influence has been immeasurably
vast, set the Philosophy of Physics,
Mathematics, and Science on a foundations
that would carry it to modern times. He was
teacher of Alexander the Great and this gave
him many opportunities and an abundance
of supplies. He founded a school called
Lyceum where he established a library
which aided in the production of many of his
hundreds of books. He was a pupil of Plato
but immersed himself in empirical studies
and shifted from the theoretical Platonism to empiricism. He believed all
peoples' concepts and all of their knowledge was ultimately based on
perception. Aristotle's views on natural sciences represent the
groundwork underlying many of his works and profoundly shaped
medieval scholarship .
He viewed the sciences as being of three types:
• theoretical (Maths, Physics, Logic and Metaphysics)
• productive (the Arts)
• practical (Ethics, Politics).
He contributed little to Mathematics however, his views on the nature of this subject and its
relations to the physical world were highly influential. Whereas Plato believed that there was
an independent, eternally existing world of ideas which constituted the reality of the universe
and that mathematical concepts were part of this world, Aristotle favored concrete matter or
substance.
Aristotle regarded the notion of definition as a significant aspect of argument: he required that
definitions reference to prior objects. For instance, the definition, 'A point is that which has no
part', was unacceptable for him, as it does not assert or deny. Instead a hypothesis asserts one
part of a contradiction, for instance that something is or is not. There are many views as to
what Aristotle's hypotheses are:
10
11. ▪ existence claims
▪ any true assumption within a science
▪ the stipulation of objects at the beginning of a typical proof in Greek
mathematics.
For instance in AB is a line stipulated to exist a hypothesis is: ‘Let there be a line AB’.
He treated the basic principles of Mathematics, distinguishing between axioms (a statement
worthy of acceptance and is needed prior to learning anything) and postulates (that need not
be self-evident, but their truth must be sustained by the results derived from them): this
distinction was later on used by Euclid who organized all Greek Mathematical (Geometrical)
knowledge in the same way we have been using nowadays.
Aristotle explored the relation of the point to the line dealing with the problem of the
indecomposable and decomposable and made the distinction between potential
infinity and actual infinity: he stated that only the former actually exists, in all regards.
He is also credited with the invention of Logic, through the syllogism (the most famous is this
one: “All men are mortal, Socrates is a man, Socrates is mortal”), the law of contradiction
(two antithetical propositions cannot both be true at the same time and in the same sense. X
cannot be non-X. A thing cannot be and not be simultaneously. And nothing that is true can be
self-contradictory or inconsistent with any other truth → ¬( X∧¬X ) ) and the law of the
excluded middle (there is no third alternative to truth or falsehood: in other words, for any
statement X, either X or not-X must be true and the other must be false). His logic remained
unchallenged until the century. Aristotle regarded logic as an independent subject that should
precede Science and Maths.
In any society at every moment all the strong and developing forces work in relation to each
other, sometimes unknowingly, sometimes quite obviously. As we look back into the past,
where there are many chapters permanently erased from our view, we tend to see books and
men and ideas as particular and individual, and it is easy to forget that intellectual fields cross
all lines of human endeavor constantly. So the study of Greek medicine is necessary for an
understanding of Greek drama and Aristotle's critique of it. Acknowledge of Greek history is
required for understanding of Aristotle's Ethics, and as we learn more about the ancient world
to know a great deal more about the nature and the influences of Greek Mathematics, which
was perhaps the single greatest achievement of the Greek mind, and one which certainly
influenced the general tenor of the society more thoroughly and in more ways than we us can
yet fathom.
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13. Maths and Philosophy in XVI and XVII centuries
In the late 16th and 17th centuries Mathematicians began to deep the question about the
continua — straight lines, plane figures, solids – For instance one question is: “Is a line
segment composed of an infinite number of indivisible points?” If so, and if these
infinitesimals have zero width, how does the line segment come to have a positive length?
And if they have nonzero widths, why isn’t the sum of their widths infinite?
Already the ancient Greeks posed questions like these and Aristotle, trying to solve it, had
argued that continua could not consist of indivisibles. but in these centuries philosophers try
to demonstrate that thinking of them this way yielded insights not easily obtained from
traditional Euclidean geometry.
In Italy, Torricelli, Cavalieri and Galileo were approached Geometry in a new way involving
infinitesimals. If classical Euclidean Geometry is conceived as a top-down approach with all
theorems following by pure logic from a few self-evident axioms, the new approaches can be
thought of as bottom-up, inspired by experience: a plane can be considered an infinite number
of parallel lines and a solid an infinite number of parallel planes.
But in some part of Europe philosophers could go on with their thinking while others couldn't
as the Jesuits and the Roman Catholic associated chaos, confusion and paradoxes with
infinitesimals and the motley array of proliferating Protestant sects.
The philosophers we are going to analyze are involved in this revolution of thinking: two of
them worked inside the Catholicism, one lived in the Protestant environment.
Descartes
René Descartes was born on March 31, 1596, in the French town
called La Haye. He was entered into Jesuit College at the age of
eight, where he studied for about eight years, then he spent several
more years in Paris studying Mathematics with friends, such as
Mersenne. He then received a law degree in 1616. At that point in
time, a man that held that type of education either joined the army
or the church: he chose to join the army of a nobleman in 1617, so
a young man he traveled around Europe. While serving, Descartes
came across a certain geometrical problem that had been posed as a
challenge to the entire world to solve. Upon solving the problem in
only a few hours, he had met a man named Isaac Beeckman, a
Dutch scientist who became a very close friend of his. Since
becoming aware of his mathematical abilities, the life of the army was unacceptable to
Descartes. His health had not been good for all his life and the soldier life was not suitable for
him. So, in 1621, Descartes resigned from the army and traveled for five years. During this
period, he continued studying pure Maths, then, in 1626, he settled in Paris where he
constructed optical instruments. In 1628 he devoted his life to seeking the truth about the
science of nature and moved to Holland where he remained for twenty years, dedicating his
time to Philosophy and Maths. During this time, Descartes wrote the book Meditations on
First Philosophy, where he introduced the famous phrase "I think, therefore I am." meaning he
wanted to find truth by the use of reason, taking complex ideas and breaking them down into
simpler ones that were clear.
Since he was young, he showed an interest in Physics, but, while he was writing his first
book, he heard that Galileo had been arrested for claiming that the Earth rotates around the
Sun, so he decided not to publish his book, and to concentrate on Mathematics instead. But
his friends wanted him to publish his ideas, so in 1637 Descartes published a book
called Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les
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14. sciences, or Discourse on methods for conducting reason and seeking truth in the sciences in
which he described the methods of studying Science.
This book has had three appendices and one was
about Optics (light and lenses), one was about weather, and one
was about Mathematics, called La Géometrie and, despite its
title, it is focused on the connections between Geometry and
Algebra. In this book we can see many modern algebraic
conventions: for example, Descartes used letters from the
beginning of the alphabet for constants and known quantities,
and letters from the end of the algebra for variables: this is the
reason why we usually solve equations for x, and not some other
symbol. Descartes believed that Mathematics was the only thing that is certain or true, so it
could be used to model the complex ideas of the universe into
simpler ideas that were true.
La Géometrie, even it is an appendix of the main Philosophy
book, is divided into several chapters: the first one explains
connections between Algebra and Geometry, and gives both
algebraic and geometric ways to solve equations, the second and
third ones investigate more complicated subjects as conic
sections and plane loci (curves at a fixed distance from different
lines in a plane): we can find general equations for certain types
of parabolas and hyperbolas and the classification of linear
function (first degree) and curves (second and upper degrees)
depending on their algebraic representation. Here he introduced
his theory about determining a point in a plane by pairs of real
numbers (ordered pairs), now known as Cartesian Plane. Although this had been done before
by other mathematicians earlier in the history of Maths, his become a standard as he had
gathered all the tools for coordinate graphing, using these reference lines to analyze the curves
he studied. Like Apollonius, he usually drew his curves first, then drew reference lines to
analyze them with but thought that negative numbers did not represent "real" physical
quantities, so he ignored negative roots of equations, and he avoided measuring in more than
one direction on a line whenever possible. Probably because of his not good health, he never
got out of bed before 11 in the morning, and it is said (although the story is probably a myth)
that Descartes came up with the idea for his coordinate system while lying in bed and
watching a fly crawl on the ceiling of his room.
Descartes had gathered all the tools for coordinate graphing.
Because of this accomplishment, he is often given credit for
inventing the coordinate plane, even though he never graphed an
equation.
La Géometrie was soon recognized as an important work of Maths,
but it still took several years for its ideas to become well known.
The information moved slowly for two reasons: Descartes wrote
his book in French, so scholars who didn't know French could not
read it and he discussed many difficult problems without giving
examples or explaining simpler cases. Eventually, Descartes' friends and students published
guides and commentaries for La Géometrie: many of these guides were in Latin, since almost
all educated people knew Latin at that time and gave examples and explained simpler
problems.
In 1649, Descartes was invited by the Queen to Sweden as Court Mathematician. It is said
that the Queen wanted to work on Maths at an early hour in the mornings. Thus, Descartes
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15. had to rise early to reach the palace. Due to the cold climate, he developed pneumonia after
just a few months and died on February 11, 1650.
Pascal
Blaise Pascal (Clermont-Ferrand 1623 – 1662 Paris) was
a French philosopher and scientist, one of the greatest and
most influential mathematical writers of all time: he was also
an expert in many fields, including various languages, and a
well-versed religious philosopher. When he was only three
years old Blaise's mother died and in 1632 the family left
Clermont and settled in Paris. His father had unorthodox
educational views and decided to teach his son
himself, deciding that Blaise was not to study mathematics
before the age of 15 and all mathematics texts were removed
from their house. Blaise however, his curiosity raised by this,
started to work on geometry himself at the age of 12. He
discovered that the sum of the angles of a triangle are two
right angles and, when his father found out, he surrendered and gave to Blaise a copy of
Euclid's book “Elements”. When he was 14 he started to accompany his father to Mersenne's
meetings: Mersenne belonged to the religious order of the Minims, and his cell in Paris was a
frequent meeting place for many mathematicians, Gassendi,
Desargues and others. When he was 16, Blaise presented a single
piece of paper containing a number of projective
geometry theorems, including his mystic hexagon to one of
Mersenne's meetings. In December 1639 the Pascal family left
Paris to live in Rouen where his father had been appointed as a tax
collector for Upper Normandy. Shortly after moving to Rouen,
Blaise published his first work, Essay on Conic Sections, in
February 1640. In this six-pages paper he wrote the fundamental
theorem that states that the three points of intersection of the opposite sides of a hexagon
inscribed in a conic section lie on a straight line. A hexagon inscribed in a conic section
essentially consists of six points (called “vertices”)
anywhere on the conic section The the segments connecting
consecutive vertices are called the "sides" of the hexagon.
The straight line on which the three points of intersection lie
is called the Pascal line. The hexagon itself is called a
Pascal hexagon.
Working three years (1642-45) Pascal invented the first
digital calculator (called the Pascaline) to help his father
with his work collecting taxes. Designing the device he had to face with three problems due to
the design of the French currency at that time: there were 20 sols in a livre and 12 deniers in a
sol. Pascal had to solve much harder technical problems to work with this division of the livre
into 240 than he would have had if the division had been 100.
The year 1646 was very significant for the young Pascal as his father
injured his leg and had to recuperate in his house. He was looked
after by two young brothers from a religious movement just outside
Rouen. They had a profound effect on the young Pascal who became
deeply religious.
In this period Pascal began a series of experiments on atmospheric
pressure: by 1647 he proved to his satisfaction that a vacuum
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16. existed. Descartes visited Pascal on 23 September: his visit only lasted two days and the two
argued about the vacuum which Descartes did not believe in. Descartes wrote, rather cruelly,
in a letter to Huygens after this visit that Pascal “...has too much vacuum in his head.”
In August of 1648 Pascal observed that the pressure of the atmosphere decreases with height
and deduced that a vacuum existed above the atmosphere. In October 1647 Pascal wrote New
Experiments Concerning Vacuums which led to disputes with a number of scientists who,
like Descartes, did not believe in a vacuum.
Pascal's father died in September 1651 and after he wrote to one of his sisters giving a deeply
Christian meaning to death in general and his father's death in particular: these ideas here are
at the basis of his later philosophical work Pensées.
In 1653 he published the Treatise on the Equilibrium
of Liquids in which he explains the law of pressure
that now we know as Pascal's law: “that pressure
exerted anywhere in a confined incompressible fluid
is transmitted equally in all directions throughout the
fluid such that the pressure variations (initial
differences) remain the same“. In other words, if no other forces are acting on a fluid, the
pressure will be the same throughout the fluid and the same in all directions In this period he
also composed, but never published, another monographs that was discovered among his
manuscripts after his death: the Treatise on the Weight of the Mass of Air. These two treatises
represent a very important contributions to the Hydraulics and Hydrostatics. It is in
recognition of his important work in the study of fluid mechanics that a standard unit of
pressure is today known as the pascal (Pa), defined as a force equal to 1 Newton per square
meter.
Between 1648 and 1654 he worked on conic sections and produced important theorems in
projective geometry. In The Generation of Conic Sections, he first part of a treatise on conics
which Pascal never completed, he considered conics generated by central projection of a
circle: the work is now lost but Leibniz and Tschirnhaus made notes from it and so now we
can have a fairly complete picture of the work.
Studying the results obtained by Tartaglia, the came out with the so called “Pascal's triangle”
and wrote about this in Treatise on the Arithmetical Triangle (1653-1654) was the most
important on this topic that lead Newton to his discovery of the general binomial theorem for
fractional and negative powers.
Pascal called the square containing each number in the
array a cell. The numeral 1’s at the top of his triangle
head perpendicular rows; those on the left side of the
triangle head parallel rows. He called the third side of the
triangle the base (diagonal) and the cells along any
diagonal row “cells of the same base”. The first diagonal
row (consisting of the number 1) was for him row 0, the
second diagonal row (1, 1) was row 1; and so on. He
constructed the triangle calculating each the number
value of each cell is equal to the sum of its immediately
preceding perpendicular and parallel cells (i.e.item 4 in
row 7 in the base diagonal 120 = 36 + 84). Furthermore, the number value of each cell is also
equal to the sum of all the cells of the preceding row (from the first cell to the cell
immediately above the target cell). For example, 126 (the number value of cell 6 in row 5) = 1
+ 4 + 10 + 20 + 35 + 56 (the sum of cells 1-6 of row 4).
Pascal explained also in detail how the Triangle could be used to calculate combinations (n
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17. things taken r at a time indicated now with the symbol Cn ,r ): we would move
perpendicularly down to the nth row and then move diagonally r cells (i.e. to calculate C5,4
we have to go perpendicularly down to row 5 and then move diagonally 4 cells and find that
the number of combinations is 5).
In the Summer of 1654 he exchanged some letters
with Fermat where they laid the foundations for the theory of
probability. They considered the dice problem (how many times
one must throw a pair of dice before one expects a double six),
already studied by Cardano, and the problem of points (how to
divide the stakes if a game of dice is incomplete) also deepened by
Cardano, Pacioli and Tartaglia Pascal and Fermat solved the
problem of points for a two player game but did not develop
powerful enough mathematical methods to solve it for three or
more players.
In a section of the Treatise, Pascal explained how to use the
Triangle to solve the Problem of Points:
“A needs 3 more points, B needs 5 more points and the game will end after seven more tries
since at that juncture one of the players must reach ten points: count 3 + 5 rows on the
Triangle; then sum the first 5 items. That sum divided by the sum of all items in the row is A’s
portion of the stakes. Then sum the remaining 3 items in the row and divide that total by the
sum of all the items in the row. That will be B’s portion.”
From the Triangle:
(1+7+21+35+35) ÷ (1+7+21+35+35+21+7+1) = 99/128 = A’s portion.
(1+7+21) ÷ (1+7+21+35+35+21+7+1) = 29/128 = B’s portion.
Expressed as a percentage, A receives 77.34375 percent of the stake; B receives 22.65625
percent of the stake.
By October 1654 he almost lost his life in an accident: the horses pulling his carriage bolted,
the carriage was left hanging over a bridge above the river Seine and he was rescued without
any physical injury but he was much affected psychologically. Not long after he underwent
another religious experience, on 23 November 1654,
and he pledged his life to Christianity.
After this time Pascal made visits to the Jansenist
monastery at Port-Royal des Champs about 30 km
south west of Paris and between 1656 and early 1657
he published anonymous 18 writings on religious
topics (Provincial Letters) being published during
1656 and early 1657 in defense of his friend Antoine
Arnauld, a Jansenist who was an opponent of the
Jesuits and who was on trial before the faculty of
theology in Paris for his controversial religious works.
In this period (late 1656 -1658) he also wrote his most famous work in Philosophy, Pensées, a
collection of personal thoughts on human suffering and faith in God : this work contains
'Pascal's wager' which claims to prove that belief in God is rational with the following
argument. He said that “If God does not exist, one will lose nothing by believing in him, while
if he does exist, one will lose everything by not believing.”: here he uses probabilistic and
mathematical arguments but his main conclusion is that “...we are compelled to gamble...”.
His last work (1658) was on the cycloid, the curve traced by a point on the circumference of a
rolling circle: he stayed up late the night unable to sleep for pain thinking about these
mathematical problems. He applied Cavalieri's calculus of indivisibles to the problem of the
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18. area of any segment of the cycloid and the centre of gravity of any segment and could
calculate the volume and surface area of the solid of revolution formed by rotating the cycloid
about the x-axis. He published a challenge offering two prizes for solutions to these problems
to many famous mathematicians, as Wren, Laloubère, Leibniz, Huygens, Wallis, Fermat and
several others, many of whom communicated their discoveries to Pascal without entering the
competition with him and could find the arc length, the length of the arch, of the cycloid.
Pascal published his own solutions to his challenge problems in the Letters to Carcavi.
After that time on he took little interest in scientific subjects and spent his last years giving to
the poors and going from church to church in Paris attending one religious service after
another. He died at the age of 39 in intense pain after a malignant growth in his stomach
spread to the brain.
Leibniz
Gottfried Wilhelm Leibniz (Leipzig July 1, 1646 - Hanover
November 14, 1716) was a German philosopher, mathematician,
physicist and statesman who occupies a prominent place in the
history of both Maths and Philosophy: noted for his independent
invention of the differential and integral calculus, is one of the
greatest and most influential metaphysicians, thinkers and logicians
in history and also invented the “Leibniz wheel” and suggested
important theories about force, energy and time. Leibniz also
developed the binary number system, which is at the foundation of
virtually all digital computers. Leibniz made major contributions
to Physics and Technology anticipating notions that were
developed much later in Biology, Medicine, Geology, Psychology,
Linguistics: his writings (many of them never published) show
major contributions to Politics, Ethics, Theology and History. Modern Physics, Maths,
Engineering would be unthinkable without his contribution about the fundamental method of
dealing with infinitesimal numbers. In Philosophy, Leibniz is mostly noted for his optimism:
our Universe is the best possible one that a God could have created. Leibniz, along
with Descartes and Baruch Spinoza, was one of the three great 17th century advocates
of rationalism but his philosophy also looks back to the scholastic tradition, in which
conclusions are produced by applying reason to “a priori” definitions rather than to empirical
evidence.
His father Friedrich was a professor of moral Philosophy at the University of Leipzig and died
when he was only six, while his mother, Catharina Schmuck, was the daughter of a rich local
lawyer who influenced his philosophical ideas. As a result, his early education was somewhat
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19. haphazard, but he became fluent in Latin and studied works of Greeks scholars so he was
fluent at them by the age of 12 and he was ready for the university at the age of 15. He
pursued the course in law in preparation for a political career and also studied Theology,
Mathematics, and the new natural Philosophy of the Enlightenment, received his Bachelor’s
Degree in Philosophy by 1662 and earning his Masters Degree by 1664. He also gained his
Bachelors degree in Law by 1665: this shows the extent and diversity of his academic
interests. After graduation, he applied for a doctorate in Law, but was refused due to his young
age, so he presented his thesis to the University of Altdorf, where professors were so
impressed that they awarded him the degree of Doctor of Laws (1667) and gave him a job of
professorship: he declined the offer and accepted instead a position in the service of the
elector of Mainz.
In this period Louis XIV's aggressive activities were a serious threat to the German states so
in 1670 Leibniz published a pamphlet proposing a defensive coalition of the northern
European Protestant countries, recently weakened by the Thirty Years War. In the mean time,
to give these principalities an economic recovery, he conceived a plan whereby Louis might
gain Holland's valuable possessions in Asia by sort of a "holy war" against non-Christian
Egypt: Leibniz was invited to Paris to present his plan that was not adopted at the end.
Despite this, stayed 4 years in the French capital, visiting twice London (1673 and 1676) and
this experience was crucial for his intellectual development.
Before going to Paris, Leibniz developed a calculating machine based on the principles of the
“Pascalina” but capable of performing much more complicated mathematical operations; his
demonstrations of this machine before the Académie Royale des
Sciences and the Royal Society of London aroused much interest
and led to fruitful relations with members of these groups and to
his election to membership in the Royal Society shortly after his
first visit in London.
In Paris he met the Dutch mathematician Christiaan Huygens who
was very important as a stimulus to Leibniz's interest in
Mathematics during the years of his residence there Leibniz
developed both the integral and the differential calculus.
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20. In 1676 Leibniz transferred his services to the house of Brunswick and moved to Hanover,
town which became his home and where he mainly lived for the rest of his life. He was sent
on important diplomatic missions, with freedom to seek out leading scholars wherever he
went and he received many honors, as well as a generous stipend, and had ample leisure for
pursuing his own interests. He was in charge to write a history of Brunswick from earliest
times and he could access not only to the resources of the ducal library but also to the
historical repositories in Germany and Italy.
He couldn't complete the history itself (at his death he had completed to the year 1005) but he
added geological data to bear for the first time on historical interpretation and used original
documents in a modern way.
In 1672 he published a pamphlet where he proposed an alliance of all the European countries
against Turkey, an example of a possible reunification of all Christians, adducing historical
evidence of this unity in correspondence with the French prelate Jacques Bossuet: he didn't
succeeded again in his attempts at mediation of differences.
In 1678 he founded journal for the publication of scholarly papers named “Acta
eruditorum”,that gained wide circulation in Europe: over the next 35 years, most of his own
published writings appeared.
He imprinted his name in the world of mechanical calculation, he was also the first to create a
pinwheel calculator (step rekoner) explaining in detail how it worked in 1685 and developed
the “Leibniz Wheel” that helped to improve the binary number system which is at the basis of
the digital computers.
In 1700 he was elected a member of the French Académie Royale and, in the same year, the
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21. Akademie der Wissenschaften was founded at Berlin because his recommendation: following
the patterns of the French Académie and the Royal Society of London, he drew up its statutes
and was its first president, maintaining that position for the rest of his life. Through his
influence similar academies were established at Dresden, St. Petersburg, and Vienna.
Because of his disposition to moderation and tolerance and his position of leadership among
European scholars, he had an active role as diplomat: the warmth and loyalty of many friends
and supporters can be seen in his enormous correspondence with them. Between these people
there were also women and he had a positive influence to develop the new "learned woman"
idea.
Leibniz' last years were overshadowed by a controversy
with the powerful president of the Royal Society, Isaac
Newton, who said he had invented calculus before Leibniz,
without publishing it, and who made friends sign texts he
wrote to support this claim: now we considered to have
been a case of independent discovery by two highly gifted
minds but at the time the quarrel was really hard for the
exchanges of accusations that lasted for more than 10 years
by partisans on one side and then the other. Newton was in
a sense essentially practical: he invented tools then showed
how these could be used to compute practical results about
the physical world, but Leibniz had a broader and more
philosophical view and saw calculus not just as a specific
tool in itself, but as an example that should inspire efforts at other kinds of formalization and
other kinds of universal tools.
Finally, an investigating commission got some results and exonerated Newton and failed to
remove the charge of plagiarism against Leibniz: the cutting off of free communication of
ideas between the English scientists and those of the Continent was ironically to the detriment
of the former: Leibniz's notation was more efficient than Newton's and facilitated the great
strides in mathematical Physics made on the Continent during the next hundred years, in
which the participation of English scientists was negligible.
The whole procedure was a crushing offense for Leibniz, who had always been a proponent of
free interchange among scholars and who was really sad when the Duke of Brunswick's
refused to include him in his entourage when, in 1714, he became England's George I, as he
considered him a controversial figure.
Leibniz died at Hanover in 1716 and his popularity with his own countrymen had waned with
his declining court favor: his only worthy eulogy was composed on the first anniversary of his
death by the French academician Bernard de Fontenelle and it was read before the meeting of
Leibniz's colleagues in Paris and recorded in their archives.
http://www-history.mcs.st-and.ac.uk/Biographies/Pythagoras.html
http://www.iep.utm.edu/zeno-par/
http://www.answers.com/topic/platonic-academy-1
http://www.ucmp.berkeley.edu/history/aristotle.html
http://www.math.wichita.edu/history/men/descartes.html
http://www-history.mcs.st-and.ac.uk/Biographies/Pascal.html
http://www.iep.utm.edu/pascal-b/
http://www.famousscientists.org/gottfried-leibniz/
http://www.dmoz.org/Society/Philosophy/Philosophy_of_Science/Mathematics/
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