**Unformatted text preview: **MATH 338 - History and Philosophy of Mathematics
Ancient civilizations used rudimentary mathematics for accounting,
engineering, land surveying, astronomy, etc. Around 2 000 BC, there was only MM BC applied math, no theoretical math. There was no progress in mathematics
between c. 2 000 BC and c. 500 BC. The Egyptians MM BC Hieroglyphic
Numerals = 3 · 1 + 4 · 10 + 6 · 100 + 2 · 1 000 = 2 643 Ex The Egyptians almost exclusively used unit fractions, with the exception of
2/3. To write a fraction, they would write the denominator under a mouth
symbol, except for 1/2 and 2/3 which had Fractions their own symbols. If the denominator
was too large to t under the mouth, it
would be placed in the top left corner 3+4
12 of the frame the number was written in. = 1
4 + 1
3 7
12 = = Non-unit fractions were written as sums
of unit fractions.
The Egyptians didn't grasp the
concept of π π, so they approximated it as π ≈3+ 1
7 when dealing with circular structures. The Babylonians MM BC Although the Babylonians were contemporary to the Egyptians, we are more
familiar with the mathematics of the former as many more Babylonian documents have survived than Egyptian ones. Babylonians wrote on clay tablets
with triangular pointed styluses, after which the tablets were baked and much
better preserved than the Egyptian paper-like papyrus, most of which disintegrated throughout the millenia.
The Babylonian number system was a mix of base 10 and base 60. Astronomy
and Angles Babylonian obssession with astronomy led to their developing an elaborate
method of calculating angles to decipher the trajectory of stars. The data they compiled was so accurate that it was used some 3 millenia later by
Kepler, a renowned astronomer of the Renaissance. The use of degrees today
and the concept of 360 degrees for a circle is a leftover from Babylonian math
(360 = 60 · 6). 1 A Babylonian tablet with numbers on
it was uncovered and deciphered, re- 3
5
7
15
21
13500 vealing a sequence of values that t the
pythagorean theorem. Early Notion
of the
Pythagorean
Theorem Note the last row which represents a case of the equation of Pythagorean triples x2 − y 2 2 2 + (2xy) = x2 + y 2 2 4
12
24
8
20
12709 5
13
25
17
29
18541 32 + 42 = 52
for x = 125 and y = 54, which must have been surprisingly dicult to nd 52 + 122 = 132 at the time, when multiplication was ··· one of the most complex mathematical
operation known. Another tablet was found with a ge- Irrational
Numbers ometric proof of the existence of the
irrational number √ 2, using two unit squares cut into four triangles and rearranged into another square. The Babylonians this for √ 3 Greek Mathematics couldn't reproduce so ignored its existence.
~500 BC 415 AD Greek philosopher and mathematician
from day Turkey. Thales is often credited with Miletus, Thales' theorem: modern A triangle inscribed in a circle with exactly one side crossing the
center of the circle is a right triangle.
Thales opinion (existence) Thales of
Miletus of on the math was ontology
wa- that ter is the arche (origin) of math,
meaning
ability that
to substances
as math relies distinguish
by their on objects our ∼ 624 BC ∼ 546 BC and measure (read size/volume/length/duration etc. Water is used as a metonym for measure, as it was used to measure many
things such as volumes, or even time in
clepsydras).
Thales emphasized the importance of deductive reasoning and proofs to
establish truths. 2 Euclid's proof of the pythagorean
theorem
Greek philosopher and mathematician,
of whom we know very little. Most of
what we know about him was written
down centuries after his death.
is credited with the He pythagorean theorem, which states that the square
on the hypothenuse of a triangle is Pythagoras
of Samos equal to the sum of squares on the
other two sides. pythagorean
movement was that math originates
from our ability to count objects and
to use arithmetics.
One of the tenets of the ∼ 570 Plato BC - ∼ 495 BC Greek philosopher who believed that deduction must be based on unprovable truths (axioms). ∼ 425 BC ∼ 348 BC Plato believed humans perceive mathematical objects incompletely, as if perceiving shadows on a wall without being able to completely make out what
objects cast the shadows. He argues that these objects exist only in a perfect, separate universe inaccessible to humans.
Aristotle Greek philosopher. A pupil of Plato and tutor to Alexander the Great. ∼ 425 BC ∼ 348 BC Aristotle believed that mathematical objects are simply abstract ideas constructed in the human mind, as opposed to reections of actual objects in a
parallel universe. Euclid of
Alexandria Greek mathematician widely known for writing The Elements , an encyclo- pedia of all geometry and math known at the time. All of his results are based
on his 5 axioms: 1. A segment can be constructed from any 2 points in space.
2. Any segment may be extended indenitely.
3. From any segment, a circle can be constructed with the segment as its
radius. 4. All right angles are equal.
5. If 2 lines are perpendicular to a 3rd, the 2 lines are parallel to each other
(parallel postulate). Alternatively, Playfair's axiom can be used
instead: given a line and a point, exactly one parallel can be constructed
through the point. 3 ∼ 340 BC ∼ 260 BC Apollonius
of Perga Greek geometer known for writing Conic sections , a text describing the prop- erties of the 4 ways to cut a cone: 262 BC 190 BC 1. A section parallel to the base is a circle.
2. A section not parallel to the base and which does not cut through the base
is an ellipse.
3. A section which cuts through the base and is parallel to any plane tangential
to the conical surface is a parabola.
4. A section which is perprendicular to the base is a hyperbola.
Apollonius showed that conic sections are dened by 2
on their major axis. If P P F1 + P F2
Pappus of
Alexandria foci, points F1 and F2 is any point on an ellipse, then the length
is constant Greek teacher from Alexandria in Egypt, who contributed to the study of
conic sections.
Euclid had discovered that, for each focus, there exists a line outside the conic
section called the directrix, which is perpendicular to the major axis. Pappus found that if the section and such that P is any point on
P D is perpendicular to the directrix, then D called the eccentricity. is a point on the directrix Ellipse Parabola Hyperbola ε<1 ε=1 ε>1 The last Greek mathematician is said to be
in 415 AD by an angry Christian mob. PF
PD = ε is a constant Hypatia of Alexandria, murdered No new math was invented under the Romans and European math did not progress at all until around 1415.
it actually regressed at times, as invaluable, unique copies of books were
destroyed under bursts of religious dogmatism.
Around 550 AD, Roman emperor Justinian, a devout Christian, closed all
universities in the empire including Plato's academy, as everything you really
needed to know was obviously in the Bible.
Luckily, copies of important Greek mathematical works survived in the remnants of Alexander the Great's empire, which stretched from Macedonia to
northern India, and were studied in these countries. Prime Numbers
Prime
Numbers Prime numbers are natural numbers that only have 1 and themselves as divisors. Because of this property, they have fascinated mathematicians for millenia, and many have attempted to nd formulae to generate them. Fermat
"Primes" Fn = 22 + 1 F5 = 22 + 1 = 232 + 1 Mersenne
"Primes" Mp = 2p − 1 M11 = 2017 = 23 · 89 n 5 4 not prime! not prime! ∼ 290 ∼ 350 Generating
Primes There is still no formula for generating primes. Theorem There are innitely many prime numbers.
Suppose not. Then there exists a largest prime number 11 · · · · · p + 1 p. Let Ep = 2 · 3 · 5 · 7 ·
p, plus be the product of all prime numbers up to and including i divides Ep because there is always a remainder
1
. We have that Ep is a prime bigger than p, or contains a prime factor bigger
i
than p, in which case we have a contradiction and there are indeed innitely
1. By construction, no prime many prime numbers. Goldbach
Conjecture Can all even numbers be written as the sum of 2 primes?
Nobody knows!
n 6= 6 :
Perfect
Number 6
n One that equals the sum of all its ∈ N = 1, 2, 3
6=1+2+3 divisors Indian Math
The VII place value numbering system, the number 0 and the negative numbers familiar to us were invented in medieval India. Indians were good at showing
how to solve problems and systems of equations but, contrarily to the Greeks,
never did any formal proofs. Brahmagupta Compiled a math book with results in arithmetic using the new Indian numbers. ∼ 600 Some of the results he found include: • −x · −y = xy
• 0
0 =1 • 0
x =0=0·x • The area (oops) A of a cyclic quadrilateral (a four sided gure inscribed in a circle) is v
u 4
uY
A = t (s − ci )
i=1
where
of the th ci is the length of the i side
ci 's equal to 0 gives a general c1 +c2 +c3 +c4
. Setting one
2
formula for the area of a triangle, and s := since all triangles are cyclic. This formula was known to the Greeks as Heron's formula.
Bhaskara From Maharashtra. Came up with most of today's known trigonometric formulas by reading Ptolemy's astronomy, itself based on Babylonian angles and ∼ 600 ∼ 680 elevations for stars and planets. Muslim Math
During the early days of the Islamic conquests, the Muslims captured Alexan- VIII 642 dria in 642 AD and, according to 13th century arabic sources, burned the
library of Alexandria, the largest known library of its time in the world. House of
Wisdom The Muslims built Baghdad and opened the House of Wisdom, where in- tellectual works and scholars converged, quickly becoming a major center of
knowledge. Important Greek mathematical books were collected by thinkers
from across the Muslim empire, and translated into arabic, thus preserving the
Greek legacy. 5 ∼ 750 Muhammad
AlKwharizmi An ethnic Persian born in Khwarezm, modern day Uzbekistan. Wrote several The Compendious Book on Calculation
by Completion and Balancing (~820), the original title of which (Al kitab al
mukhtasar hisab al-gabr wa'l muqabala ) gave rise to the word algebra, and
books on Indian numerals, including ∼ 780 ∼ 850 where he gives basic results like factoring and multiplying out binomials: x3 − 1 = (x − 1) x2 + x + 1 algebraically as
geometrically, as the Greeks did. Khwarizmi wrote down He is one of the rst mathematicians to study mathematics
opposed to purely the general quadratic formula: c2 x2 + c1 x + c0 = 0
and proved that there was no general solution that does not involve negative numbers. The word algorithm comes from Al-Khwarizmi's name. (alkhwarizmi algwarithmi algwarithm algorithm whooaaaaa) Umar AlKhayyam An ethnic Persian astronomer, mathematician and poet, born in Nishapur,
modern day Iran. Known as 1 048 - 1 131 Omar Khayyam in the West. Khayamm lived in Baghdad and is considered to be the most inuential mathematician of his era,
making major contributions such as a geometric solution to a cubic equation
of the form AlSamawal
alMaghribi x3 + c1 x = c. A moroccan Jew mathematician (Samawal = Samuel) who converted to Islam,
keeping his faith secret for fear of oending his father.
Samawal wrote The Brilliant in Algebra (∼1 At 19 years old, ∼ 1 130 ∼ 1 180 180), in which he introduces polynomials of degree greater than 3, fractional powers and negative
powers. He too broke from the Greek geometric approach in favor of algebra.
Samawal grasped the notion of the n binomial theorem (x + 1) =
and of what we call X n
xm
m
m=0 Pascal's triangle, 600 years before Pascal did. Samawal proved that between any two rational numbers, there exists an irrational, and that between any two irrational number, there exists a rational
number.
The muslims preserved and expanded on Greek math and codied Indian arithmetic, but like the Indians, neglected formal proofs. Rebirth of European Mathematics
Adelard of
Bath An English philosopher.
Muslim rule from ∼ 720 Adelard lived in Toledo, which had been under (Conquest of Spain by the Umayyads lasted from 711 to 788) until 1085 when Christian Spain conquered it back. He learned arabic
and found an arabic version of Euclid's Elements, which he translated into Latin (the lingua franca of Western Europe) and reintroduced it in Europe.
This marks the beginning of the rediscovery of Greek math in Europe, which took many centuries.
Adelard also translated arabic versions of works of Aristotle which were thought
to have disappeared. 6 XII ∼ 1 080 ∼ 1 152 Leonardo
di Pisa
Fibonacci An Italian mathematician, son of a wealthy merchant who directed a trading
post in modern day Algeria. It was there that Fibonacci learned about the ∼ 1170 Hindu-Arabic numeral system used by north african merchants.
Inspired by Khwarizmi's work, he nished writing the Liber Abaci (The Book - ∼ 1240 of Calculations) in 1202, which introduced and popularized the Hindu-Arabic
numerals in Europe, where Roman numerals were still the norm. printing press wasn't invented until ∼ Since the 1 450, the popularization of this system in Europe took about a century to spread. Gerolamo
Cardano Italian polymath who found the general solution to the cubic equation: 1 501 - 1 576 0 = x3 + x2 x2 + x1 x + x0
x2
=⇒ y 3 + y1 y + y0 = 0
x := y −
3
y1
y3
y := z −
=⇒ z 3 − 1 3 + y0 = 0
3z
27z
3
y
2
0 = z 3 + y0 (z3 ) − 1
27
Then solve for this quadratic in Development
of Symbolic
Notation − Robert
Recorde + Christoff
Rudolff
√ = × ∼1360 1489 1525 1557 1618 > < James
Hume
y John
Wallis Johann
Rahn 1637 1637 1659 ∞ x 1631 ÷ A French mathematician. Viète introduced the use of letters to represent ables (Viète used the letter
letter Galileo
Galilei William
Oughtred Johannes
Widmann Nicole
Oresme Thomas
Harriot François
Viète z3. x c vari- 1540 - 1603 to represent a generic variable instead of the used nowadays). An Italian polymath and widely considered to be the 1st modern physicist. 1564 - 1642 Up until him, physics had not progressed past Aristotle's ndings (including
Aristotle's 1st law: a moving object has a force pushing it). acceleration (which is the
rate of change of velocity). This required the use of a mathematics of
motion, which is the origin of calculus. Galileo introduced new laws for physics involving This deed the Greek philosopher Zeno's paradox, the idea that at any given point in time, a moving object travels no distance, which means its speed is
always 0, making the mathematical study of motion impossible! Johannes
Kepler German mathematician, astronomer and astrologer. Used a primitive form of 1571 - 1630 calculus to describe the motion of planets around the sun and calculated the
area of the arc traveled by the Earth around the Sun in a month. René
Descartes Another French mathematician. Introduced Cartesian coordinates: graphs,
algebra and geometry. equations of curves. This marks the union of 7 1596 - 1650 Pierre de
Fermat Yet another French mathematician. metica Arith- Found a copy of Diophantus' 1607 - 1665 which he transcribed into symbolic notation, adding notations, extend- ing proofs... Major contributions include: • Fermat's Little Theorem: (ap − a) ÷ p ∈ N, ∀p
m • Fermat primes: 22 + 1 prime intended to be a general formula to generate primes, but turns out this doesn't work. Oops! • Proved we can't solve • Speculated that x3 + y 3 = z 3 xn + y n = z n in N. couldn't be solved for in In the margin of his book, wrote: N for all n > 3. I have a marvellous proof but the margin of this book is too small for me to write it. The proof was found
in 1994 by Andrew Wiles, and is posthumously called Fermat's Last Theorem.
Calculus
Isaac
Newton XVII English physicist and mathematician who developed calculus with a geometric 1642 - 1726 approach (dicult to read), and applied it in physics. Newton used the term
"uxion" to refer to quantities that approach 0 (the precursor of limits) instead of the concept of innitesimally small quantities which are already
arbitrarily close to 0, and used a dot notation to denote derivatives: f (x + σ) + f (x)
f˙ (x) =
σ
Gottfried
Leibniz fluxion σ German mathematician who developed calculus with an algebraic approach 1646 - 1717 (easier to read than Newton's geometric approach), and applied it in pure
mathematics and algebra. Used the d
dx notation for derivatives: df
dx
Leonhard
Euler Swiss mathematician, one of the most eminent in the history of mankind, and
the most productive, with around 70 original mathematical volumes to his
name. Credited with contributions to: • Innitesimal calculus • Graph theory • Topology • Analytic number theory • Mathematical notation, especially for functions • Found power series for sin (x)
• ex cos (x) Euler's identity: eiπ = −1 8 cosh (x) etc. 1 707 - 1 785 Finding
P
∞
1
k=1 k2 Euler was interested in nding the values of innite sums of the form
Here is how he proceded to nd this sum for the case 1. Recall sin (x) = ∞
X (−1) n n=0 2. Divide by x to nd P∞ 1
k=1 ki . i = 2: x1+2n
(1 + 2n)! x−1 sin (x)
∞
sin (x) X
x2n
n
=
(−1)
x
(1 + 2n)!
n=0 3. Factor x−1 sin (x) by keeping in mind that its roots are x∗ = nπ, n ∈ Z\ {0}
∞
∞
Y
Y
sin (x)
x
x
1+
=
(nπ − x) (nπ + x) =
1−
x
nπ
nπ
n=1
n=1 4. Plug in x = πθ 2 !
Y
∞
∞
Y
θ
θ
θ
sin (πθ)
1−
12 −
=
1+
=
πθ
n
n
n
n=1
n=1
∞
∞
Y
X
θ2
1
=
1 − 2 = 1 − θ2
− θ4 (· · · ) − θ6 (· · · ) − · · ·
2
n
k
n=1
k=1 = ∞
X ∞
2n
X
(πθ)
n (πθ)
(−1)
=1+
(1 + 2n)!
(1 + 2n)!
n=1
2n (−1) n=0 n 5. Compare the coecients for 1 − θ2 θ2 : ∞
∞
2n
X
X
1
n (πθ)
4
6
(−1)
−
θ
(·
·
·
)
−
θ
(·
·
·
)
−
·
·
·
=
1
+
2
k
(1 + 2n)!
n=1 k=1 =⇒ −θ2 ∞
∞
2·1
2
X
X
1
(πθ)
1
1
2
2π
=
(−1)
=⇒
−θ
=
−θ
k2
(1 + 2 · 1)!
k2
3! k=1 k=1 ∞
X
1
π2
=
k2
6 k=1 Q∞ 1
k=1 ki for even i, but could not
nd it for odd i. We don't know a procedure to nd this for odd i to this day. Euler followed the same procedure to nd 9 Reinventing
Calculus Using similar methods, the Bernoullis, Laplace, Lagrange and others devel- XIX opped all calculus known today: dierential equations, multiple integrals, surHowever, their ignorance of the concepts of continuity,
limits, dierentiability and integrability led to shortcomings and incon- face integrals, etc.
sistencies such as: x := ∞
X 2n =⇒ 2x = 2 n=0 2x − x = ∞
X 2n = n=0 2n − n=1 ∞
X ∞
X 2n+1 = n=0 2n = −20 = −1 = x < n=0 The fallacy lies in the use of x= ∞
X ∞
X 2n n=1
∞
X 2n ⊥ n=0 2x: ∞
X 2n = ∞ =⇒ 2 · x = 2 · ∞ n=0
and there is no such thing as innity times 2. More and more shortcomings
became evident by the 19th century. A completely new formalization of calculus
using analysis and limits occured in the 20th century to remedy this.
Non-Euclidean Geometry XIX So far, mathematicians had developped geometry without entirely questioning
Euclid's 5 axioms, upon which all geometry known at the time was based.
However, the parallel postulate, Euclid's 5th axiom stating that given a line and a point there exists exactly 1 parallel to the original line going
through the point, seemed impossible to prove, despite attempts by reputed
mathematicians such as Gauss, Bolyai and Lobachevsky to do so.
It turns out the parallel postulate can fail in 2 types of universes that completely break away from euclidean geometry: • In hyperbolic geometry, there are innitely many parallels. Angles of triangles sum up to less than • In 180◦ . elliptical geometry, there are no parallels. Angles of triangles sum up to more than 180◦ . All 3 types of geometry are equally valid but modern physics tends to indicate
that our universe is elliptical and not euclidean in nature. Further Developments
Niels
Henrik
Abel Norwegian mathematician who, still in his 20s, proved that it is impossible to XIX 1 802 - 1 829 solve the general quintic equation 5
X ci xi = 0 i=0
a problem unresolved for 250 years. Evariste
Galois 1 811 - 1 832 French mathematician who • found which quintics are solvable • proved that none of the equations of order higher than 5 could be solved He did so between the ages of 19 and 21. He died in a duel over a dispute he'd
had in a tavern. 10 Pierre
Wantzel ...

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