This is a summary of the book of Mathematical Worlds written by Danny Beckers. The summary contains all the highlights of the book including thoughts about certain events and notes from the lectures. Every chapter is clearly stated with subtitles, important people from that time and a little summar...
SUMMARY HISTORY OF SCIENCE
Introduction
Mathematics as a subject in its own right, was born in Western Europe in the sixteenth
century
Latin: “mathematics” or Greek: “mathesis” ⟶ part of philosophy that served as a
prerequisite to the higher studies of medicine, Theology, and law.
A practice, in this book, is an institutionally, or by habit, sanctioned way of working, by a
group of people. Whether by common background, an educational system or shared
experiences, this group, geographically, or by some form of communication culturally
bound, shares a similar background.
In this book, a practice is called “mathematical” when the people active within that
practice, or the people seeking help from that practice, consider it to be mathematical.
Three different realities of education:
- Present-day school, both as a building and an institute
- Double meaning:
The first one is the actual formal education taking place in school, instruction
regarding such subjects as languages, geography, history and (of course)
mathematics.
The second one derives its meaning from becoming a proper man or woman,
believing, and behaving in the right way, according to custom or formalised
etiquette.
- It has to be noted that education is shaped by social, cultural, and political ideals.
Chapter 1. Geometry, philosophy, and world order: the episode
before 1200
No mathematical practices until 1200
One constant that drove mathematical sparks: esoteric knowledge
- Denote knowledge known only to a limited group of people who engage in it
Motivations for attempting to describe in number or lines:
- Mysticism
- Trying to understand the intention of the Gods or God
- Finding out what the future held
- i.e.: Babylonian and Egypt priests and scribes, Gilgamesh epos
⟶ In general people regarded knowledge as wisdom, as something that had been
revealed.
During ancient Greek civilisation a practice rose that did view these subjects as
something special: group of philosophers —> interests was much broader than
mathematics alone.
- Geometria: measuring the world ⟶ first step towards astrology/astronomy
- Arithmetica: associated with merchants (=kooplui) and administrating ⟶
philosophers thought there was wisdom to be found in numbers
Wisdom was mundane but also esoteric
⟶ Not regarded as true philosophy but did make up the simplest examples of
philosophising and were thereby at the very least of pedagogical importance to the
philosophers.
,Mathematics and myths: knowledge of the world
The text discusses the role of esoteric knowledge, numerology, and astrology in ancient
cultures, which contained the seeds of what later would become mathematics. It also
mentions how Greek philosophers, interested in understanding the world in physical,
political, and mystical senses, introduced a practice that viewed mathematics as something
special, and had separate words for geometria and arithmetica. These subjects were
considered important by a small group of culturally and political privileged people, who
could afford themselves for contemplation, and were of pedagogical importance to the
philosophers.
Thales of Myletus (ca. 624 – ca. 548): start of our mathematics story
Called the first mathematician
Rich merchant who therefore could afford himself some education
First to formulate a proof for Thales’ theorem
Greek philosophy was motivated in two ways:
- Esoterically: geometria was a part of philosophy and knowledge on how the world
was ordered ⟶ knowledge of divine reasons or interventions
- Practically: measuring distance of object, dealing with coins
To many of the ancients, however, the ideas of geometria were first and foremost part
of, or even originated from, myths and beliefs. The ideas were associated with astrology
and numerology.
Pythagoras (ca. 571 – ca. 500) and the Pythagorean order
Geometria and arithmetica were the essence behind a divine order
Don’t know if Pythagoras came up with theorem himself or obtained by his master
Everything could be expressed as a number:
- Connection between musica and reality ⟶ length of snare representative of
musical tone it played (theory of ratios)
- Numbers are thought of as in counting objects: pebbles or shards
- Counting objects used in voting: in that sense numbers decided upon matters of
politics
⟶ Special numbers (equal to divisors, triangular, prime) became interesting
⟶ Reality was numerological in nature
Plato (424 – 348)
Some philosophers didn’t trust arithmetica as a basis for true philosophy.
- To them, deriving knowledge from geometria became more common.
Real knowledge was not derived from senses but from logos (reason: in logos order was
found)
People in the cave only see the projections of objects outside the cave —> the
projections were created by the ideai (“real” world: perfect and external ideas)
- Senses reveal truth but not what is behind it
Arithmetica and geometria were important in trade and warfare, but most notably
geometria also allowed the philosopher to gain knowledge of the eternal ideai.
,Aristotle (384 – 322): essence of arithmetica and geometria was much more mundane.
Arithmetica and geometria were abstractions of the ideas given by our senses.
Arithmetica and geometria use was limited to trade and warfare.
World is made of earth water fire air (his book physics)
Geometria as part of philosophical practice
Greek philosophers that were trying to understand the eternal ideai:
Motivation: geometria was more solid than arithmetica, closer to astronomia
Assumption: world is based on divine straight lines and circles, flat surfaces, and orbs
Strengthening the mysticism of circles and squares:
Delian riddle: doubling the cube.
Quadrature of circle: moons of Hippocrates ⟶ he tried to solve but didn’t manage.
Euclid’s elements
Stunning example of geometria, although:
- To Euclid it might have been a textbook or not, but it wasn’t a textbook as we
know it ⟶ according to Euclid, geometria is strictly about reasoning
- It was written and copied (including commentaries written by the philosopher) not
printed.
It presents the reader to the elementary parts of geometria:
- Starts with definitions, referring to the objects the reader knows about from
drawing figures in the sand.
- Presents axioms and hypotheses which are the basis of his reasoning.
- Never calculates.
- Straight lines and circles are the only objects that exist at the start, the rest is to be
constructed. This reflects the divine origin of these objects.
- They are important steps in reasoning and in that sense the explicit mention of
these reasons is stunning: equal things, with equal things added, are equal.
- From reason comes a list of definitions, theorems and proofs, constructions, and
proofs.
Book 1: Consists of elementary triangle geometry (construction of equilateral triangle,
proof of triangle, proof Pythagorean theorem and proof of its reverse)
Book 2: Number of theorems about quadrangles (not refer to area, construction of a
square n-gon)
Books 7-10: Number theory (number was magnitude (= absolute waarde)) ⟶ number
is an amorphous (= vormloos) subject to Euclid.
Books 11-13: About the solids, only 5 building blocks of universe
Archimedes (300 BC)
De Methode (“on the method”) he uses the above to illustrate how he got his idea for
the ratio between sphere, cylinder, and cone.
Sand reckoner: in trying to count number of grains of sand he introduced an abstract
number system, which allowed him to express numbers of arbitrary value.
, The practicalities in war and horoscopes were normal. Archimedes could write about
geometria and arithmetica. Wisdom highly regarded, that regard also is seen in the
anger of the soldier who stabbed Archimedes.
Apollonius and conic sections
Apollonius of Perga (262-190) wrote eight books, named De Konika
The conic sections (parabola, hyperbola, ellipsis) interesting:
1. Not trivial to construct (intersection solid with plane) ⟶ fascinating properties
could be proved about these figures (new ground to discover).
2. Could be used to solve some of the more remarkable problems that had proven
difficult to solve ⟶ doubling of cube with the help of hyperbola and parabola.
Intermediary between straight line, circle, and curvilinear figures.
Roman and Hellenistic culture
Eager to absorb Greek culture: no appetite for geometria, but for astrological
predictions.
Geometria was viewed from an Aristotelian perspective instead of platonic.
Education in arithmetica was practiced both in Greek and Roman numerals and every
well-educated man (which was only a small percentage) learned elementary arithmetic
⟶ counting board or abacus was used for everyday reckoning
Less emphasis on proof, more on practicality ⟶ philosophical machines:
- Astrolabe: used to calculate local time based on geographical latitude or vice versa
- Antikythera: used to determine position of heavenly bodies.
Mathematics within Christian Philosophy
With the fall of the roman empire, Greek ideas were less supported. The early Christians
distanced themselves from the pagan Greek philosophers. The ideas behind the Greek world
order were not consistent with the notion of an omnipotent God. The new Christian ideas
proved to be much more supportive for mathematical practices, since the Christian God was
a deity who favoured rules and laws. These divine rules and laws reflected favourable upon
the ideas within geometria or arithmetica. Christian thinkers thought God was well versed in
geometria, since Creation was so harmonious. Interests in quadrivium was mostly because
of the connection to divine. It was valued because it was not trivial contrary to trivium.
Christian thinkers trying to connect fields of theology and mathematical practices:
Saint Augustine Boethius (perpendicular to Augustine)
Autodidact Raised in a wealthy and politically
influential family
Not versed with Greek philosophers ⟶ Well versed in Greek philosophy.
later generations considered him one of the
churches fathers
De libero Arbitrio ⟶ compared knowledge De institution arithmetica: rules for
of arithmetica to divine revelation. computes ⟶ calculate easter
Focused on divine nature of number. Focused on practical nature of number.
Number is as far from origin as its double is Build on Pythagorean number theory,
from the original number arithmetica, musica
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