Samenvatting
History of Science Extensive Summary CH1-7
- Instelling
- Vrije Universiteit Amsterdam (VU)
This is an extensive summary of the History of Science Chapter 1-7 covering all major happenings and people throughout CH1-7.
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Chapter 1 (before 1200)Mathematics and myths: knowledge of the world:
Red line
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 politically privileged people, who could afford themselves time for contemplation, and were of
pedagogical importance to the philosophers.
Important people:
Thales of Myletus: the start of our story with the first mathematician (620 - 580):
- First to formulate a proof for Thales’ theorem
- Called first mathematician
- Motivated in two ways:
- Motivated esoterically: Geometria was a part of philosophy and knowledge on how the
world was ordered: knowledge of divine reasons or interventions
- Also motivated practically: measuring distance of object, dealing with coins
Pythagoras (571 – 500) and the Pythagorean order:
- Geometria and arithmetica were the essence behind a divine order
- 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
- Don’t know if Pythagoras came up with theorem himself as obtained by the master
- Special numbers (equal to divisors, triangular, prime)
- Everything could be expressed as a number: connection between musica and reality →
length of snare representative of musical tone it played (theory of ratios)
Plato (424 – 348): (Geometria helps understand knowledge of real things (ideai))
- Real knowledge → not from senses → order was in logos
- People in the cave only see the projections of objects outside the cave → the projections
were created by the ideai
- Senses reveal truth but not what is behind it.
- Geometria helps understand knowledge of real things (ideai), also useful in war but not that
useful
Aristotle: (384 - 322) (arithmetica and geometria were abstractions of the ideas given by our senses)
- 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 practise
Motivation: Geometria 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
1. To Euclid it might have been a textbook or not, but it wasn’t a textbook as we know it.
2. It was written and copied not printed. Copied and including commentaries
In Euclid’s elements:
- 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
- The axioms reflect the necessities and possibilities of constructing objects in an ideal world
- 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
are stunning: equal things, with equal things added, are equal
- From reason comes a list of definitions, theorems and proofs, constructions and proofs
Book1: Book I consists of elementary triangle geometry (Construction of equilateral triangle, proof of
triangle inequality, proof Pythagorean theorem)
Book2: Number of theorems about quadrangles (not refer to area, construction of a square n-gon)
Books 7-10: Number theory (number was magnitude,)
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
- In trying to count number of grains of sand he introduced an abstract number system
- The practicalities in war and horoscopes was 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
De Konika:
- The conic sections (parabola, hyperbola, ellipsis) Interesting ->
• 1: not trivial to construct (intersection solid with plane)
• 2: Fascinating properties could be proved about these figures (new
ground to discover)
- Could help to solve doubling of cube
- Intermediary between straight line, circle and curvilinear figures
, Roman and Hellenistic culture
No appetite for geometria, but for astrological predictions. Aristotelian perspective instead of
platonic
- Less emphasis on proof more on practicality:
1. Astrolabe (used to calculate local time) and the
2. Antikythera (used to determine position of heavenly bodies)
Abacus for arithmetic
Mathematics within Cristian Philosophy (300 AD)
Red line
With the fall of 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. Cristian thinkers thought God was well versed in geometria, since Creation was so harmonious.
Interests in quadrivium was mostly because of connection to divine. It was valued because it was not
trivial contrary to trivium.
Christian Thinkers Trying to connect fields of theology and mathematics:
Saint Augustine:
- Autodidact → not versed with Greek philosophers, one of the church fathers.
- In his De Libero Arbitrio, he compared knowledge of arithmetica to divine revelation.
- Number is as far from origin as its double is from the original number
- Focused on divine nature of number
Boethius(perpendicular to Augustine):
- Raised wealthy → Was well versed in greek Greek philosophy
- Build on Pythagorean number theory, arithmetica, musica
- De institutio arithmetica: Rules for computes → calculate easter
- Focused on practical nature of number
Sait Augustine Boethius
Autodidact Raised wealthy
Not versed with greek philosophers well versed in greek Greek philosophy
De libero Arbitrio → compared knowledge of De institutio arithmetica: Rules for computes →
arithmetica to divine calculate easter → practical
Focused on divine nature of number Focused on practical nature of number
Christians: Followed one god → strict rules for numbers → interpretation of numbers came from the
bible → they used powers of 2, 3 often because used in bible
Greek: more open minded about numbers → more god’s
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