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MATH 5400, History of MathematicsLecture 1: Introduction
Professor: Peter Gibson
http://people.math.yorku.ca/pcgibson/math5400
September 8, 2016
The history of...what, exactly?
What is mathematics?
P. Gibson (YorkU) Math 5400 8.9.2016 2 / 16
The history of...what, exactly?
What is mathematics?
P. Gibson (YorkU) Math 5400 8.9.2016 2 / 16
A proposed answer
1 Mathematics is the science of the abstract.
2 It is a scholarly pursuit conducted with precision and rigour.
3 Mathematical ideas have been—and continue to be—central tocultural developments that have dramatically altered human society.
P. Gibson (YorkU) Math 5400 8.9.2016 3 / 16
A proposed answer
1 Mathematics is the science of the abstract.
2 It is a scholarly pursuit conducted with precision and rigour.
3 Mathematical ideas have been—and continue to be—central tocultural developments that have dramatically altered human society.
P. Gibson (YorkU) Math 5400 8.9.2016 3 / 16
A proposed answer
1 Mathematics is the science of the abstract.
2 It is a scholarly pursuit conducted with precision and rigour.
3 Mathematical ideas have been—and continue to be—central tocultural developments that have dramatically altered human society.
P. Gibson (YorkU) Math 5400 8.9.2016 3 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...,groups, Riemannian manifolds,measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...,groups, Riemannian manifolds,measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...
limits, differential equations...,groups, Riemannian manifolds,measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...
,groups, Riemannian manifolds,measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...,groups, Riemannian manifolds,
measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...,groups, Riemannian manifolds,measure spaces, categories etc.
, etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
The science of the abstract
An abstract painting:
Mark Rothko, Magenta, Black, Green on Orange, 1947
Abstract refers to precisely defined abstract structures, such as numbers,arithmetic operations, lines, curves,...limits, differential equations...,groups, Riemannian manifolds,measure spaces, categories etc., etc...
P. Gibson (YorkU) Math 5400 8.9.2016 4 / 16
Rigour and precisionPrecision and rigour confer a kind of timelessness that makes mathematicsunique among scholarly pursuits.
An example of some old mathematics: al-Karaji (c. 972)
This is a area of current research!
P. Gibson (YorkU) Math 5400 8.9.2016 5 / 16
Rigour and precisionPrecision and rigour confer a kind of timelessness that makes mathematicsunique among scholarly pursuits.
An example of some old mathematics: al-Karaji (c. 972)
This is a area of current research!
P. Gibson (YorkU) Math 5400 8.9.2016 5 / 16
Rigour and precisionPrecision and rigour confer a kind of timelessness that makes mathematicsunique among scholarly pursuits.
An example of some old mathematics: al-Karaji (c. 972)
This is a area of current research!P. Gibson (YorkU) Math 5400 8.9.2016 5 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
Some changes in human society...
the first city states (civic administration)
industrialization
electricity (motors, artificial light, radio)
computers (so much for the slide rule...)
the iPhone (the information age in full swing)
These all have mathematical underpinnings.
P. Gibson (YorkU) Math 5400 8.9.2016 6 / 16
History
History is bunk.
—Henry Ford
How can we understand the history of mathematics?
P. Gibson (YorkU) Math 5400 8.9.2016 7 / 16
What is the purpose of history?
History tells us not only about the past, but also the present.
P. Gibson (YorkU) Math 5400 8.9.2016 8 / 16
What is the purpose of history?
History tells us not only about the past, but also the present.
P. Gibson (YorkU) Math 5400 8.9.2016 8 / 16
An analogy: the study of genetics and evolution relates to both the pastand present structure of organisms, simultaneously.
(Ernst Haeckel)
Similarly, the history of human culture refers simultaneously to the pastand present. (Consider language: current usage embeds remnants of pastculture.)
P. Gibson (YorkU) Math 5400 8.9.2016 9 / 16
An analogy: the study of genetics and evolution relates to both the pastand present structure of organisms, simultaneously.
(Ernst Haeckel)
Similarly, the history of human culture refers simultaneously to the pastand present. (Consider language: current usage embeds remnants of pastculture.)
P. Gibson (YorkU) Math 5400 8.9.2016 9 / 16
An analogy: the study of genetics and evolution relates to both the pastand present structure of organisms, simultaneously.
(Ernst Haeckel)
Similarly, the history of human culture refers simultaneously to the pastand present.
(Consider language: current usage embeds remnants of pastculture.)
P. Gibson (YorkU) Math 5400 8.9.2016 9 / 16
An analogy: the study of genetics and evolution relates to both the pastand present structure of organisms, simultaneously.
(Ernst Haeckel)
Similarly, the history of human culture refers simultaneously to the pastand present. (Consider language: current usage embeds remnants of pastculture.)
P. Gibson (YorkU) Math 5400 8.9.2016 9 / 16
The historical method
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 10 / 16
The historical method
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 10 / 16
The historical method
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 10 / 16
The historical method
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 10 / 16
The historical method
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 10 / 16
Some examples from the history of mathematics
The dawn of civilization:
A clay tablet from Uruk (c. 31st century BC).
P. Gibson (YorkU) Math 5400 8.9.2016 11 / 16
Some examples from the history of mathematics
The dawn of civilization:
A clay tablet from Uruk (c. 31st century BC).
P. Gibson (YorkU) Math 5400 8.9.2016 11 / 16
Some examples from the history of mathematics
The dawn of civilization:
A clay tablet from Uruk (c. 31st century BC).
P. Gibson (YorkU) Math 5400 8.9.2016 11 / 16
A much more recent tablet:
Examples of Pythagorean triples (c. 18th century BC) in sexagesimalnotation.
P. Gibson (YorkU) Math 5400 8.9.2016 12 / 16
A much more recent tablet:
Examples of Pythagorean triples (c. 18th century BC) in sexagesimalnotation.
P. Gibson (YorkU) Math 5400 8.9.2016 12 / 16
Al-Karaji, again:
On the construction of water wells (c. 10th century AD).
P. Gibson (YorkU) Math 5400 8.9.2016 13 / 16
Al-Karaji, again:
On the construction of water wells (c. 10th century AD).
P. Gibson (YorkU) Math 5400 8.9.2016 13 / 16
An eighteenth century example:Euler’s translation of Benjamin Robins’s work on ballistics.
Sfltut
bet
ARTILLERIE
einer Unterfucfcuntj{i>D(^^te
(am] famen
bem ^nglif^en t>ea ^)rn. Benjamin 9vobin
ton
Seon^arb u(er
364 BAS ZWEYTE CAPITEL VON DEM WIEDEKSTANDE DEE LUFT [636637
wovon das Integrale 1st
Setzt man nun Kurze halber =z, und e fur die Zahl, deren hyperbolischer
Logarithmus == 1, so wird
Die andere Aequation
dt~*~^
giebt
r//_~ 4wc
.t(, f ~~
7
3 (& -h ) v yt?
Man setze, um die Irrationalitat zu heben, h = aa, und t; = uu, so wird
-4^c $aadii Snc/ du du\3 uu(aa-{-uu) 3 \aa + ww w^^/
wovon das Integrale zum Theil auf der Quadratur des Zirkuls beruht. Dennes ist
/adu. , w== A. tang. ,aa + 'uu D a
das ist einem Zirkul-Bogen, dessen tangens = , wenn der Eadius = 1 ge-
nomruen wird. Also bekommt man
, 8c/i A , w.l ^\t -= - A. tang. C } .
3 \a & a ' u J
Man setze nun wiederum a = Yh9und u = Yv, und bestimme die GrroBe C
dergestalt, daB t; = b wird, wenn t == 0, so wird man finden
(r- A. tang. ^~ ~ A. tang. -- A ; r )
\yjt8Yh yh
*Yh YV yvyv
oder
P. Gibson (YorkU) Math 5400 8.9.2016 14 / 16
Summary
1 Mathematics is the science of the abstract.
2 It is a scholarly pursuit conducted with precision and rigour.3 Mathematical ideas have been—and continue to be—central to
cultural developments that have dramatically altered human society.
I the first city states (civic administration)I industrializationI electricity (motors, artificial light, radio)I computers (so much for the slide rule...)I the iPhone (the information age in full swing)I and much more...
P. Gibson (YorkU) Math 5400 8.9.2016 15 / 16
Summary
1 Mathematics is the science of the abstract.
2 It is a scholarly pursuit conducted with precision and rigour.3 Mathematical ideas have been—and continue to be—central to
cultural developments that have dramatically altered human society.I the first city states (civic administration)I industrializationI electricity (motors, artificial light, radio)I computers (so much for the slide rule...)I the iPhone (the information age in full swing)I and much more...
P. Gibson (YorkU) Math 5400 8.9.2016 15 / 16
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 16 / 16
We distinguish several types of sources:
primary sources
secondary sources
tertiary sources
Three key questions:
1 What do we know?
2 How do we know it?
3 What don’t we know?
P. Gibson (YorkU) Math 5400 8.9.2016 16 / 16