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History of Mathematics: Stories Retold, Stories Forgotten

Lecturer: David Sweeney

Originally Taught: Summer School 2012

This course will look at the history of mathematics. At stories that have been retold so many times that they have lost all but the barest of meanings and at some stories that have not been told often.

Each day we will look at a different event or time in the History of Mathematics. Different stories will be looked at and compared. I will attempt to tell new stories about these events, or at least stories that do not figure too prominently in most history of mathematics text books. We will to do some of the mathematics, we will count and draw. We will look at the different ways in which mathematics is performed in the different times and in the different stories.

Course Schedule:
Monday: Number in Classical Greece
Here we will get a feeling for the Greek concept of Number. This is a very different conception to our own. For the Greeks a number was always a definite number of definite things. Thus five sheep, 120 boats. We will work with pebbles to gain a consciousness of number as performative and embodied. We will see how the relations between numbers (20 sheep and 4 sheep are 24 sheep, one quarter of 100 stones is 25 stones) prefigure the mathematical operations of addition and multiplication, subtraction and division. We will look at the Greek concept of the kinds of numbers, the even and the odd. This classical attempt to classify all numbers and so move away from number as embodied.
Readings: Shorts texts will come from Plato, Aristotle, Nicomachus, and Jacob Klein “Greek Mathematical Thought and the Origin of Algebra.”

Tuesday: Geometry and the Angles of a Circle
Euclid’s The Elements of Geometry is probably the most successful text book of all time. Much of the modern idea of not only what mathematics should be, but also what logic and science should be, stems from an attempt to emulate the deductive reasoning of this work. However the success of this work has meant that very little remains of the works of Geometry as it was practiced before The Elements. What has been lost? On this day we will look at one idea in Geometry that Euclid cut out. The angle between a circle and a straight line. At least one proof using this concept remains in the work of Aristotle. I will give some of my ideas on the use of this concept and we will draw some basic proofs. Comparing the proof in Aristotle to Euclid’s proof of the same theorem which does not use such angles will give an idea of how mathematics can change and what can be lost and gained.
Readings: Short texts will come from Euclid and Aristotle.

Wednesday: Numbers for the Moderns
With the Renaissance the Greek idea of number was changed and interpreted and the modern ‘symbolic’ number was born. I will tell both the ‘normal’ story that is told regarding this development and I will give an outline of Klein’s telling of these events. We will try to get at the different feeling that results from thinking in terms of the embodied numbers looked at in the first day and the symbolic numbers that now surround us.
Readings: Shorts texts will come from Jacob Klein “Greek Mathematical Thought and the Origin of Algebra.”

Thursday: Calculus and Infinitesimals
Calculus is the study of, among other things, how to find the area of a curved object. For instance, how many square units are there in half a circle? The classical approach to such problems gave a different procedure for each shape. During the Renaissance a method was sought that would answer all such problems of area and volume with one single procedure.  The key was the idea of Infinitesimals. Infinitesimals are infinitely small geometrical objects or infinitely small numbers. For about two hundred years infinitesimals were the foundation of this new procedure which was called the Infinitesimals Calculus, or The Calculus. We will look at some of Kepler and Newton’s simpler proofs and have a go at drawing and using infinitesimals. Then we will look at Lagrange’s Calculus without infinitesimals and my own work with calculus with and without infinitesimals.
Readings: Short texts will come from Kepler, Newton, and Lagrange.

Friday: The Modern World
What stories are we to write about Modern Mathematics? Why is the world split into those who feel an affinity for mathematics and those who do not? What is modern mathematics? How big is it? Why does modern mathematics look so different to its predecessors? What difficulties face modern mathematics? What is the modern mathematician afraid of?
Readings: Shorts texts will come from Jacob Klein “Greek Mathematical Thought and the Origin of Algebra.”

One of my questions of the History of Mathematics
Mathematics has always, it seems, been both a way to do something and a certain ‘something else’. There has always been a story with the doing, whether that story comes from Plato as ‘mathematics is the way to the thought of the gods’ or whether that story is ‘man, I don’t want to do maths, everyone says its sooo hard’. It is a story that marks out mathematics as something special either greater than or worse than many other human activities.

Why do the different stories and retellings of mathematics and its history so often point to mathematics as in some way special?