as fast so the Tortoise is given a lead of 100 paces. By the time Achilles has run 100 paces the Tortoise is 10 paces ahead. By the time that Achilles has made up these 10 paces the Tortoise is one pace ahead, and so on. Achilles can never catch up with the Tortoise.
Both these paradoxes cut space up into infinitely many pieces. In mathematics it is possible for infinitely many numbers to have a finite sum. This was shown by Archimedes in about 210 BC and formed an important part of mathematics in the 17th century. The two paradoxes can be written thus:
The Dichotomy:
1/2 + 1/4 + 1/8 + 1/16 + … = 1
Is it possible to cross the field?
Achilles and the Tortoise:
100 + 10 + 1 + 1/10 + 1/100 + … = 1111/9
Therefore, Achilles will catch up with the Tortoise after running 1111/9 paces – a finite distance.
Although this explanation satisfies mathematicians, it does not answer the question of how it is possible to accomplish infinitely many tasks in a finite period of time. Zeno thought it impossible and hence that any sort of motion is an illusion.
4th century BC Greece
Plato and Platonism
Plato (428–348 BC)
A philosophy of mathematics that claims that mathematics exists outside the human mind and that it is essential for the education of enlightened people.
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Plato was one of the most important philosophers of all time. His name has even entered our language: a “platonic” friendship is one without any sexual content.
He was also a mathematician and held the discipline of mathematics in very high regard. Above the gate of his academy was written: No one ignorant of geometry can enter here.
Plato’s own achievements in this field were minor – one (disputed) story is that he invented a device with moving rods for the problem of “doubling the cube.” On the other hand, his influence on the philosophy of mathematics was enormous.
In mathematics, Platonism is the theory that mathematical objects – numbers, triangles, and so on – have an existence independent of the human minds that think about them. It is a theory that is very difficult to justify without extra philosophical assumptions. Where do these abstract ideas exist? Is there another universe, completely different from our material universe, which contains abstract objects? Plato seems to have thought so. The essence of his philosophy is that the material world is but an inferior copy of the world of abstract forms.
According to Plato, the work of the mathematician involves discovery rather than invention. Mathematicians investigate the universals which exist independently of mankind, rather than create ideas from their own minds.
The other justification is theological. Abstract ideas exist in the mind of God. Plato said: “In mathematics, men think the same thoughts as gods.”
4th century BC Greece
Conic Sections
Menaechmus (380–320 BC)
These are a set of curves that are exposed when a cone is cut.
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Conics occur in many places in nature as well as in mathematics.
Imagine a double cone (similar to the one in the diagram below). This cone is cut by a plane. The exposed surface is a conic section.
Double cone.
The type of conic depends on the angle of the plane. The numbers in the diagram indicate the following:
1. The plane is horizontal: a circle.
2. The plane is slightly tilted: an ellipse.
3. The plane is parallel to the side of the cone: a parabola.
4. The plane is steeply tilted: a hyperbola. Notice there are two parts to this curve, both labeled number 4.
There are many ways of defining conics without involving a three-dimensional cone. Take a fixed point F and a fixed line d as in the diagram shown on the following page. Suppose a variable point X moves so that the ratio XF:Xd is constant. Then X moves in a conic.
Focus and Directrix.
The ratio is called the eccentricity, the value of which tells us the sort of conic that is defined. The fixed point F is called the focus. The Earth moves round the Sun in an ellipse with the Sun at the focus. The eccentricity is about 1/90.
There are many other real-life examples of conics. A wheel seen at an angle is an ellipse. Cut a cucumber, and the exposed surface forms an ellipse. Throw an object in the air and you will see that its path is a parabola. The reflecting surface of a car headlight is formed from a parabola.
The Long Range Navigation (LORAN) system used to guide ships involves intersecting hyperbolas.
3rd century BC Greece
Euclid’s Elements
Euclid (c. 325–265 BC)
These are a compilation of geometric theorems.
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Euclid’s Elements lists and proves the geometrical results of his day. It has survived as a model of logical reasoning and as a basis for learning mathematics for 2,000 years.
Euclid’s Elements of Geometry consists of 13 books that cover the mathematics known during the time of writing. It is largely a compilation of other mathematicians’ results rather than original work by Euclid himself, but its structure, a chain of theorems proceeding logically from clearly stated definitions and axioms, is due to Euclid.
The collection contains all traditional school geometry. The fifth proposition is that the base angles of an isosceles triangle are equal, the 32nd is that the sum of the angles of a triangle is 180° and the 47th is Pythagoras’s Theorem.
It is very much pure mathematics. When a pupil at Euclid’s academy asked what gain he could make from it, Euclid contemptuously told his slave to throw the student a coin.
The theorems are all phrased in terms of geometry, as was nearly all Greek mathematics, but a large part of the work lays the foundation for number theory.
The book was used continuously for teaching mathematics for 2,000 years. British intellectual Bertrand Russell was introduced to the Elements at the age of 11 and recorded the impression it made on him:
This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.
3rd century BC Greece
The Fifth Postulate
Euclid (c. 325–265 BC)
The postulate, proposed by Euclid, which gives the properties of parallel lines.
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Euclid’s