history

Did Pythagoras Really Exist?

Pythagoras is probably one of the most famous names in mathematics; almost all high school students will know the beloved Pythagoras Theorem. But did he really exist?

“Sadly, it is now almost universally assumed by classical scholars that Pythagoras never existed. It seems that there was a group of people in southern Italy called Pythagoreans who invented a “Founder” for their beliefs who, accordingly, lived and died in a manner consistent with those beliefs.”

–The Book of Dead Philosophers by Simon Critchely

There are, in fact, no primary sources about Pythagoras which have survived. The only piece of literature that we have of him is from a small set of texts that was written between 150AD and 450AD. That is, 600 to 1000 years AFTER Pythagoras is said to have lived. So, their accuracy is questionable.

The existence of a group called the Pythagoreans is undebatable. They were an orphic-like cult, obsessed with numerology. There are indeed many stories of Pythagoras, however most include supernatural occurrences rather than significant accounts of his life. For example. one tale describes him as possessing a golden thigh; another declares he was the son of the god Apollo. This turns Pythagoras into a rather mystical figure in ancient history and “for some, these lies and contradictions hint that Pythagoras was simply an exaggerated or even fictional leader concocted by the members of a religious sect.”

Furthermore, many historians suggest that Pythagoras was not even the first to develop his famous theorem, but that in fact the Egyptians discovered it long before he did. It can be argued that the mathematical significance of the early Pythagoreans (pre-450 BC) has been exaggerated, apart from their contribution to harmonics, and that they only later evolved into serious mathematicians as geometry became popular across Greece.

What do you think? M x

MATHS BITE: Infinity

Infinity is a difficult concept to grasp for more practical subjects such as physics, however it has become vital in mathematics where it is treated as if it were a number (but it is not the same as a real or natural number).

The earliest recorded idea of a mathematical infinity comes from the Greek philosopher Zeno of Elea, who came before Socrates. The Ancient Greeks distinguished the ‘potential infinity’ from the ‘actual infinity’, meaning that instead of saying that there is an infinite number of primes, they preferred saying that “there are more prime numbers than contained in any given collection of prime numbers”.

Indian mathematicians also touched on the concept of infinity; Surya Prajnapti classified all numbers into three sets and then subdivided these groups into three further orders:

Enumerable: lowest, intermediate and highest;
Innumerable: nearly innumerable, truly innumerable, and innumerable innumerable;
Infinity: nearly infinite, truly infinite, infinitely infinite.

The well known symbol of infinity: , was coined by John Wallis when he used it for area calculations by splitting a region into an infinite number of strips with a width equal to ${\frac {1}{\infty }}$ .

Although the concept of infinity was known to the Ancient Greeks, defining infinity mathematically was not an easy task. In fact, it was not done until the 1800s. As a result, many 19th century mathematicians spoke violently against it; “infinity was something for philosophers to discuss”. This skepticism was called ‘finitism’, a movement heralded by mathematicians such as Leopold Kronecker. When Georg Cantor published the first formal proof of the existence of infinity in 1874, Kronecker spoke out against it:

“I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.”

It is a tricky concept to explain, or even believe in its existence. Despite these obstacles, infinity has made its way into mathematics and is used in a large variety of fields, including in the calculus developed by Leibniz and Newton (both real and complex analysis), set theory, geometry and fractals. Without it, there would have been a halt in the development of certain areas in maths. What is the limit of 1/x as x approaches 0? Mathematicians answer this simply using infinity.

It helps mathematicians answer the otherwise unexplainable – if indeed infinity is an adequate explanation.

What are your thoughts? M x

Forgotten Mathematicians: Egyptian Mathematics

Upon request, I have decided to continue this series on Forgotten Mathematicians.

In around 6000 BCE the ancient Egyptians settled along the Nile valley due to its fertile land. For both religious and agricultural reasons, they began to record the patterns of lunar phases and the seasons. In very early Egyptian history, the Pharaoh’s surveyors used their body parts to take measurements of the land and buildings (for example the cubit), and consequently a decimal numeral system was developed based on our ten fingers.

It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system – at least as early as 2700 BC. A stroke was used for units, a heel-bone symbol for tens, a coil rope for hundreds, etc.

The oldest mathematical text is thought to be the Moscow Papyrus, which dates to the 12th dynasty in 2000-1800 BC. The Moscow Papyrus and The Rhind Papyrus (dating from around 1650 BCE) contain mathematical problems, which are mostly practical, but a few are posed to teach the manipulation of the number system without a practical application. The Rhind Papyrus contains evidence of other mathematical knowledge such as unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means and arithmetic and geometric series.

Furthermore, the ancient Egyptians were the first civilisation to develop and solve quadratic equations, as revealed by the Berlin Papyrus.

Unit Fractions

Practical problems of trade and the market led to the development of fractions. They could also be used for the simple division sums.

For example is they needed to divide 3 loafs among 5 people:

they would first divide 2 of the loaves into thirds;
then divide third loaf into fifths;
they would then divide the third that was left over into 5 pieces;
Therefore, each person would get 1/3 + 1/5 + 1/15 = 3/5.

Multiplication

This was achieved by the process of repeatedly multiplying the number in question and 1 by two on two separate sides. The corresponding numbers could then be used as a sort of multiplication reference table:

the combination of powers of two which add up to the number to be multiplied was isolated;
the corresponding numbers on the other side add up to give the answer.

Ancient Egyptian method of multiplication

Sources: 1 | 2 | 3

Hope you enjoyed! M x

Maths and Philosophy #3: Aristotle

Aristotle was a Greek philosopher, born around 382 BC. At age 18, he joined Plato‘s Academy in Athens, remaining there until the age of 37. Although he was mainly a philosopher – his writings constitute the first comprehensive system of Western philosophy – Aristotle made important contributions to mathematics by systematising deductive logic.

With the Prior Analytics, Aristotle is credited as the first to formally study and formalise logic. His concept of logic was to dominate the West until the 19th century advances in mathematical logic.

Aristotelean Logic

Aristotelean Logic begins with the distinction between subject and predicate:

Subject: an individual entity, or class of entities
Predicate: a property, attribute or mode of existence that the subject may or may not possess

The fundamental properties of predication were:

Identity: everything is what is it and acts accordingly
Non-contradiction: it is impossible for a thing to both be and not be
Either-or: everything must either be or not be

Aristotle also developed rules for chains of reasoning that would, if followed, never lead to false conclusions; this is the basic principle of mathematics, where the ‘rules’ are axioms. In this Aristotelean form of reasoning, the links are syllogisms: pairs of propositions that, taken together, give a new conclusion:

Some A is B.

All B is C.

Therefore, some A is C.

In his logic, Aristotle distinguished between dialect and analytic: “dialect only tests opinions for their logical consistency; analytical works deductively from principles resting on experience and precise observation.”

This was very different from Plato’s Academy’s teachings, where dialect was supposed to be the only proper method for both science and philosophy.

Sources: 1 | 2 | 3

Hope you enjoyed this series! M x

Maths and Philosophy #1: Socrates

I have come up with the idea to start themed weeks (where all posts during that week follow a specific theme). This week I will be talking about ancient Greek mathematicians that delved in the fields of Maths and Philosophy, starting with Socrates.

Socrates is credited as one of the founders of modern Western philosophy. His student’s, Plato, dialogues are one of the most comprehensive accounts of Socrates to survive. In today’s post I wanted to discuss Plato’s dialogue entitled “Meno” in particular, as here we can see Socrates’ teaching of mathematics.

“All I know is that I know nothing”

In this dialogue, Socrates’ considered that all the knowledge we can possess is already within us, and that the process of reasoning is merely an act of recollection.

Socrates questions an un-schooled servant boy about a geometrical problem. He begins by drawing a square, and asks the student to construct a square twice as large.

The boy initially says that he does not know, but after further questions he thinks that the answer is to make the edges twice the size of the original square, as shown below.

However, the area of this new square is greater that twice the original square; the student correctly observes that it is four times the area. Socrates reiterates the question, asking “how would you construct a square with just twice (not four times) the area of the original?”. We need a square with half the area of the one the student just constructed. Socrates asks the boy if we can cut each of the four squares in half by drawing a line connecting opposite corners and the boy answers yes, producing the final result below.

The square formed is called Plato’s Square.

“Socrates: What do you think, Meno? Has he, in his answers, expressed any opinion that was not his own?
Meno: No, they were all his own.
Socrates: And yet, as we said a short time ago, he did not know?
Meno: That is true.
Socrates: So these opinions were in him, were they not?
Meno: Yes.
Socrates: So the man who does not know has within himself true opinions about the things that he does not know?
Meno: So it appears.
Socrates: These opinions have now just been stirred up like a dream, but if he were repeatedly asked these same questions in various ways, you know that his knowledge about these things would be as accurate as anyone’s.
Meno: It is likely.
Socrates: And he will know it without having been taught, but only questioned, and find the knowledge within himself?
Meno: Yes.
Socrates: And is not finding knowledge within oneself recollection?”

For a full excerpt click here.

In this dialogue, Socrates joins mathematics and philosophy together by using mathematical proof to demonstrate a philosophical belief.

Stay tuned for the post on Wednesday! M x

VIDEO: History of Zero

I wanted to quickly share a video, which I stumbled upon on the history of zero. A video by The Royal Institution and narrated by Hannah Fry, it describes how the history of mathematics closely mirrors the history of zero – first as a concept, and then ultimately as a number. The short animation shows how the evolution of the understanding of zero has helped shape our minds and our world.

I especially love the video due to its accessibility – anyone can understand it. It is in this way that mathematics can be popularised and loved by all!

Hope you enjoy! M x

Ada Lovelace

Ada Lovelace, born December 10th 1815 in England, was a mathematician who is considered to have written instructions for the first computer program.

Her mother, who had a passion for mathematics, raised Lovelace with an excellent education in all subjects, particularly maths. At that time, it was extremely rare for mother to provide such extensive education, particularly a scientific one, to a daughter, as women weren’t allowed to go to university or join learned societies.

Lovelace excelled at mathematics and was encouraged by her tutor Mary Somerville, who was another notable female mathematician. At age 17, Lovelace met Charles Babbage, known as ‘the father of computers’, and they quickly became lifelong friends. Babbage described her as

“that Enchantress who has thrown her magical spell around the most abstract of Sciences and has grasped it with a force which few masculine intellects could have exerted over it”.

The Analytical Engine

The Analytical Engine was a machine that would be able to store data and perform sequences of instructions defined on punch cards and fed into the machine. Although it was never built, the Analytical engine was a direct forerunner of the computers we use today.

Lovelace became deeply intrigued with this machine and when, in 1842, she was asked to translate an article describing the Analytical Engine by the Italian mathematician Luigi Menabrea, Babbage asked her to expand the article “as she understood the machine so well”. The translated article became three times longer and Lovelace added a description of a method for the engine to repeat a series of instructions – a process known as looping, which is used in computer programs today. Due to this, she is often referred to as the ‘first computer programmer’.

Ada Lovelace sadly died of cancer aged 36, a few years after her groundbreaking publication of “Sketch of the Analytical Engine, with Notes from the Translator”. Although her contributions to computer science were not discovered until the 1950s, long after their publication, her passion and vision for technology has made her a powerful symbol for women in science.

Sources: 1 | 2 | 3 | 4

Hope you enjoyed reading about a very important woman in our mathematical history! Would you be interested in more posts about the field of computer science? M x

Forgotten Mathematicians: Babylonian Maths

The Babylonians lived in a region of Mesopotamia, which is now known as Iraq, as shown in the map below.

As with most ancient civilisations, a mathematical system developed as the bureaucratic need for a system to measure plots of land, tax individuals etc. arose due to the settlement of the civilisation.

There is evidence that from around 2600 BCE onwards the Babylonians produced multiplication and division tables, tables of squares, square and cube roots and worked on geometrical exercises and problems involving division. Furthermore, later tablets from 1800 to 1600 BCE show topics such as fractions, algebra and methods for solving linear, quadratic and cubic equations being tackled. The Babylonian mathematicians also produced a few approximations, including √2 which was accurate to five decimal places!

Unique Number System

The Babylonians used an advanced number system with a base of 60, rather than base 10, which is the base system in widespread use today. Counting physically in this base system was done using 12 knuckles on one hand and five fingers on the other. Unlike the other number systems used at the time by the Egyptians, Greeks and Romans, it was a true place-value system, where the digits in the left column represented larger powers of 60. The numbers 1-59 were given using two symbols that were combined in distinct ways, as shown below.

Babylonian Numerals | Source: MacTutor History

Base 60 was a wise choice of base system, as it has been conjectured that the advances the Babylonians made in mathematics were greatly facilitated by the fact that 60 has such a large number of factors; 60 is the smallest number to be divisible by all numbers from 1-6. The remnants of this number system can still be seen today. It was the Babylonians that divided the day into 24 hours, with 60 minutes in an hour and 60 seconds in a minute, a system still in use today.

Another great mathematical advance by the Babylonians was the concept of the number zero, which was represented by a circle character, something that had not been recognised by the Egyptians, Greeks or Romans. However, they are not necessarily credited with its discovery as it was used more as a placeholder, rather than being a number used in calculations.

Construction of Tables

One of the most astonishing aspects of Babylonian mathematics and their calculating skills was their construction of tables to facilitate their calculations. For example, two tablets that were found in Euphrates give squares of the numbers up to 59 and cubes of the numbers up to 32. To construct these tables Babylonians used formulae to make the calculations easier. For instance, to compute square numbers the following formula was used to make the multiplication easier:

ab = [(a + b)² – (a – b)²]/4

Pythagoras’ Theorem

On the Plimpton 322 clay tablet, which now resides in the British museum, the following is written:

“4 is the length and 5 the diagonal. What is the breadth ?
Its size is not known.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9 ?
3 times 3 is 9.
3 is the breadth.”

Due to this tablet, many claim that the Babylonians may have had an understanding on Pythagoras’ theorem before the Greeks. This claim is fortified by the fact that the Babylonian’s understanding of quadratics was extensive. However, there is a lot of controversy over this as due to the damage and age of the tablet, interpretations of the writings vary greatly.

Recent News

A newly deciphered Babylonian tablet reveals the path of Jupiter. Babylonians were not only fantastic at pure mathematics, but were very competent in astronomy and placed great value in its study due to the fact that they believed that if they understood what happened in the skies they would know what would happen on Earth as they were connected. They have recently been in the news over the deciphering of a clay tablet, by Ossendrijver, that reveals an early form of integral calculus to calculate the path of Jupiter, a technique which was thought to have been invented in Medieval Europe.

Although it was previously believed that the Babylonians did not use geometry for their astronomical calculations, this tablet undeniably shows that the Babylonians had geometric understanding, allowing them to develop a geometric technique to make arithmetic calculations.

To read more, I suggest this article!

I have started to link my sources within the text so if you want to read more, have a look at the links! M x

Forgotten Mathematicians: Mayan Maths

The Mayan civilisation settled in the region of Central America from around 2000 BCE. At its peak, it was one of the most densely populated and culturally dynamic societies in the world.

Due to the importance of astronomy and calendar calculations in Mayan society, their mathematicians constructed a very sophisticated number system. They used a vigesimal number system (based on base 20, and to some extent 5), which is said to have originated from counting on fingers and toes. Their numerals consisted of only 3 symbols: a dot representing 1, a bar representing 5 and a shell representing 0. The fact the the Mayans knew the value of 0 is incredible, as most of the world’s civilisations had no concept of 0 at that time.

These three symbols were used in various combinations so that even uneducated people could do simple arithmetic for trade and commerce, as addition and subtraction was done by adding up the dots and bars, which is relatively simple. Numbers larger than 19 were represented by the same kind of sequence, but a vertical place system format was used to show each power of 20, as shown in the image below.

Mayan numerals

In the Mayan society, mathematics was an extremely important discipline and appears in Mayan art, such as wall paintings, where mathematics scribes or mathematicians are depicted by number scrolls which trail from under their arms. Interestingly, the first mathematician identified in one of these paintings was a female figure.

Furthermore, despite not having the concept of a fraction, Mayan mathematicians produced extremely accurate astronomical observations using no tools other than sticks. For example, they were able to measure the length of the solar year to a far higher degree of accuracy than that used in Europe – their calculations gave an answer of 365.242 days, compared to the modern value of 365.242198 – as well as the lunar month – their value was 29.5308 days, compared to the modern value of 29.53059. That’s incredibly accurate!

However, due to their geographical disconnect from other civilisations, their mathematical discoveries had no influence on the European and Asian numbering systems or mathematics.

Let me know what you think! M x