III The Background Story of Math.

III.The Background Story of Math.
[Contd. A journey to the wonderland of math.by Ajay Kumar Chaudhuri]

The background story of something is nothing but its "history" to us. But has the subject of Mathematics any history? If so, how and when did history come to it?
 In reply, it may be said that, in the same way in which history comes in everywhere else. A human being is a historical creature; it is that history which makes the difference between man and every other creature on the Earth. An elephant and an ant have no history, or if they have, then no single elephant or ant is aware of it, it is imposed on them by human beings – just as he, in a more tangible way may be, has imposed history on each and every creature and creation in the nature. Each of every man and every woman have a personal  history and also a family history – and it’s a pity if one does not know about one’s parents, grandparents, great grandparents and if possible further back than that. A nation becomes a nation by knowing about its past.

So, for better understanding and learning mathematics with joy and pleasure we should know the history of it. But we have no faintest idea how did the concept of mathematics, as we now call today, creep and impinge the minds of our farthest ancestors, about of whom we know nothing.In fact,the history of mathematics is nearly as old as humanity itself.

From multifarious evidences strewn in different parts of all over the Earth in different forms of scratches, sketches, figures, language etc., it may be guessed that the origin of mathematical thought lies in the concept of number, magnitude and form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of “number” concept evolving gradually over time is supported by the existence of languages which preserve the distinction between “one”, “two”, “many”, but not of numbers larger than two.

Almost recently, in 1960 a Belgian explorer while exploring what was then Belgian Congo, discovered a dark brown fibula (lower leg bone) of a baboon with some striking features, near the area of Ishango near the Semiliki River where Lake Edward empties into Semiliki which forms the part of the headwaters of the Nile River, now a border between Uganda and Congo. The bone was found among the remains of a small community that fished and gathered in this area of Africa. The settlement had been buried in a volcanic eruption.

This bone with a sharp piece of quartz affixed to one end, perhaps for engraving, was first thought to be a tally stick as it has a series of notches what has been interpreted as tally marks covered in three columns running its length. Some scientists have suggested that the grouping of notches on this bone tool indicate a mathematical understanding that goes beyond counting or it was for a better grit on the handle. It is believed that the bone is perhaps more than 20,000 years old and dated to Old Stone Age. If someone is very interested to see it with own eyes, he may visit the Royal Belgian Institute of Natural Sciences, Brussels, Belgium where this Ishango bone is on permanent exhibition.

In this context, it will be worthy to mention that Africa is widely accepted as place of origin of humans from their ancestors ‘great apes’, as evidenced by the discovery of hominids (great apes) and their ancestors, as well as later ones that have been dated to around seven million years ago. As Africa is considered to be the oldest inhabited territory on Earth so, it is probably that many evidences like this Ishango bone had been buried under the earth of this continent which will be unearthed one day to endow us with clues of our advent,subsequent evolution and long survival till now.

Numbers and counting have become an integral part of our everyday life. It is an interesting story how these digits have come to dominate our world. We should remember that the number sense is not the ability to court, but the ability to recognize that something has changes in a small collection. Some animal species are also capable of this. For example, the number of young that the mother animal has, if changed, will be noticed by all mammals and most birds. Mammals have more developed brains and raise fewer young than other species, but take care of their young for a much longer period of time. It was noticed that many birds have a good number sense. If a nest contains four eggs, one can safely be taken, but when two are removed the bird generally deserts. The bird can distinguish between two and three.

An experiment done with a goldfinch (a very small song bird of America) showed the ability to distinguish piles of seeds : three from one, three from two, four from two, four from three and six from three. The goldfinch almost always confused five and four, seven and five, eight and six and ten and six.

In the insect world the solitary wasp (a flying insect, often black and yellow that can sting) seemed to have the best number sense. The mother wasp lays her eggs in individual cells and provide each egg with a number of live caterpillars (which are worm like larva of butterflies or moths) on which the young feed when hatched. Some species of wasp always provide five, others twelve and others as high as twenty four caterpillars per cell. The solitary wasp in the genus Elmenus will put five caterpillars in the cell if it is going to be a male (the male is smaller) and ten caterpillars in a female cell. This ability seems to be instinctive and not learned since the wasp’s behavior is connected with a basic life function.

One might think people would have a very good number sense, but as it turns out, people do not. Experiments have shown that the average person has a number sense that is around four.

I think, it will be worthy to ask, in this context, why the Automated Teller Machine (commonly known as ATM) PINs are mostly of four digits? Wouldn’t be wiser if the PIN was longer, so that no one could guess it? Isn’t why our e-mail passwords are also expected to be six letters or more?

Let us recall an interesting story of invention of ATM and reason for four digit PINs of it, to be contended ourselves with the answers of those queries.

John Shepherd Barron (1925 – 2010) was a Scottish national born in India. Later he relocated to Britain and pursued his education from the University of Edinburgh and at Trinity College, Cambridge. After returning empty handed from the bank, as it was closed by that time Shepherd Barron was disappointed to have no other solution to wait till the bank would open next day. And thus in a similar fashion like Archimedes, Shepherd Barron claims to have hit his Eureka moment while taking a bath. A self-sufficient cash dispensing machine was what he was thinking about. And soon the ATM was invented in the early 1960 s. The first ATM was installed at a bank in Enfield, London in 1967.

Now, let us come to the question of four digit PINs of ATM. Well, the Scottish inventor of ATM, Shepherd Barron also proposed a six digit PIN. But the idea of six digit PIN had to be rejected because of Scot’s wife Caroline. Funnily enough, longest string of numbers she could remember was four. Although, there are few banks that do offer six-digit PINs for security purposes. Shouldn’t those of us using four digit PINs be thinking of the woman? It is get tough to recall those four digits at a time, imagine what six or more would do to us?

So what separates people from the rest of animal kingdom? It may include many things, but the ability to count is very much one of them. Counting, which usually begins at the end of our own hands or fingers, is usually taught by another person or possibly by circumstance. It is something that we should never take lightly, for it has helped advance the human race in countless ways.We are born with number sense,but we are to learn how to count.

Now let us see why, how and where the ideas of numbers, numerals* and counting systems were born in the minds of our most distant ancestors. For this, we are to explore the ancient civilizations of peoples of different places of the world throughout a very long passage of time which are obscured under thick dust of time in the unfathomed darkness of history.

Presently, the earliest known archaeological evidence of any form of writing or counting is scratch marks on a bone from 150,000 years ago. But the first earliest solid evidence of counting, in the form of the number one, is from a mere twenty thousand years ago. An Ishango bone, which I have mentioned earlier, was found in Congo with two identical markings of sixty scratches each and equally numbered groups on the back. These markings are a certain indication of counting and they mark a defining moment in Western civilization.

* Difference between numbers and numerals :

A number is a thing that we talk about in mathematics such as “four”, which is hard to define exactly. It may be said, it is an abstract property or “fourness” that is shared by any set of four things.

A numeral, on the other hand, is any name or symbol for that number, such as “4” or “IV”, or 100 (binary) etc.

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