Paul P. Mealing

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Friday, 25 December 2009

The Origins of Mathematics

Note the plural in the title, because the mathematics we use today comes from a number of different sources, geographically and culturally. We have a very Eurocentric outlook on mathematics that belies its global heritage.

I came across a small, well-presented volume in my local bookshop: The Bedside Book of Algebra by Michael Willers, a Canadian high school teacher of my vintage, going by the pop-cultural references he sprinkles throughout. When I thumbed through it, my first impression was that it contained nothing I didn’t already know, but I liked the presentation and I realised that it gave a history of mathematics as well as an exposition. It would be an excellent book for anyone wanting a grasp of high school mathematics, as it covers most topics, except calculus and matrices. The presentation is excellent, as he delivers his topics in 2 page bites, and provides examples that are easy to follow. I read it from cover to cover, and learnt a few new things as well as reacquainting myself with old friends like Pascal’s triangle. In fact, Willers revealed a few things about Pascal’s triangle that I didn’t know, like its relationship with the Fibonacci sequence and its generation of fractal patterns using a ‘tiling’ algorithm developed by Polish mathematician, Waclaw Sierpinski in 1915.

I already knew that the Chinese had discovered Pascal’s triangle some 500 years before Pascal (11th Century, Jia Xian), but I didn’t know that the earliest known reference was from an Indian mathematician, Varahamihira, in the 6th Century, or that it appeared in 10th Century Persia, thanks to Al-Karaji. (Blaise Pascal lived 1623-1662.)

Fibonacci (1170-1250) is most famously remembered for the arithmetic sequence that bears his name and also the ‘Golden Ratio’, which can be generated from the sequence. Both the Fibonacci sequence and the Golden Ratio can be found in nature – for example, flower petals are invariably a Fibonacci number and the height of a person’s navel to their height is supposedly the Golden Ratio, but I’m unsure if that is true or just wistful thinking on the part of renaissance artists. Because the Fibonacci sequence is derived by the sum of the previous 2 numbers in the sequence, there are some natural events that follow that rule, like the unchecked population growth of rabbits (that is provided as an example in Willers’ book) and was apparently the original example that Fibonacci used to introduce it.

But we all owe Fibonacci a great debt, because it was he who introduced the Hindu-Arabic numeral system to the Western world in a popular format that has made life so much easier for accountants, engineers, economists, mathematical students and anyone who has ever had to deal with numbers, which is all of us. When I was a child I was told that we used ‘Arabic’ numerals, and I only learned recently that they originated in India. The 7th Century Indian mathematician, Brahmagupta, formulated the first known mathematical concepts that treated zero as a number as well as a place holder (to paraphrase Willers).

Zero and negative numbers were treated with suspicion by the ancient Greeks and Romans, as they preferred geometrical over arithmetical analysis. Because there were no negative areas or negative volumes, the idea of a negative number was considered ‘absurd’. (I have to admit I had the same problem with ‘imaginary’ numbers, when I first encountered them, but I’m getting off the track, and I’ll return to imaginary numbers later.) Likewise, there was no place for a number that represented nothing, but once one introduces negative numbers, zero becomes inevitable, because a negative plus a positive of the same amount must give zero. But zero as a place holder is even more important, because it facilitates all arithmetical computations. As Willer says, imagine trying to do basic arithmetic with Roman numerals, let alone anything esoteric.

Willers quotes Pierre-Simon Laplace (1749-1827): “It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.”

Willers is the first author I’ve read who makes a genuine attempt to give the Indians and the Persians their due credit for our mathematical heritage. Like the Chinese, the Indians discovered Pythagoras’s triangle before the Pythagoreans, though many people believe Pythagoras actually learnt it from the Babylonians. The Indians also investigated the square route of 2, as well as π (pi), around the same time as the ancient Greeks. In the middle ages, a succession of Indian scholars worked on quadratic equations.

But it is Brahmagupta (598-670), who lived in northwestern India (now Pakistan), to whom Willers devotes one of his 2-page treatises, because he argues that Brahmagupta had the biggest influence on Western mathematics. He lists Brahmagupta’s 14 laws, all dealing with the arithmetic ‘rules’ applicable to zero and negative numbers that, in modern times, we all learn in our childhood.

Willers also gives special attention to two Islamic mathematicians, Al-Khwarizmi (born around 780) and Omar Khayam (1048-1122). Al-Khwarizmi came from Khwarezm (present-day Uzbekistan) and worked in the ‘House of Wisdom’ (see below). He gave us two of our most common mathematical terms: algebra and algorithm. Algebra came from the title of a book he wrote, Hisab Al-jabr w’Al-Muqabala, derived from the word, ‘Al-jabr’. Significantly, he developed methods for deriving the roots of quadratic equations.

The word, algorithm, also comes from the title of a book, Algoritmi de Numero Indorum, which is a Latin translation of one of Al-Khwarizmi’s Arabic texts, now lost. And, according to Willers, algorithm ‘means a number of steps or instructions to be followed.’ Of course, this word is now associated with computer programmes (software). This modern incarnation began with Alan Turing’s famous ‘thought experiment’ of a ‘Universal Turing machine’, as the first iconic example of the modern use of algorithm, which is literally a set of instructions, otherwise known as ‘code’. All modern computers are universal Turing machines, by the way, so it’s much more than a thought experiment now, and algorithms are the code, or software, that drives them.

Omar Khayam is probably better known as a poet, from his authorship of The Rubaiyat, a collection of 600 quatrains (4 line poems). But he also authored a number of books on mathematics, including Treatise on Demonstration of Problems of Algebra (1070), in which he solves cubic equations geometrically by intersecting conic sections with a plane. If one cuts a cone with a plane it describes a curve on the plane. Depending on the angle of the plane with the cone, one can get a circle, an ellipse, a parabola, or, using two cones, two mirror hyperbolae. In another text, that Khayam references, but has since been lost, he writes about Pascal’s triangle, though, obviously, he called it something else.

Omar Khayam provides the best quote in Willers’ book, taken from the above text:

The majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes.

As Willers points out, Plato’s academy closed in 529 and Fibonacci came on the scene in Pisa at the end of the 12th Century, and it wasn’t until the renaissance that Western science, art and philosophy really gained the ascendancy again. The interim period is known colloquially as the ‘dark ages’, because knowledge and scientific progress seemed to stagnate. As Willers says: “From that point [the closure of Plato’s Academy] until the thirteenth century the mathematical centre of the world was in the East.”

According to Willers, The House of Wisdom was established in Baghdad by Harun Al-Rashid (763-809) and translated works from Persia, Greece and India. It was a centre for education in humanities and sciences until it was destroyed by the Mongols in 1258. Without this Islamic connection over that period, the Greek and Roman knowledge in the sciences, philosophy, mathematics and literature, which, today, we consider to be our Western heritage, may have been lost.

As well as providing this historical context, in more detail than I can render here, and that most of us don’t even know about, Willers gives us excellent exposition on a number of topics: permutations and combinations, probability theory, logarithms, trigonometry, quadratics, complex algebra, the binomial theorem, and others.

He treats all these exemplarily, but I would like to say something about complex arithmetic and imaginary numbers, because it was a personal stumbling block for me, and, in hindsight, it shouldn’t have been. The set of imaginary numbers contains only one number, i, which is the square route of -1 (some texts say the set contains 2 numbers, i and –i, but, being pedantic, I beg to differ). Now all through my childhood, the square route of -1 was considered an impossibility like dividing by zero. So when someone finally came up with i, I believed I’d been conned – it was a convenience, invented to overcome a conundrum, and, from my perspective, it should have remained an impossibility. Part of the problem, as Willers points out, is that it’s called an ‘imaginary’ number, when it’s just as real as any other number, and I think that’s a very good point.

When one thinks that the Pythagoreans had serious problems accepting irrational numbers, and then the Greek and Roman mathematicians who followed them, had conceptual issues with zero and negative numbers, the concept of i is no different. It’s a number and it opens up an entirely new world in mathematics that includes fractals, the famous Mandelbrot set and quantum mechanics. If one doesn’t explain complex numbers using the complex plane (or Argand diagram) then it won’t make sense, but if one does, everything falls into place. In particular, multiplying by i rotates any graph on the plane through 90 degrees (a right angle), and by i2 (-1) by 180 degrees. In an ordinary number line with positive numbers running right and negative numbers running left, multiplying a positive number by -1 rotates the number through the origin (0) by 180 degrees to its negative equivalent. If you have an i axis running vertically through 0 then multiplying a number by i just rotates it by 90 degrees (half way). If you draw the graph it makes perfect sense.

Imaginary numbers, like multiple dimensions, demonstrate that the mathematical world can go places that the physical world doesn’t necessarily follow, yet these esoteric mathematical entities can have applications in the real world that we don’t anticipate at the time of their discovery. Reimann’s geometry giving us Einstein’s General Theory of Relativity and imaginary numbers giving us the key to quantum mechanics are two cases in point, both barely a century ago.

I’m one of those who sees mathematics as an abstract territory that only an intelligent species can navigate. Personally, I would like to think that we are not the only ones in the universe who can, and maybe there is at least one other species somewhere who can navigate even further than we can. It’s a sobering yet tantalising thought.


Addendum: I've since written an exposition on imaginary numbers and the complex plane, for those who are interested.


2 comments:

Timmo said...

Paul,

Willers is the first author I’ve read who makes a genuine attempt to give the Indians and the Persians their due credit for our mathematical heritage.

If you are interested in a similar lesson, but, in the case of science, you might check out Clifford Conner's A People's History of Science.

Paul P. Mealing said...

Thanks Timmo,

We have a Western bias when it comes to science and maths, and tend not to give credit where it's due.

Regards, Paul.