Where did we go wrong
Why we calculate from right to left, when everything else we do is the other way around
TL;DR — Our natural way of learning, reading and writing has been from left to right. It’s a wonder how simple mathematics has evolved to be from right to left and using intensive carryovers. The benefits of natural rhythm and intuition in maths far outweighs its costs and it would be worth to consider the same for our future generation. Examples within the article.
It is around 6000BC and we are in east China. Civilisation has started to evolve here owing to nearness with a waterbody, viz. east China Sea. With incremental progress towards the concept of nation-states, though its maturity is still far into the future, there is a real need for some form of tool, even if rudimentary, to manage the treasury for the king-the protector of the realm. Military rationing and predictive astronomy, have also come to an unmanageable head without a way of counting objects and people. It is still the world where pictograms rule the roost. We have not yet developed a way to store our texts, and the only recourse we have is raw earthenware tablets. As the time goes by, we have found a more reproducible way of counting than using our fingers, and a more stable way at that — beads. And, we seem not to get enough of it. As we transition from Xia to Shang dynasty, we see the emergence of minor trade routes, along with a rudimentary earthenware Abacus (Saunpan).
Abacus has gradually been developed and refined over the next few centuries and has become intuitive enough to be used easily, and portable owing to our expertise with woodwork. As our languages and scripts, flow naturally from top to bottom, and/ or left to right, counting on Abacus develops the same way. It is natural for us to use left to right in arithmetic in general, and on Abacus in particular.
It is now around 2000 BC, and we are in India, since it has become a hotpot of cultural and social activities, which will define the way forward for the world. Observation of natural phenomenons have been crystallised into authoritative manuscripts and scrolls. Numerous philosophers and saints have further refined these observations, into what will come to be known as Vedas. We have not seen paper yet, we use leaves and increasingly parchments. Hence, we have to rely on oral traditions to promulgate the natural laws, methods and means. The best way forward on this path for us is through rhythms and rhymes, and that’s what we structure the Vedas into. Our arithmetical methods evolve into sutras and shlokas, for easy recall and poetic appeal.
The shlokas are in Sanskrit, one of the oldest of languages — read and written from left to right, in line with our natural inclination and intuition. Therein, the calculations and arithmetic involved, flow from left to right, such as the derivation of π (pi) above. Unfortunately, oral traditions demand higher memory retention thus limiting the spread and ubiquity of arithmetic in general, and these methodologies in particular. Dark clouds loom large, since we are not sure how far would this tradition go. We have hence started to experiment with shortcuts for the numeral in mathematics. Brahmi script is gradually gaining ground, especially across mathematics, the importance of which we have not yet come to fathom just yet.
It is 100BC and we are opening up to the idea to trade over long distances. We have most things in abundance but the explorers in us have to venture out, to see other places, to exchange things with other areas. It is an era when civilisations will come together. A new class evolves, that of merchants and tradesmen. An entrepreneurial machinery is born, that of primitive guilds (collegium). These guilds will become stronger as we move into our future. We are sellers and buyers of goods, rather than the connoisseurs. We need bookkeeping and accounting, and the shlokas, sutras, or tools such as Abacus pose a challenge. We need shortcuts. The cuneiform way of writing has become more condensed by now across civilisations, and we are able to express ourselves fully using very few “glyphs” — elemental symbol for the purpose of writing.
Pictograms are the relics of lore. One of the variants of this new form of writing has become, what is now known as, Old Persian.
Surprisingly it has started to flow from right to left. We don’t know whether it has been because of religious stimulations, or for some other reasons. And there are still some confusions wherein we have texts written both from left to right and vice versa. But overall, Persia has given a new direction to the scripts-right to left.
The bastion of arithmetic has fair chance of crumbling now to lead way for new method of counting and calculation. It’s a balancing act for now, but one or the other would have to give in.
By around 600–700 CE, for the first time in over millions of years, we see ancient equivalents of modern day numerals, inspired from Brahmi numerals which we saw earlier. Silk Road, and associated expansion of trade take these method further west to cover most of present-day Middle East. The man often attributed to bringing it to fore at the global stage, as we would come to see shortly, is Al-Khawarizmi.
Silk Road pushes the boundaries further towards Europe, leading to their own numerals, viz. the Roman numeral relegated mostly to the Church and its archdioceses. Simple, as the Arabic numerals were, are accepted widely across the bastions of European societies, morphing themselves on the way to emerge as the modern decimal system numerals.
These simple numerals will change the world as we know it.
With consolidation of monarchies across European nations, advent of 15th century pushes the explorers and navigators, through sea and oceans, to crisscrossing the globe with their Niña, Pinta, and Santa María. Starting off with exchange and trade of goods, 16th and 17th centuries further entrench the “import” of European nations across major landmasses. Along with, starts the induction of European traditional mathematics into the curriculum of colonised nations. The impact is substantial enough that within two generations, its overweening influence has relegated the native and more intuitive arithmetic methods to the corners of old, now ancient, textbooks housed in museums with longings for footfall. Owing to further development in technology over 19th and 20th centuries, our communications systems, the computers, transport and almost all applications of mathematics, are fundamentally defined in units, tens and hundreds, i.e., from right to left, and not the other way around. We’ve lost our touch with intuition, we’ve lost our touch with the natural.
Organic evolution of our language has always been based observations of natural phenomenon, more specifically the phonemes, or sound elements, as I like to think of them as. The phonemes in turn have originated from vocal folds and hence the structure of our vocal organs. Similar is the case with arithmetic and its rules. Naturally, there was a strong case why simple arithmetic methods should have been from left to right. But, unfortunately — for the lack of a better word — they did not.
There is a swift and spiffy Scientific/ Social experiment that can more or less shed light on why left to right, following our natural rhythm, is superior, and more importantly faster, than those which are currently in place.
Let’s sum up two random numbers, 495 and 167. Using “traditional mathematics”, we start from the ones, 5 + 7 = 12, forget 5 and 7; and keep 2 as the temporary sum along with 1 as a carryover in our cache.When I say cache, it’s the cache in our own head, temporary memory if you will.Moving on to tens’ place, our carryover has become 1*10 = 10. Subsequently, we get another two numbers to store in our cache, i.e., 90 and 60. Till now, we have 4 numbers in our cache, viz. the temp. sum 2, new numbers 90, 60 and carryover 10. We condense 90 + 60 as 150, with which we still have three numbers in our cache — 150, 10 as a carryover and 2 as our temp. sum. 150 and 10 makes 160, from which we use 60 to make the temp. sum 62. We empty our cache and store 100 remaining from 160. (We’ve used 60 to update our sum till now). Finally, we add 400 and 100, to get 500 and add our 100 from our cache to it. We have the sum as 662.
At any point in time we are keeping two or more digits in our cache, compounded with the fact that going from right to left, sans the habit part since we have been taught to do it this way, goes against our path of least resistance.
Let’s now look at method which relies on our intuition, on our natural inclination. We add first two sums, 400 and 100, makes 500, which is our temp. sum. One number in cache till now — 500. Second numbers 90 and 60, makes 150, two numbers in cache for now. 500 and 150 makes 650 and empties our cache to have a single number, i.e., the temp. sum — 650. Finally, 5 + 7 = 12, again two numbers in cache — 650 and 12.650 + 12 maketh 662, and we are done. At any point in time, we keep a maximum of two numbers.
We are traversing the path of least resistance and it comes to us naturally, with practice.
Granted, when you would start off using the intuitive method rather than traditional, it will be difficult. Since we have not developed a habit for it, since we have not been using it day in and day out since childhood. However, given a few weeks of usage, the time taken are dramatically lower than the traditional methods. It is no wonder, classes on mental mathematics increasingly are inching towards intuitive methods for faster calculations.
Summarily, our natural inclinations have been refined over centuries of civilisational growth and development and have been a result of numerous experimentations. It might just as well be worth the effort for us to relook the way education is being imparted currently to our upcoming generations. We want the best for our kids, probably it is a bit late for us to go back and reorient ourselves with the more natural ways of learning, but it is not, in the least, for our future.
