“India explains Zero and Math better than the West”

Jonathan J Crabtree, Elementary Mathematics Researcher from Melbourne, Australia says that it was in 1968 that he noticed that his teacher's explanation of multiplication was wrong. Her definition/explanation of multiplication was missing India's zero.

Jonathan says that he seemed to notice more problems with Maths not very obvious to others. “So, in 1983 when I smashed my back and was told I might never walk again, I made a promise to God. Let me walk and I'll fix Mathematics. I regained my health and so I just felt I had to keep my promise. I began by exploring Vedic Maths in 1983, yet the speed calculation tricks weren't the answer to the paradoxes and inconsistencies in the western foundations of Maths. So, I just kept looking!”

After about 25 years, in 2008, he says he began to wake from dreams about Maths. “I began to rewrite maths as I understood it should be structured, yet who would listen to a nobody like me? I had to research Mathematics as deeply as anybody ever had. That meant one of my first peer-reviewed papers, on how the English definition of multiplication had been wrong since 1570 took years to write as I double-checked things over and over, spanning 16 languages.”

In his article, The Lost Logic of Elementary Mathematics and the Haberdsher who Kidnapped Kaizen, Jonathan wrote: Euclid’s multiplication definition from Elements, (c. 300 BCE), continues to shape mathematics education today. Yet, upon translation into English in 1570 a ‘bug’ was created that slowly evolved into a ‘virus’. Input two numbers into Euclid’s step-by-step definition and it outputs an error. Our multiplication definition, thought to be Euclid’s, is in fact that of London haberdasher, Henry Billingsley who in effect kidnapped kaizen, the process of continuous improvement. With our centuries-old multiplication definition revealed to be false, further curricular and pedagogical research will be required. In accordance with the Scientific Method, the Elements of western mathematics education must now be rebuilt upon firmer foundations.”

In this interview with CSP Jonathan shares his ideas on how Maths needs to be learnt and taught:

You have written about the value of Zero as opposed to the place value of zero. Is zero India's greatest contribution to Mathematics?

I would say its India's definition of zero as the sum of equal yet opposite quantities that is important. Zero as a placeholder is almost trivial in comparison. The medieval Arabic world and later Renaissance Europe understood zero as a placeholder, yet not as the pivotal tipping point that lies between equally numerous positive and negative quantities and numbers, that resonates with modern laws of physics.

 Why is it that the Sulbha sutra which is older than Pythagoras is not mentioned at all by the West?

Well, I've mentioned the Sulba Sutras. https://www.linkedin.com/pulse/cutting-onion-makes-you-cry-least-its-area-easy-pi-jonathan-crabtree/ and the paper at http://www.bit.ly/squareO has references that do deep-dives into the Sulba Sutras.

For 2500 years, mathematicians had fun trying to square the circle using only a compass and straightedge. Yet in 1882 this classic Greek problem was proven to be impossible. So, what do we do? Before the compass, in both India and Egypt, circles were drawn via ‘peg and rope/ cord’ methods. The Egyptians, long before Pythagoras, used a rope with 12 evenly spaced knots to form right angles by making triangles with sides 3, 4 and 5. So, along with foundations for pyramids, came foundations of geo (earth) metry (measurement).

In India, from 800 BCE, when the Śulba Sūtras, (rules of cord/ rope) were written, ropes were often used for altar construction. Much later, the Hebrew Mishnat ha-Middot, (Treatise of Measures), dated 150 CE by its translator, has; ‘The circle has three aspects: the circumference, the thread [diameter] and the roof [area]. Which is the circumference? That is the rope surrounding the circle…’ Notably, ‘line’ came to us via the Latin, ‘linea’, meaning ‘a linen thread, a string, line.’

If we use ropes, like these ancient cultures, before the Greeks added the problematic restrictions of ‘compass and straight edge’, we may find an interesting way to square the circle. So this outdoor project for students uses pegs and ‘curved edge’ (rope), like Egypt, in order to square the circle. From squaring the circle, mathematics teachers can disintegrate curves, to reveal the ancient origins of integral calculus.

Ahmes, the scribe who wrote the Egyptian Rhind papyrus around 1650 BCE, created squares with sides equal to eight ninths the diameter of a circle. Thus, an Egyptian circle, with diameter 9 units, had an area of 63.617 square units, while the square had a greater area of 64 square units. In India, from around 800 BCE, a solution was to create squares with sides equal to thirteen fifteenths the diameter of a circle. Thus, an Indian circle, with diameter 15 units, had an area of 176.715 square units, while the square had a lesser area of 169 square units. Edward de Bono, who coined the term lateral thinking, created po as an alternative to linear ‘yes/ no’ thinking. Since we are presenting a lateral solution to an ancient problem, we define po as the ratio of a circle’s area to the area of the square derived from the circle, the Egyptian po, at 0.994, is better than the Indian po, of 1.046. Our goal is to achieve po = 1.

How can one rewrite Mathematical history?

With painstaking detailed research and references. Plus we must realise the importance of NOT superimposing our current knowledge onto that of ancient writers. Many people for example, say Euclid wrote about geometric algebra in his Book II of Elements. He didn't. However, we can take what Euclid did and use it for modern-day purposes. People often rewrite history by assuming the ancients and the moderns are/were on the same page, so to speak. They rarely are.

Do you think historical redress is important for India to feel a renewed sense of pride in her achievements?

 Yes and No. I think India has a false sense of pride and an unhelpful arrogance when it comes to mathematics education. “We invented zero” seems to be an excuse for national laziness when it comes to critical thinking. Where is the redress to come from? The British aren't going to apologise any time soon or provide some remedy or compensation for their prior wrongs and grievances. What is important is to educate India's children on the correct laws of elementary maths as I reveal in my various presentations. India's children should be made aware that Āryabhaṭa gave us base ten positional notation and Brahmagupta gave us the arithmetical laws of positives negatives and zero. But rather than dwell on the history, educators need to make changes to the primary level curriculum to make it both Indian and correct mathematically to clear up all the confusion and fear regarding negative numbers in particular.

Euclid laid axioms for Euclidian Geometry, are there parallels in Indian Geometry. Or was our focus on Algebra and arithmetic?

India's focus was on the ground and on the heavens. Construction of sacrificial altars for rituals was why the Sulba Sutras were required. Āryabhaṭa, Bhāskara and Brahmagupta were astronomers first and mathematicians second. More importantly, they were empirical scientists and the maths they used was for the purpose of understanding the universe around us. Compare that with the approach of the Greeks whose mathematics was largely based on the philosophy of Aristotle and Plato, and I'm on the side of the scientists! The Indians, like the Chinese, are usually said to be more focused on 'applied' rather than 'pure' mathematics.

Bhaskaracharya's method of solving mathematical problems included examples using from daily life and puzzles. Can we use his pedagogy today to make Math more interesting to children? 

I like the eBook the Illustrated Lilavati. However, I'm an elementary maths historian. Would children find the sorts of word problems involving monkeys or bees irrelevant to them today? I suspect so. However, if the problems could be meaningful and grasp the child's imagination, then maybe all that's needed is to update the word problems of Bhāskarācārya so they relate to traffic or climate change issues, for example.

In Western Mathematics, there is only one way of multiplying and most of our tables are learnt by rote learning. How is the Indian method different?

I disagree with the premise that there are more ways to multiply in India than in the West. However, it is worth noting that with base ten all you need do is learn your multiplication table up to 9 x 9. To go higher is not necessary in a metric country.

How have your ideas been received in universities abroad? Do Western Mathematicians agree with what you say?

Western mathematics education professor are mostly defensive. They don't seem to care about elementary maths. All they seem to care about is higher maths, from algebra up. It just happens that most of the problems are in our concepts of zero and negative integers. Because I am so well-equipped with historical evidence, the question usually becomes one of 'Does it really matter?' In my opinion, educators who feel that way do not reflect the scientific method or the philosophy of kaizen, or continuous improvement. They also have either no recollection of what it’s like to be learning the language of maths, or have no empathy with the child's experience.

How can Brahmgupta's 18 laws help in understanding elementary Math better?

Have a look at slides 130 - 134 in the slideshow at www.j.mp/IndiasMaths and you'll see common sense returned to integer ordering. Yes, 7 negatives are greater than 4 negatives and a debt of ₹7000 is greater than a debt of ₹4000 so -7 > -4 according to both Brahmagupta and common sense!

Have a look at slides 208 - 214 and you will see with your eyes why -1 x -1 must equal +1. It's a consequence of Brahmagupta's definition of zero and has nothing to do with the 'distributive property of multiplication' as claimed by so many.

The Kerala school of Math predates Newton, says a Manchester study. Was the way India did Math different from the West?

For much of the past 2000 years western mathematics has been dominated by the maths of Euclid. This meant number theory only emerged on the whole numbers. It also meant zero and one were both excluded from the set of integers as were negative numbers. Also, the Church in Europe did not want people discussing India's ideas of zero (the void) and infinity. The void was the realm of the devil and only God was infinite.

What does your research on Indian Mathematics, reveal about the way Indians thought about their world, the universe?

The Indians seems to have enjoyed a child-like wonder at the marvels of the universe that were not held under lock and key by organised religion, as was the case in Europe. Whether infinitely small or infinitely large, Indians appeared comfortable with the concept.