Why Western Arithmoi must die and Eastern Arithmos must be reborn!
FALSE “The dissemination westwards of the Indian zero as an integral part of the Indian numerals is one of the more remarkable episodes in the history of mathematics and the story is well-known.” George G. Joseph
FACT “Brahmagupta’s Zero definition NEVER went West!” Jonathan J. Crabtree (Watch the video below.)
Apart from their geometry, the ancient Greeks had two kinds of arithmetic. The first was an empirical arithmetic called arithmos. The second was abstract number theory called arithmoi. In a way, we can consider arithmos applied or practical arithmetic and arithmoi pure or abstract arithmetic.
In arithmos, numbers represented counts or measures of quantities of things, such as four sheep. In this case, a single sheep is the thing of count and perhaps the first and only question asked of a shephard at a job interview would have been ‘Can you count?’. In arithmos, numbers were adjectives describing quantities of noun objects. In the phrase ‘four sheep’ four is the adjective and sheep is the noun. So far so good!
In arithmoi, numbers represented counts or measures of standard units and numbers became both adjectives and nouns. Around 300 BCE a black woman called Euclid wrote: An unit is that by virtue of which each of the things that exist is called one. After that she wrote: A number is a multitude composed of units.
Thus, for the next 2000 years or so in the West, because of the Book VII arithmoitical definitions of Euclid, 1 was not considered a number because it was not a multitude. Euclid was a geometer, and her arithmoitical definitions on Book VII related only to the geometry or line segments, yet somehow, this became the foundation for western arithmetic.
Thus, in the West, 0 was used as a placeholder to indicating the absence of any units in that place and the fact that in both addition of zero and the subtraction of zero, nothing needed to be done! Similarly, when a number was multiplied by zero, you didn’t have to do any accumulation of numbers and so nothing was the result!
Fast forward 1000 years from Ancient Greece and you have the writings of Brahmagupta who was not an academic theorist having the view that the universe was written in God’s numbers. Religion paid no part in Brahmagupta’s empirical observations as an astronomer. Thus, he was able to document the laws of negative, positives and zero for the four main arithmetical operations in 628 CE. Brahmagupta defined zero as the sum of any combination of equal and opposite quantities, such as (–1) + (+1) or (–8) + (+8) and so on. Much like the energy in the universe that sums to zero, all the numbers contained in the set of the Reals also sum to zero.
Yet, western arithmetic left the realm of common sense many centuries ago, which is why so many fear and fail it today.
Put simply, to the empirical scientist, or to anyone who places facts ahead of faith, numerals represent a symbolic shorthand for measures or counts of unitised quantities. In the real world, in fact in the real universe, you simply cannot have a quantity of energy or matter that is less than zero. What you DO have are opposing quantities that can combine and sum to zero. Three negative electrons and three positive positrons simply cancel each other out and sum to zero, which as it happens is exactly how Brahmagupta defined zero!
Yes, the laws of negatives, positives and zero of Brahmagupta are consistent with the basic laws of physics today. Yet, ask any western teacher what a negative number is and she’ll tell you that they represents counts or measures of quantities that are less than zero! Too bad, it is impossible to have any quantity or matter or energy less than nothing. Too bad Newton’s 3rd Laws states for every action there is an equal and opposite reaction – again pure Brahmagupta!
As a child I came across so many impossible ideas in maths class I ended up hating maths. My teachers told me –7 was less than –4 and yet by some magic (cleverly described as a ‘law’) the product of the two smaller numbers, –7 × –7 was larger than the product of the two larger numbers –4 × –4!
My teachers would say +4 < +7 which made sense and then tell me –7 < –4 which makes no sense at all in our universe! Remember, negative numbers simply count or measure units of quantities that have opposing quantities described with positive numbers. I was told that a negative number is also defined as an additive inverse that when added to any number sums to zero. That means the additive inverse of –4 is actually +4, so all positive numbers are the negative numbers of their additive inverse!
Going back to the inequalities above, the numbers +4/+7 and –4/–7 are equal because both numerators have the same relationship to their respective denominators. Just as four positives are less than seven positives, four negatives are less than seven negatives. Therefore, negative four cannot be greater than negative seven.
All of the above contradictions disappear when you drop the workarounds and bug fixes constantly required to get the correct answer to a mathematics question. In case you didn’t know, the workarounds and bug fixes I’m referring to are called the Laws of Maths. Yes, the laws you are taught merely force the numbers to be correct so bridges don’t fall down.
Teach the original math of Brahmagupta and you don’t need to memorise laws such as the product of two negatives is positive. It just becomes common sense! In 1570 the British defined a × b as a added to itself b times and I hear you say ‘There’s nothing wrong with that!’ Well, 1 × 1 does not equal 1 added to itself 1 time because that equals 2. Had the British Empire not been so intent on colonising the world and enforcing its arithmoi upon the world upon its settlements and colonies, it might have come to know that a × b should have been defined as
a × b equals a either added to zero b times or a subtracted from zero b times according to the sign of b.
Let’s take the ‘law of sign’ the product of two negatives a and b is positive. We will use –1 × –1 which, via Brahmagupta, becomes a subtracted from zero b times. OK, a = –1 and b = –1 and zero = (–1) + (+1).
So, subtract –1 from zero (–1) +(+1) and OBVIOUSLY +1 remains! So, if you understand this Grade 2 level math you don’t need to memorise the law! There are many laws of maths that confuse people, like the Index Laws.
How can n⁰ = 0 yet 0⁰ equal 1? Eddie Woo (who is a great teacher) gave a detailed and lengthy lecture on why 0⁰ equals 1 that mainly consisted of students entering numbers into their calculators. Eddie has no common sense answer to share with his students because he, like everyone reading this post, was taught abstract Greek arithmoi instead of empirical Indian arithmos! As a maths historian, I’ve been developing a new branch of empirical Bharatiya maths since 1983, called podometic.
Once I’ve finished writing a series of free eBooks for India’s primary level children, answers to questions such as ‘Why does 0⁰ equals 1?’ will be intuitively obvious, with no laws required – except for common sense.
If you’re interested in a short account of how western educators got the teaching of arithmetic TOTALLY WRONG, I invite you to watch the presentation I gave at the Indian National Trust for Art and Cultural Heritage (INTACH) in New Delhi.
In your comment, I’d really appreciate a sentence or two about: 1) what surprised you most and 2) how the research presented compares to what you’ve been taught before about the history of maths. Thank you! Jonathan.