ZERO Priyedarshi Jetli longer version february 21 2010

ZERO Try viewing the gargantuan Universe through it (zero); and you’ll find everything beautifully in focus; marvellously harmonious and synchronized like never before, (Parekh 2004, 124) I ask: how many pens are on my desk? I start counting one, two, three, until I have counted all, let us say 8. The origin of numbers is often taken to be a construction by humans for counting. This explains why 0 was not posited as a number in most cultures. Neither the number 0 nor the numeral ‘0’ were understood in ancient times, hence posing a huge hurdle in the development of mathematics. Let me remove the pens one by one and count backwards, 7, 6, 5, 4, 3, 2, 1. Now, I remove the last pen from the table and ask: How many pens are on the table now? The answer demands a quantity. ‘Nothing’ is not an indicator of quantity. So, I will answer ‘0’ where 0 is a number. So, we get to the number 0 by counting. It is the reverse process of counting that perhaps led to the concept of 0 as a number as far back as 6th century BCE. The common sense notion of 0 as nothing and the mathematical notion of zero as a quantity, a number, must have coexisted from ancient times. What is 4 – 4? Any school child would answer ‘0’. But if we don’t have 0, then we have to say either that it is not possible or that it is nothing. But if nothing is not a number then subtraction is not closed over the set of numbers. Similarly, 4 – 5 = –1. If we don’t have negative numbers again subtraction would not be closed. Hence, we must admit both negative numbers and 0 as a number to maintain closure. After common sense let us now turn to a historical account. First, we must distinguish between zero as a number and 0 as a numeral. The latter involves the concept of 0 as a place holder to represent large numbers like ‘405,067,890’. The former is the mathematical notion or meaning of zero as a number. From the 3rd century BCE to the 9th Century CE zero as a numeral was discovered independently in three different regions of the world. First, in 3rd century BCE Babylonia where base 60 was used, zero was employed as a place holder. Hence, the number 124 was written as: (McQuillin 1997) Here the left hand group indicates two 60s to be added together and the right hand group indicates four units or ones to be added to the two 60s, so the number is 124. The number 1856 was written as: (Ibid.) The first grouping indicates 30 60s which add up to 1800, the second grouping 5 tens which add up to 50 and the third group adds up 6 units. Hence, the total is 1856. The Babylonians had a notion of zero because they grouped together ten units as one. The problem with such place holders was that some numbers could not be distinguished except by context (Ibid.). 1800 would be written with just the first column of 1856 above, but since there would be no columns after that 1800 would be indistinguishable from 30. Next, zero appeared as a placeholder among the Mayans in Central America around 350 CE. Based on their calendar a month (unial) was 20 days (kins), a year (tun) was 360 days, a katun was 7200 days, a baktun was 144,000 days. So when a date that was 8 baktuns, 14 katuns, 3 tuns, 0 unials and 12 kins (Ibid.) (which would be the 12th day of the first month of the year 3483 on their calendar) was to be written, the problem was how to represent no unials, so they used the shell as the zero place holder: (Ibid.) The Babylonian and Mayan systems of numbers may be called ‘sexagesimal’ which are based on time keeping or arc measures (Christopher 2007). What was required for the concept of zero as a placeholder was the decimal system. Even though some Indian historians claim that the decimal system was employed by the Harappa civilisation around 2000 BCE (Barua et al. 2009, 7), it is confirmed that it originated in India in the fifth century CE in the Jain text Lokavibhâga (458 CE). The symbol used for zero was a dot or a small circle, however even this use is not documented until 628 (McQuillin 1997) and what seems to be confirmed by consensus among historians of zero is that zero as an explicit place holder with a proper symbol for it was definitely there in 876 in an inscription in Gwalior. As 0 as a placeholder had existed before and there is no convincing proof that the invention of the numeral 0 in India was independent of the Babylonian and Maya inventions earlier, the significance of 0 being used as a place holder in India is that it was preceded by a developed understanding of the concept of zero as a number by Āryabhata (476–550) and Brahmagupta (598–668) whereas an understanding of what zero as a number was at best undeveloped in the Babylonian and Mayan scenarios. Āryabhata is often credited with the invention of zero. However, in The Aryabhatia, though the notion of a place hoder is designated by the word ‘kha’ (O’Conner and Robertson 2000a), however ‘his writing does not clearly indicate the knowledge of zero as a number in its own right. […] “Aryabhatia” […] does not contain zero as a number’ (Ranatunga 2009, 11). Nonetheless, some argue that Āryabhata must have had a clear conception of zero because without the concept of zero and the place value system he could not have constructed his method of square roots of perfect squares (O’Connor and Robertson 2000b). (Āryabhata 1930, 54) Even though this is a reconstruction using the modern ‘0’ we can see that 0 is used both as a place holder as well as the concept of the number of 0 is used in 9 – 9 = 0, in which 0 is not nothing but a number as when the number 0 is the remainder then we started with a perfect square. Whatever may have been implicit in Āryabhata is made explicit by Brahmagupta, who may be acknowledged as the ‘man who found zero, addition, subtraction and multiplication’ (Ranatunga 2009, title page). The ancient Egyptian, Babylonian, Greek, Chinese and other civilisations not only had and used the arithmetical operations but they also had developed algorithms. However, they lacked simple algorithms and rules for these operations, both of which were constructed by Brahmagupta. These rules require the concept of zero as a number. Brahmagupta gave the rules for zero such as: The sum of zero and a negative number is negative, the sum of zero and a positive number is positive, the sum of zero and zero is zero. (O’Connor and Robertson 2000a) Zero was hence properly integrated by as a number and as an integral part of defining the basic operations of arithmetic. This developed notion of zero was exported eastward to China and westward to Persia and the Arabs. Abū 'Abdallāh Muhammad ibn Mūsā al-Khwārizmī (780–850) brought the concept of zero as number to the Arabs. The word ‘zero’ is derived from the Arabic word ‘sifr’. Abraham ibn Ezra introduced zero to the Europeans in the 12th century. (Ibid.) The Italian Fibonacci (1170–1250) wrote: ‘Compared to the methods of the Indians, everything else is a mistake. […] The nine Indian figures are 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures and with the sign 0 … any number may be written.’ (Ranatunga 2009, 36–8) Fibonacci hence embalmed the significance of 0 as a place holder and as a number that makes both counting and arithmetical operations immensely more simplified. When arithmetic was finally axiomatised, Peano (1858–1932) gave as its first axiom: ‘Zero is a number’. Frege (1848-1925) (established the definition of number first as: ‘The number which belongs to the concept F is the extension of the concept “equal to the concept F”’ (Frege, 1884, #68). Then he gives the definition of zero as: ‘0 is the Number which belongs to the concept “not identical to itself”’ (Ibid., #74). Putting the two definitions together we get: ‘0 belongs to the concept F as the extension of the concept “equal to the concept F” which belongs to the concept “not identical to itself”’’. This formal definition is the key to defining numbers in terms of sets. 1 is the number which is the set of all singletons: {(x1), (x2), (x3)…}, 2 is the number which is the set of all doubletons: {(x1, y1), (x2, y1), (x3, y3)…}, and so on. 0 then is the set which is the set of all 0tons, but there are no 0tons, so 0 is the set containing nothing or the empty set: {{}}. The distinction between the null set and zero is essential not only for mathematics but for the scientific and philosophical concept of zero as well. Whereas the null set may be identified with nothing, zero is not nothing. Rather 0 is the beginning. If 0 is considered a natural number then it is the beginning of numbers. The point (0,0) in the Cartesian coordinate system is appropriately called the ‘origin’ as it is the beginning from which you can go out in any direction in one of the four quadrants. In the periodic table, hydrogen is the first element with the atomic number 1. Andreas in 1926 assigned the atomic number 0 to the theoretical element tetraneutron which consists solely of a cluster of four neutrons (Wikepedia 2010). However, this is not the notion of a beginning but is the notion of nothing in the sense that it is an element that has no chemical properties and hence cannot technically be placed in the periodic table. In linguistics Panini and Saussure used “zero” to designate ‘a term for which, unlike the other terms, no phonological bit is statable’ (Allen 1955, 112). This again is not the concept of beginning but of nothing as it specifies the use of ‘null operators’ (Duller 1996, 125). It would hence be a mistake to consider Panini’s zero as the mathematical zero as ‘what is generally called “zero” in linguistics is, by comparison with the mathematical use, inappropriately so called’ (Allen 1955, 113) because the mathematical zero is not nothing but the set containing nothing. Hence, Brahmagupta’s concept of zero as a number is similar to that of Frege rather than to that of Panini. In order for Brahmagupta’s rules of 0 to apply 0 must be a number and not nothing which is not a number. Frege’s definition does the trick of tying down 0 as a number which is the set containing the null set. In common usage we find support for 0 as the beginning. At midnight the time turns to 00:00, marking the beginning of another day. At the moment a task begins we use the term ‘zero hour’. When a spaceship takes off the countdown goes ‘7, 6, 5, 4, 3, 2, 1, 0’. Instead of 0 sometimes they say ‘takeoff’. Hence, 0 marks the beginning of the journey of the astronauts in the spaceship. Nagarjuna (150–250 CE) identified the authentically real with ‘śūnya’, which is usually taken to designate nothingness or emptiness. P. T. Raju however claims that Nagarjuna meant to designate the mathematical zero by ‘śūnya’ (Raju 1954, 701). The realists claimed that only that was real which satisfied the disjunction: p, or ~p, or p and ~p, or neither p nor ~p, where p was the proposition stating the esistence of something. Nagarjuna claimed that what is real is what denies all four of the possibilities in the disjunction and this he called ‘śūnya’. If this is the mathematical 0 it can be written as: 0 = ~[p  ~p  (p & ~p)  ~( p  ~p)], which can be rewritten as a conjunction: 0 = ~p & ~~p & ~(p & ~p) & ~~( p  ~p). If p is ‘0 is a positive number’ and ~p is ‘p is a negative number’ (presupposing here that all numbers are either positive or negative which begs the question of zero, but serves here as a methodological device to establish 0 as a number), then each of the four corners are denied. 0 is not a positive number; 0 is not a negative number; 0 is not both positive and negative as positve and negative are opposites and cannot be attributed to the same number; 0 is not neither positive nor negative since there is no definite number that is neither positive nor negative (Raju 1954, 699–700). So the definition of 0 as the denial of all four corners is satisfied, yet 0 is a number. Nagarjuna establishes this by the claim that 0 is in the middle of the number line that extends to infinity to the right and left. This is why 0 is the most real as it is in the middle hence establishing consistency with Nagarjuna’s philosophy of the middle path. Even if we don’t accept Nagarjuna’s anti-realism, we can at least say that 0 is more real than the other numbers as it is the beginning, and in modern arithmetic everything begins with 0 which is taken to be the first number and all other numbers henceforth are defined on the basis of 0 and the relation of successor. Some of the followers of Sankara, interpreting Nagarjuna’s reality or zero as a negation thought that they could get an affirmation and the positive existence of Brahman by negating Nagarjuna’s four-cornered negation. If we interpret Nagarjuna’s śūnya as nothing or the null set, then the Shakraites would succeed as the complement of the null set is the universal set, hence by negating that reality is nothing they would obtain that reality is everything which is Brahman. However, if Nagarjuna’s śūnya is the mathematical zero which is not the empty set, then negating the proposition that reality is zero will not get to Brahman. Historically, since zero as a number did not emerge implicitly until the fifth century and explicitly until the seventh century, it is unlikely that Nagarjuna would have thought of śūnya as a number. Yet in defence of Raju we might say that Nagarjuna was close to the concept of zero as a number but would have to wait for the development of mathematics to make it explicit. It is extremely difficult to conclusively establish when a concept like that of zero as a number really emerged. We have claimed that it was with Brahmagupta, but then some may argue that zero was not defined as a number until Frege at the end of the nineteenth century. Yet, once we have the definition, we can say that Brahmagupta had the same notion of zero as a number as Frege even though he had not offered the explicit definition. Going backwards we could argue that the seeds were sown long long ago, that the Egyptians, Mesopotamians, Babylonians, Chinese, Mayans, Phoenicians, Indians, and many other ancient cultures, all had the concept of zero as nothing, but they also had some notion though undeveloped of zero as something which would eventually evolve into the notion of zero as a number. References and Bibliography Allen, W. S. 1955. ‘Zero and Panini’, Indian Linguistics, 16: 106–113. Āryabhata, The Āryabhatīa. English Translation by W. E. Clark, 1930. Chicago: Chicago University Press. Barua, Ankur, N. Testerman and M. A. Basilio. 2009. The Concept of Emptiness (Śūnyatä) in Modern Mathematics (ebook). Hong Kong: Buddhist Door, Tung Lin Kok Yuen. Christopher. 2007, June 27, available at http://www.exisle.net/mb/lofiversion/index.php?t46490-50.html. Deen, S.M. 2007. Science Under Islam: Rise, Decline and Revival. Lulu Press. Duller, Anthony. 1996. ‘New Zero and Old Khmer’, Mon-Khmer Studies Journal, 25: 125–132. Frege, Gottlob. 1884. Foundations of Arithmetic (translated by J. L. Austin, 1950). Revised edition, 1953, 1960, New York: Harper & Brothers. Krebs, Robert E. and Carolyn A. Krebs. 2003. Groundbreaking Scientific Experiments, Inventions and Discoveries of the Ancient World. Westport, CT: Greenwood Press. Mathworld. 2010. ‘Peano’s Axioms’, available at http://mathworld.wolfram.com/PeanosAxioms.html. McQuillin, Kristen. 1997. ‘A Brief History of Zero’ (revised 2004), available at http://www.mediatinker.com/blog/archives/008821.html O’Connor, J. J. and E. F. Robertson. 2000a. ‘A History of Zero’, Ancient Indian Mathematics Index, available at http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html. ———. 2000b. ‘Aryabhata the Elder’, Ancient Indian Mathematics Index, available at http://www-history.mcs.st-and.ac.uk/Biographies/Aryabhata_I.html. P. C. World.com. 1999. ‘The Digital Century: Computing Trhrough the Ages’, available on http://www.dougengelbart.org/press/archives/991124/index.html Parekh, Nikhil. 2004. ‘Just a Big Zero’, available at http://www.nikhilparekh.com/Poem.asp?page=124 Raju, P. T. 1954. ‘The Principle of Four-Cornered Negation in Indian Philosophy’, Review of Metaphysics, 7(4): 694–713. Ranatunga, Sanjaya. 2009. Life and Mathematics of Brahmagupta: Man who Found Zero, Addition, Subtraction, Multiplication and Addition. Available at at http://fathersofmathematics.com/Brahamagupta.doc. Swanson, Jennifer. 2008. ‘When did the Number Zero Come to Be’, available at http://math.cudenver.edu/~jloats/sum08%20Student%20wrk/UP%20Loaded/1oral.pdf. Wikepedia. 2010. Zero (Number), available at http://en.wikipedia.org/wiki/0_(number) Priyedarshi Jetli Department of Philosophy, University of Mumbai January 31, 2010 PAGE 10