## “How can nothing be something?”

May 29, 2012 § Leave a comment

십진법 숫자 중 가장 늦게 발명되고, 0이라는 개념을 발견해 놓고도 ‘아무것도 없다’는 것의 형태를 인정하는 데 또 한참이나 걸렸다.

기원전 2000년대 중반, 바빌로니아인들이 ‘없다’라는 숫자 개념을 사용했고, 기원전 4세기부터 중국에서도 0과 음수의 개념이 나타난다. 중국인들은 8세기 인도에서 0이라는 기호가 들어오기 전까지는 글자나 네모 형태의 기호를 사용했다. 인도인들의 기록엔 원형태의 0이라는 기호가 876년 유물에 보인다. 아랍에서는 0이라는 개념과 기호가 5세기경에 나타나고, 825년에 페르시아의 과학자, al-Khwārizmī는 그리스와 힌두인들의 지식을 정리하고, 0의 개념을 설명한 산수책을 출간하였다. 그리고 이 책이 12세기에 라틴어로 번역되면서, 서양세계에 0을 비롯한 아라비아 숫자가 소개되었다. 한편, 고대 그리스, 로마인들은 0이라는 숫자 개념에 확신이 없었던 것 같다. 어떻게 ‘없다’는 것이 ‘어떤 것’이 될 수 있는지에 관한 논쟁이 철학으로 연결되었고, 중세에 이르러서는 0의 존재와 ‘아무 것도 없다’는 상태에 대한 신학적인 논쟁으로 이어졌다. 기원전 130년경 바빌로니아의 영향을 받은 Ptolemy가 현대의 0에 해당하는 개념과 기호를 *Syntaxis Mathematica*라는 책에 소개하였고, 이것이 0을 사용한 그리스 숫자에 대한 최초의 기록일 것이다. 하지만 0이라는 개념과 기호는 부분적으로만 사용되었을 뿐 그리스 수학계에 일반적으로 통용되지는 않았다. 525년경에 이르러 달력표기에 로마숫자와 함께 글자로 0이라는 개념을 표기하기도 했다. 기원전 마야인들과 중미대륙 사람들도 0이라는 개념과 기호를 사용했지만 다른 고대사회에 큰 영향을 끼치지는 못했다. 유럽에서는 15세기 후반까지 아라비아숫자체계는 수학자들 사이에서 쓰이고, 일상적으로는 로마자가 통용되다가, 16세기에 이르러서야 널리 쓰이게 되었다.

‘없다’라는 개념이 형태를 갖추고서야 십진법이 완성되었고, 지금은 o과 1일 지배하는 컴퓨터의 시대이다.

없다는 것의 형태를 인정한 것처럼, 비물질도 물질의 한 형태로 편입시킬 수 있을까?

물질과 비물질의 구분이 애매모호해지는 증감현실 시대엔 이러한 접근방법이 더 타당하지 않을까?

The word *zero* came via French *zéro* from Venetian *zero*, which (together with *cypher*) came via Italian *zefiro* from Arabic صفر, *ṣafira* = “it was empty”, *ṣifr* = “zero”, “nothing”. This was a translation of the Sanskrit word *shoonya* (śūnya), meaning “empty”.

### Mesopotamia

By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a *space* between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.

The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.

### India

The concept of zero as a number and not merely a symbol for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division. The Indian scholar Pingala (circa 5th-2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code. He and his contemporary Indian scholars used the Sanskrit word *śūnya* to refer to zero or *void*. The use of a blank on a counting board to represent 0 dated back in India to 4th century BC. In 498 AD, Indian mathematician and astronomer Aryabhata stated that “Sthanam sthanam dasa gunam” or place to place in ten times in value, which is the origin of the modern decimal-based place value notation.

The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the *Lokavibhâga*, dated 458 AD, where *shunya* (“void” or “empty”) was employed for this purpose. The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 AD. There are many documents on copper plates, with the same small *o* in them, dated back as far as the sixth century AD, but their authenticity may be doubted.

### China

Since the 4th century BC, counting rods were used in China for decimal calculation s including the use of blank spaces. Chinese mathematicians understood negative numbers and zero, some mathematicians indicated the for the latter with *wúrù* (無入 “no entry”), *kōng* (空 “empty”) and the frame-like symbol 口/囗, until Gautama Siddha introduced the symbol 0 in the 8th century.

Prior to that, *The Nine Chapters on the Mathematical Art*, composed in the 1st century AD, had already explicitly stated “[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number.”

### The Arab world

The Hindu-Arabic numerals and the positional number system were introduced around 500 AD, and in 825 AD, it was introduced by a Persian scientist, al-Khwārizmī, in his book on arithmetic. This book synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero. It was only centuries later, in the 12th century, that the Arabic numeral system was introduced to the Western world through Latin translations of his treatise *Arithmetic*.

### Greeks and Romans

Records show that the ancient Greeks seemed unsure about the status of zero as a number. They asked themselves, “How can nothing *be* something?”, leading to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.

*number*zero in the Old World. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number. In later Byzantine manuscripts of Ptolemy’s

*Syntaxis Mathematica*(also known as the

*Almagest*), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, *nulla* meaning “nothing”, not as a symbol. When division produced zero as a remainder, *nihil*, also meaning “nothing”, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). The initial “N” was used as a zero symbol in a table of Roman numerals by Bede or his colleague around 725.

### The Americas

^{[22]}Since the eight earliest Long Count dates appear outside the Maya homeland,

^{[23]}it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates.s lacked a final sexagesimal placeholder. Only context could differentiate them.

Although zero became an integral part of Maya numerals, it did not influence Old World numeral systems. Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.

Here Leonardo of Pisa uses the phrase “sign 0”, indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called *algorismus* after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and was one of the earliest scientific books to be *printed* in 1488. Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe.

## Zero : The biography of a dangerous idea

May 29, 2012 § Leave a comment

SEIFE, Charles. Zero : The biography of a dangerous idea. Penguin books, 2000.

## Eléments d’histoire des mathématiques

May 29, 2012 § Leave a comment

BOURBAKI, Nicolas. *Éléments d’histoire des Mathématiques*. Paris : Masson, 1984

Nicolas BOURBAKI est un mathématien imaginaire, sous le nom duquel un groupe de mathématiciens francophones, formé en 1935.

0의 역사

BOURBAKI, Nicolas. trad. John MELDRUM.* Elements of the History of Mathematics*. Germany : Springer-Verlag Berlin Heidelberg New York, 1994.