The history of math is murky1murky模糊不清。,predating2predate早于。any written records.When did humans first grasp the basic concept of a number? What about size and magnitude3magnitude巨大 。,or form and shape?

數(shù)學(xué)的開端早于所有現(xiàn)存的書面記錄，因而其歷史模糊不清。人類何時(shí)開始掌握基本的數(shù)字概念？何時(shí)開始掌握大小和度量概念，或外形和形狀概念？

[2]In my math history courses and my research travels in Guatemala,Egypt and Japan,I’ve been especially interested in the commonality4commonality共同點(diǎn)。and differences of mathematics from various cultures.

[2]我教授數(shù)學(xué)歷史課程，為了做研究，去過危地馬拉、埃及和日本。在授課和研究過程中，我對(duì)各個(gè)文化間數(shù)學(xué)的異同尤其感興趣。

[3]Although no one knows math’s exact origins,modern mathematicians like myself know that spoken language precedes written language by scores of5scores of許多。millennia.Linguistic clues show how people around the world must have first developed mathematical thought.

[3]數(shù)學(xué)的確切起源無人知曉。但是，現(xiàn)代數(shù)學(xué)家（比如我自己）都清楚，口頭語言的發(fā)端比書面語言要早幾千年。從語言學(xué)中能找到線索，說明世界各地的人們最初是如何發(fā)展出數(shù)學(xué)思維的。

Early clues

早期線索

[4]Differences are easier to comprehend than similarities.The ability to distinguish more versus6versus（比較兩種不同想法、選擇等）與……相對(duì)，與……相比。less,male versus female or short versus tall must be very ancient concepts.But the con-cept of different objects sharing a common attribute7attribute特征。—such as being green or round or the idea that a single rabbit,a solitary8solitary孤零零的，離群的。bird and one moon all share the attribute of uniqueness—is far subtler.

[4]數(shù)學(xué)中的“不同”概念，比“相似”概念容易掌握?！岸唷焙汀吧佟?，“男”和“女”，“矮”和“高”，人類很早就能區(qū)別這些概念。而“不同的物體擁有共同特征”這一概念則復(fù)雜得多——比如，都是綠色、都是圓形，或單只兔子、一只孤鳥和一輪明月都擁有“獨(dú)一無二”這一特征。

[5]In English,there are many different words for two,like “duo,” “pair”and “couple,” as well as very particular phrases such as “team of horses”or “brace of partridge.” This suggests that the mathematical concept of twoness developed well after humans had a highly developed and rich language.

[5]英語中有很多詞表示“雙”的概念，例如duo、pair和couple，還有些很特別的短語，如team of horses（一同拉車的兩匹馬）、brace of partridge（打獵時(shí)射下的一對(duì)鷓鴣）。這表明，在人類語言高度發(fā)展和極大豐富很久以后，才有了“二”這個(gè)數(shù)學(xué)概念。

[6]By the way,the word “two” probably was once pronounced closer to the way it’s spelled,based on the modern pronunciation of twin,between,twain(two fathoms9fathom英尋（測(cè)量水深用的長度單位，合6英尺或1.829米）。fathom來自古英語f?thm，表示“伸展開的雙臂”，因而1英尋也就是兩臂之長。),twilight (where day meets night),twine (the twisting of two strands)and twig (where a tree branch splits in two).

[6]順帶一提，two一詞原先的發(fā)音，很可能跟拼寫方式相近。這一猜測(cè)，乃是基于現(xiàn)代英語中以下這些詞的發(fā)音：twin（雙胞胎）、between（在……之 間 ）、twain（ 水 深 兩 尋 ），twilight（日夜之交）、twine（合股絞線）、twig（樹枝分岔處）。

[7]Written language developed much later than spoken language.Unfortunately,much was recorded on perishable10perishable易損的，易腐的。media,which have long since decayed.But some ancient artifacts that have survived do exhibit some mathematical sophistication.

[7]書面語言比口頭語言出現(xiàn)晚得多。不幸的是，很多書面記錄都寫在易損的介質(zhì)上，腐爛已久。不過，也有些古代人工制品得以留存至今，可以看出數(shù)學(xué)的成熟發(fā)展。

[8]For example,prehistoric tally11tally（符契上的）刻記；計(jì)數(shù)的簽籌。sticks—notches12notch（刻在棍子等上的）計(jì)數(shù)刻痕。incised13incise（在表面）雕；切入。on animal bones—are found in many locations around the world.Though these might not be proof of actual counting,they do suggest some sense of numerical record keeping.Certainly people were making one-to-one comparisons between the notches and external collections of objects—perhaps stones,fruits or animals.

[8]比如，全球很多地方都出土了史前文物“記數(shù)棒”——刻有計(jì)數(shù)痕跡的動(dòng)物骨頭。這些或許算不上真正計(jì)數(shù)的證據(jù)，但確實(shí)表明，史前人類有了保存數(shù)字記錄的意識(shí)。當(dāng)時(shí)的人類，肯定是拿著記數(shù)棒，與某種外部收集的物體（也許是石頭、水果或動(dòng)物）一對(duì)一比照計(jì)數(shù)。

Counting objects

計(jì)算物體數(shù)量

[9]The study of modern “primitive”cultures offers another window into human mathematical development.By“primitive,” I mean cultures that lack a written language or the use of modern tools and technology.Many “primitive”societies have well-developed arts and a deep sense of ethics and morals,and they live within sophisticated societies with complex rules and expectations.

[9]對(duì)現(xiàn)代“原始”文化的研究，也提供了一扇了解人類數(shù)學(xué)發(fā)展的窗口。我說的“原始”文化，是指沒有書面語言或沒有現(xiàn)代工具和技術(shù)的文化。很多“原始”社會(huì)有發(fā)達(dá)的藝術(shù)，深刻的倫理道德觀，社會(huì)發(fā)展成熟，具備復(fù)雜的規(guī)范和要求。

[10]In these cultures,counting is often done silently by bending down fingers or pointing to specific parts of the body.A Papuan14巴布亞族，新幾內(nèi)亞的原住民。tribe of New Guinea15新幾內(nèi)亞，太平洋西南島嶼，澳大利亞以北。can count from 1 to 22 by pointing to various fingers as well as to their elbows,shoulders,mouth and nose.

[10]在這些文化中，人們計(jì)數(shù)時(shí)通常默不作聲，只彎曲手指，或者指向身體的某個(gè)部位。新幾內(nèi)亞的某個(gè)巴布亞部落，計(jì)數(shù)時(shí)指向各個(gè)手指、手肘、肩膀、嘴巴和鼻子，用這種辦法，能從1數(shù)到22。

[11]Most primitive cultures use object-specific counting,depending on what’s prevalent in their environment.For example,the Aztecs16Aztec阿茲特克人。阿茲特克是14—16世紀(jì)的墨西哥古文明。would count one stone,two stone,three stone and so on.Five fish would be “five stone fish.”Counting by a native tribe in Java begins with one grain.The Nicie tribe of the South Pacific counts by fruit.

[11]很多原始文化使用“物體計(jì)數(shù)法”。其中的“物體”各有不同，取決于在當(dāng)?shù)丨h(huán)境中，什么東西最為普遍。例如，阿茲特克人計(jì)數(shù)時(shí)，會(huì)數(shù)“1石頭、2石頭、3石頭”，等等?！?條魚”在阿茲特克人說來，就是“5石頭魚”。另外，爪哇的某個(gè)原住民部落，從“1谷”開始計(jì)數(shù)；南太平洋的尼西部落用水果計(jì)數(shù)。

[12]English number words were probably object-specific as well,but their meanings have long been lost.The word “five” probably has something to do with “hand.” Eleven and 12 meant something akin to “one over” and “two over”—over a full count of 10 fingers.

[12]英語中的數(shù)詞很可能也曾特指物體。不過，如今數(shù)詞中的物體含義早已喪失。比如，five（5）一詞很可能跟hand（手）有關(guān)。eleven（11）和twelve（12）則接近于“多1”和“多2”——十個(gè)手指都數(shù)完后，還多了“1”和“2”。

[13]The math Americans use today is a decimal,or base 10,system.We inherited it from the ancient Greeks.However,other cultures show a great deal of variety.Some ancient Chinese,as well as a tribe in South Africa,used a base 2 system.Base 3 is rare,but not unheard of among Native American tribes.

[13]美國人的數(shù)學(xué)系統(tǒng)使用十進(jìn)制，也就是以10為基礎(chǔ)的系統(tǒng)，這是從古希臘人那兒繼承下來的。不過，以哪個(gè)數(shù)字為基數(shù)，在各個(gè)文化間差異很大。有些古代中國人，還有南非的某個(gè)部落，使用以數(shù)字2為基數(shù)的二進(jìn)制。三進(jìn)制很少見，但偶爾也會(huì)有人使用，例如某些美洲土著部落。

[14]The ancient Babylonians used a sexagesimal17sexagesimal六十的；六十進(jìn)位的。,or base 60,system.Many vestiges18vestige遺跡，遺痕。of that system remain today.That’s why we have 60 minutes in an hour and 360 degrees in a circle.

[14]古巴比倫人用的則是六十進(jìn)制，也就是以60為基數(shù)的系統(tǒng)。這種計(jì)數(shù)系統(tǒng)的影響一直遺留至今，所以60分鐘才會(huì)等于1個(gè)小時(shí)、圓周才會(huì)有360度。

Written numbers

數(shù)字的書寫

[15]What about written numbers?

[15]那么，數(shù)字如何書寫呢？

[16]Ancient Mesopotamia19古美索不達(dá)米亞，西南亞古文明，位于底格里斯河和幼發(fā)拉底河之間，現(xiàn)伊拉克境內(nèi)。had a very simple numerical system.It used just two symbols: a vertical wedge (v)to represent 1 and a horizontal wedge (＜)to represent 10.So ＜＜vvv could represent 23.

[16]古美索不達(dá)米亞人的書面數(shù)字系統(tǒng)很簡單，只有兩個(gè)符號(hào)：豎的楔形（v）代表1，橫的楔形（＜）代表10。所以＜＜vvv就代表 23。

[17]But the Mesopotamians had no concept of zero either as a number or as a place holder.By way of analogy,it would be as if a modern person were unable to distinguish between 5.03,53 and 503.Context was essential.

[17]但是，美索不達(dá)米亞人沒有“0”的概念?！?”既沒有被列為數(shù)字，也不占位。打個(gè)比方，這就好像現(xiàn)代人沒法區(qū)分5.03、53和503一樣。沒有“0”，數(shù)字前后的上下文就成了區(qū)分的關(guān)鍵。

[18]The ancient Egyptians used different hieroglyphs20hieroglyph象形文字。for each power21power乘方，冪。of 10.The number one was a vertical stroke,just as we currently use.But 10 was a heel bone,100 a scroll or coiled rope,1000 a lotus flower,10,000 a pointed finger,100,000 a tadpole and 1,000,000 the god Heh22赫神，古埃及神靈，是“無限”或“永恒”的人格化。holding up the universe.

[18]古埃及人用不同的象形文字表示10的冪。數(shù)字1就是直直的一豎，跟我們目前使用的數(shù)字差不多；不過，古埃及人的10卻用踵骨來表示。100用一個(gè)卷軸或一盤繩子，1000用蓮花，10000用伸出的手指，100000用蝌蚪，1000000則是赫神托起整個(gè)宇宙。

[19]The numerals most of us know today developed over time in India,where computation and algebra were of utmost importance.It was also here that many modern rules for multiplication,division,square roots and the like were first born.These ideas were further developed and gradually transmitted to the Western world via Islamic scholars.That’s why we now refer to our numerals as the Hindu-Arabic numeral system.

[19]大多數(shù)現(xiàn)代人熟知的數(shù)字系統(tǒng)起源并發(fā)展于古印度。在古印度，計(jì)算和代數(shù)這兩門學(xué)問，占據(jù)最重要的地位。現(xiàn)代數(shù)學(xué)的很多規(guī)則，如乘、除、平方根之類，也誕生于古印度。后來，伊斯蘭學(xué)者們進(jìn)一步發(fā)展了古印度的數(shù)字系統(tǒng)和數(shù)學(xué)規(guī)則，并漸漸傳播到西方世界。由此，我們的數(shù)字系統(tǒng)才得名“印度-阿拉伯?dāng)?shù)字系統(tǒng)”。

[20]It’s good for a young struggling math student to realize that it took thousands of years to progress from counting“one,two,many” to our modern mathematical world.■

[20]從1、2數(shù)到許多，再到我們今日的數(shù)學(xué)世界，其間經(jīng)歷了幾千年的艱辛發(fā)展歷程。知曉這段歷史后，同樣艱辛掙扎的數(shù)學(xué)專業(yè)年輕學(xué)子，一定會(huì)覺得安慰吧。□

（譯者曾獲第三屆“《英語世界》杯”翻譯大賽優(yōu)秀獎(jiǎng)）