The Development of Mathematics
by Rit NosotroChange Over Time essay
Describe the development of mathematics.
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Summary:
With our modern understanding of mathematics, its historical development may seem straightforward, and the ideas we have inherited about mathematics appear obvious. When asked about the major developments of mathematics, the average person would likely mention the calculator or the computer. Now that these modern advances have become so ingrained in our lives we hardly give a thought to the most fundamental developments that truely are the greatest discoveries, such as multiplication, the ten numerals we use to specify numbers, or even the idea that we can consider "zero" an actual number. One possible reason for why we view certain historically difficult concepts as simple is the progression in which we learn mathematics in the culture and time we live in; we learn mathematics in an order that makes it easiest to master in a short amount of time. On the other hand, mathematicians throughout history discovered these principles in a much different order. They discovered them by building their knowledge off of the base of wisdom that they had inherited from those before them, the mathematical and numerical systems that they found themselves in, and the motivations that their cultures provided. We learned mathematics from books and teachers; they developed mathematics through tedious trial and error. Small steps by numerous people from numerous cultures over thousands of years directed the development of mathematics, not giant leaps by brilliant individuals, as we may have expected.
What seems most obvious to us is that ten numerals are used in various combinations to express all possible real numbers; this is a base ten system with place values. Yet early cultures did not use this system. Even in America we have more than one way of writing numbers. Of course we have the numerical system with 0 1 2 3 ... 9, but we also sometimes use a "tallying" system when counting things such as scores; we write a set of four verticle lines and then write a fifth diagonal line through them to indicate a group of five. Many other cultures developed their own numerical systems; the Romans were one such culture. Their use of consecutive symbols implied a meaning of simple addition (one-one-one or III meant three); our consecutive symbols mean to what power of ten the numerals should be multiplied by before adding (one-one-one or 111 means one hundred eleven, implied by 1 x 102 + 1 x 101 + 1 x 100). Many other cultures created systems similar to the Romans with varying levels of placement importance and with several other base systems, including the Islamic nations, the Chinese, and the Mayans. For simple counting and addition, any of these numerical systems work perfectly fine. Yet the Roman system, and others like it, make multiplication very difficult if even possible; without multiplication much of modern mathematics would not exist. For mathematics to develop further a more advanced system had to be found. The development of this advanced system that we find in modern mathematics began even before the arrival of, and was improved by, the Babylonians (they used a base sixty system with minimal placement importance). Many other cultures later influenced this development which the Indian mathematicians finally completed with the numerals and place values that we use today.
In America and many other modern nations, the idea of zero being a number is extremely ingrained and used in every day life. It is found as part of telephone numbers and street addresses, it is used to specify dollar amounts at grocery stores, and it is taught early on in childhood education. This mathematical idea seems natural, yet the idea of a number that literally represents "nothing" was not an obvious necessity throughout much of the history of mathematics. John O'Connor and Edmund Robertson (from the University of St Andrews in Scotland) mention that numbers in general took on a very literal meaning, not the abstract meaning that we give them today:
"There are giant mental leaps from 5 horses to 5 'things' and then to the abstract idea of 'five'. If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer"1.How then did zero develop? First off we should realize that zero in our modern notation has two important purposes, both developing individually from one another. Of primary importance is its use as a type of place holder: without a zero or some other similar symbol it would be impossible to clearly indicate if you meant 11 or 101 or 10012. The Babylonians eventually saw this need and developed a place holding symbol before they imaged the idea of zero as a number. The Mayans also developed a symbol, but (unlike the Babylonians) they used it as both the number "zero" and as a placeholder. ("We do not know the date of these [Mayan] mathematical achievements but it seems certain that when the system was devised it contained features which were more advanced than any other in the world at the time."3) In their article "A History of Zero", O'Connor and Robertson show the progression of the modern concept of zero from ancient Babylon to India, then to Islamic nations and finally to Europe through the famous Fibonnaci4 (From Pisa in modern day Italy, c.1170-c.1250 AD5). More key figures in numerous locations and times show up in this historical progression including Ptolemy (Egypt, 85-165 AD) and Brahmagupta (India, 598-670 AD), as well as Greek astronomers6. Thus we begin to see a progression of how numeric notation was shared amongst nations and, in the process, became more advanced and developed. Yet under what circumstances did the first inquisitive minds begin to see a need to develop the basic mathematical principles as the foundation upon which huge amounts of mathematics and science itself now rests?
As necessity is the mother of invention, so was necessity the mother of mathematics. This necessity expressed itself in numerous areas. First and foremost, the need to accurately record transactions of goods and money forced the development of the early numeric notation. Quantities were measured with weights of particular sizes, and the amount of goods bought, sold, or stored was recorded. As advances to civilization progressed, other influences arose including navigation and complex architecture (as well as the geometry necessary for these architectural constructions). Astronomy, and with it the astrology of certain religious societies, also became a motivation to further the development of mathematics; from this grew the development of calendars based around the movements of the heavenly bodies. O'Connor and Robertson claim that the Egyptians' need to accurately calculate when the Nile would flood was one motivation for developing a calendar--both an astronomical and mathematical venture. They go on to explan how this one event in one location influenced world history: it was a very important development because it became the early basis for the later developments of the Julian and Gregorian calendars7.
On the other side of the world the development of a calendar also influenced the Mayan culture's motivation for mathematic development: ". . . astronomy played an important part in the religion which underlay the whole life of the [Mayan] people. Of course astronomy and calendar calculations require mathematics . . . "8 Similar motivations (calendars, trade, astronomy etc.) applied to all people, and numerous cultures likewise developed similar mathematic ideas, regardless of their isolation from each other due to geographic barriers (such as the Atlantic ocean). This is naturally the case, for the development of math (regardless of the location of the individuals) is simply a discovery of the non-chaotic order of God's creation, and thus uniform principles would apply as much to America as it would to the Eurasian continent. Nonetheless, only a few cultures directly impacted the progression that we can now trace backwards from modern mathematics.
Mathematics, as we know it, can be traced through stages of development within four distinct cultural groups--Mesopotamian/Babylonian, Greek, Indian, and Arabic/Islamic--before finally making its way to the European/American cultural group. J. Melville, professor of mathematics at St. Lawrence University in New York, mentions other Babylonian/Mesopotamian developments, besides the previously mentioned numeric system, such as mathematical tables (for "multiplication, roots, powers, reciprocals, [and] coefficients") from the Old Babylonian period approx. 2000-1600 BC9. Also mentioned by Melville is their computation of simple geometric problems10. The Babylonians had seen mathematics as very concrete, describing actual physical objects or weights or measures: ". . . most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones . . . ."11 It is here that the Greeks diverged and began to delve into more theoretic and abstract ideas, though still often couched in the ideas of geometry. Many scientific giants came from this Greek culture and with them their great discoveries: Pythagoras (c. 570-c. 49012) who formalized the "Pythagorean theorem" that describes the relationships of the angles and sides of a right triangle; Euclid (fl. c. 29513) who wrote the famous book "Elements" that establishes logical arguments concerning geometry and mathematics; and Hipparchus (c. 180-c. 12514) who made significant and early contributions to trigonometry. Next in this line of significant discoveries came the Indian mathematicians to whose credit the current numerical system we use today should be given (regardless of the name we have labeled it with: "Arabic numerals"). O'Connor and Robertson mention that an outside force, the printing press, encouraged the developed Indian symbols to become standarized15. Finally came the Arabic/Islamic influence: the connection of Indian mathematics to Europe, the saviour of Greek mathematic writings, and the source of algebraic thought16. The word "algebra" itself even originates from a single word (al-jabr) from the title of a book written by the Arabic mathematician Muhammad ibn Musa al-Khwarizmi (ca. 800-ca. 847)17. All these developments finally arrived in Europe, in part, due to the influence of Leonardo Fibonacci who suggested using the "Arab method of reckoning" instead of the abascus which was in use in Italy at the time18.
A ten year old child could easily master the mathematical exercises that ancient Babylonians could not even imagine. Nonetheless this does not imply that these mathematic principles are obvious. On the contrary, these principles took years of painful experimentation and careful thought which developed first numeric systems, then principles of simple arithmetic, the development of the number zero, geometric calculations, calendar and astronomic observations and measurements, and algebraic principles. These developed in numerous places around the globe due to similar or identical motivations such as record keeping and construction. Even now, mathematicians continue to expand our extent of knowledge concerning mathematics, building upon those ideas that have been handed down to them from mathematicians of the past. We sometimes see the calculator and the computer as the most incredible discoveries in mathematics, and to some extent this is true. Yet we must remember what these tools truely are: only a quicker way to do the exact same equations that took centuries for diligent men to discover.
Quick Quiz:
Endnotes:
1 John J O'Connor and Edmund F Robertson. "A history of Zero." University of St Andrews, Scotland. November 2000. <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html> (May 27, 2005)
2 Ibid.
3 John J O'Connor and Edmund F Robertson. "Mayan Mathematics." University of St Andrews, Scotland. November 2000. <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Mayan_mathematics.html> (May 27, 2005)
4 O'Connor and Robertson, "A history of Zero."
5 John J O'Connor and Edmund F Robertson. "Leonardo Pisano Fibonacci." University of St Andrews, Scotland. October 1998. <http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Fibonacci.html> (May 27, 2005)
6 O'Connor and Robertson, "A history of Zero."
7 John J O'Connor and Edmund F Robertson. "An overview of Egyptian mathematics." University of St Andrews, Scotland. December 2000. <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_mathematics.html> (May 27, 2005)
8 O'Connor and Robertson, "Mayan Mathematics."
9 Duncan J. Melville. "Chronology of Mesopotamian Mathematics." St Lawrence University. May 30, 2001. <http://it.stlawu.edu/%7Edmelvill/mesomath/chronology.html> (May 27, 2005)
10 Duncan J. Melville. "Old Babylonian Mathematics." St Lawrence University. September 3, 1999. <http://it.stlawu.edu/~dmelvill/mesomath/obsummary.html> (May 27, 2005)
11 Ibid.
12 David E. Joyce. "Chronological List of Mathematicians." Clark University. September 19, 1995. <http://aleph0.clarku.edu/%7Edjoyce/mathhist/chronology.html#toc> (May 27, 2005)
13 Ibid.
14 Ibid.
15 John J O'Connor and Edmund F Robertson. "Indian numerals." University of St Andrews, Scotland. November 2000. <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html> (May 27, 2005)
16 "Algebra and Trigonometry." PBS. <http://www.pbs.org/empires/islam/innoalgebra.html> (May 27, 2005)
17 Ibid.
18 Ibid.
Sources for Further Research
Photographic gallery of historic mathematical (and related scientific) documents - http://www.loc.gov/exhibits/vatican/math.html
Full text of (and commentary on) Euclid's "Elements" - http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
Lengthy articles on nine cultures mathematics - http://www-history.mcs.st-andrews.ac.uk/Indexes/HistoryTopics.html
Biographies of over 1600 mathematicians - http://www-history.mcs.st-andrews.ac.uk/BiogIndex.html
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