essay on mathematics a mirror of modern civilization

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The Art of More: How Mathematics Created Civilization

An illuminating, millennia-spanning history of the impact mathematics has had on the world, and the fascinating people who have mastered its inherent power, from Babylonian tax officials to the Apollo astronauts to the eccentric professor who invented the infrastructure of the online world Counting is not innate to our nature, and without education humans can rarely count past three—beyond that, it’s just “more.” But once harnessed by our ancestors, the power of numbers allowed humanity to flourish in ways that continue to lead to discoveries and enrich our lives today.     Ancient tax collectors used basic numeracy to fuel the growth of early civilization, navigators used clever geometrical tricks to engage in trade and connect people across vast distances, astronomers used logarithms to unlock the secrets of the heavens, and their descendants put them to use to land us on the moon. In every case, mathematics has proved to be a greatly underappreciated engine of human progress.     In this captivating, sweeping history, Michael Brooks acts as our guide through the ages. He makes the case that mathematics was one of the foundational innovations that catapulted humanity from a nomadic existence to civilization, and that it has since then been instrumental in every great leap of humankind. Here are ancient Egyptian priests, Babylonian bureaucrats, medieval architects, dueling Swiss brothers, and renaissance painters. Their stories clearly demonstrate that the invention of mathematics was every bit as important to the human species as was the discovery of fire. From first page to last,  The Art of More  brings mathematics back into the heart of what it means to be human.

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Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

Mathematics in Civilization

• First Online: 29 October 2021

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• Gerard O’Regan 4  

Part of the book series: Texts in Computer Science ((TCS))

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This chapter considers the contributions of early civilizations to the computing field, including the achievements of the Babylonians, Egyptians, Greeks and Romans, and the Islamic world. The Babylonian civilization flourished in Mesopotamia (in modern Iraq) from about 2000 B.C. until about 500 B.C., and they made important contributions to mathematics. The Egyptian Civilization developed along the Nile from about 4000 B.C., and their knowledge of mathematics allowed them to construct the pyramids at Giza. The Greeks made major contributions to Western civilization including mathematics, logic and philosophy. The Golden Age of Islamic civilization was from 750 A.D. to 1250 A.D., and enlightened caliphs recognized the value of knowledge, and sponsored scholars to come to Baghdad to gather and translate the existing world knowledge into Arabic.

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Computing in Early Civilizations

Of course, it is essential that the population of the world moves towards more sustainable development to ensure the long-term survival of the planet for future generations. This involves finding technological and other solutions to reduce greenhouse gas emissions as well as moving to a carbon neutral way of life. The solution to the environmental issues will be a major challenge for the twenty-first century.

Tutankhamun was a minor Egyptian pharaoh who reigned after the controversial rule of Akhenaten. Tutankhamun’s tomb was discovered by Howard Carter in the Valley of the Kings, and the tomb was intact. The quality of the workmanship of the artefacts found in the tomb is extraordinary, and a visit to the Egyptian museum in Cairo is memorable.

The origin of the word ‘democracy’ is from demos ( \(\updelta \upeta \upmu {\text{o}}\varsigma\) ) meaning people and kratos ( \({\upkappa \uprho \upalpha \uptau }{\text{o}}\varsigma\) ) meaning rule. That is, it means rule by the people and it was introduced into Athens following the reforms introduced by Cleisthenes. He divided the Athenian city-state into 30 areas, where 20 of these areas were inland or along the coast and ten were in Attica itself. Fishermen lived mainly in the ten coastal areas; farmers in the ten inland areas; and various tradesmen in Attica. Cleisthenes introduced ten new clans where the members of each clan came from one coastal area, one inland area on one area in Attica. He then introduced a Boule (or assembly) which consisted of 500 members (50 from each clan). Each clan ruled for 1 / 10 th of the year.

The Athenian democracy involved the full participations of the citizens (i.e. the male adult members of the city-state who were not slaves), whereas in representative democracy the citizens elect representatives to rule and represent their interests. The Athenian democracy was chaotic and could be easily influenced by individuals who were skilled in rhetoric. There were teachers (known as the Sophists) who taught wealthy citizens rhetoric in return for a fee. The origin of the word ‘sophist’ is the Greek word \(\upsigma{\text{o}}\upvarphi {\text{o}}\varsigma \) meaning wisdom, and one of the most well known of the sophists was Protagoras. The problems with the Athenian democracy led philosophers such as Plato to consider alternate solutions such as rule by philosopher kings. This totalitarian utopian state is described in Plato’s Republic.

The Elgin marbles are named after Lord Elgin who was the British ambassador to Greece (which was then part of the Ottoman Empire), and he removed them (at his own expense) over several years from the Parthenon in Athens to London during the first decade of the nineteenth century. The marbles show the Pan-Athenaic festival that was held in Athens in honour of the goddess Athena after whom Athens is named.

The origin of the word Hellenistic is from Hellene (‘ \({\text{E}}{\uplambda \uplambda \upeta \upnu }\) ) meaning Greek.

The hanging gardens of Babylon were one of the seven wonders of the ancient world.

Henry Rawlinson played an important role in the decipherment of cuneiforms, and especially for making a copy of the large Behistun inscription (the equivalent of the Rosetta stone for Assyriologists) that recorded the victory of the Persian king Darius I over those who rebelled against him in three languages. Edward Hicks and others played a key role in deciphering the inscription.

A positional numbering system is a number system where each position is related to the next by a constant multiplier. The decimal system is an example: e.g. 546 = 5* 10 2  + 4* 10 1  + 6.

The decorations of the tombs in the Valley of the Kings record the life of the pharaoh including his exploits and successes in battle.

The cartouche surrounded a group of hieroglyphic symbols enclosed by an oval shape. Champollion’s insight was that the group of hieroglyphic symbols represented the name of the Ptolemaic pharaoh “Ptolemy”.

The Rhind papyrus is sometimes referred to as the Ahmes papyrus in honour of the scribe who wrote it in 1832 B.C.

The length of a side of the bottom base of the pyramid is b 1 , and the length of a side of the top base is b 2. .

Plato’s Republic describes his utopian state, and seems to be based on the austere Spartan model.

The Pythagoreans took a vow of silence with respect to the discovery of incommensurable numbers. However, one member of the society is said to have shared the secret result with others outside the sect, and an apocryphal account is that he was thrown into a lake for his betrayal and drowned. The Pythagoreans obviously took Mathematics seriously back then.

The library in Alexandria is predated by the Royal library of Ashurbanipal which was established in Nineveh (the capital of the Assyrian Empire) in the seventh century B.C. The latter contained over 30,000 cuneiform tablets including the famous “Epic of Gilgamesh”, and the laws in Hammurabi’s code.

The town of Aswan is famous today for the Aswan high dam, which was built in the 1960s. There was an older Aswan dam built by the British in the late nineteenth century. The new dam led to a rise in the water level of Lake Nasser and flooding of archaeological sites along the Nile. Several archaeological sites such as Abu Simbel and the temple of Philae were relocated to higher ground.

Syracuse is located on the island of Sicily in Southern Italy.

The origin of the word “odometer” is from the Greek words ‘ \({\text{o}}\updelta {\text{o}}\upzeta \) ’ (meaning journey) and \({\upmu \upvarepsilon \uptau \uprho }{\text{o}}{\upnu }\) meaning (measure).

The figures given here are for the distance of one Roman mile. This is given by \(\uppi {4}*{4}00 = {12}.{56}*{4}00 = {5}0{24}\) (which is less than 5280 feet for a standard mile in the Imperial system).

Socrates was a moral philosopher who deeply influenced Plato. His method of enquiry into philosophical problems and ethics was by questioning. Socrates himself maintained that he knew nothing (Socratic ignorance). However, from his questioning it became apparent that those who thought they were clever were not really that clever after all. His approach obviously would not have made him very popular with the citizens of Athens. Chaerephon (a friend of Socrates) had consulted the oracle at Delphi to find out who the wisest of all men was, and he was informed that there was no one wiser than Socrates. Socrates was sentenced to death for allegedly corrupting the youth of Athens, and the sentence was carried out by Socrates being forced to take hemlock (a type of poison). The juice of the hemlock plant was prepared for Socrates to drink.

Chrysippus was the head of the Stoics in the third century B.C.

Aquinas’ (or St. Thomas’) most famous work is Summa Theologica.

The Aeneid by Virgil suggests that the Romans were descended from survivors of the Trojan war, and that Aeneas brought surviving Trojans to Rome after the fall of Troy.

Carthage was located in Tunisia, and the wars between Rome and Carthage are known as the Punic wars. Hannibal was one of the great Carthaginian military commanders, and during the second Punic war, he brought his army to Spain, marched through Spain and crossed the Pyrenees. He then marched along southern France and crossed the Alps into Northern Italy. His army also consisted of war elephants. Rome finally defeated Carthage and destroyed the city.

The Celtic period commenced around 1000 B.C. in Hallstatt (near Salzburg in Austria). The Celts were skilled in working with Iron and Bronze, and they gradually expanded into Europe. They eventually reached Britain and Ireland around 600 B.C. The early Celtic period was known as the “Hallstaat period” and the later Celtic period is known as “La Téne”. The later La Téne period is characterised by the quality of ornamentation produced. The Celtic museum in Hallein in Austria provides valuable information and artefacts on the Celtic period. The Celtic language would have similarities to the Irish language. However, the Celts did not employ writing, and the Ogham writing used in Ireland was developed in the early Christian period.

The origin of the word ‘Calculus’ is from Latin and means a small stone or pebble used for counting.

Modular arithmetic is discussed in chapter seven.

Augustus was the first Roman emperor, and his reign ushered in a period of peace and stability following the bitter civil wars. He was the adopted son of Julius Caesar and was called Octavian before he became emperor. The earlier civil wars were between Caesar and Pompey, and following Caesar’s assassination civil war broke out between Mark Anthony and Octavian. Octavian defeated Anthony and Cleopatra at the battle of Actium, and became the first Roman emperor, Augustus.

The origin of the word ‘Moor’ is from the Greek work \(\upmu \upupsilon {\text{o}}\uprho {\text{o}}\upzeta \) meaning very dark. It referred to the fact that many of the original Moors who came to Spain were from Egypt, Tunisia and other parts of North Africa.

The Moorish influence includes the construction of various castles ( alcazar ), fortresses ( alcalzaba ) and mosques. One of the most striking Islamic sites in Spain is the palace of Alhambra in Granada, and it represents the zenith of Islamic art.

The Catholic Monarchs refer to Ferdinand of Aragon and Isabella of Castille who married in 1469. They captured Granada (the last remaining part of Spain controlled by the Moors) in 1492.

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O’Regan, G. (2021). Mathematics in Civilization. In: Guide to Discrete Mathematics. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-81588-2_1

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