CAM180/AAM180
History of Mathematics General Elective
Semester One AY 2005-2006

A list of suggested mini-project titles is given below.
Or you are most welcome to suggest your own title.
You may work singly or in pairs but not in larger groups.
Please email your choice/suggestion to shutler@nie.edu.sg
You are also welcome to email me for advice e.g. if you have
a rough title but would like some help firming it up a bit.

Some of the mini-projects below could sustain two people, or two
pairs, working on the same mini-project because the scope is broad,
but in many cases the lack of books in the library make it difficult
for more than one person or pair to be working on the same title.
[As people express their preferences I will enter their name(s) against the
titles, but note that it will not necessarily be first come first served!]
If too many people want to do the same mini-project, or the titles
are too similar, I will contact you via email and try to resolve
the clash in the most amicable way possible.

1) To what extent is Mathematics influenced by Culture?[Eileen Poon][Ivy Ho]
Is Mathematics a collection of absolute truths independent of culture, or have different people developed different kinds of mathematics depending upon their socio/economic/cultural context? For example, many of the Egyptian geometrical formulae were strictly speaking "wrong" but were almost as accurate, and much more practical, than our modern "correct" formulae.

2) Did the Egyptians know Pythagoras' Theorem?
Is it possible that the Egyptians did know Pythagoras' Theorem, but that they could not write it down in their papyri because they lacked the algebric symbols needed to express it? Their architectural achievements were astounding, and suggest that they knew perfectly well how to construct very accurate right-angled triangles.

3) Would children understand modern numeration better if they learnt Egytian numeration first?
[Shaoyang & Thilak]

Some of the manipulations which we use in our modern number system, such as borrowing and carrying, are quite difficult to explain because the explanations themselves have to be expressed in the very same number system. Would the Egyptian numeration system provide a convenient place to step back to, so as to get a better perspective on place value?

4) What is Plimpton 322? [Ng Hanjun & Tan Wei Ching]
Plimpton 322 is clearly a mathematical table of some sort, but what exactly is it tabulating? There is reason to believe that the missing portion contains the answer, but how can we know what it says if it is missing? There are in fact two rival theories, and if the missing portion were recovered (it still exists somewhere) that would settle one of the longest running controversies in Babylonian Mathematics.

5) Why did the Babylonians choose base sixty? [Huang Weiting]
Was it something to do with the hexagon? Or perhaps to do with calendars and seasons. We know that the Babylonians were very capable astronomers. Does the measurement of time have anything to do with it? Come to think of it, why do we use base ten? How many other choices of number base are there/were there? Why did any of these get chosen in the first place? Is any particular choice necessarily the "best"?

6) Is our modern (Hindu-Arabic) numeration the best?[LIM ENG TANN]
How many other different numerations are there? (Note that this is really a different question from the choice of number base). Did they evolve one out of the other, or arise separately. How did Hindu-Arabic numeration evolve? Is it really that different from say Babylonian numeration? Could modern numeration be improved?

7) What was Euclid's "The Elements"? [Gay Yu Ting & Zhuo Ziwei]
How do we come to posess copies of it? What purpose was it intended to serve when Euclid wrote it? How much do we know about Euclid? What impact did it have on the subsequent development of Mathematics researc? What impact did it have on the way in which Mathematics is taught?

8) What was Archimedes "Method"?
Why did he go to such great lengths to conceal it? Why did he decide to reveal it near the end of his life? Why were all original copies lost? To what extent did subsequent mathematicians have to "rediscover" the method for themselves? What would the world be like today if Archimedes' "Method" had not been lost?

9) To what extent are we taught by Euclid at school but by Archimedes at University?
Until quite recently (even within living memory) the teaching of Mathematics in schools was dominated by Euclid's elements, yet by the time we reach University, or even when we start to study Calculus at A-level, we have to adopt a completely different approach i.e. infinitesimal analysis, in order to make progress, especially in the physical sciences. Is this a good thing or a bad thing? To what extent have recent educational reforms corrected this problem? Or have they just disguised it by removing explicit mention of Euclid from school textbooks? Could it be argued that much of the school geometry taken from Euclid's "Elements" is hard to motivate because the purposes which Euclid had in mind have now been deleted from school syllabuses?

10) What is the Parallel Postulate?
To what extent was Euclid's "Elements" based on the project of reducing all of Mathematics to simple truths which no one would doubt? Was he successful in this? Why did the Parallel Postulate not fit in with this scheme? How many great Mathematicians of history tried their hand at resolving this problem? Who ultimately was responsible for the solution? How did this give rise to non-Euclidean geometries?

11) Where does the notion of "proof" originate?
The Egytians and Babylonians appear not to have had a notion of "proof" but the Greeks made a very big deal about it. Were they right? Are there an alternatives to "rigorous proof"? Can we always prove everything? Is "proof" always a good thing, especially in an educational context? What is the significance of G"odel's Theorem?

12) Origin and computation of the value of Pi.
Where does the constant Pi originate from? Did the Egyptians really know of its existence in the way we understand it now? Did the Babylonians? Why does Euclid's elements not mention it? What motivated Archimedes to compute its value? Have there been better values since (i.e. more accurate approximations)? What kind of number is it anyway? e.g. it is certainly "irrational" but is it more than this? Why have people computed its value to a million decimal places? What did they hope to find?

13) Where do fractions come from?[Yee Kheng Wah]
The Egyptians were probably the first to have a notion of "fraction" e.g. 1/2, 2/3, etc. but their system was excruciatingly awkward to use in practice. Why? How did later Mathematicians improve on the situation? When did our modern fractional notation first arise? What about "decimal" fractions?

14) Where do negative numbers come from?
What problem were Mathematicians trying to solve for which negative numbers were a useful tool? Do negative numbers really exist in their own right, or could you argue that they are just a construct of mathematical notation? Did the original dispute over their existence run along these lines, or were there other issues at stake? In fact where does most of our modern algebraic notation (e.g. +,-,x, etc.) come from?

15) Where do complex numbers come from?
Most people think of the formula for the solution of the quadratic equation, but historically speaking this is incorrect. It was not until Mathematicians tried to solve the cubic equation that coplex numbers really came into their own. Even then many Mathematicians would not accept them, hence the name "imaginary numbers". Why?

16) How did ancient Mathematicians compute square roots?
Or cube roots? Or even reciprocals? In fact how did ancient Mathematicians perform any of the numerical computations which these days we simply punch into our electronic pocket calculator without a second thought? What about Mathematical Tables? How were they computed in history? Or in the Modern World (i.e. 16th-20th Centuries) Have we abandoned mathematical tables entirely? (no! not even the modern microprocessor!!)

17) Where do trigonometric functions come from?
Even just the etymology of the names "sine", "tangent" "secant" can tell you a great deal. Who first discovered them? Or was it a gradual process? What were the specific problems which motivated the Mathematicians who worked on them? How were the first trigonometric tables computed? where do our modern trigonometric formuale come from? (e.g. sin(A+B)=sinA.cosB + cosA.sinB ) How does your electronic pocket calculator compute triginometric functions? What is the difference between degrees, radians, and gradiants?

18) Where do logarithms and the exponential function come from?
Were they discovered by many people or just one person? What was the specific problem which logarithms were designed to overcome? How did people try to overcome these problems without logarithms? (e.g. "prostaphaeresis"). How was the first table of logarithms computed? What is the differnce between "natural logarithms" and logarithms to the base ten? Why are natural logarithsm so "natural"?

19) Who built the first computer?[Imam Kartasasmita]
How do you define "computer" anyway? If it has to be "electronic" then the first computer was constructed during World War II. What was it used for? How did it differ from a modern PC? If a "computer" can be mechanical, however, then the first "modern" computer was constructed almost 100 years earlier by Charles babbage. What did he build his computer for? It could even be argued that the ancient Greeks invented the first mechanical computer, if you define it to be any mechanical device designed to solve a mathematical problem.

20) Where does "standard deviation" come from?
What problem was it created to solve? Where do our modern ideas about statistics come from anyway? Does standard deviation have any flaws? Are there better alternatives? It could be argued that contemporary uses of statistics, especially in the social sciences, have tended to obscure the origins of statistics, i.e. in the physical sciences, and that many of the problems associated with teaching statistics could be resolved if it were put back into its original context.


THE END :)