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 :)