Course Reflection

The course started with a discussion on “why do we teach mathematics history?” and we are ending with a group art project presentation on a historical mathematics topic not covered in the course. This course is an interesting guided journey that gradually answers the question of why we should teach mathematics history.

In terms of developments, we come to understand the hard works and smart ideas of the mathematicians of the long past. We come to feel the details of how ideas were developed, and the difficulties faced by ancient peoples. Our story of mathematical development started around Mesopotamia/Babylon (today Syria, Iraq, Iran, Eastern Turkey), Egypt, and then Greek, Roman, and Islamic empires, Asia, Europe, and other places. In terms of the time period, we have attempted to cover perhaps four thousand years. If the set of all known and unknown mathematicians who have contributed to our body of knowledge is a bird, the journey of this bird is thousands of years in time and tens of thousands of kilometers in width. The journey is filled with progress, stagnation, difficulties, improvements, and multitudes of applications both happy and sad. Yet, the journey is probably still at the beginning. The bird has to continue flying into the long future.

In class, we wrote blogs out of readings, imitated ancient methods, developed ideas based on old mathematics. Understanding some details of the development certainly will help us gain more maturity in basic mathematics. This in turn, together with our continued study, would assist us in assisting our students to become responsible citizens equipped with necessary skills. Perhaps, few of our students would become contributing mathematicians. 

The course has a lot of room for flexibility. In addition to numbers, geometry, and algebra, ancient people also developed logic and logical reasoning which are inseparable from math and everyday matters. Logic is an unbiased method of reasoning toward a conclusion that can be tested. We can also learn how mathematics has been applied in various industries. In fact, mathematics was responsible for not only the building of civilization but also for warfighting, injustice, and environmental destruction. Educators’ goals in teaching math should include “peace”, “justice”, “diversity” and “environment”. In democratic education, we should enjoy the freedom to discuss what happened (both good and bad) in the past with respect to the use of mathematics (technology). There are plenty of lessons to learn from. By understanding and knowing all forms of the application, we would be more concerned about our future, our students’ future, and the future of human beings on this planet Earth. We can pass on this concern and need for development toward a peaceful society to our students. Overall, as a class, we overcame the obstacles given by the current situations. The flow of the course is great and I have witnessed many incredible projects/research from my peers. I would like to say thanks to Susan who tried her best to ensure the quality of this course (offered online) as well as her support and valuable feedback. I wish everyone a safe Christmas and a happy new year.

Assignment 3 Reflection

 

It was a pleasure doing this research. Time is one of the most important variables in this universe. Keeping track of time accurately has been therefore crucial to people's day to day lives. The sun is the source that provides life on this planet. Due to its property of being steady and consistent (relatively speaking), the sun is what we can count on to understand time. Hence, we looked into the history of time telling among several geographical locations. The sundial from different places may look different and function differently. That is why it was fascinating to see different perspectives. We are absolutely impressed by the genius of our ancestors. And making a sundial can be used as an activity in a high school classroom! We did run into some technical issues when putting together the art piece. Since we are doing this virtually, it is really hard to exchange and edit drawings.  It would be much easier if we can draw on actual paper or a poster. However, this is a good learning experience for all of us because we will likely get benefit from this in the future.     

Assignment 3: History of Sundials

Our group chose to represent our topic, the history of the sundial, through this piece of drawing. This is because sundials are usually artistic in their designs, and visual representation can easily differentiate the various types of sundials. In history, many nations have individually developed and used sundials to keep track of time. Since there was no direct linkage between all the nations in using sundials in history, we have decided to combine all our findings together in one drawing.

In this drawing, we have put the large sundial in the center with cardinal directions pointing at the geographic location of different regions. Although being a sundial, the large sundial tells a different story from the time. We are focusing on the history of sundial in ancient China, ancient Greek, Renaissance Europe, and Medieval Islam. For each region, we put down the most typical representative sundial used in the era by the mentioned nations.  On top of that, we are representing our findings with drawings that we think are symbolic of the history and development of corresponding sundials. We decided to place the sun at the east where it rises, and the shadow of the gnome separates the three regions that we are going to introduce in detail. 

In teaching, we can show this drawing to the class, and ask students to discuss the history and relations to the given topics. The topic of sundials can be used to explore how trigonometry was used to tell time and improve the accuracy and precision of sundials from different periods of time. This artwork can also be combined with geography or physics classes where it is relevant. We can also include a hands-on activity in class to engage students in making sundials. 

 

Group Project: History of Sundial (Draft)

 Topic: The history of sundial

Art format: One painting and one hand-made sundial 

Reference list:

[1] 2,000-year-old sundial unearthed in southern Turkey's Denizli, Daily Sabah, 20 March 2020

[2]: Archaeologists find Bronze Age sundial dating back more than 3,000 years Ancient Origins, 07 Oct 2013

[3]: Sundials: An Introduction to Their History, Design, and Construction From Hands-on history, a resource for teaching mathematics,  2007 J. L. Berggren, Simon Fraser University

[4]: Ancient Chinese Sundials Kehui Deng, 2015

[5]: A brief history of time measurement Feb 2011, University of Cambridge, By Leo Rogers

[6]: Short history of sundials European association for astronomy education

[7]: The mathematics of sundials Australian senior mathematics journal 22(1) Jill Vincent University of Melbourne

[8] http://cultureandcommunication.org/deadmedia/index.php/Sundial  (sundial timeline)

[9] https://equation-of-time.info/sundials-with-shaped-styles

3 things from "An Introduction to The Mathematics of The Golden Age of Medieval Islam"

 1.  “In the tenth century the scholar al-Sijzī, writing from an unnamed locality, complained that where he lived people considered it lawful to kill mathematicians. (Perhaps this was because most mathematicians were also astronomers, and hence astrologers.)… … mathematicians and astronomers in Islam could expect both honor and support… … ’’.

This is interesting to know. I think the development of any subject would require support from various channels. I cannot believe that during the tenth century, it is considered lawful to kill mathematicians. People at that time relate mathematicians to astronomers to astrologers to fortune tellers and then perhaps to witchcraft. This indicates that the general public at that time was scarcely exposed to any type of science, even the policymakers (who made it lawful to kill mathematicians). On the other hand, I do see that many examples of great mathematicians died either prematurely or from unnatural causes, namely Fourier, Archimedes, Galois, Gödel, Cardano, Abel, Ramanujan, Riemann, Ramsey, and more. This makes me wonder about the reasons for such a visible trend because mathematics is considered the backbone of all science subjects.   

2. “Al-Khwārizmī’s achievements in geography earn him a place among the ancient masters of that discipline… … Al-Khwārizmī’s contribution went beyond this to assist in the construction of a map of the known world … … Among al-Khwārizmī’s achievements in his geographical work The Image of the Earth were his correction of Ptolemy’s exaggerated length of the Mediterranean Sea and his much better description of the geography of Asia and Africa’’.

 I always know that Al-Khwārizmī contributed hugely to mathematics and astronomy, however, I never knew that Al-Khwārizmī also made a huge contribution to the subject of geography. It makes total sense since math can be essential to geography because map-making involves relating spherical geometry and plane geometry. Even small tasks such as mapping a portion of the surface of a sphere onto a plane need theories to support and needs time to verify (for example by travelers and sailors) the precision and accuracy. Having an accurate map at that time would strategically put the ruler/caliph at an advantage when expanding/controlling his/her territories.   

3. “‛Umar is admired more as a poet than as a mathematician, and yet his contributions to the sciences of mathematics and astronomy were of the first order… … he was able, in 1079, to present a plan to reform the calendar then in use… … and produced a length for the year closer to the true value than does the present-day Gregorian calendar”.

‛Umar al-Khayyāmī is the only poet–mathematician I have ever encountered in the literature. I think normally people don’t associate something so sentimental (such as poetry) with something so rational (such as mathematics). I guess what poetry and math have in common is the requirement of being imaginative and creative. Mathematics is not just crunching numbers and doing computations, it is the way for which “logic” expresses itself. Similarly, poetry is how a poet expresses him/herself.

Another thing I was surprised about is that ‛Umar al-Khayyāmī produced the length for the year closer than the present-day calendar. This was quite impressive since it was almost 500 years earlier than the Gregorian calendar!!   

Trivium & Quadrivium

 Three quotes from the article that made me stop and that surprised me in some way.

(1) “Logistic was practical and utilitarian, a study for children and slaves; logic was a liberal art, a study for free men” (Dorothy V. Schrader , 2018, Page 266).

 This tells us something about the social classes of medieval Europe. The slaves (I guess peasants and other workers) worked to produce food, did the construction and maintenance of buildings, etc. These workers studied logistics. The free men studied logic and arithmetic not for the purpose of practical works but philosophical.

Those who prayed (clergy), those who fought (knights), those who worked (peasants)

According to Wikipedia, the period between 1000AD and 1347 AD was an expansion of the population. By estimate, the population grew from 35 million to 80 million. Around 90% of the population was rural peasants, many of them settled into villages called manors. Peasants paid noble overlords rent for places and for services. So one wonders who went to the schools or universities to study logic and arithmetic to know some properties of numbers, proofs and some formal demonstrations, etc (Dorothy V. Schrader , 2018, Page 266)? Meanwhile, good nutritious food production by peasants might be a contributing factor in population expansion.

 (2) In the first arithmetical period (5^th to 10^th century approximately) , the emphasis was on the art of computation especially on the method of establishing the date of Easter (Dorothy V. Schrader , 2018, Page 267).

 Free men went to study logic in the lower level, and then arithmetic in the upper level. Almost all they did, in terms of applications, was to learn how to correctly calculate the date of Easter.  The date of Easter was one very important matter during that period. Many important days of the Christian Church were dependent on the date of Easter. We can imagine the difficulties these free men faced partly due to the Roman numeral system they were using. 

 Only in the second arithmetical period (end of 10^th to end of 12^th century) all four arithmetic operations on abacus were possible due to improvement made by Gerbert (Dorothy V. Schrader , 2018, Page 269). Compared to the scribes doing math on base 60 positional system in Babylonian period more than 2500 years earlier (~ Hammurabi in old Babylon, 1700 BCE), it seems that development of mathematics went through several darknesses.

 (3) The third arithmetic period (end of 12^th century to the end of middle ages ~ late 15^th century) was one of great activity and great change, an almost intellectual revolution in Europe. The Hindu Arabic number (positional decimal number system used today) was introduced and zero was added to the previously zero-less world. It was possible by translation into Latin from Arabic and Syriac (old Syrian language) (Dorothy V. Schrader , 2018, page 267).

We have learned that, by evidence, positional system with base 60 was used in Babylon of Mesopotamia (current Iraq) in 1700-300 BCE (Victor J. Katz, 2009, page 10-12).  Greeks used this base 60 positional system for astronomy but it is not clear if they used this system in other situations. Even though earlier ancient Chinese used base 10 system, the clear evidence of positional system including zero in China came from 12^th century (Victor J. Katz, 2009, page 198). In India, Syria and Cambodia earliest evidence of the use of decimal place value system came from 7^th century.  Meanwhile, until the third arithmetic period, European were using the cumbersome Roman numeral which requires a great number of symbols to write a large number. Calculations with Roman numerals are difficult.

 It took more than 2500 years for human being to finally adopt the positional numeral system after first  discovering it in Babylon. If the positional numeral system was a bird species, it was an amazingly difficult journey dotted with deaths and rebirths for that species to spread its seeds around the world for humans to make good use of.

Reflection on Assignment 1

 In our first assignment for this course, we talked about Babylonian arithmetic using base 60 sexagesimal system. We learned about the history and Babylonian way to multiply and divide in base 60 system as well as the modern way interpretations of multiply and divide in the sexagesimal system. Overall, I think the organization of our presentation was reasonable. We covered some basics at the beginning so that audience have good background for the later content. One thing we would consider to improve in the future is to include a slide for summary and wrapping up. We shouldn't have ended our presentation abruptly. We had rehearsed several times before the presentation to make sure we had good pacing. We didn't use much of materials from the media but we did have activities to engage the class with some interactions. I would say that we did a satisfactory job and we will take what we have learned for the betterment of our future studies. 

 

Number with personality

Ramanujan considered that each of the positive integers was one of his personal friends. According to Major’s paper, this type of individuals who have the tendency to put personal characteristics to numbers are thought of having Ordinal Linguistic Personification (OLP). Comparing numbers as his personal friends, Ramanujan associated not only the personalities to numbers, but also possibly ages, genders, looks, and etc. Since OLP is one type of synaesthesia, one can presume that Ramanujan can make connections between many different kinds of mental experience. 

I would consider introducing the concept of linking characteristics and numbers to students at lower grades. The reason is that I have seen young kids who are emotionally scared of mathematics. Some may call this the “math anxiety”. Asking students to give numbers personalities and characteristics might help them “dedemonize” math. For example, a student can think of himself as “Zero”. “One” is the guy living next door. “One” is a weird old man who only makes friends with people who are “primes” such as “Two”, “Three”, or “Five”. “Three”, “Four” and “Five” are living down the street and they formed a band called “P-Triple”. They are proud to be the band with the youngest average age. However, “Six”, “Eight”, and “Ten” are in another competing band called “Nobody is Odd”. The story can go on with more characters and more plots. The point is to make students more familiar with numbers and their properties. Also, some important attributes of math such as creativity and imaginations are promoted in this type of learning. 

As a math student, numbers always means more than just numbers to me. For example, I really like the number “seven”. For some reason, “seven” is disliked by many people. But to me, seven is an interesting number. It is single digit, prime, odd, and “hard to get along with”. However, it is not only a Mersenne prime, but also a double Mersenne prime. It has the highest probability of being rolled with two standard dice. It is, outside of three, the only other dimension that a vector cross product can be defined. And one thing everyone knows, we have seven days in a week!! So, to me, “seven” is like a low-key and odd person that is often underappreciated by the others. Many of my favorite people are like that. Hence, if I have to pick a favorite number, it will definitely be seven.

The dancing Euclidean proof

 1. “embodying mathematical concepts and relationships through dance and movement”

Dance and movement can be done by person(s) with or without an object such as a stick or an umbrella in an environment in which the action is possible. These actions can represent a mathematical or physical process such as drawing a line, a circle, marking a length on the line. Even the speed of a moving particle can be represented by adjusting the speed of the dancer movement. I see a lot of opportunities not fixed to euclidean proofs. For example, we can have two concentric circles with different diameters. Two dancers move along the circles at two different speeds such that they complete one round trip at the same time. The audience can clearly see that the dancer on the larger circle moves faster (greater linear speed) to keep the same angular speed as the other dancer on the smaller circle. 

2. “potential to make the beauty of Euclid’s proofs accessible to mainstream audiences”

Dancing and moving make people feel good, happy, and engaged. By allowing learners to participate or even watch others doing the dance we create a happy learning environment. This is a good benefit of dancing and movement whenever possible in helping people learn through doing something that makes them happy.

3. “Help students understand and appreciate the beauty of Euclid’s proofs in new multisensory, experiential ways”

 

By dancing to construct a mathematical process students are part of the construction tools and objects (pencil, divider, ruler, etc.). Before the dance, students need to plan each move and entire process - that gives the exterior point of view. During the process students manage the movement, direction, magnitude and speed. Therefore they get the whole interior view of the process. 

Explication and commentary on a poem about Euclid

 Poem 1: Euclid Alone Has Looked on Beauty Bare by Edna St. Vincent Millay

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

Speculation

Euclid alone saw the beauty of geometry (or geometry is the beauty) clearly (in his time). Euclid displayed the beauty he saw to the rest of the people. That made people who talked foolishly about the ‘geometry’ going silent. Geometry made people come face to face with the world they live in and not to be selfish in the old way in which they learned nothing. After passing through many different generations, many people still did not understand geometry. But those who understood geometry escaped from the dark world and came to the world in which they used geometry to build civilization and beauty.

The old days before the geometry were primitive and terrible. At the first opportunity of the beauty of geometry in details, Euclid was alone who saw it clearly.

When Euclid first understood the concept of geometry, it was him alone who saw the beauty in details and clear. The event that Euclid managed to compile a complete beauty of geometry happened once, but far away from our time. It was fortunate that people re-discovered the beauty. We have heard the great importance of geometry as a massive sandal set on stone more than 2000 years ago! (metaphor, according to wikipedia Eda was a feminist)

Poem 2: The Euclidean Domain

…Euclid alone

Has looked on beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

—Edna St. Vincent Millay, Sonnet

Euclid alone has looked on Beauty bare?

Has no one else of her seen hide or hair?

Nor heard her massive sandal set on stone?

Nor spoken with her on the telephone?

Proud poets, as you penned your paeans to Beauty,

Did you not think it was your bounden duty

(Though it were one that any might have loathed)

To tell that you have only seen her clothed?

And as you sang praise, Orpheus, of Eurydice,

Your mouth became the orifice of your idiocy!

For Beauty bare you never yet had seen,

’Twixt Hades’ depths and lofty Hippocrene.

O Beauty! Would you, for this mathematician,

Remove (if it would cause to give permission

To look on Beauty bare too great a scandal),

Once only, and then but far away, your sandal?

Speculation

Repeat the second part of the poem by Edna St. Vincent Millay.

Is it true Euclid alone saw the beauty of geometry bare (clearly, completely)?

Has no one else seen her (geometry) hiding or even just her hair? (metaphor)

Has no one heard of the importance of geometry? Has no one heard that geometry set her sandals on stone (and sent out the loud sound wave) more than 2000 years ago?  

Has no one worked with geometry? Has no one spoken to geometry on the phone?(Sarcasm)

You (Edna) were a proud poet. You wrote your poem of praise to geometry (beauty). Did you think that it was your burden (duty) to tell that you only saw the surface of the beauty of geometry? It looked like this is something anyone was unwilling to do.

(Edna might not be trained as mathematician according to wikipedia)

As you sang the praise (Edna’s poem) - at this point David used the comparison with Orpheus mourning, praising his late wife Eurydice (both from ancient Greek mythology) - your mouth became the pipe producing your idiocy (stupidity). Because, you have never seen the bare beauty of geometry. It was between the depths of the Hades (god of the dead) and self imposed inspiration.

O beauty (geometry), would you, for this mathematician (is it David?) remove the event that ‘Euclid or someone discovered you in whole’ (which is taken as the beauty set her sandal on stone with a loud sound) into our world more than 2000 years ago? Would you (beauty) please, for once, go back in time and did not set your sandals on stone? To look at that beauty could come with a great scandal.

Notes: According to wikipedia, Euclid presented the already discovered geometries into one single logically organized work. His contribution seems to be the organization including writing axioms, and putting proofs in the collection.

Eye of Horus and Unit Fractions

 The ancient Egyptian symbol “eye of Horus” (from the sky god Horus who was usually depicted as a falcon) was considered as a protective amulet to the Egyptian. It has been commonly painted on the bows of boats both protected the vessels and "saw" the way ahead. Moreover, the eye is constructed in six fractional parts, representing the shattering of the eye of Horus into six pieces. According to historical documents, these six parts also represent six senses of human. The inner corner of the eye indicates one half, the iris is one fourth, the eyebrow is one eighth, the outer corner of the eye is one sixteenth, and the decorations below the eye are one thirty-second and one sixty-fourth respectively. 

Note: The infinite geometric series with ratio less than one is:

The unit fraction numbers in the Eye of Horus are the first six terms of the geometric series:

Hence, it looks like Egyptian scribes would approximate one using the first six terms of this series because the sum: 1/2 + 1/4 +...+ 1/64 = 63/64.

In Milo Gardner paper (The Arithmetic used to Solve of an Ancient Horus-Eye Problem, 2006), explanations were given on how Egyptian scribes divide one (64/64) by number. In ancient Egypt hekat was a volume unit. One hekat can be considered 64/64 hekat. There are also other volume units such as hin, dja and ro where

1 hin = 1/10  hekat

1 dja = 1/64 hekat

1 ro = 1/320 hekat

Scribes would express the division of a hekat in terms of Horus Eye unit fractions and these smaller units of hekat. Here are some examples from Milo Gardner paper(2006).

In addition to Horus Eye unit fractions, Egyptians also used other unit fractions. All fractions can be represented as a sum of unit fractions (1/n, n is any natural number). According to (Katz, 2008, pg.5) the Egyptians expanded the fraction with non-unit numerators into the sum of unit fractions. Whenever a number of objects need to be equally divided into a number of receivers, this expansion method can be useful. For example, suppose we need to equally divide five loaves of bread to seven people. Then one can expand 5/7 into sum of unit fractions as follows: 

5/7  = 1/2 + 1/7 + 1/14 

This expansion also gives the plan of how to cut the bread. That is each person will get half of a bread loaf plus one seventh of a bread loaf plus one fourteenth of a bread loaf. So it is very interesting to know that some ancient peoples made good use of unit fractions in their daily business when they did not have the luxury of modern mathematics.

Many numbers have special properties, meaning or applications. Cultures are also connected to some numbers. In this Wichita State University website (http://www.math.wichita.edu/history/Topics/snumbers.html), there is some information on special numbers such as perfect numbers (a positive integers that is equal to sum of its divisors excluding itself such as 6 = 1+2+3). 

Amicable numbers are the pairs of numbers with the following properties:

1. Sum of divisors of first number (excluding the first number) equals the second number

2. Sum of divisors of second number (excluding the second number) equals the first number

An example is 220 and 284. Sum of divisors of 220 is:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

Sum of divisors of 284 is:

1 + 2 + 4 + 71 + 142 = 220.

There are also special numbers in cultures. Ancient Greeks are said to have believed in four elements (earth, water, air, and fire). Native American culture also talked about four directions (east, west, north, south) from where the wind came, and each of these winds had accompanying stories.

Constructing a magic square

 

Three by three magic square: Each square has a number from 1, 2, ..., 9 used once. Sum along any row, any column, and diagonal are all 15. What are the numbers in each square?

                                                                        1 + 2 + ... + 9 = 45

Write the square as 3 by 3 matrix.

The method of 'false position'

 The method of false position is suitable for solving an equation with one unknown. Usually the equation contains proportionality w.r.t. the unknown. Chinese dish problem discussed last week can be solved by this method.

Response on " Was Pythagoras Chinese?"

•  Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

Since I was born and raised in China before coming to Canada at age of 12, the foundation of my mathematical knowledge was learned in China. This includes arithmetic operations, integer operations, fraction operations, solving algebraic equations and system of equations. Because of the difference in mathematical curriculums, I was ahead of my class for several years since grade 7. In Canada, I learned pre-calculus in high school and more advanced mathematics in university. Hence, I have the experience learning math from two different cultures.

In China, elementary school math is taught with a blend of Chinese and Western sources. I remember learning how to use an abacus and we have our own version of Pascal’s triangle we call it “Yang Hui triangle”. Also, as mentioned in Gustafson, we learned right triangle theory as “gou-gu” theorem. Nonetheless, we also use Arabic numerals, and we call our variables x, y, z. Even though we have Chinese names for sine, cosine, and tangent, we write using the Western notations in problem solving. As a matter of fact, using various sources of mathematics have little impact on student’s learning. What made a huge difference between Canadian school and Chinese school is the way of teaching the knowledge. As mentioned in Gustafson’s work, memorization serves as a ground for knowledge building in Chinese education. Students are required to memorize definitions, theorems, as well as ways to solve problems. Moreover, no calculator is ever allowed in exams. In my opinion, this way serves well on building a good foundation but when students become more mature, I’d like being taught by Western approach more. When in advance level, students should use tools to aid achieving the purpose of understanding. Time should not be spent on doing arithmetic by hand, yet knowing how to do arithmetic by hand is also important. 

•  What are your thoughts about the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle...check out its history.)

I would say, the way of naming mathematical theorems after the founder is sometimes bit misleading. Nowadays with the advancement in technologies, the development of knowledge are watched closely by academic society across the world. It would be easier to identify the source of a new idea. However, when communications cannot be done efficiently in the past, the validity on awarding one (or more, for co-founding) person for the contribution is questionable. Different people can independently come up with breakthroughs (e.g. Newton and Leibniz). In my opinion, the naming should be referring the knowledge instead of referring the person who come up with it (e.g. intermediate value theorem). Of course the acknowledging of the inventor is also important, but that can be done separately. Another benefit for not naming after a person is to remove the culture bias and barrier in knowledge. For example, when we look at the theorems in math courses, we get the Eurocentric feeling in mathematics because most of the theorems are named after European mathematicians. However, mathematics has been developed across different countries and cultures for quite a long time. Instead of fighting for who did it first, people should pay more attention on the evolvement of the knowledge. 

Babylonian word problems

 Word problems can be both practical and imaginary (pure math). For example, the first problem in our reading is “I have added seven times the side of my square to eleven times its area, and it is 6;15”. If we think from plane geometry point of view with assumption that the numbers ‘seven’ and ‘eleven’ do not have dimension (unit-less), then we would say there is no practical value.

7(side of square) + 11(Area of the square) = 6;15

It is because the equation representing the statement does not satisfy dimensional equality to be practical. The left hand side is the sum of two objects with the first object in dimension of [length] and the second object in the dimension of [area] or square of [length]. Some people would treat it as a puzzle. In fact, the term ‘square’ is not necessary. ‘Square’ and ‘side’ can be replaced by ‘circle’ and ‘radius’ or, more generally, ‘f(x)’ and ‘x’. Thus, if we use symbolic algebra, it can be written as:

7x + 11 f(x) = 6;15

In this statement the function f can be an abstraction such as a measure of an n-dimensional structure built from a component object of value x. We can further generalize it by replacing the constants with other objects (words, parameters) such as:

ax + b f(x) = c

On the other hand, we can assign the length dimension (7 meter, 7 feet, etc.) to the number seven; assign no dimension (unit-less) to the number 11. Then the problem becomes “the area of the rectangle made of seven meter side and the side of my square added to eleven times my square is 6;15”. 

The problem now has a practical value. Therefore, when we talk about a word problem or a set of word problems such as writings on clay tablets found in Babylon archaeological extraction from the so-called lens of ‘practicality’, ‘generality’, or ‘abstraction’ while sitting at our time, we would see that they are connected or we can build the connections. In other words, in the language of contemporary (symbolic) algebra in which we have a store of not only natural language but also mathematical symbols, we see the connection. We can see them separate, and we see them together because we can specialize, generalize, and replace some objects with abstract objects. Any applied problem can be generalized, then add abstraction to it. When we studied ‘vector space’ (abstract idea) in linear algebra, we could also transform in opposite direction.

 However, if we can transport ourselves to the time of Babylonia, what would be the case? We would not have symbolic algebra. We could start from ‘generality’. Generalization is possible using words. The above same example can be reworded into: “I have added seven times the size of a side of my object to the eleven times the size of my object. It is 6;15”. Here the word ‘object’ can be square, circle, cube, etc.

For abstraction without symbols, it is possible but it would be limited by the skills in using words and sentences, and experiences either real or imaginary. In above rewritten word problem, we could build the ‘object’ as an abstract object. In “Babylonian algebra (Crest of the Peacock)” we learned that Babylonians did power of four and eight even though they could only have power of two (area) and three (volume) in applications. Therefore, it is safe to assume that limited abstraction is possible without using symbolic algebra.

By this discussion, we find that by learning the history of mathematics (Babylonian and Egyptian), we are studying applied and pure mathematics of those days, and we could learn how generalization and abstraction can be possible and to what extent without using symbolic algebra.