The Fascinating Plimpton 322

Mathematics is one of the most powerful tools in human existence. From basic counting and grocery shopping to advanced applications such as in computing and aerospace, math plays important roles and can be found everywhere. However, the math system was not built in one day. All the current have their past. Believe it or not, some archaeological findings in mathematics are still fascinating to us today, even though they were from hundreds or even thousands years ago. The clay tablet named Plimpton 322 is one of these valuable and interesting findings (Fig. 1).

Figure 1. The clay tablet Plimpton 322[1].

Plimpton 322 is a great presentation of how advanced Babylonian mathematics was. The clay tablet was written around 1800 BC. The math represented on the table is now known as Pythagorean triples. It was long before the development of Greek mathematics. The clay tablet was purchased by George Arthur Plimpton from an archaeological dealer in 1923. It now lies in the G.A Plipton Collection at Columbia University numbered 322[2]. This is also how the tablet was named Plimpton 322.

The table on the tablet was inscribed with four columns and fifteen rows. In the rightmost column, which is the 4th column, it includes the number of rows which is numbered from 1 to 15. Before we jump into the math, we need to know how numbers were recorded and how to translate these symbols to the math we understand. In the table, 1 is symbolized as which also called a “stroke”[1]. From 2 to 9, the numbers were inscribed by combining multiple of strokes. Figure 2 shows how numbers 2 through 9 were written in the table. The number 10 was written as.The same combination fashion works for numbers 10, 20, 30, 40, and 50 (Fig.3). How to write numbers between 10 and 20 or 20 and 30? the number system on the tablet still relys on combining “strokes”. 11 was written aswhich is an addition or combination result of 1 and 10. 0 is recorded as blank space. At this moment, you might think it is the same system as the system similar to the one we use today. However it is not quite. The math system we use now is based on 10. However, the system the Babylonian used was based on 60[1]. It means that if the number is written as, which tansclates to 1;2;1. It mathmatically represents 1*(60*60)+2*(60)+1=3721.

Figure 2. How the numbers from 2 to 9 were inscribed on Plimpton 322[1].

Figure 3. How the numbers 10, 20, 30, 40 were inscribed on Plimpton 322[1].

With the basic translation we just learned, now we can translate those symbols to the numbers we understand for the 2nd, 3rd and 4th columns. For the 4th column, it is listed 1 to 15 for each rows. The numbers here represent the row numbers (Fig. 4). Now we can also understand the 2nd column by using the addition base 60 number system (Fig. 5). You can go back to the earlier mathematical example of calculating 1;2;1 if you have any difficulties. When we have the basic understanding of how numbers are inscribed, we now need to know what these numbers mean or how the math works on the tablet. Archeologists translated the names on the top of each column which offers more information to the later discovery. On the 2nd column it has the word “width” and the 3rd it says “diagonal”. Mathematicians thought it might relate to Pythagorean triples. The results showed the guess was right if you ignore the error the tablet maker had (the corrected numbers are marked as red in Figure 5). For each row, the diagonal d, width w, and the unknown length of the last side of triangle l, make an equation w2+l2=d2. It is also true that all the ls are integers. So now the question remains on what does the 1st column do? It is actually the fraction of d^2/l^2. From Figure 1, every single row in the 1st column consists a series of numbers. The numbers can be translated into decimal numbers. For instance, in the first row where d=169 and l=120. Therefore, the fraction of d2/l2 should be about 1.9834. The first column has numbers 1;59;0;15. How is that 1.9834? On our normal 10 based system, for instance, 1.234 can be written as 1+2/10+3/(10*10)+ 4/(10*10*10). The same method also holds for the tablet decimal numbers. However, the only difference is that the numbers are 60 based. Therefore, 1:59:0:15, which is the decimal number inscribed on the fourth row of the 1st column can written as 1+59/(60)+0/(60*60)+15/(60*60*60)=1.9834.

Figure 4. The 4th column of Plimpton 322 (From the top to the bottom row, number of rows were inscribed which is 1 to 15)[1].

Figure 5. The 2nd and 3rd columns of Plimpton 322 (On each picture’s left side, the numbers on the left are the numbers decrypted from the inscribed symbols, the numbers in blue are the results after addition)[1].

Who made Plimpton 322 and what was the purpose of making it? Eleanor Robson, a Plimpton 322 researcher, believes that it was made by a male due to geographical reason. All the female scribes from ancient Mesopotamia lived much further north to where this tablet was made. Robson also does not agree that the tablet author was a professional mathematician because the professional academic disciplines is a phenomenon of the very recent past. There are two possibilities to the author’s identity which Robson brought up. First is that the author could have been a trainee scribe or a teacher. He might have taught simple math techniques in scribal schools. He also knew the document format used by the temple and palace administrators of Larsa [4]. Therefore, he was not a student. This leaves him highly likely to be a professional bureaucratic scribe. Robson also believes that the tablet served for teaching purposes. The tablet is similar to problem lists. This is likely to explain why the l numbers are missing on the tablet [4]. They are all integers and waiting to be solved from the known values. The fifteen rows with the same method also indicate that the teacher might offer this tablet for students to practice repeatedly on those problems.

Now the math puzzle of the ancient tablet Plimpton 322 is finally solved. However, the exact answers for how the tablet was created or who was the author still remain elusive. It is fascinating for us to know how incredibly advanced Babylonian math was back into the time around 1800 BC. They knew Pythagorean triples which today is still super commonly used in math and its fields of application. We can clearly see that the history of mathematics is also a part the history of human existence. Mathematics was created, developed and is still developing and helping human beings to reach new and more advanced eras in human history.

Reference:
[1] Casselman, Bill. “The Babylonian Tablet Plimpton 322.” The University of British Columbia. Web. 8 Feb. 2015.
[2] Neugebauer, O.; Sachs, A. J. (1945), Mathematical Cuneiform Texts, American Oriental Series 29, New Haven: American Oriental Society and the American Schools of Oriental Research, pp. 38–41.
[3] Joyce, David. “PLIMPTON 322.” Clark University. Web. 8 Feb. 2015.
[4] Eleanor Robson. Words and Pictures: New Light on Plimpton 322, American Mathematics Monthly 109, 2002.

A New World of Thought in Fractions

I never thought about math as a fluid subject that would be approached differently depending on the needs of a culture.  We are taught from a young age how to do math and what it applies to, but I never thought that in different cultures, in a different time, people’s application of math could be so different from ours.

Let me explore my thought this way.  When I think of a math problem I need to solve on a daily basis, it is usually based around money, or computer memory.  I have five dollars and I want to buy a sandwich and fries.  Do I have enough money?  Or I have 3 GB of data for my phone.  Can I watch my favorite TV show on 4G or do I need to wait until I get home?  Many of the math problems we run into on a day-to-day basis support a monetary society where I have money, so I use that money to get more things.

This is starkly different from what we learned of different cultures in class.  The culture in particular I think of is Egyptian culture.  From the book, Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics, I learned that in ancient Egypt they had a very different fractional system than we do.  When we think of fractions we have a denominator and a numerator.  I like to think of the denominator as a bucket with n places for an object. The numerator is how many objects we have in the bucket, or how full it is.  In Egypt they had a very different system.  They represent fractions as a single number with a line over it.  The number represented an amount and the line indicated that it was a fraction.  It is a little foreign to us, but if the Egyptians wrote 6, with a line over the six, then we could think of it as 1/6.  It is important to note, and I will discuss later that this system can represent any fraction.  It would just take a bit more work in order to do that.

Why did they do it that way and why do we do things the way we do?  I think in our society our number system fulfills our desire to fill and make sure we have enough.  Our fractional system is very supportive of the money we use, because money is how we support ourselves. To elaborate on this I will use my bucket analogy for fractions. If you recall I said the denominator in our fractional system is like a bucket with room for n amount of objects.  The numerator tells us how many objects are in the bucket.  If the numerator and denominator are the same then we have filled one bucket. We use this all the time when we go shopping food.  When we shop the denominator is the price of what we want to buy.  The numerator is how much money we have.  If we have enough money, which represents the numerator, to fill the price, which represents the denominator, then we can get that item.  Or using my analogy you could say we filled the bucket.

In ancient Egypt their fraction system also could have supported their means of commerce.  They could have had more of a barter system were you worked for food and your master or boss gave you a portion of what you helped him produce.  For example if ten workers made 12 loaves of bread the boss possibly would have taken four for himself, leaving eight for the workers.  Using the Egyptian fractional system he could quickly and efficiently think, I have eight loaves for ten workers.  I will divide 5 loaves into halves giving me 2, with a line over the 2(1/2), which is, half a loaf for each worker.  Now I still have 3 loaves left.  If I divide those into fourths or 4, with a line over the 4(1/4), then I will have given each worker ½ a loaf and ¼ a loaf and there will be ½ a loaf left.  Know all I need to do is break that into tenths, but since it was a half loaf that would actually be twentieths.  That gives me 2 4 20, all with a with a line over them, which to us would be ½+1/4+1/20.  Now the boss had divided his loaves evenly and easily.

Learning this got me wondering.  How was it that other cultures did math?  Could I find another unique way fractions were used?  In the article, History of Fractions, Liz Pumfry talks about a Babylonian method of writing fractions.  Their method was also very simple and a little confusing.  To explain the Babylonian fraction system I first have to explain that the Ancient Babylonian number system was base 60.  The Babylonians wrote their fractions as demonstrated in the image below.

Image from nrich.maths.org.

The first grouping of symbols is for twelve and the second is for fifteen.  In Babylonian fractions this would come out to be 12 + (15/60).  That is, if this was a fraction.  Because the ancient Babylonians did not have a symbol for zero or a decimal point this grouping of numbers could have other meanings.

 x60 Units Sixtieths Number 12 15 12+15/60=12 12 15 720+15

But why did the Babylonians use this method for fractions?  It seems to me that this method kept their number system simple, but in my quick analysis of this study I have found Babylonian fractions to be limited. In the article Babylonian Mathematics, it says, “Irregular fractions such as 1/7, 1/ 11, etc were not normally not used. There are some tablets that remark, ‘7 does not divide’, or ‘11 does not divide’, etc.” From this it seems we can deduce that the Babylonians disregarded some fractions.  It seems to me this could have limited them in some regards. For example, if they only used sixty as a denominator then the smallest number they could represent would be 1/60.  Let’s say they could increase this, though, using 602 or 60n.  If this is the case they could represent much smaller numbers, but it would still be difficult to represent values such as 1/7 or 1/13.  They might be able to do this by adding different values together, but that sounds very difficult and sloppy.   From this I conclude that they would lose accuracy with their fractional system.

Despite losing accuracy, addition and subtraction of fractions would be made much easier if the denominator was always sixty or a power of sixty.  This would make it easier for them to teach fractions as well as learn fractions.  They would not have to worry so much about finding a common denominator as we do.

It is very interesting to note that the Babylonian representation of fractions is very similar to our representation of decimals.  It is, in fact, so similar that I feel it is important to point out.  Our decimal system is base 10.  If we write 0.1 we are essentially writing 1/10.  This is essentially the exact same as the Babylonian system except they are using base sixty.  This is in fact so similar that it causes us to have the same problem representing some fractions.  If you want to write 1/7 in our decimal system you will quickly realize that this is impossible.  There is no concrete way to represent 1/7 in a base ten system.

It is very interesting to think about these different methods for representing fractions.  The Egyptians, if you think about it, actually had a very strong system that, while confusing, could represent all fractions very accurately.  It also suited their needs very well.  The Babylonians had a system very similar to our decimal system.  It seems foreign to us to use base sixty, but base ten is all we know and if they were confronted with our decimal system they would probably find it similarly confusing.  Our fractional system is, I believe, very strong, but in some ways I look at the Egyptian method and think it could have been very helpful when I wanted to share candy with friends or at the dinner table when everyone is fighting to make sure they get their equal portion of food.

In my studies I could not find any other Ancient Babylonian use of fractions.  I am also left to speculate on their reasons for their fractional system.  It is interesting to consider what a different cultures mathematical systems might have been and speculate as to why they used the method they did.

Sources

http://nrich.maths.org/2515

Count Like an Egyptian: A hand-on Introduction to Ancient Mathematics by David Reimer

http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf

Exploring Limits

In calculus, a limit is defined as the value of a function as it approaches some point. Sometimes, a function has no finite limit at a point because it just keeps growing, and we say the limit is infinite. In this case, the function never reaches the limit but the value grows arbitrarily large as it gets nearer and nearer to the limit. In our reading, I have been considering limits in a different light. I have been thinking about the limits of civilizations as they progress in their development of mathematics. Some civilizations seem to reach a limit of understanding and because of cultural restraints, their limited number systems, or even because they outwardly reject an idea, they stop progressing. Fortunately, sometimes their discoveries shape and influence other cultures and, as a whole, progression continues. I would like to explore different limits in the progress and development of mathematics and consider what limits us today.

Plimpton 322. Image: Public domain, via Wikimedia Commons.

In ancient Mesopotamia, more than 4000 years ago, the Babylonians used the base of 60 to develop a high level mathematical system. They developed positional notation and could use fractions as well as whole numbers. They developed systems to figure square roots. Clay tablets from that time show tables with logarithms, multiplication facts and reciprocal pairs. There is information about calculating compound interest and solving quadratic equations. Writings on the tablets suggest that math was a subject that was taught and studied. In many ways, they seem to have exceeded the capabilities of other civilizations that came much later in history. No one can question that their accomplishments were amazing, to say the least, and perhaps influenced other cultures. However, because most of their mathematics were only for very practical purposes like conducting business, surveying land and constructing buildings, they stopped short of exploring some of the deeper meanings of things. For example, our text points out, “In the Babylonian square-root algorithm, one finds an iterative procedure that could have put the mathematicians of the time in touch with infinite processes, but scholars of that era did not pursue the implications of such problems.” (Merzbach and Boyer, pg. 26) What might have been the implications if they had? As they approached the limit, they stopped rather than exploring the infinite possibilities. They stood on the brink of even greater discovery, but did not pursue it.

Pope Sylvester II. Image: Public domain, via Wikimedia Commons

One of the most dramatic examples of cultural influences limiting the progress of mathematics is the example of the progression of Indian positional decimal arithmetic to Europe. Mathematicians in India had developed a number system with ten digits, including zero, and used it to develop methods of computing fractions, square roots and π. In the tenth century, Gerbert of Aurillac attempted to introduce the system to Europe. He had learned the system first hand from Arab scholars in Spain.   However, he was rejected and during this time of the Crusades in Europe, he was rumored to be sorcerer. He died after a short reign as Pope Sylvester II. “It is worth speculating how history would have been different had this remarkable scientist-Pope lived longer” (Bailey and Borwein, 6).” The Indian system was reintroduced 200 years later by Leonard of Pisa, but was rejected again and considered “diabolical”. It wasn’t until the beginning of the 1400’s that scientists began using the system. “It was not universally used in European commerce until 1800, at least 1300 years after its discovery” (Bailey and Borwein, pg. 6). While many other areas of the world were able to do complicated computations using the Indian system, Europe, because of its cultural restraints, was still laboring with Roman numerals. Imagine what the brilliant minds of the Europeans might have discovered or developed if they had the ease of the Indian number system? In this case their culture may have created a limit that kept them from infinite discoveries.

Today in our world we have amazing tools to help us progress. Not only do we have the combination of a well-developed number system, thousands of theorems and laws and the knowledge of centuries of learning, we also have technology that assists in remarkable ways. Indeed we have all the tools of the past plus the technology of our day. However, are there things yet to be discovered, or have we reached a limit? Are there obstacles in our society or ways of thinking that limit us? As recently as the early 1900, women had a difficult time pursuing their mathematical interests. Even today, women and minorities continue to be underrepresented in the math and science fields. What might have been the result if woman had been afforded the same educational opportunities as men over the years? Do we limit ourselves by the way we approach math? Are there different number systems or “languages of math”? In recent years, computer scientists have given us other “languages” for coding. Are there similar languages for math? The challenge for our day is to not be content and accept that what has been learned is all there is.

In our reading for class I have been amazed at how often a group or civilization is on the brink of great mathematical discovery, but because of varying reasons they stop short of the mark. Sometimes cultural influences limit the progress and other times it seems individuals do not look far enough to find deeper meaning or answers. It is true that hindsight may be twenty/twenty, but I can’t help wondering what future civilizations may look back on and see that we barely missed. What are we on the brink of discovering if only we would look forward and push closer and closer to the undefined limits?

Sources:

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Bailey, David H., and Johnathan M. Borwein, “The Greatest Mathematical Discovery?,” 2011.

Faith and Math: the Origins of Math

Religions. Image: Randall Munroe.

Religion and math are oft thought of as being separate and often in opposition, at least within western society. We recently learned about the connections between math and religion in India (http://www.bbc.co.uk/programmes/b03c2zvr) but did not explore where else faith has had an impact on mathematics.

Where does math come from?

The main two answers to this are as follows: humans discover math, or humans create math. In the case of the first, it is accepted that all of math exists, has existed, and will always exist, regardless of whether or not we are aware of it. Even though the ancient Greeks were unfamiliar with negative numbers, negative numbers existed, but had simply yet to be discovered. This mode of thought is described as mathematical realism, and can be defined as the belief that our mathematical theories are describing at least some part of the real world (http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf pg. 36). There are several subdivisions among this group and more detail is given to this later. The second statement, that humans create math, is characteristic of mathematical anti-realism. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true.

The Realists

The realists should probably be subdivided into two main groups: Platonists and everyone else, with the “everyone else” being a minority, so we should probably have a definition for mathematical Platonism. According to both Stanford’s and the internet’s encyclopedias of philosophy, mathematical Platonism is based on the following theses: Existence, Abstractness, and Independence. Basically, mathematical objects exist they are also abstract, and your language, thoughts, religion, or anything else doesn’t change what they are.  I should probably also mention that there are also subcategories amongst the Platonists, like traditional Platonism, full blooded Platonism, and some others, but I don’t want to get into that.  There are, however, mathematical realists who do not subscribe to Platonism. One such group is the physicalists. A strong proponent of physicalism was John Stuart Mill. The argument for this is that math is the study of ordinary physical objects and is therefore an empirical science. According to this mathematics is basically meant to discover laws that apply to all physical objects. For instance, 1+1=2 gives us the law of all physical objects that when you have 1 of the object and you and another of the object you have 2 of the object instead. This differs from Platonism in that these objects are no longer abstract, but rather describe all objects. These are not the only two categories of realists. The main problem I have with this is that it means if all objects were to vanish math would cease to be true. This is because physicalism is not based on the abstractness of mathematical objects which means that the objects themselves must exist.

The Anti-realists

Anti-realism is in general the belief that Math does not have an ontology. As with mathematical realism there are a lot of different subcategories of mathematical anti-realism. I’ve chosen to talk a bit more about conventionalism and fictionalism because they seemed interesting.

Conventionalism holds that mathematical statements are true only because of the very definitions of the statements. By this mode of thought, math does not necessarily have any connection to the real world; it exists because we create it and it is true because we have made it to be true. The statement “pi is the ratio between the circumference of a circle and its diameter” is true only because we define a circle as being a shape with a radius r, a diameter 2r, and a circumference 2*pi*r, and not because the universe made it so. In this sense, the above statement makes about as much sense as “all bachelors are unmarried”; both are obviously true, however this is because of their definitions rather than being the result of some universal laws.

Fictionalism argues that statements like 1+1=2 make about as much sense as “Harry Potter’s owl was Hedwig”. Yeah it’s true, but only within its given context. It is important to note that statements such as 1+1=3 make about as much sense as “You’re NOT a wizard, Harry”, because given the context of the story, or fiction, these statements simply make no sense. There are some interesting similarities about Fictionalism and Platonism. The biggest one is that both of them take mathematical statements at face value. This is to say that both of them take 1+1=2 to mean that to add the mathematical object 1 and adding it to another mathematical object 1 will result in the mathematical object 2. The difference then is that where Platonism takes this to also mean that these abstract objects exist, Fictionalism does not accept that these objects exist. This is different from conventionalism in that conventionalism doesn’t even accept that you are referring to objects, regardless of their existence. The thing about Fictionalism is that the subject doesn’t technically actually even a little exist. By this I mean that Harry Potter doesn’t actually exist (probably) and that therefore he isn’t actually a wizard (probably) and that since he doesn’t exist he doesn’t actually own an owl named Hedwig (probably), and that by that same logic 1 doesn’t actually exist, and neither does 2, and 1+1 doesn’t equal 2 because none of them exist.

Implications of these schools of thought

Mathematical realism, in a certain sense, seeks to prove truths about the universe. This is most obvious when you consider modes of thought like physicalism, under which math would be a really general science, but even under Platonism you are seeking to find laws that govern these abstract objects you are finding. So for instance, when you have one of some object, and you add another of that object to that first object, you now have two of that object and according to mathematical realists, this is true. It is a fact. According to mathematical anti-realists, if you remove the humans, or whatever it is that is observing this addition, then there is no longer a group, one of the things, or two of the things. These concepts existed only because the humans said they existed, and when the humans stopped existing and thus stopped observing this these things lost the properties of being one, being grouped, and finally being two. The exact way in which this is argued depends on what subcategory one subscribes to. (https://www.youtube.com/watch?v=TbNymweHW4E&list=UU3LqW4ijMoENQ2Wv17ZrFJA)

How this relates to faith

Regardless of whether you believe that the statement “pi is the ratio between the circumference of a circle and its diameter” is true because of universal laws or because of human created definitions, the statement is still true. The importance of this is that it means that there, at least at this point in time, is no way to verify whether the reason for math existing is tied to the very nature of the universe or whether it is simply the product of the human mind. As a result of this, the belief in either of these theories is, at least in a certain sense, a leap of faith.

My thoughts on this

My personal opinion on this leans towards mathematical realism and more specifically Platonism. I agree that mathematical objects exist, but that they do not by necessity have a real world counterpart and thus are abstract, and I believe that regardless of whether or not humans exist, the mathematical concepts we have found to be true will still be true, even if no-one is around to appreciate, understand, or use them. One big reason I have for thinking this way is because of how various isolated cultures ended up discovering the same mathematical principles. By this I mean that counting systems, simplistic though they may have been, were not a unique event to just one area, but rather a common feature. I mean the Mayans had a counting system, so did the Greeks, Egyptians, Babylonians, Indians, etc. It seems somewhat unlikely to me that all these isolated cultures would create a method for defining something that doesn’t exist.

Idea channel’s episode titled “Is Math a Feature of the Universe or a Feature of Human Creation?”

Mark Balaguer’s “Realism and Anti-Realism in Mathematics”

http://web.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-Realism%20in%20Mathematics.pdf

Stanford’s Encyclopedia of philosophy entry on Platonism and mathematics

http://plato.stanford.edu/entries/platonism-mathematics/

The Internet’s Encyclopedia of philosophy entry on Platonism and mathematics

http://www.iep.utm.edu/mathplat/#H1

Wikipedia’s entry on philosophy of mathematics. No, this was not used as a source; it is however, useful for additional reading.

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Mayans count as well

http://maya.nmai.si.edu/maya-sun/maya-math-game

Greeks count as well

A History of Mathematics, Merzbach and Boyer, pages 52-55

For Babylonian counting see

Plimpton 322

For Indians having a number system click the bbc  thing below

For the link to the bbc story thing

http://www.bbc.co.uk/programmes/b03c2zvr

for the comic

http://xkcd.com/900/

Is Math Culturally Independent?

Is Math culturally independent?   Eleanor Robson asked this question regarding Plimpton 322. She wrote, “We tend to think of mathematics as relatively culture-free; i.e., as something that is out there, waiting to be discovered, rather than a set of socially agreed conventions.  If a simple triangle can vary so much from culture to culture, though, what hope have we in relying on our modern mathematical sensibilities to interpret more complex ancient mathematics?”  And yes, this was a homework question, but for some reason this question stuck with me, and I went looking a bit further.  For those of you who may not know, Plimpton 322 is an ancient Mesopotamian tablet around which there is some controversy. ¹  Scholars have claimed that Plimpton 322 is anything from a set of Pythagorean Triples, to a table of reciprocal numbers, or that it is possibly a trigonometry table.  The truth is, we just don’t know for sure; but whatever it is, it is definitely  more complex than the tax forms or accounting forms we typically expect the Mesopotamians to have left lying around.  (I just put it down a second ago, where did it go?)  Robson’s comment about the triangle mentioned refers to the difference between how we normally picture or represent a triangle and the standard Mesopotamian way of representing a triangle.  We have a tendency to depict a flat side facing down (for example Δ). The Mesopotamians, however, tended to represent their triangles pointing to the right similar to our play symbol. (Emblem unavailable at this time. Please consult your mp3 or video player, sorry for any inconvenience).

This question from Robson brought to my mind the idea of Musica Universalis.  Musica Universalis2 is a philosophical concept that is based on some assumptions made by the Pythagoreans, namely the combination of math and theology.  The Pythagoreans belived that everything had a numerical attribute,³ and they also found an appeal in certain symbols, such as the tetraktys and the Harmony of the Spheres (another name for the Musica Universalis).4  The concept of Music of the Spheres concerns the movements of the Planets, the Stars, the Moon and the Sun. (Remember, the thought at this time was that they all revolved around the Earth.)  One way of interpreting this was that there was some vast Celestial Orrery or Machine that had been set into motion. This Orrery controlled not only the motion of the celestial bodies but also the affairs of men. During these millennia there was no distinction between astronomy and astrology.

An example of an orrery. Image: Sage Ross, via Wikimedia Commons.

Johannes Kepler is a well know and still revered astronomer.  Kepler also believed there to be no distinction (at least it is not recorded) between astronomy and astrology and as an adviser and astronomer to Emperor Rudolph II he made horoscopes for not only the Emperor but also various allies and foreign leaders.  Johannes Kepler believed he had worked out much of the celestial orrery in his Mysterium Cosmographicum.5  The commonly held belief of the time was that all things could be understood by observing natural motions; whether those motions were of the planets, the stars, or in some cases the patterns of other natural phenomena.

Since all patterns can be represented mathematically, math then becomes the language of the universe. This idea can also be traced back to the Pythagoreans.  The concept that everything is a piece of celestial machinery that can be understood through math is still around us to this day, or at least it seems that the repercussions of it are. After all, if everything is patterns, and patterns can be interpreted wonderfully though Math, then Math must therefore be the Language of the universe. (That’s logical, that is.)  This seems to be the idea that Eleanor Robson is arguing against.  (Frankly, I agree with her.) This concept of a pure language of math is rather a strange convention that our society has if you really think about it. After all, the argument could be made that English (or any language really) is some sort of divine language because we can use it to so eloquently describe the world around us. Or perhaps Music is our divine language. It is pattern based, after all. So is this idea of everything being describable through math a belief we have found to be true, is it a truth that we somehow stumbled upon millennia ago, or is it a conceit of our culture?

1 A History of Mathematics by Uta C. Merzbach & Carl B. Boyer

Archetypes of Wisdom: An Introduction to Philosophy , Douglas J. Soccio

Musings: Ancient Thinking

If there’s one realization that’s guaranteed to send a mind reeling, it’s getting a glimpse at just how different human beings from one culture to another are at thinking. Even in modern day, the contrasts between two societies can be striking and staggering. Envision an American visiting Japan for the first time with no clue of just how different Japanese culture is from American culture. Coming from a culture that prizes individual accomplishment and direct communication, the tendency for Japanese culture to assign worth based on relationships and to consider addressing a subject bluntly as ‘clumsy’ would be shocking, if not dazzling (Western Washington University).

Archeologists deal with the difference of thought on a regular basis. In studying now-extinct civilizations, much of their deciphering of the artifacts they discover cannot be done by relying on modern ideas and understandings. Even in the subject of math, perhaps the most definite rules of the universe understood by mankind, ideas and understanding of mathematical concepts and uses have drastically evolved.

When I began this class, I was excited for the topics that I expected would be covered. I’ve still enjoyed the classes thus far, but I can say with absolute certainty that I was not prepared for the culture shock that would come with dealing with mathematical concepts that, in our modern society, are so basic and fundamental.

Mesopotamia. Image: Giusi Barbiani, via Flickr.

Zero. Zero is perhaps the most important digit in our system of numerals. It’s a place holder; it’s a starting point; it’s the middle of a number line that goes on for infinity in each direction. But go back to the inception of mathematics beyond 2+2=4, and you will find zero is as nonexistent as that which it represents. And boy, what a difference NOT having a zero makes. Perhaps the first function of zero that one misses when working without it is its job as a placeholder. How does one write 10, or 100, or 1000 without a zero? The Mesopotamians used a base 60 system, which meant instead of 10, 100, etc. their digits went up 60, 3600, 21600, and beyond. But still, the problem becomes: How does one write those without a zero?

Like this: 1. That’s it. 1. When the Mesopotamians landed on a power of 60, they wrote it as 1, because just as if you took the zeroes out from behind 10, 100, and 1000, all that’s left is a 1. This both creates a problem, but at the same time it provides a fascinating workaround. Since any power of 60 can be written as 1, the numbers prior to them can treated as fractions. For instance–and it’d be useful to think of a clock for this–if you wanted to write ‘half’ using Mesopotamian numbers, you would not write 1/2, but rather 30. Think minutes; thirty minutes is half of an hour, which is 60 minutes. 1/4 would be 15. 1/8 is a little more complex, as it comes out as 7;30 (that’s 7 sixties and 30 ones), but it’s still exactly like 7 minutes and thirty seconds is one eighth of an hour.

I could be a millionaire! If this was Mesopotamian. Image: David Guo, via Flickr.

Which brings the subject to reciprocals; reciprocals are fractions that, when multipled to a number, produce a 1. Again, because the Mesopotamians didn’t have a zero, their representations of 60, 3600, 216000, etc. all appear as 1. Because of this, reciprocals in the Mesopotamian numerals are sets of numbers that multiply not just to one, but any power of 60. Some examples would be 4 and 15, which multiply to 60, or 16 and 225, which produce 3600. Because these powers of 60 appear as 1, these sets count as reciprocals. It’s truly staggering what not having a zero does to math.

But when you consider the applications these ancient civilizations, such as the Mesopotamians, used math for, it does not make much sense to have a zero. For their purposes, a representation for zero would be irrelevant. Using a base 60 system, they could count to far higher with their digits before needing place holders. And when you’re counting cattle or grains or simple transactions at the market, zero is the last number you want to see on your accounting clay. This was a society that dealt entirely in positive numbers and practical, tangible concepts. We can look back at the Mesopotamian number system now and think, “Look at how hard it was for them to do even basic operations like completing the square,” but in a time when each man could only farm as much land as they and their family could do themselves and the technology for giant architectures was not common, completing the square was about as advanced in mathematics as any one person ever needed to attempt.

Like writing, mathematics is a largely intangible concept, and thus got off to slow start purely for practical purposes. Archaeological evidence indicates it would have simply started in counting animals and crops for the purposes of trade, or perhaps for counting people in some sort of rudimentary census. It wasn’t until humanity’s capability for the written language had advanced enough to express and record complex ideas that math began to see use for architecture and infrastructure. For the Mesopotamians, perhaps one of the most important uses of mathematics was in irrigation. Mathematical standards enabled uniform construction of materials, which was essential for carrying water the long distances necessary to hydrate the numerous farms of Mesopotamia. Advanced accounting and inventory ensured that construction had all of the materials a project would require without being oversupplied, as well as pay and supply the necessary manpower to work the construction project (Melville, Robson).

The mathematical system of the Mesopotamians can be quite a culture shock for American students. I myself was lost on the concept for the first week. It took a lot of practice for me to understand and comprehend the ‘reciprocals’ required due to the lack of a Mesopotaian zero. But it’s truly fascinating, regardless of its difficulty. It’s amazing to think this is one of the first advanced number systems to exist in human history. Its differences are shocking; for the unprepared mind, they can leave one feeling numb and lost. But once one manages to cross the bridge from the present to the past, the concepts ready to be rediscovered  are truly staggering.

Why base 60?

Plimpton 322, a text that uses the Babylonian base 60 number system. Image: Public domain, via Wikimedia Commons.

There always seems to be so many “why” questions in mathematics. One of the largest mistakes made by educators today is brushing over those “why” questions. As a future educator I wanted to dissect the reason for base 60, so that I can explain to my future students exactly why or at least give them a better answer than “because I said so”. This is why I have chosen to research the theories behind why the Babylonians chose to work in base 60.

What were they thinking when they chose that base? We are not the first ones to question the Babylonians’ use of base 60. Theon of Alexandria, fourth century AD, and Otto Neugebauer of the 1900s also tried to answer this question.4 The struggle comes with the uncertainty of the past. No historian has been able to present such a convincing theory that it dismisses all other theories. With that being said, read the theories and pick the one that makes most sense to you and just go with it!

Theory 1: Maximization of Factors

Theon of Alexandria originally presented this theory for the reason of base 60.4 This theory states that 60 was chosen because it was divisible by 1, 2, 3, 4, 5, and 6. Therefore, 60 is the smallest number that maximized divisors. Because of the vast number of factors 60 is “easy” to work with.1 I put easy in quotation marks because it is only easy if that’s what you were taught from the beginning! Imagine being taught to work in base 10 all your life and all the sudden you switched to working in base 60 when you got to college. I’m sure you would have some other choice words to describe base 60 that did not include the word easy. The theory of maximization of factors is the most popular of the theories. I am assuming this is the case because it is fairly straight forward in its explanation.

Theory 2: Weights and Measures

This theory was presented by Neugebauer.4 He proposed that the Babylonians chose base 60 based on the weights and measures adopted from the Sumerians. The overall idea behind his theory was that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. After research we can find that the system of weights and measures of the Sumerians did use 1/3 and 2/3 as basic fractions. My complaint with this theory is that the systems of weights and measures would have come prior to determining the base. I feel as if the determination of a base would have come prior to the system of weights and measures. But that’s just my humble opinion.

Theory 3/4 : Geometric shapes with 60°

These theories have to do with basic geometry. The Babylonians knew that there were 365 days in a year. However, when creating a circle, they chose to have the degree of a circle equal 360°. This was a choice made out of convenience because 360° was simpler to work with than 365°. The Babylonians then used a standard ruler and compass to construct a hexagon inscribed in a circle. This hexagon allowed for 6 partitions measuring 60° each.3 This theory was presented by the historian of mathematics Moritz Cantor.3 The reasoning behind choosing a hexagon is still fuzzy to me. The other coordinating theory was based on the importance of the equilateral triangle to the Sumerians. Sumerians considered the equilateral triangle the “fundamental geometric building block”.4 Perhaps the reason the Sumerians thought the equilateral triangle was so important was because of the connection between it and the hexagon inscribed in a circle discussed earlier. An equilateral triangle has angle measures of 60°. Was the choice of base 60 as easy as that?

Theory 5: Astronomy

The astronomical theory was quite simple. The number 60 is the product of 5 and 12. Babylonians believed there were five planets at the time; Mercury, Venus, Mars, Jupiter, and Saturn.4 They also believed there were 12 months in a year. Could base 60 have been the obvious choice because of those two important numbers?

Theory 6: Fingers and Part of Fingers

Could it be that they counted the segments of their hands? We are all familiar with the idea of counting fingers and even toes, but counting pieces of fingers? That seams odd. This theory was very confusing for me. When you count the segments you get 12 on each hand. That gives you 24 pieces in total. That doesn’t tell me anything about 60. But apparently there is some further explanation. “One can count up to 60 using your two hands. On your left hand there are three parts on each of four fingers (excluding the thumb). The parts are divided from each other by the joints in the fingers. Now one can count up to 60 by pointing at one of the twelve parts of the fingers of the left hand with one of the five fingers of the right hand. This gives a way of finger counting up to 60 rather than to 10. Anyone convinced?”4 I am most certainly not convinced. Unfortunately, I was unable to identify who originated this method.

Theory 7: Joined Forces

Joined forces is the name I gave to a theory proposed by several historians. This theory states that there were two civilizations prior to the Babylonians. One civilization worked in bases 5 and the other civilization worked in base 12. Another possibility is that one civilization worked in base 6 and the other in base 10. Either way, when the Babylonians joined with these other civilizations, they decided to compromise with the previous civilizations. The compromise was decided by multiply to two previously used bases together to get base 60.3 “One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture.”3 I will not dive into how they chose base 5 or base 6 or base 10 or base 12 in this post.

Theory 8: Committee

The last intriguing possibility is that either a ruler or a committee made the decision to use base 60. I think this theory along with a combination of another theory is very plausible. I believe that there was a committee of scientists and mathematicians that researched base 60. After the research of base 60 and other basses, the committee met with the hierarchy. The hierarchy could have been a political leader or the leader of the educational system at the time. After comparing the pros and cons of base 60 along with other bases, the hierarchy and committee chose the base that would be used in mathematics from then on. There is the argument that changing a civilizations number structure by committee creates a mess. Remember, America trying to switch to the metric system? It didn’t end well.

Nevertheless, math didn’t come from this magical land. It has origins and theories discovered by real people. I believe that if we discuss these origins and the thought process behind the theories more students will have an interest in mathematics.

*Superscripts within the text refer to the corresponding links and resources

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