## Internal Assessment

## Mathematics

### Topic: Is Math Invented or Discovered

## Word Count: 2153

## Table of Contents

TOC o “1-3” h z u Introduction PAGEREF _Toc15436628 h 3History PAGEREF _Toc15436629 h 3Discovery PAGEREF _Toc15436631 h 7Invention PAGEREF _Toc15436632 h 7Arguments from two sides PAGEREF _Toc15436633 h 7Realism PAGEREF _Toc15436634 h 7Anti-Realism PAGEREF _Toc15436635 h 8Conclusion PAGEREF _Toc15436636 h 8Bibliography PAGEREF _Toc15436637 h 10

The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

-Eugene Wigner

IntroductionThe word mathematics, derives from the Ancient Greek. In the Ancient Greek language, it means what one learns, what one gets to know, and in Modern Greek just lesson. Though in some parts of the world, it means astrology. Mathematics includes a handful of studies and topics, such as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). There is no general definition for mathematics, every countries or culture has their own definition or meaning. The highly respected Greek philosopher Aristotle defined mathematics as “the science of quantity”, this definition prevailed until the 18th century. And The German mathematician Carl Friedrich Gauss referred to mathematics as “the Queen of the Sciences”. It is certain that mathematics not only helped our society to grow as a whole, but it can be found in every little thing. From a particle as small as sand to a thing as big as a universe. Among all of the complex numbers, formulas, and equations. What lies underneath is the biggest questions that even math cant solve, is mathematics invented or discovered?

HistoryThe history of mathematics can be seen as an ever-increasing series of abstractions. Discoveries and inventions were slowly surfacing. At first, people only have extremely limited knowledge of the concept of numbers like one and two. For example, the realization that a collection of two apples and a collection of two oranges have something in common. Later on, as evidenced by tallies found on bone, people starting to gain knowledge of how to count physical objects. Moreover, prehistoric peoples may have also recognized how to count abstract quantities, like time days, seasons, years. Though the pieces of evidence for more complex math does not emerge till around 3000 BC. It was around the time when the Babylonians and Egyptians began to use arithmetic, algebra, and geometry for taxation and other financial calculations. Not only that, but they are also used for buildings and astronomy. In Mesopotamia and Egypt, the most ancient texts about math are from 20001800 BC. Throughout those early texts, Pythagorean triples were mentioned numerous times and thus it shows that the Pythagorean Theorem was the most ancient and widespread mathematical development after basic arithmetic and geometry.

## left5715

This mathematical tablet was recovered from an unknown place in the Iraq desert. It can be determined, apparently from its style, that it was written originally sometime around 1800 BCE. It is now located at Columbia University.Figure 1: Plimpton 322, Babylonian tablet listing Pythagorean triples.

Figure 1 shows a tablet listing Pythagorean triple that is created by the Babylonians. They will write on wet clay by pressing symbols into in which symbolizes different numbers. In order to get a better understanding of this tablet, figure 2 is a contrast between their numbers and our modern numbers.

3972214230360lefttopAs shown in figure two, the number 1 is written in a single stroke which look like this , and the number 2 to 9 were all written in the form of combining multiples of a single stroke.

2 3 4 5 6 7 8 9

1909397950900

258338119804800Same as the number 10, it was written in a single character . The number 20, 30, 40, and 50 were also all written in the form of combining multiples of a single character.

2030 40 50

19814744445

## Figure 2

This discovery not only shows that the Babylonians around that time had already discovered and gained the concept of singles and multiples but also invented a system of numbers in order for them to understand and use this as a tool for communication.

However, this section is only a portion of the whole tablet, and the other parts of the tablet were even more fascinating.

Because this table contains part of a list of Pythagorean triples, that is to say, integers w, l, d with

## w? + l? = d?

So the w (width), d (diagonal), and l will be presented below.

As mentioned earlier, that part was only the last column in the whole tablet. The third column shows another list of numbers that is formed with the symbols previously mentioned.

## left7805

The heading of the third column includes the word for diagonal.

The order for the image is first the origin column, and then the symbols written in modern numbers for better understanding, and at last will be the decimal equivalents.

*Error corrections are in red.

## Figure 3

## left182365

This image shows the second column in the tablet, the heading of the second column includes the word for width.

The order is the same as column three which is mentioned above.

## Figure 4

## left5715

And at last, this is the first column on the tablet, though unfortunately the heading of this column has not been translated.

### Interpolations are in green.

left1000961Regarding the hints given by the words diagonal and width, mathematician Neugebauer and Sachs discovered that if w is the entry in the second column and d is the entry in the third column, then in all but a few cases d? -w? turned out to be a perfect integer square l?. Here is the resulting table of calculations, in modern notation.

The tablet created by the Babylonians was a strong example of how people discovered the concept of math and recreating or reinventing it into a tool that is usable for human. The question is math invented or discovered can hardly be answered. The question itself was quite similar to the chicken or egg first question. No one or nothing can have solid evidence on what comes first. Math is everywhere and can be in everything, you can say it belongs to a part of the universe, or you can say it is simply a system created by human in order for us to understand the world around us.

DiscoveryDespite the ongoing discussion and debate about whether or not math is invented or discovered, we need to first define the terminology that we use. Discovery, according to the Cambridge dictionary, is defined as the process of finding information, a place, or an object, especially for the first time, or the thing that is found. For instance, the discovery of electricity. In which the concept of discovering a certain thing certainly relies on the fact that something that was already there and already exist. Math, however, can be seen in both ways. Some parts of math were surly discovered by humans. For instance, the concept of the Fibonacci sequence found on flowers pedals and pine cones. Though at the same time, why is it called the Fibonacci Sequence? And why is there a list of numbers that are defined as the Fibonacci sequence? Is that discovered too?

InventionNumbers are math, the modern numbers like 1, 2, 3, and 4, it is no doubt the fundamental mathematics which everybody is aware of today. It is a part of math created by people in order to comprehend the math that is more complex and intense. Invention, according to the Cambridge dictionary, is defined as something that has never been made before, or the process of creating something that has never been made before. Numbers are an invention, and although people may argue that numbers are simply a language used in math, similar to the language we use to communicate on a daily basis. However, as mentioned earlier, math can be seen in both ways. If numbers were never created, there will be no Fibonacci sequence, because the sequence itself, is made out of a list of numbers. So if math invented? Yes. And is math discovered? Yes. These types of philosophical question really depend on how you see things. In this case, how you see math and how you define it. If you think it is simply a language created for comprehension, then for you, math is an invention. But if you think math is something connected and embedded in nature, then maybe, math is just a discovery of the logics and truth of this universe.

Arguments from two sidesThis philosophical question has brought together scientists, mathematicians, and philosophers. And they are mainly divided into two sides, though there may have different names to define these two groups, they are most commonly known for mathematical realism and anti-realism.

RealismPhilosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it

### Galileo Galilei in The Assayer

The quote written by Galileo Galilei represents realism, they believe mathematics is the language of the universe. And math is only a tool for us to get a better understanding of our universe. They believe that the universe is a formal axiomatic system, and we exist as part of those axioms. All the cases of our mathematical system matching how the planet circles, or how the flowers grow, is not a coincidence. And math is an independent property, it does not by any means, need human to survive or to exist. There are many variations on realism, but the most popular form is Platonism: a metaphysical position which explains that mathematical entities are abstract, have no spatiotemporal or causal properties and are eternal and unchanging. Although on the other hand, anti-realis may say otherwise.

Anti-RealismAnti-realisms believes that mathematics depends on humans to exist, it is made purely out of our creativities and understandings. They believe that only the things we can touch and feel physically can be discovered. And math is not a physical object, it is rather an idea created and made in our minds to aid our understanding of the behavior of the universe. Same as the idea of death and alive, they are not fundamental to the universe, they only exist as terminology to separate two very different things. Same as the Realism, there are many variations in anti-realisms as well, and the most well-known one is Formalism, which proposed the idea that all mathematics can be derived from a set of axioms or self-evident assumptions. Axioms usually cannot be proven true due to the fact that they are very basic or self-explanatory, thus must be assumed. We do not discover axioms in any real sense; we decide upon them. Anti-Realisms denies the fact that we created mathematics, yet at the same time, never denied its usefulness to our existence.

ConclusionMathematics has come a long way. Millions and millions ago, human has no knowledge and understanding of math. However, thanks to the work of all those intelligent and respected mathematicians, math was able to thrive till this very day. Today, everything we see and touch has something to do with math. Tools and transportation that gives us convenience in life, as well as the technologies that bring us joy and entertainment. To attempt to conclude the debate and seize an absolute answer is seemingly beyond the bounds of possibility. Mathematics can certainly be argued in both ways, each and every one of us has our own concept and definition of what counts as truth, knowledge, or existence. Perhaps the belief that whether or not mathematics is discovered or is invented is just a belief and cannot be said to be right or wrong. So what kind of evidence would it take to resolve this debate? As far as we can tell, humans are the only species who owns a mathematical system. Thus it would be difficult for us to know whether math is indeed the language of the universe. However, if we ever meet an alien race, they would probably have a completely different system of their own that bears no resemblance to our mathematics. If so, the anti-realism wins, since it would completely overthrow the belief of math being the language of the universe and therefore math is very much created by humans. But what if they do have the same mathematic systems as us? Then it would seem to act as very strong evidence that math is indeed the native language of the universe. At last, is math invented or discovered? Perhaps invented, perhaps discovered, or perhaps both. Philosophical questions are not and will never be a simple yes or no, the philosophy of mathematics and its consequences are indeed daunting for even the most brilliant minds. And as far as for what we can do, we must all learn to gain a greater appreciation for the complexity and intricacy of mathematics and philosophy.

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Greek Mathematics. Wikipedia, Wikimedia Foundation, 22 July 2019, en.wikipedia.org/wiki/Greek_mathematics.

History of Mathematics. Wikipedia, Wikimedia Foundation, 15 June 2019, en.m.wikipedia.org/wiki/History_of_mathematics.

Math: Discovered, Invented, or Both? PBS, Public Broadcasting Service, www.pbs.org/wgbh/nova/article/great-math-mystery/.

Mesopotamia. Wikipedia, Wikimedia Foundation, 9 July 2019, en.m.wikipedia.org/wiki/Mesopotamia.

Pythagorean Theorem. Wikipedia, Wikimedia Foundation, 19 July 2019, en.m.wikipedia.org/wiki/Pythagorean theorem.

The Babylonian Tablet Plimpton 322. Plimpton 322, www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html#meaning.

Unknown. Mathematical Realism and Anti-Realism: All the Way down to the Nature of the Universe Itself. Mathematical Realism and Anti-Realism: All the Way down to the Nature of the Universe Itself, 1 Jan. 1970,