 # How did mathematics begin?

How did mathematics begin?

Mathematics is a fundamental tool for understanding the world around us, and it has a long and complex history dating back to ancient times. The earliest known examples of mathematical activity can become traced back to the ancient Sumerians of Mesopotamia. Who developed a system of counting using a sexagesimal (base 60) number system more than 5,000 years ago.

The Egyptians also developed a sophisticated system of mathematics. Which became used to solve practical problems such as measuring land, building structures, and conducting trade. They developed a decimal system for counting, which was based on hieroglyphic symbols representing the numbers one to nine and multiples of ten.

#### The Greeks made significant contributions to mathematics, particularly in the areas of geometry and number theory. Euclid, who lived in the 4th century BCE, wrote the seminal work “Elements,” which laid out the foundations of Euclidean geometry and provided a comprehensive overview of mathematical knowledge at the time.

Other civilizations, such as the Chinese and Indians, also made important contributions to the development of mathematics. In China, for example, mathematicians developed a sophisticated system of arithmetic, algebra, and geometry, and made important discoveries in the areas of number theory and probability.

Over time, mathematicians built upon the work of their predecessors, developing new mathematical techniques and ideas that have allowed us to better understand the world around us. Today, mathematics is a broad and diverse field that encompasses many different areas of study, including algebra, geometry, calculus, and statistics, and it continues to play an important role in our understanding of the universe.

#### How did mathematics begin?

Starting in September, NFL teams begin to face each other on a weekly basis. In order to outcompete the other team, players must rely not only on peak physical maintenance, but also a strong game plan. Over time, strategies become increasingly complicated and sophisticated. Furthermore, new game plans and solutions became created by coaches in an effort to best prepare their team.

This process is comparable to mathematics, where finding ways to solve ever-growing and ever-diversifying sets of problems, requires that mathematicians find new ways to solve these new problems. Creating new methods of explanation, would mean producing entirely novel solutions to problems. Discovering methods of explanation, on the other hand, would mean that the solution had already existed, and that we had just finally found it. Throughout history, ancient civilizations proved over and over again that math was and currently became created.

#### Throughout history, there has always become a need for the quantification of objects and ideas. In Donald Allen’s “The Origin of Mathematics”, he describes ancient examples of the need for quantification.

For example, in pecking orders, civilizations used numerical order to construct hierarchical rankings. Similarly, with the worth of a herdsman, a shepherd would need to count and quantify exactly how many sheep he had, to measure how much he was worth.  In order to fulfill the need to answer the questions: “how many?” and “how much?”, ancient civilizations created their own unique counting systems (1).

Beginning even without writing, counting systems materialized in the form of finger numbers (Eves 13). Finger number systems expressed “numbers by various positions of the fingers and hands” (Eves 13). Later, these numerical systems would progress into a simple grouping system such as the Egyptian hieroglyphics. In a simple grouping system, a symbol would be “selected for [a] number base,” and other symbols would be adopted to represent subsequent powers of that number base (Eves 14).

#### Moreover, to express specific values, symbols would become used  “additively, each symbol repeated the required number of times” (Eves 14). For example, if: a = 1, b = 10, and c = 100, in order to write “113,” the expression would read: cbaaa. In other cultures, more complex numerical systems developed, such as the Chinese multiplicative system (Eves 17). Multiplicative systems use symbols to designate a base and it’s subsequent powers, and other specific symbols to designate all individual values leading up to the initial base value.

For example, if: a=1, b=2, c=3, d=4, e=5, f=25, “107,” would become depicted. As  d(f)a(e)(a)(b), with “e” being the base value, “f” a power of the base value, and “a,” “b,”. And “c” as the individual unit values. Another example of an ancient numerical system is the positional system. Which the Babylonians used. In which the position of each symbol designates to which base power the value becomes applied (Eves 20). This system is where the ideas of a “ten’s place, hundred’s place, and thousand’s place” originated.

#### Moreover, all of these numerical systems were entirely new, innovative creations, that civilizations used, to help them quantify values. “The variation in forms of numerical systems supports the notion that math became created, in that each civilization created their own unique methods of numerating, while became geographically isolated. Another example of differentiation between each culture’s numerical system is their variation in number bases. While the Egyptians used base 10, the Mayans used base 20, and the Babylonians used base 60 (Eves 20).

Ancient mathematics saw the development of various, unique numerical systems. However, some remarkable consistencies across different cultures. Suggesting that there might have become some pre-existing template that civilizations followed in developing numerical systems. Perhaps these counting systems became discovered and not created.

For example, both the Greek and Roman civilizations developed simple grouping number systems and both the Babylonian and Mayan developed positional grouping number systems (Eves 20). The Greeks and Roman systems were extraordinarily similar as they both used base 10. But these shared ideas can become explained by geographic proximity and information sharing through trade networks.

Regarding the Babylonians and Mayans, it simply seems far more probable that these systems were developed in similar ways due to their similar intentions, purposes, and interests, rather than through discovery.

#### The Babylonians made significant advances in mathematics, specifically in geometry. One of their most prominent contributions to mathematics was their development of the “method of means”; specifically, their use of the method to approximate the square root of two.

As Donald Allen describes in his “Babylonian Mathematics”, the method of means was based around a general equation that averages the quantities of two perfect squares using the equation B = (A + 2/A)/2. The more times you ran your values through this equation, the more accurate your approximation would end up being. This method of means was an entirely novel practice. There are no other civilizations that developed a similar method of approximating square roots like the Babylonians. Because of its uniqueness, the Babylonian method of means provides evidence that math became created.

Similar to the Babylonians, the Egyptians discovered a means of multiplication and division, creating a “template” for efficiently multiplying and dividing. As described in Tamu’s “Egyptian Mathematics”, these operations centered around a strategy of doubling.

This doubling strategy that the Egyptians created was unlike any other civilizations’ method of multiplying or dividing. This novelty of this operation bolsters the argument that math created rather than discovered. Thus, the method of multiplying and dividing that the Egyptians created, provides evidence that math became created. Rather than discovered.

#### Others might argue that while the Egyptians created this method of multiplication in an effort to maximize efficient calculation, the multiplicative properties themselves had already existed, and thus, were merely discovered. Similarly, some would argue that the Babylonian method of means became merely a discovery. In that the inherent mathematical properties already existed. While the processes of mathematics became expanded through this creation. Thus, the theoretical aspects of math simply uncovered by the Egyptians and Babylonians.

The issue with the opposing argument is that multiplication itself is not an intrinsic property. Multiplication is a tool that mathematicians use to simplify quantification. If one was to argue that numbers had the intrinsic ability to become multiplied. Thus, the argument would simplify to become “large numbers exist”. Which is not a sound argument for why these Babylonian and Egyptian processes discovered rather than created. Regardless, the development of new methods of multiplication, division, and approximation, by the use of intrinsic mathematical properties in unique ways is inherently a novel extension of mathematics, thus deeming them creations.

One of Euclid’s most significant contributions to mathematics was his proof of infinite primes by use of a strategy of contradiction. Euclid first considered the idea that there exists a finite set of numbers that includes all prime numbers, which he then proved through computations, to be unfeasible. This proof of infinite primes provided evidence for his argument against other mathematicians who believed that all quantities and values were finite. By proving that primes were infinite, Euclid created mathematics, because he developed an entirely novel idea and method to support it.

#### Others might argue that Euclid’s proof of infinite primes by contradiction was a discovery of mathematics in that prime numbers themselves were infinite. Conceptually, the idea of infinity had always existed, but was for many years shrouded in mystery. In JJ O’Connor and E F Robertson’s “History Topic: Infinity”,  they describe how “from the time people began to think about the world they lived in… questions about time… was there a finite end … was this space finite or does it go on forever” had existed (1). Thus, there always existed some theory regarding infinity, despite the idea becoming constantly rejected, due to its obscure nature.

As a result, many mathematicians classify mathematical developments regarding infinity as discoveries of distinct characteristics. However, even if the idea of limitlessness existed in nature, it had never become symbolically represented or conceptually integrated in mathematics. Thus, Euclid’s proof of infinite primes became really a creation of mathematics. In that it conceptually defined a property of nature. Furthermore, implemented it into the mathematical scene. The first occurence ever!

#### How to generalize the diffusion model to solve high-dimension problems?

Later along the timeline, Georg Cantor would become well-known for his mathematical contributions. Specifically, Cantor would create many new classifications of numbers, categorizing them by examination of their properties through a method called diagonalization. In Donald Allen’s “The History of Infinity”, he summarizes how “using his diagonalization method, [Cantor] was able to demonstrate orders or powers of infinity of every order” (16).

Thus, through diligent study, Cantor found that infinite sets had unique powers, or quantities. For example, the set of infinite rational numbers are countably infinite, while the set of infinite real numbers is not countable, meaning infinity could come in different sizes, and have multiple values (Allen 16). Cantor’s contributions to mathematics in the genre of sets of infinite numbers can thus be summarized as the creation of the method of diagonalization to serve as a strategy in proving mathematical relationships.

#### Some might consider Cantor’s contributions discoveries as they expanded our knowledge of the many unique properties between sets of numbers. This argument is centered around the idea that these properties that Cantor described were hidden, but inherently existent in our numerical systems; he “discovered” new relationships between the units of the previously developed counting systems through the method of diagonalization.

However, there is no natural or intuitive explanation of the idea that certain sets of numbers have intrinsically greater power or different order than others. It is true that the concept of limitless quantities existed. But at the least, Cantor created a portion of mathematics by creating the method of diagonalization which assisted him in his explanations for conceptual questions.

At a point in time, ancient Grecian society was dominated by the mathematical and philosophical ideology of Pythagoras and his cult. They believed that everything in nature was finite. However, a follower of Pythagoras uncovered the falsity of this notion, as Donald Allen in “Pythagoras and the Pythagoreans” finds, “ One account gives that the Pythagoreans were at sea at the time and when Hippasus produced (or made public) an element which denied virtually all of Pythagorean doctrine, he was thrown overboard” (Allen 32). In William Dunham’s Journey through Genius, he details these specific contributions.

Hippasus summarized that commensurability is the idea that a ratio of two things is a rational value (Dunham 8). And while the Pythagoreans believed that “any two magnitudes were commensurable,” and that “whether in music or astronomy,”  everything could be understood in whole numbers (as discussed in the previous paragraph), Hippasus found that “the side of a square and its diagonal are not commensurable” (Dunham 9).

#### As a result, this means that no value can divide evenly into both the lengths of a side of a square and its diagonal, dismembering the Pythagoreans’ claims that whole numbers could account for all quantities. Hippasus discovered the concept of incommensurability by disproving the original concept that was developed by Pythagoras. However, this realization also qualifies as a creation, as Hippasus created the means for us to explore other, more nuanced realms of theoretical mathematics.

In addition to Hippasus’ contributions regarding commensurability, he also researched the concepts of different types of numbers. For example, he created two classifications of real numbers: the algebraic, and the transcendental. What distinguishes the two, is that algebraic numbers are any potential “solution[s] to … polynomial equation[s],” and transcendentals are numbers that are “not solution[s] of any polynomial equations” (pg. 24).

Hippasus also created a classification of “constructible numbers,” which are numbers that can be drawn with a compass and straightedge and are strictly algebraic (Dunham 24). Contrarily, transcendental numbers, as the name might imply, are numbers that cannot be constructed using a compass and a straightedge (Dunham 24). For example, pi is transcendental and thus not constructible. By creating new classifications of numbers, Hippasus’s contributions provide evidence that math, the symbolic representation and expression of concepts, was created.

#### While Hippasus created these classifications of numbers, these works can be considered discoveries in that he discovered these unique properties of whether or not numbers could be constructed, or serve as polynomial solutions, and their implications for other branches of mathematics. For example, in terms of constructible numbers, he discovered that certain numbers maintained the intrinsic property of not being able to be constructed with only the aid of a straight edge. However, what makes his contributions useful and applicable are his creative classifications of numerical values. The Tsinghua Bamboo Slips, containing the world’s earliest decimal multiplication table, dated 305 BC during the Warring States period

In ancient India, Brahmagupta attempted to develop rules for arithmetic involving zero. In JJ O’Connor and EF Robertson’s “History topic: A history of Zero,” they detail the rules that Brahmagupta proposed, and his struggles with dealing with zero. For example, he struggled with division, especially when zero existed as the denominator because no-one was able to grasp these properties of zero when it came to division ( O’Connor 3).

Moreover, believed that if, for example, you had an unknown divided by zero and set that value equal to another unknown, and you multiply both sides by zero in accordance with algebraic law, you would get that unknown equal to zero ( O’Connor 3). Thus he believed that dividing by zero would result in the quotient of zero, however, we know today that that is not true ( O’Connor 3).

#### Aside from division, Brahmagupta laid down fundamentals for arithmetic, including rules for subtraction and multiplication with zero ( O’Connor 3).

The issue with Brahmagupta’s rules for division with zero would later be explored by other medieval Indian mathematicians. In Boyer’s “An early Reference to Division by Zero,” Bhaskara, a leading Indian mathematician, attempted to solve this division problem.  He would arrive at the conclusion that any value divided by zero would be infinity (Boyer 224). However, the proof for Bhaskara’s solution relies on the assertion that a/0 X 0 = a, which is mathematically untrue. Despite their lack of success, Brahmagupta and Bhaskara’s contributions can be considered creations in that they created and standardized rules for arithmetic.

One could argue that through Brahmagupta and Bhaskara’s development of arithmetic rules, they discovered uncertainties and inaccuracies in their own works and thus in the arithmetic properties themselves. However, neither mathematician really uncovered any uncertainties when creating the basic rules for arithmetic, they merely provided the most logical solution to a problem that neither of them fully understood.

#### Chinese ancient civilizations made significant algebraic contributions, specifically by their development of the “Fair Distribution of Goods” and “Excess and Deficit” methods. In JJ O’Connor and EF Robertson’s “History topic: Nine Chapters on the Mathematical Art,” they outline and explain these Chinese developments, and their significance.

The “Fair distribution of Goods method”, was an algebraic concept, used to calculate “ratio and proportion… problems about traveling, taxation, sharing etc” (O’Connor 2). Similarly, the “Excess and Deficit” provides a method of solving “essentially linear equations” by a strategy similar to that of false position, in which one makes”two guesses at the solution”, and then computes “the correct answer from the two errors” (O’Connor 2). These Chinese contributions can be considered creations in that they created novel methodology for algebraic implementation

Some may also classify these developments as discoveries. The reason being, that  the fundamental mathematical properties already existed within the numbers of the problem. For example, the basic additive nature of quantities is what allows for these calculations. Thus the argument would be that the Chinese discovered a way to manipulate the nature of multiple quantities to solve complex algebraic problems.  However, even if it is true that the Chinese only discovered a way to manipulate pre-existing mathematical relationships, they still created novel methods of solving algebraic equations.

#### In the medieval middle east, Arabic mathematicians made considerable strides in developing algebraic expression. O’Connor details in his “History Topic: Arabic Mathematics: Forgotten Brilliance,” “Al-Karaji is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today” (3).

The very nature of Al-Karaji’s contributions provide evidence that math was created in that he created an entirely new genre of mathematics, algebra. Nothing was discovered as the use of variables in the form of letters in mathematics already existed, only it was used exclusively for geometric purposes. Al-Karaji’s distinguishing of algebra from geometry is an explicit demonstration that math is created.

With an endless desire to find certainty and understanding regarding the laws of nature, we as humans are constantly attempting to develop new systems of symbolic representations, or forms of mathematics, to help us prove and justify our claims. Mathematics is in a way, very much an art.

Our colors can be mixed and matched to create new colors, and we can develop different styles of drawing and painting to express objects and emotions in unique ways, much like how we mix and match our numbers to calculate quantities and solve algebraic expressions. Across the world and throughout history, the gradual development and growth of mathematics has provided evidence that math is a creation.

History Of Mathematics : Who first invented mathematics? Was Math Created or Discovered?

Written by Mitchel Wang History Of Mathematics : Who first invented mathematics? Was Math Created or Discovered?Written by Mitchel Wang

#### Bibliography

Allen, Donald. “Pythagoras and the Pythagoreans.” 6 Feb. 13AD,

www.math.tamu.edu/~dallen/history/pythag/pythag.html.

Allen, Donald. “The Origin of Mathematics.” 5 Jan. 1997,

Allen, Donald. “Babylonian Mathematics.” 28 Jan. 1997,

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

Allen, Donald. “Egyptian Mathematics.” 21 Apr. 2001,

www.math.tamu.edu/~dallen/history/egypt/egypt.html.

#### Allen, Donald. “The History of Infinity.” www.math.tamu.edu/~dallen/history/infinity.pdf.

Dunham, William. “Journey Through Genius.”

Boyer, C. B. “An Early Reference to Division by Zero.” American Mathematical Monthly, 50

(1943), 487- 491.

Eves, Howard. The History of Mathematics: An Introduction. 6th ed., vol. 1, McGraw-Hill, 2006.

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www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html.

O’Connor, J J, and E F Robertson. “History Topic: Infinity.” Feb. 2002,

www.math.tamu.edu/~dallen/history/egypt/egypt.html.

O’Connor, J J, and E F Robertson. “History Topic: Arabic Mathematics: Forgotten Brilliance?”

Jan. 4, 2016.

O’Connor, J j, and E f Robertson. “Nine Chapters on the Mathematical Art.” Dec. 2003,

History Of Mathematics : Who first invented mathematics? (rebellionresearch.com)

How did mathematics begin?