what is mathematics essay

What is mathematics?

Mathematics is at the heart of science and our daily lives.

Inventor of mathematics
Ancient Greek Mathematics
Importance of mathematics

Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, computers , software, architecture (ancient and modern), art, money, engineering and even sports.

Since the beginning of recorded history, mathematical discovery has been at the forefront of every civilized society, and math has been used by even the most primitive and earliest cultures . The need for math arose because of the increasingly complex demands from societies around the world, which required more advanced mathematical solutions, as outlined by mathematician Raymond L. Wilder in his book " Evolution of Mathematical Concepts " (Dover Publications, 2013).

The more complex a society, the more complex the mathematical needs. Primitive tribes needed little more than the ability to count, but also used math to calculate the position of the sun and the physics of hunting. "All the records — anthropological and historical — show that counting and, ultimately, numeral systems as a device for counting form the inception of the mathematical element in all cultures," Wilder wrote in 1968.

Who invented mathematics?

Several civilizations — in China , India, Egypt , Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians, who lived in the region that is now southern Iraq, were the first people to develop a counting system with a base 60 system, according to Wilder.

This was based on using the bones in the fingers to count and then use as sets, according to Georges Ifrah in his book " The Universal History Of Numbers " (John Wiley & Sons, 2000). From these systems we have the basis of arithmetic, which includes basic operations of addition, multiplication, division, fractions and square roots. Wilder explained that the Sumerians' system passed through the Akkadian Empire to the Babylonians around 300 B.C. Six hundred years later, in Central America, the Maya developed elaborate calendar systems and were skilled astronomers. About this time, the concept of zero was developed in India.

As civilizations developed, mathematicians began to work with geometry, which computes areas, volumes and angles, and has many practical applications. Geometry is used in everything from home construction to fashion and interior design. As Richard J. Gillings wrote in his book " Mathematics in the Time of the Pharaohs " (Dover Publications, 1982), the pyramids of Giza in Egypt are stunning examples of ancient civilizations' advanced use of geometry.

Statue of Muhammad ibn Musa al-Khwarizmi

Geometry went hand in hand with algebra . Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī authored the earliest recorded work on algebra called "The Compendious Book on Calculation by Completion and Balancing" around 820 A.D., according to Philip K. Hitti , a history professor at Princeton and Harvard University. Al-Khwārizmī also developed quick methods for multiplying and dividing numbers, which are known as algorithms — a corruption of his name, which in Latin was translated to Algorithmi.

Algebra offered civilizations a way to divide inheritances and allocate resources. The study of algebra meant mathematicians could solve linear equations and systems, as well as quadratics , and delve into positive and negative solutions. Mathematicians in ancient times also began to look at number theory, which "deals with properties of the whole numbers, 1, 2, 3, 4, 5, …," Tom M. Apostol, a professor at the California Institute of Technology, wrote in " Introduction to Analytic Number Theory " (Springer, 1976). With origins in the construction of shape, number theory looks at figurate numbers, the characterization of numbers, and theorems.

Mathematics in ancient Greece

The word mathematics comes from the ancient Greeks and is derived from the word máthēma, meaning "that which is learnt," according to Douglas R. Harper, author of the " Online Etymology Dictionary ." The ancient Greeks built on other ancient civilizations’ mathematical studies, and they developed the model of abstract mathematics through geometry.

Greek mathematicians were divided into several schools, as outlined by G. Donald Allen, professor of Mathematics at Texas A&M University in his paper, " The Origins of Greek Mathematics ":

In addition to the Greek mathematicians listed above, a number of other ancient Greeks made an indelible mark on the history of mathematics, including Archimedes , most famous for the Archimedes' principle around the buoyant force; Apollonius, who did important work with parabolas ; Diophantus, the first Greek mathematician to recognize fractions as numbers; Pappus, known for his hexagon theorem; and Euclid, who first described the golden ratio .

The golden ratio is one of the most famous irrational numbers; it goes on forever and can’t be expressed accurately without infinite space.

During this time, mathematicians began working with trigonometry , which studies relationships between the sides and angles of triangles and computes trigonometric functions, including sine, cosine, tangent and their reciprocals. Trigonometry relies on the synthetic geometry developed by Greek mathematicians like Euclid. In past cultures, trigonometry was applied to astronomy and the computation of angles in the celestial sphere.

The development of mathematics was taken on by the Islamic empires, then concurrently in Europe and China, according to Wilder. Leonardo Fibonacci was a medieval European mathematician and was famous for his theories on arithmetic, algebra and geometry. The Renaissance led to advances that included decimal fractions, logarithms and projective geometry. Number theory was greatly expanded upon, and theories like probability and analytic geometry ushered in a new age of mathematics, with calculus at the forefront.

Development of calculus

In the 17th century, Isaac Newton in England and Gottfried Leibniz in Germany independently developed the foundations for calculus, Carl B. Boyer, a science historian, explained in " The History of the Calculus and Its Conceptual Development " (Dover Publications, 1959). Calculus development went through three periods: anticipation, development and rigorization.

In the anticipation stage, mathematicians attempted to use techniques that involved infinite processes to find areas under curves or maximize certain qualities. In the development stage, Newton and Leibniz brought these techniques together through the derivative (the curve of mathematical function) and integral (the area under the curve). Though their methods were not always logically sound, mathematicians in the 18th century took on the rigorization stage and were able to justify their methods and create the final stage of calculus. Today, we define the derivative and integral in terms of limits.

In contrast to calculus, which is a type of continuous mathematics (dealing with real numbers), other mathematicians have taken a more theoretical approach. Discrete mathematics is the branch of math that deals with objects that can assume only distinct, separated value, as mathematician and computer scientist Richard Johnsonbaugh explained in " Discrete Mathematics " (Pearson, 2017). Discrete objects can be characterized by integers, rather than real numbers. Discrete mathematics is the mathematical language of computer science, as it includes the study of algorithms. Fields of discrete mathematics include combinatorics, graph theory and the theory of computation.

Why mathematics is important

It's not uncommon for people to wonder what relevance mathematics serves in their daily lives. In the modern world, math such as applied mathematics is not only relevant, it's crucial. Applied mathematics covers the branches that study the physical, biological or sociological world.

"The goal of applied mathematics is to establish the connections between separate academic fields," wrote Alain Goriely in " Applied Mathematics: A Very Short Introduction " (Oxford University Press, 2018). Modern areas of applied math include mathematical physics, mathematical biology, control theory, aerospace engineering and math finance. Not only does applied math solve problems, but it also discovers new problems or develops new engineering disciplines, Goriely added. The common approach in applied math is to build a mathematical model of a phenomenon, solve the model and develop recommendations for performance improvement.

While not necessarily an opposite to applied mathematics, pure mathematics is driven by abstract problems, rather than real-world problems. Much of the subjects that are pursued by pure mathematicians have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities.

These abstract problems and technicalities are what pure mathematics attempts to solve, and these attempts have led to major discoveries for humankind, including the universal Turing machine, theorized by Alan Turing in 1937. This machine, which began as an abstract idea, later laid the groundwork for the development of modern computers. Pure mathematics is abstract and based in theory, and is thus not constrained by the limitations of the physical world.

According to Goriely, "Applied mathematics is to pure mathematics, what pop music is to classical music." Pure and applied are not mutually exclusive, but they are rooted in different areas of math and problem solving. Though the complex math involved in pure and applied mathematics is beyond the understanding of most people, the solutions developed from the processes have affected and improved the lives of many.

Originally published on Live Science .

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Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

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Reuben Hersh

Table of Contents

This is an interesting, important, ambitious, and infuriating book, one that deserves both attention and response from the mathematics community. It has many good things to say, and it has the ambition to reshape the debate on the philosophy of mathematics.

There was a time, early in the twentieth century, in which mathematicians were passionately interested in the philosophy of mathematics. People were deeply concerned about what mathematics is, what sort of existence mathematical objects have, and their opinions on these questions actually influenced the mathematics they did.

There were three major points of view in the debate about the nature of mathematics. The formalists argued (roughly: the short summaries that follow are really caricatures) that mathematics was really simply the formal manipulation of symbols based on arbitrarily-chosen axioms. The Platonists saw mathematics as almost an experimental science, studying objects that really exist (in some sense), though they clearly don't exist in a physical or material sense. The intuitionists had the most radical point of view; essentially, they saw all mathematics as a human creation and therefore as essentially finite. The intuitionists refused to have any dealings with completed infinite sets, rejected the "law of the excluded middle" (i.e., the claim that a mathematical statement is always either true or false), and were willing to give up large tracts of classical analysis that didn't fit this point of view.

The fact that the debate never really got resolved, together with the complicating factor of Gödel's incompleteness theorems, seem to have caused most mathematicians to lose interest. In recent years, most mathematicians seem to have been content with an attitude best described by Jean Dieudonné. In everyday life, we speak as Platonists, treating the objects of our study as real things that exist independently of human thought. If challenged on this, however, we retreat to some sort of formalism, arguing that in fact we are just pushing symbols around without making any metaphysical claims. Most of all, however, we want to do mathematics rather than argue about what it actually is. We're content to leave that to the philosophers.

Reuben Hersh wants to change this. His book is an attempt to get us all involved in the debate about the nature of mathematics. To this end, he does a number of things. First, he argues that most writing on the foundations of mathematics is woefully ignorant of actual mathematical practice. Second, he tries to break the three-way tie by making a new proposal as to what mathematics really is. Third, he runs through the history of the philosophy of mathematics to argue that (a) his position is not really new, but has a distinguished pedigree, and (b) that all the other positions are clearly wrong. Finally, he connects philosophical positions on the nature of mathematics to broader philosophical and political issues.

The foundational debates of the early twentieth century turned on the issue of certainty . Everyone agreed that mathematical statements were true in an absolute sense, that one could be certain that they were true. The issue was to find a philosophy of mathematics that guaranteed that certainty. Hersh's position is that the desire for certainty is simply a mistake. In fact, he argues, regardless of our ideals, mathematics is done by fallible people, and so the traditional philosophies cannot really guarantee certainty. So let's give it up: mathematics is a human endeavor, and mathematical truths are uncertain like any other truths.

But if we locate mathematics as a human construction, we need to account for the very strong feeling that mathematical objects have some sort of independent existence. The number pi (or the number two) is not just something in my head! Hersh agrees, and proposes that mathematical "objects" are really socio-cultural constructs. As such, they really do transcend individual minds even while remaining human creations. Hersh calls this view of mathematics humanism .

So far so good: it's an interesting proposal, and has much in its favor. It makes modest claims for mathematics which actually correspond to our human experience as mathematicians, and it takes seriously the fact that mathematics is learned and taught. (It is also, which Hersh does not observe, completely compatible with a Platonist understanding of mathematical objects; all that one needs to do is to delete the (implicit) "merely" in Hersh's claim that mathematical objects are socio-cultural constructs. Our mathematical concepts can certainly be socio-cultural constructs that attempt to grasp and understand Platonist mathematical objects...) The humanist view allows us to escape from metaphysics and to focus attention on things we can actually observe first-hand: the mathematical community and how it learns, teaches, and develops mathematics.

There are other persuasive things about "humanism." For example, it is deeply aware that mathematics has a history . Both formalism and Platonism often give the impression that they deal with mathematics as a completed product, when in fact mathematics is produced by people working in socio-cultural contexts. A look at the history of mathematics certainly seems to undermine a naive formalism (since "formal proof" is a relatively recent phenomenon). History also asks difficult questions for Platonism; does it really make sense to claim that, say, Galois groups "really exist", and that they existed even in Euclid's time?

On the other hand, the humanist view reduces our feeling that mathematical truths are really true to a social consensus, something we learn. This opens the possibility that Little Green People from Mars, if they exist, have a mathematics that is not only different from, but actually contradictory to ours. This is a position that many mathematicians find extremely hard to take.

One should emphasize that the evidence for some sort of "certainty" in mathematical truths goes beyond our intuitive feelings about them. The history of mathematics also points in this direction. We may no longer accept some of Euclid's arguments as rigorous, but we do think every one of his theorems is still true ! What other science can make such a claim? Or consider Fermat's Last Theorem: is there any other field of human endeavor in which a question posed in 1636 can still make sense, in exactly the original terms, 350 years later?

In arguing for his humanist philosophy of mathematics, Hersh has some very good points to make, but he seems to spend much more time attacking the rival positions of formalism and Platonism. Against formalism, he follows and develops the criticisms of Imre Lakatos: he argues that no one actually writes formal proofs of anything, and that the view that mathematics is simply meaningless symbol-pushing is impossible to believe. Against Platonism, he argues that believing in eternal mathematical objects existing independently of human thought is only possible if one believes that God exists, which, he says, no one does anymore.

While there is something to both arguments, neither of them really takes the opposing philosophical position seriously. Formalism is more solid than Hersh makes it seem, Platonism has been the position of serious philosophers who were not theists, and, after all, many philosophers do believe that God exists. So do many mathematicians. This points up the basic problem: all too often, Hersh is willing to dismiss the opinions of eminent thinkers after only a very shallow interaction with their thought.

As a result, Hersh's historical survey of the philosophy of mathematics is, to my mind, the weakest part of the book. The quick sketches of the thought of various philosophers really do justice to no one. (They remind me of John Rist's description of Bertrand Russell's History of Western Philosophy : "sophistic and simplistic misinterpretations of most Western ethics and metaphysics.") An example is his treatment of George Berkeley's very complex philosophy of mathematics, which gets reduced to "using the deficiency of mathematics to support religion." He then adds "His attack on mathematicians is unique since St. Augustine," which is hard to understand, since in his section on St. Augustine Hersh explains that Augustine's "attack on mathematicians" is really nothing of the sort!

Even less convincing is Hersh's attempt, in the last chapter, to correlate philosophers' positions on mathematics with their political stances, in order to reach the conclusion that "the Platonist view of number is associated with political conservatism, and the humanist view of number with democratic politics." Aside from the fact that he takes as axiomatic that being associated with the latter is better, the whole argument is based on a painfully arbitrary distinction of left versus right. (David Hume as a leftist?!).

The book concludes with an invocation of the story of the blind men and the elephant "as a metaphor for the philosophy of mathematics, with its Wise Men groping at the wondrous beast, Mathematics." All except, it seems, the humanist, who can see the whole elephant and laugh at the fallibility of all the others. Hersh doesn't seem to realize how arrogant this attitude is.

What I think is missing from the book is a realization that the philosophy of mathematics is indeed philosophy, and not science, and that therefore it cannot ignore the overarching philosophical issues that relate to it. It is clear that one's beliefs on metaphysics and epistemology, and particularly one's stance with respect to the notion of truth , are going to have an enormous impact on one's positions on the nature of mathematical objects and mathematical truths.

However, despite the serious limitations of Hersh's treatment of other thinkers, and despite the fact that he does not argue for his "humanist" position as forcefully as he might have, this is still a book that deserves attention. It should be widely read, discussed, and argued with. I hope it stimulates many responses, and most of all that it manages to convince more mathematicians of the importance of the questions with which it deals.

Fernando Gouvêa ( [email protected] ) is chair of the Department of Mathematics and Computer Science at Colby College and editor of MAA Online.