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History of mathematics
The availability of availability of mathematics could be realized since ancient times in the form of counting. It is a very simple form of mathematics where ancient people use their thumb or some figures to count any item or things. It was the most basic form of mathematics that lay down emphasize on its availability since humanity exists. Since starting the mathematics, have become the basis of scientific inventions, calculations, new development and advancements, systematic study of shapes and objects. The huge revolution that has been faced in the history of mathematics is related to the invention of zero. It was something that had changed the dimensions of this subject. After zero, it becomes possible to focus on wide scope of mathematics and its spectrum also increases at very large scale. The role of Indian mathematicians like Aryabhata, Pinangla and Brahmgupta was immense within the invention of Zero. They provide huge level of contribution towards the use of Zero in their theorems. During their work, these philosophers and mathematicians provide the use of zero in the decimal value system. However, the role of Greek mathematicians is immense in the history of mathematics. They provided the mathematics rules that become the subject of the entire subject. It is to acknowledge that at the time of Alexander the great the period was called as Hellenistic mathematics. It is clear that Greek mathematicians used all the logics to propound specific set of rules in the history of mathematics. Babylonian mathematics is also very crucial with the perspective of understanding the history of mathematics. Mesopotamian people gave their huge contribution into the Islamic mathematics. The early civilization of Mesopotamian people provides the evidence about metrology system that is termed as highly complex in nature. Further, the multiplication calculations along with geometrical and divisional problems were also grew in the era of 2500 BC to 3000 BC. Some clay tablets have been recovered which provided the evidence about using the algebra, quadratic and cubic equations. The numeral system for Babylonian was based on sexagesimal numerical system that is generally based on 60. Thus, it is clear that current time system like 60 seconds in one minute, 60 minutes in one hour, etc. is based upon the Babylonian mathematics. The Babylonian mathematician’s principles were also similar to Romans and Greeks mathematicians. They just used the same place value system in decimal value. However, there were certain controversies into the principles and concepts given by the Babylonian mathematicians that are still required to be proved.
Ahead the Egyptian work is also equally useful with the perspective of the history of mathematics. The major work of Egyptian mathematicians is the Rhind papyrus and Moscow Papyrus. The Rhind papyrus is the work related to the formulas and methods related to division and unit fractions. Further, the multiplications and rules related to composite and prime numbers also come into the category of Rhind Papyrus. The solutions of linear series as well as the arithmetic and geometric solutions is also a part of Egypt mathematician works. In China also the role of mathematics was immense with respect to the deliver lots of contribution into such subject. There are proofs that in China the value of pie was known to the seven places to the decimal that was highly more than the west mathematicians. The pattern to solve the linear equations in China was also advanced as compare to western countries. The Gaussian reduction is something that has a high level of relevancy into the mathematics history of China. Further, the role of India is also immense within the development of mathematics. But Indians use the calculations in astronomy just to predict the stars consequences. Thus, it is clear that the calculation is very old in the Indian number system.
The zero has been propounded by the Indian mathematician and most importantly it was passed on to the western countries through Islamic mathematician. Thus, it is clear that the basis of modern mathematics has been put by the Indian mathematician. The Maya civilization is something that has huge level of relevancy with the development of Zero into the numerical value system. They number base was twenty and most significant contribution is related to providing the calendar dates. Thales, Pythagoras, Euclid, Apollonius, Archemedes, etc. these are certain mathematicians who have provided their significant contribution into the field of math over the different period of time. In different centuries, these mathematicians show their relevance to the math and its related concepts. The Pythagoras theorem is something that has changed the dimensions completely. However Greek claim to be the inventory of this principle but evidence were clear that Mesopotamian and Babylonian have also contributed in Pythagoras theorem far earlier than the Greek mathematicians.
The work of Fibonacci propounded into the 1175-1250 AD was the most crucial work into the field of mathematics. It tools place in European mathematics. At the end of Roman Empire in the west the mathematicians into the west region come to an end. But Arabic, chine, and Indian mathematicians continued their practices and increased their understanding towards its principles and concepts. The Fibonacci sequence has brought down the transformation into the field of math. The sixteenth century onwards the huge changes into the field of mathematics have been experienced. It was the era when the foundation for modern mathematics had been put into practice. The development in equation solving issues, Arabic numerical and algebra was huge into the sixteenth century. Girolamo Cardano was the one who change the dimensions for the cubic equation and provide a solution for the fourth degree polynomial equation. Francois viete developed the + and – sign during his practice period. The invention of logarithm has also done by the Francois Viete, which was used in bringing the transformation into the multiplication process. Ahead the development of rules and formulas of logarithm has been done by John Napier in the year 1614. He advances the field of the logarithm and definitely made it more interesting. Other than John Napier the seventeenth century belongs to the work of Kepler, Newton and Galileo, Descartes and Leibniz. The Logarithm actually reduces the calculation time and gives birth to new mathematics diverse ideas. Isaac Newton gave the most significant principle for the mathematics named calculus. It was one of the most important inventions into the field of math. The ground work for the probability theory was set by the Blaise Pascal and Fermat that later become the foundation for the game of gambling through combinatory.
Later it becomes the foundation of utility theory when the Pascal and Wager gave new dimensions to the probability theory. In the era of the eighteenth century again the dimensions were set by Leonhard Euler when he propounded the graph theory and most importantly the complex analysis and multitude of analysis were also noticeable work of Leonhard Euler. Next change or newly pioneered work did by Laplace as his work shed light on celestial mechanics and statistics. Further, the Nineteenth century took the mathematics up to the next and highest level. This century was most revolutionary century ever in the history of mathematics. In this century, the Carl Guass changed the thought process towards the math and provided lots of stunning and amazing piece of works. His work included the functions of complex variables, geometry and most importantly the convergence of series. The fundamental theorem of algebra and quadratic reciprocity law was also firstly propounded by the Carl Guass. His contribution into this field was immense. The birth of ideas related to the curves and surfaces was given by the German mathematician Bernhard Reimann. He has re shaped the understanding about the geometry and gave new dimensions to this field of mathematics. Hermann Grassman and William Hamilton provided the new facts related to the abstract algebra. At one hand, Grassman provided the knowledge about vector spaces, and Hamilton shed light on noncommunicative algebra. This was the time when the foundation of computer science came into existence. The George Boole, a British mathematician, actually propounded the Boolean algebra that includes the 0 and 1 only numbers. Nineteenth century actually reveals so many principles and fundamentals which were completely unknown or forced to be unsolved since last so many centuries. The most noticeable fact was that so many mathematical societies come into existence to teach the principles of math to students and scholars. Thus in this way the historical development of mathematics continued till the twentieth century Rmanujan, Alan Turing and recently Andrew wiles that has changed the dimensions of mathematics and brings some noticeable changes. The Andrew wiles has solved the last theorem of Fermet which remain unsolved since three centuries. In the twentieth century, it actually becomes the profession and a systematic way to focus on mathematics and its related concepts. Now challenges for upcoming mathematicians are 23 (10 have been solved,7 are partially solved, 2 are remaining and 4 are under arguable status regarding their solutions) unsolved problems which are still the challenge in the field of math and needs more concentration by the future mathematicians. Thus, it explains the brief historical development of mathematics from its distorted inception to recent concrete development.
History of Mathematics Essay Questions and Answers
These are some essay questions that I had to answer for my History of Mathematics class that I took during the summer of 2001. They're very short and to the point since the only requirement was to write at least a paragraph to answer the question. (And you may notice how brief they get towards the end. This is due to a bunch of them being written on the same night before I left for Prague for a week.) I debated on whether or not to even post them on my website. Finally decided to in view of all the weird things that people search for that lead them to my site. This'll add just that much more whackiness.
Essay 1) Discuss Euclidean constructions, the three classical problems, and their role in the history of Greek mathematics.
Euclidean constructions are the shapes and figures that can be produced solely by a compass and an unmarked straightedge. Although these tools were indeed simple, their range of abilities seemed unlimited. Not only could they produce a multitude of angles and lengths, but also elegant-looking regular polygons and a wide variety of 2-D shapes with desired area. These basic tools seemed to be able to do or produce anything. When, after countless attempts, they were unable to solve the three classical problems of trisecting an angle, doubling the cube, and squaring the circle, the Greeks were forced to reach out to new and more complicated instruments. It was the inadequacy in these three problems that helped make mathematicians realize that some aspects of mathematics could not be done with real world instruments and that sometimes it must rely on purely symbolic methods. Yet, some still held on to the belief that the traditional compass and straightedge could answer all their questions and were persistent in their efforts to find the means. However, after almost 2000 years of use, the limitations of the compass and unmarked straight-edge were discovered by one of the most abstract and symbolic areas of mathematics: field theory and abstract algebra. These subjects showed, once and for all, exactly what the time-honored tools could and could not do. And amongst the things they could never do were the three classical problems.
Essay 2) Discuss Euclid's Elements.
Greek mathematics is thought to have reached one of its highest points in the form of Euclid's Elements. For, even though, it is disputed as to how much of the work contained therein is actually original ideas and proofs by Euclid, he did manage to gather together a wide range of knowledge at his time on such topics as planar and solid geometry and number theory. His scheme of organization was a very concise and straightforward format that allowed the ideas to be seen as a logical progression. Further, it was the aesthetic beauty and surety of this logical progression that paved the road to future mathematicians demanding that everything be proved using well-defined objects and agreed-upon axioms.
Essay 3) Give an outline of the history of Greek mathematics from the time of Thales to the collapse of the University of Alexandria.
Starting with Thales in the 6th century BC, Greek mathematics was founded in proof. Soon after Thales in the 5th century BC, the Pythagoreans created a whole society dedicated to mathematics and the logic of numbers. Afterwards, one could say mathematics went into some sort of a cocoon, with it only having minor advances between 500 BC and 300 BC. However, it was after this period that the University of Alexandria was founded and mathematics began to flower starting with Euclid's Elements around 300 BC. Another boost of knowledge came from Archimedes, possibly the greatest mathematician of antiquity, in the 3rd century BC. In addition, the centuries to follow would bring forth other great mathematicians such as Eratoshenes, Appolonius, and Hipparchus. However, the great era of Greek mathematics would come to end somewhat abruptly when in 46 BC Caesar accidentally has the university extremely damaged and badly burned. Moreover, even though the university continued until the Arabs destroyed it in 410 AD, it would only produce commentators of mathematics and ceased to produce new and fruitful ideas.
Essay 4) Briefly give the history of logarithms. How has the use of logarithms changed? Why is it still important for students to be familiar with logarithms?
The first evidence of simple logarithms was in the Jain's Dhavala commentary where it suggests that this Indian culture may have had logarithms but never put them to any practical use. The next indicator of logarithms came in Michael Stifel's Arithmetica integra in 1544 where he suggested the use of arithmetic progressions to understand geometric ones. However, nobody truly invented the logarithm until Napier did in the 17th century. Shortly thereafter Napier and Briggs gave the logarithm its familiar base of ten and proceeded to produce large tables of logarithms of natural numbers. Originally, logarithms were invented as a way of working with large numbers and geometric sequences. It wasn't until much later that they were thought to be the inverse of exponentiation. Logarithms are still important today in the most basic analyses of natural sciences such as biology and medicine.
Essay 5) What were some lost mathematical texts? Why was this possible in the 17th century and before?
There are plenty of examples of lost texts through the ages. Most of these texts, such as all the Elements prior to Euclid's and a large portion of Archimedes's works, come out of antiquity and we only know of them through some of the commentators of the works in later years. In addition, there were probably a myriad of texts burned at the Library of Alexandria that we don't even know existed. Texts continued to be lost right up until the late 15th century when, for example, some of Robert Recorde's texts were lost. The loss of texts, for the most part, can be blamed on the lack of the printing press. Before this invention, books had to be hand-copied and so texts of a sub-par nature were simply not good enough to have multiple copies made. But after the coming of the printing press, it was just as easy to make 100 copies as it was to make 5 copies and so it became less likely for texts to be lost.
Essay 6) Could Fermat, along with or instead of Newton and Leibniz, be considered to be an inventor of calculus?
I believe that Fermat could easily be considered as an inventor of calculus. His ideas on differentiation were new and creative, in contrast to Newton and Leibniz building integration on ideas around since Archimedes. He was the first to use the idea of something that was so close to zero it practically was yet was still okay to divide by. Sure, as rigorous logic goes, this is completely ridiculous, but no one at that time had the key idea of a limit and Fermat's tiny quantities allowed mathematicians to at least build a working model of calculus. Further, he also was to have found integrals of some simple expressions. In a way, I would say that Fermat is an inventor of calculus but not a founder in that he came up with a lot of the key ideas but did not lay them on solid logical footing.
Essay 7) Since so much of calculus was developed by others before Newton and Leibniz, why are they considered to be the inventors of it?
I don't think that Newton and Leibniz are deemed the inventors of calculus for their few creative contributions. Instead, I think Newton and Leibniz are similar to Euclid in that their fame comes from their organization and use of the ideas available. Newton used the calculus to derive an assortment of physics equations, while Leibniz bestowed upon calculus its splendid notation. It is for these reasons, in addition to there original contributions to integration that earned them the recognition they receive today.
Essay 8) Trace the history of integration from the Greeks up to 1700.
The earliest signs of integral calculus came during antiquity with Euxodus' method of exhaustion and then Archimedes' method of equilibrium. The ideas then lay dormant for centuries before Stevin and Valerio used methods similar to Archimedes' in the 16th century. Soon after, Kepler used crude methods of integration to compute the volumes of a number of solids, as well as in the areas involved with his second law. Early in the 17th century, famous mathematicians such as Fermat, Pascal, and Wallis made small advances in the area of integral calculus, along with Barrow proving the Fundamental Theorem of Calculus. Finally, in the late 17th century, both Newton and Leibniz invented what we know today as integral calculus.
Essay 9) How has challenging the axioms lead to mathematical discovery?
Challenging the axioms has lead to some of the major discoveries in mathematics. The oldest challenge was that against Euclid's fifth postulate. This of course gave way to non-Euclidean geometry that was actually later found to be physical reality. Other challenges have led us to ideas of noncommutative algebras and indefinable objects. Now, it seems that challenging the axioms is one of the more popular thing to do. A mathematician will decide to change one key idea and see how long he or she can run with it without reaching a paradox.
Essay 10) Which, if any, of the three philosophies of mathematics listed in the text do you believe?
I personally cannot say that I believe whole-heartedly in any of the three theories. Upon first look, intuitionism seems very attractive. I like the idea of mathematics built upon reality, i.e. starting from the fact that we can see that one thing and another thing is two things and so on. It gives mathematics an overall purpose of understanding the world around us. However, it seems vastly limited in its approach. Logicism also seems appealing since it puts mathematics as a subset of logic and gives it the ability to go to any extent it wishes. Yet, it seems like mathematics should encompass logic and not the other way around since what often seems logical can be shown to be mathematically incorrect. Finally, formalism is neat in that it treats mathematics as a game played out on paper with a bunch of nifty symbols and rules of using them. This idea is attractive, but once again, we find a hole in the game in that there are never enough rules to completely describe mathematics. Therefore, in the end, I find none of the philosophies particularly motivating. They all seem to have their plusses and negatives and that some day man will come to the realization that all philosophies are founded on axioms and mindsets that can be chosen to be true or false and the philosophy will still be sound.
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