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## Example #1 – Mathematics Is A Universal Language

Mathematics is a universal language. Mathematics has made it possible for great advances in science and technology. It has become so important to so many fields, including those as diverse as Psychology, Economics, Medicine, Linguistics, and even History. Mathematics was invented by the Babylonians around four thousand years ago.

Knowing mathematics means being able to use it in a purposeful way. To learn mathematics, students must be engaged in exploring, conjecturing, and thinking rather than only in routine learning procedures. Students construct personal knowledge that derives from meaningful experiences. Therefore, they are much more likely to retain and use what they have learned. Teachers need to understand ways to engage students actively in learning mathematics.

Psychology contains several major propositions and is of value to the mathematics educator. It suggests many general principles of teaching. Mathematics has been influenced by psychology over the years. The methods of teaching have changed as a result of psychology. The Psychology of mathematics is focusing directly on the processes of mathematical thinking and on the ways in which people come to understand the structures. Mathematics is used in empirical science like psychology.

A very important step in general movement with all its push toward more potent rational and creative powers has been the strong focus on identifying the broad understanding, concepts, and generalizations that permeate a given discipline. Mathematics was considered useful in developing powers of logical reasoning, and it has become the major justification for teaching it. Human mental ability is developed, through experience that has developed in the human intellect of certain cognitive structures.

The school has become meaningful because the school of learning takes place only to the degree that new learning is related to existing cognitive structures. New learning can b attached to the new idea if it is incorporated into the learner’s cognitive structure by attaching it to some existing structure and modifying it to accommodate the new idea. Teachers must carefully evaluate children’s concepts and intuitive notions so that new learning can be an extension of existing ideas.

Teachers will probably find that one point of view is more successful than another. Teachers can produce the best results by teaching in a way in which they will believe will be comfortable with them and the students. It is important for the teacher to use the correct strategy. A good method is more important than the topic itself. Teachers do not need to apologize for the way they teach mathematics. The test of the quality of their teaching is a result.

The area of cognitive psychology has a great impact on elementary school and mathematics instruction. It gives an indication of how children are thinking during different periods of mental development. The mental development of a child involves a phenomenon that occurs in similar stages. Children could attain these stages at different ages, but the important point is that every child goes through each stage.

Cognitive development takes place through the mechanism of equilibration. When it is presented with a new idea, the cognitive equilibrium of a child is disturbed. In other words, a child could learn through a process that would him or her expands his or her ability in different situations.

This is when the state of disequilibrium and cognitive development in learning takes place. There are two concepts of equilibration; Assimilation and accommodation. Assimilation is the cognitive activity through which a new idea is brought into a person’s cognitive structure by applying existing cognitive structure to an unfamiliar situation. Accommodation requires the development of schemata. The schemata are developed after a child develops and prepares himself to assimilate a new problem and solution into the cognitive structure. Knowledge of the stages of development is important to understanding.

This gives us more indication of how the child is thinking in a given stage. Teachers must be concerned about the way in which they introduce a new area of study. Teachers need to improve their pedagogical skills. They have to reveal an understanding of how learning takes place. Conceptual development involves intrinsic human needs and behavior may be used to motivate advantage. Teachers should be more concerned with how children learn than they are with how they teach.

Learning mathematics involves much more than an active intellectual process that has become typically the case today. Learning must be meaningful, which means that the learner must be ready to learn. The teacher has the responsibility to build reading, motivate the learner, and constantly encourage the effort of each and every student.

## Example #2

Mathematics has been a part of life since life began and is not about to go away. Rosie O’Donnell has been quoted recently in Newsweek as saying that math should no longer be taught because “we have computers (and) no longer need to know why 3×2y/4.” Although not all mathematics is useful in everyday life there still are an indefinite amount of examples of it being practical and informative.

From tennis, government, and even to a student’s car buying experience, math is an essential skill for everyone. Does it make advancement in today’s marketplace easier and can lead to a more satisfying life. Those with a mathematic type major learn to think more abstractly and find ways to adapt easier. These critical thinking skills are essential to achieve where one wants to go. Our society is the one that?s losing if it loses mathematics. Nonetheless, O’Donnell has a point.

Society is changing to the point where people need to adapt to it. I’m not saying that O’Donnell is right, however, maybe it?s time to change the way society teaches our students. Computers are becoming more and more a part of our lives, whether we like it or not, they’re here to stay.

And why not let them do their jobs? Why not save money that is spent on, sometimes, useless education and spend it on preparing our students for the future. They’re in control, they’re going to be running the country and they deserve all the advantages given to them. Prepare for the future my friends; it?s going to be a world with more reliance on the written word rather than the written number.

## Example #3

Mathematics as we know it is a discipline that each and every one of us requires in our daily lives. Without it, the world would simply be a dangerous place to live. Would it even be possible to carry out most of the currently achieved inventions and discoveries? The answer would simply be a no! Maths has played a big part in shaping the world into its advancements today (Benacerraf, P. 1991). As interesting as it sounds, it has proven to also be challenging to some people but this greatly depends on an individual (Benacerraf, P. 1991).

### Body Paragraph One

To begin with, I would like to greatly expound on how I have been positively influenced to developing an interest in Mathematics. Numbers have always set an ideal environment for me ever since I started learning it. Whenever I first started my basic education, the first thing that influenced me was being taught how to count.

It was very intriguing knowing that I had the ability to count just like every other adult that taught me how to do so. Not too forget, my first mathematics exam was a great success. Being the very best in my math class made me fall deeply in love with maths. I could not take my mind off it. Whenever it comes to this discipline, I have always had a great sense of belonging.

This is mostly because I have always found it to be very unique compared to too many subjects and I have always performed well since the very first time I was introduced to it. Ever since I have always been quite confident, knowing in mind I only need to improve on my speed thinking.

### Body Paragraph Two

Secondly, as I pursue my major in Telecommunications Engineering, it is also important to note the units I will take in Mathematics such as Analytical Geometry as well as Calculus (Puttaswamy, T. K. 2012). These two units will play a vital part in ensuring I safely complete my major and gradually elevate into success. Analytical Geometry is very vital in order to get a greater understanding specifically in Trigonometry as such will influence the sine and cosine form of waves studied in the major. Furthermore, calculus will greatly help me to clearly know how to solve equations given different types of expressions (Stillwell, J. 2010).

In my career, Mathematics will be very important as it will be used to ensure there is accuracy during long mathematical calculations and solutions. Also, it will be used to approach several problems at different angles without limiting oneself to a specific discipline. Despite my studies and career, Mathematics also plays a vital role in my daily life. Such include, carrying simple calculations accurately, future financial planning as well as basic counting that can be applied in the house.

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Last but not the least, I used the internet that offered great assistance when it came to utilizing my Mathematical skills. The site referred to as EdReady has proven to be quite helpful and even a future solution for anybody’s Mathematical problems. The site offered several study topics in well-explained form and in addition to the brilliance was carefully illustrated examples with step by step guidance so that even the least knowledgeable student would be able to grasp some knowledge. In addition to this, there were study questions offered for further practice even when one was offline.

I found this to be quite helpful since it taught me how to know when to think out of the box or think directly as per the requirements of a question. This being one of my weaknesses, I was finally able to put an end to the embarrassing issue and now I can breathe freely and with a clear mind as well. Not to forget to mention, the site was quite easy to navigate even from the very start, guiding one to sign up for an account and up to how to tackle the lessons.

### Conclusion

In conclusion, I would confidently like to mention that Mathematics is a vital discipline in every person’s life. It enables one to have an open mind on how to solve problems because one can approach a problem in math using very many different ways. It also enables one to be alert so as not to commit unnecessary errors and to only aim for accuracy.

To be honest, Mathematics can be the best thing that has ever happened to anyone. One step at a time and it will not take one forever to approach success in Mathematics. Without this discipline, the world would certainly not reap the benefits and would not be a better place to live.

## Example #4 – Mathematical Realism And Its Discontents

Reuben Hersh, a mathematician and mathematics philosopher, believes humans created math. He reasons that math is all in the heads of humans, and is a social phenomenon. According to Hersh math is not physical, not mental, but social. Math to Hersh is a creation of humans that would not be found in other regions of the universe. According to Hersh if there were other life forms out there in the universe they would not have the same math that we have.

Hersh agrees that there could very well be aliens out there in the universe who use mathematics, but he feels that their math would much different than ours. Others would argue and say that math is the same all around the universe. Mathematicians find it mined boggling that Hersh believes math is created. A mathematician can t imagine an alien not knowing two plus two is four.

Hersh argues that all math isn’t based on facts. He says there is no proof that two plus two are four, and when a mathematician argues with him by giving him math rules Hersh replies by saying I find it so astonishing that a good mathematician would so misunderstand the nature of proof.

Hersh goes on and uses many analogies that attempt to disprove mathematics, but mathematicians believe the math is something to be found, not created. Mathematicians describe math as something to observe and find not something to create and construct.

I thought this article to be very interesting. While reading this article I was swayed from one side to the other, both sides had reasonable arguments that kept me doubtful. Math can very much is considered a created thing because we have no way of knowing if it actually applies to areas out of our sight. But as long as math is in our minds we will never get away from it.

As we travel the universe we will still see everything as math, whereas an alien might see some kind of different explanation for everything around them. Altogether math is an explanation for the universe that was created by humans, aliens in the universe might have a better or worse system of explanation but no judgment should be cast on their system. The system of mathematics is similar to the calendar system; there are different systems, but they all deal with the same principles.

## Example #5

Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra.

Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of relations, or as the science that draws necessary conclusions. This latter view encompasses mathematical or symbolic logic, the science of using symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates, and rules for combining and transforming primitive elements into more complex relations and theorems.

This brief survey of the history of mathematics traces the evolution of mathematical ideas and concepts, beginning in prehistory. Indeed, mathematics is nearly as old as humanity itself; evidence of a sense of geometry and interest in the geometric pattern has been found in the designs of prehistoric pottery and textiles and in cave paintings. Primitive counting systems were almost certainly based on using the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today.

### Ancient Mathematics

The earliest records of advanced, organized mathematics date back to the ancient Mesopotamian country of Babylonia and to Egypt of the 3rd millennium BC. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs.

The earliest Egyptian texts, composed about 1800 BC, reveal a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, and so forth), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number.

For example, the symbol for 1 was written five times to represent the number 5, the symbol for 10 was written six times to represent the number 60, and the symbol for 100 was written three times to represent the number 300. Together, these symbols represented the number 365. The addition was done by totaling separately the units-10s, 100s, and so forth-in the numbers to be added. Multiplication was based on successive doublings, and the division was based on the inverse of this process.

The Egyptians used sums of unit fractions (a), supplemented by the fraction B, to express all other fractions. For example, the fraction E was the sum of the fractions 3 and *. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. In geometry, the Egyptians calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of figures such as bricks, cylinders, and pyramids. To find the area of a circle, the Egyptians used the square on U of the diameter of the circle, a value of about 3.16-close to the value of the ratio known as pi, which is about 3.14.

The Babylonian system of numeration was quite different from the Egyptian system. In the Babylonian system which, when using clay tablets, consisted of various wedge-shaped marks-a single wedge indicated 1 and an arrowlike wedge stood for 10 (see table). Numbers up through 59 were formed from these symbols through an additive process, as in Egyptian mathematics. The number 60, however, was represented by the same symbol as 1, and from this point on a positional symbol was used.

That is, the value of one of the first 59 numerals depended henceforth on its position in the total numeral. For example, a numeral consisting of a symbol for 2 followed by one for 27 and ending in one for 10 stood for 2 ? 602 + 27 ? 60 + 10. This principle was extended to the representation of fractions as well, so that the above sequence of numbers could equally well represent 2 ? 60 + 27 + 10 ? (†), or 2 + 27 ? (†) + 10 ? (†-2). With this sexagesimal system (base 60), as it is called, the Babylonians had as convenient a numerical system as the 10-based system.

The Babylonians in time developed a sophisticated mathematics by which they could find the positive roots of any quadratic equation (Equation). They could even find the roots of certain cubic equations. The Babylonians had a variety of tables, including tables for multiplication and division, tables of squares, and tables of compound interest.

They could solve complicated problems using the Pythagorean theorem; one of their tables contains integer solutions to the Pythagorean equation, a2 + b2 = c2, arranged so that c2/a2 decreases steadily from 2 to about J. The Babylonians were able to sum arithmetic and some geometric progressions, as well as sequences of squares. They also arrived at a good approximation for ?. In geometry, they calculated the areas of rectangles, triangles, and trapezoids, as well as the volumes of simple shapes such as bricks and cylinders. However, the Babylonians did not arrive at the correct formula for the volume of a pyramid.

### Greek Mathematics

The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century BC with Thales of Miletus and Pythagoras of Samos, the latter a religious leader who taught the importance of studying numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras.

In the 5th century BC, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle-that is, constructing a square equal in area to a given circle.

Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube-that is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone.

In the latter part of the 5th century BC, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, ?, would today be called irrational.)

As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century BC mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations.

Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century BC on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes.

The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes’ writings on them are the earliest to survive.

Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist Ren? Descartes in the 17th century.

After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century AD show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers.

Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century AD. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis).

### Applied Mathematics in Greece

Paralleling the studies described in pure mathematics were studies made in optics, mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system for recording fractions and, at about the same time, composed tables of chords in a circle. For a circle of some fixed radius, such tables give the length of the chords subtending a sequence of arcs increasing by some fixed amount. They are equivalent to a modern sine table, and their composition marks the beginnings of trigonometry.

In the earliest such tables-those of Hipparchus in about 150 BC-the arcs increased by steps of 71?, from 0? to 180?. By the time of the astronomer Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his Almagest a table of chords in a circle for steps of 3?, which, although expressed sexagesimal, is accurate to about five decimal places.

In the meantime, methods were developed for solving problems involving plane triangles, and a theorem-named after the astronomer Menelaus of Alexandria-was established for finding the lengths of certain arcs on a sphere when other arcs are known. These advances gave Greek astronomers what they needed to solve the problems of spherical astronomy and to develop an astronomical system that held sway until the time of the German astronomer Johannes Kepler.

### Medieval and Renaissance Mathematics

Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centers of Greek learning. The preservation of such works as have survived to modern times began with this tradition. It was continued in the Islamic world, where original developments based on these masterpieces first appeared.

### Islamic and Indian Mathematics

After a century of expansion in which the religion of Islam spread from its beginnings in the Arabian Peninsula to dominate an area extending from Spain to the borders of China, Muslims began to acquire the results of the ?foreign sciences.? At centers such as the House of Wisdom in Baghdad, supported by the ruling caliphs and wealthy individuals, translators produced Arabic versions of Greek and Indian mathematical works.

By the year 900 AD the acquisition was complete, and Muslim scholars began to build on what they had acquired. Thus mathematicians extended the Hindu decimal positional system of arithmetic from whole numbers to include decimal fractions, and the 12th-century Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the algebra of polynomials of Muhammad al-Khwarizmi.

Al-Karaji included polynomials with an infinite number of terms. (Al-Khwarizmi’s name, incidentally, is the source of the word algorithm, and the title of one of his books is the source of the word algebra.) Geometers such as Ibrahim ibn Sinan continued Archimedes’ investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve optical problems.

Using the Hindu sine function and Menelaus’s theorem, mathematicians from Habas al-Hasib to Nasir ad-Din at-Tusi created the mathematical disciplines of the plane and spherical trigonometry. These did not become mathematical disciplines in the West, however, until the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. The Latin West acquired much of this learning during the 12th century, the great century of translation. Together with translations of the Greek classics, these Muslim works were responsible for the growth of mathematics in the West during the late Middle Ages. Italian mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for their knowledge.

### Western Renaissance Mathematics

Although the late medieval period saw some fruitful mathematical considerations of problems of infinity by writers such as Nicole Oresme, it was not until the early 16th century that a truly important mathematical discovery was made in the West. The discovery, algebraic formula for the solution of both the cubic and quartic equations, was published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The discovery drew the attention of mathematicians to complex numbers and stimulated a search for solutions to equations of degree higher than 4.

It was this search, in turn, that led to the first work on group theory (Group) at the end of the 18th century, and to the theory of equations developed by the French mathematician “variste Galois in the early 19th century.

The 16th century also saw the beginnings of modern algebraic symbolism (Mathematical Symbols), as well as the remarkable work on the solution of equations by the French mathematician Fran’ois Vi’te. His writings influenced many mathematicians of the following century, including Pierre de Fermat in France and Isaac Newton in England.

### Mathematics Since the 16th Century

Europeans dominated in the development of mathematics after the Renaissance.

### 17th Century

During the 17th century, the greatest advances were made in mathematics since the time of Archimedes and Apollonius. The century opened with the discovery of logarithms by the Scottish mathematician John Napier, whose continued utility prompted the French astronomer Pierre Simon Laplace to remark, almost two centuries later, that Napier, by halving the labors of astronomers, had doubled their lifetimes. (Although the logarithmic function is still important in mathematics and the sciences, logarithmic tables and their instrumental form-slide rules-are of much less practical use today because of electronic calculators.)

The science of number theory, which had lain dormant since the medieval period, illustrates the 17th-century advances built on ancient learning. It was Arithmetica by Diophantus that stimulated Fermat to advance the theory of numbers greatly. His most important conjecture in the field, written in the margin of his copy of the Arithmetica, was that no solutions exist to an + bn = cn for positive integers a, b, and c when n is greater than 2. This conjecture stimulated much important work in algebra and number theory but is still unproven.

Two important developments in pure geometry occurred during the century. The first was the publication, in Discourse on Method (1637) by Descartes, of his discovery of analytic geometry, which showed how to use the algebra that had developed since the Renaissance to investigate the geometry of curves. (Fermat made the same discovery but did not publish it.) This book, together with short treatises that had been published with it, stimulated and provided the basis for Isaac Newton’s mathematical work in the 1660s.

The second development in geometry was the publication by the French engineer G’rard Desargues in 1639 of his discovery of projective geometry. Although the work was much appreciated by Descartes and the French philosopher and scientist Blaise Pascal, its eccentric terminology and the excitement of the earlier publication of analytic geometry delayed the development of its ideas until the early 19th century and the works of the French mathematician Jean-Victor Poncelet.

Another major step in mathematics in the 17th century was the beginning of probability theory in the correspondence of Pascal and Fermat on a problem in gambling, called the problem of points. This unpublished work stimulated the Dutch scientist Christiaan Huygens to publish a small tract on probabilities in dice games, which was reprinted by the Swiss mathematician Jakob Bernoulli in his Art of Conjecturing.

Both Bernoulli and the French mathematician Abraham De Moivre, in his Doctrine of Chances in 1718, applied the newly discovered calculus to make rapid advances in the theory, which by then had important applications in the rapidly developing insurance industry.

Without question, however, the crowning mathematical event of the 17th century was the discovery by Sir Isaac Newton, between 1664 and 1666, of differential and integral calculus (Calculus). In making this discovery, Newton built on earlier work by his fellow Englishmen John Wallis and Isaac Barrow, as well as on work of such Continental mathematicians as Descartes, Francesco Bonaventura Cavalieri, Johann van Waveren Hudde, and Gilles Personne de Roberval. About eight years later than Newton, who had not yet published his discovery, the German Gottfried Wilhelm Leibniz rediscovered calculus and published first, in 1684 and 1686. Leibniz’s notation systems, such as dx, are used today in calculus.

### 18th Century

The remainder of the 17th century and a good part of the 18th were taken up by the work of disciples of Newton and Leibniz, who applied their ideas to solving a variety of problems in physics, astronomy, and engineering. In the course of doing so, they also created new areas of mathematics. For example, Johann and Jakob Bernoulli invented the calculus of variations, and French mathematician Gaspard Monge invented differential geometry.

Also in France, Joseph Louis Lagrange gave a purely analytic treatment of mechanics in his great Analytical Mechanics (1788), in which he stated the famous Lagrange equations for a dynamical system. He contributed to differential equations and number theory as well, and he originated the theory of groups. His contemporary, Laplace, wrote the classic Celestial Mechanics (1799-1825), which earned him the title the French Newton, and The Analytic Theory of Probabilities (1812).

The greatest mathematician of the 18th century was Leonhard Euler, a Swiss, who made basic contributions to calculus and to all other branches of mathematics, as well as to the applications of mathematics. He wrote textbooks on calculus, mechanics, and algebra that became models of style for writing in these areas. The success of Euler and other mathematicians in using calculus to solve mathematical and physical problems, however, only accentuated their failure to develop a satisfactory justification of its basic ideas.

That is, Newton’s own accounts were based on kinematics and velocities, Leibniz’s explanation was based on infinitesimals, and Lagrange’s treatment was purely algebraic and founded on the idea of infinite series. All these systems were unsatisfactory when measured against the logical standards of Greek geometry, and the problem was not resolved until the following century.

### 19th Century

In 1821 a French mathematician, Augustin Louis Cauchy, succeeded in giving a logically satisfactory approach to calculus. He based his approach only on finite quantities and the idea of a limit. This solution posed another problem, however; that of a logical definition of “real number.” Although Cauchy’s explanation of calculus rested on this idea, it was not Cauchy but the German mathematician Julius W. R. Dedekind who found a satisfactory definition of real numbers in terms of the rational numbers. This definition is still taught, but other definitions were given at the same time by the German mathematicians Georg Cantor and Karl T. W. Weierstrass.

A further important problem, which arose out of the problem-first stated in the 18th century-of describing the motion of a vibrating string, was that of defining what is meant by function. Euler, Lagrange, and the French mathematician Jean Baptiste Fourier all contributed to the solution, but it was the German mathematician Peter G. L. Dirichlet who proposed the definition in terms of correspondence between elements of the domain and the range. This is the definition that is found in texts today.

In addition to firming the foundations of analysis, as the techniques of the calculus were by then called, mathematicians of the 19th century made great advances in the subject. Early in the century, Carl Friedrich Gauss gave a satisfactory explanation of complex numbers, and these numbers then formed a whole new field for analysis, one that was developed in the work of Cauchy, Weierstrass, and the German mathematician Georg F. B. Riemann.

Another important advance in the analysis was Fourier’s study of infinite sums in which the terms are trigonometric functions. Known today as the Fourier series, they are still powerful tools in pure and applied mathematics.

In addition, the investigation of which functions could be equal to Fourier series led Cantor to the study of infinite sets and to the arithmetic of infinite numbers. Cantor’s theory, which was considered quite abstract and even attacked as a “disease from which mathematics will soon recover,” now forms part of the foundations of mathematics and has more recently found applications in the study of turbulent flow in fluids.

A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry. In non-Euclidean geometry, more than one parallel can be drawn to a given line through a given point, not on the line. Evidently this was discovered first by Gauss, but Gauss was fearful of the controversy that might result from publication.

The same results were rediscovered independently and published by the Russian mathematician Nikolay Ivanovich Lobachevsky and the Hungarian J?nos Bolyai. Non-Euclidean geometries were studied in a very general setting by Riemann with his invention of manifolds and, since the work of Einstein in the 20th century, they have also found applications in physics.

Gauss was one of the greatest mathematicians who ever lived. Diaries from his youth show that this infant prodigy had already made important discoveries in number theory, an area in which his book Disquisitiones Arithmeticae (1801) marks the beginning of the modern era. While only 18, Gauss discovered that a regular polygon with m sides can be constructed by straightedge and compass when m is a power of 2 times distinct primes of the form 2n + 1.

In his doctoral dissertation, he gave the first satisfactory proof of the fundamental theorem of algebra. Often he combined scientific and mathematical investigations. Examples include his development of statistical methods along with his investigations of the orbit of a newly discovered planetoid; his founding work in the field of potential theory, along with the study of magnetism; and his study of the geometry of curved surfaces in tandem with his investigations of surveying.

Of more importance for algebra itself than Gauss’s proof of its fundamental theorem was the transformation of the subject during the 19th century from a study of polynomials to a study of the structure of algebraic systems. A major step in this direction was the invention of symbolic algebra in England by George Peacock. Another was the discovery of algebraic systems that have many but not all, of the properties of the real numbers.

Such systems include the quaternions of the Irish mathematician William Rowan Hamilton, the vector analysis of the American mathematician and physicist J. Willard Gibbs, and the ordered n-dimensional spaces of the German mathematician Hermann G’nther Grassmann. A third major step was the development of group theory from its beginnings in the work of Lagrange. Galois applied for this work deeply to provide a theory of when polynomials may be solved by an algebraic formula.

Just as Descartes had applied the algebra of his time to the study of geometry, so the German mathematician Felix Klein and the Norwegian mathematician Marius Sophus Lie applied the algebra of the 19th century. Klein applied it to the classification of geometries in terms of their groups of transformations (the so-called Erlanger Programm), and Lie applied it to the geometric theory of differential equations by means of continuous groups of transformations known as Lie groups. In the 20th century, algebra has also been applied to a general form of geometry known as topology.

Another subject that was transformed in the 19th century, notably by the Laws of Thought (1854), by the English mathematician George Boole and by Cantor’s theory of sets, was the foundations of mathematics (Logic). Toward the end of the century, however, a series of paradoxes were discovered in Cantor’s theory. One such paradox, found by English mathematician Bertrand Russell, aimed at the very concept of a set ( Set Theory). Mathematicians responded by constructing set theories sufficiently restrictive to keep the paradoxes from arising.

They left open the question, however, of whether other paradoxes might arise in this restricted theories-that is, whether the theories were consistent. As of the present time, only relative consistency proofs have been given. (That is, theory A is consistent if theory B is consistent.) Particularly disturbing is the result, proved in 1931 by the American logician Kurt G?del, that in any axiom system complicated enough to be interesting to most mathematicians, it is possible to frame propositions whose truth cannot be decided within the system.

### Current Mathematics

At the International Conference of Mathematicians held in Paris in 1900, the German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at G?ttingen, the former academic home of Gauss and Riemann. He had contributed to most areas of mathematics, from his classic Foundations of Geometry (1899) to the jointly authored Methods of Mathematical Physics. Hilbert’s address at G’ttingen was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century.

These problems have indeed stimulated a great deal of the mathematical research of the century. When news breaks that another of the “Hilbert problems” has been solved, mathematicians all over the world await the details of the story with impatience.

Important as these problems have been, an event that Hilbert could not have foreseen seems destined to play an even greater role in the future development of mathematics-namely, the invention of the programmable digital computer (Computer). Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the 17th century, it was Charles Babbage in 19th-century England who designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape.

Babbage’s imagination outran the technology of his day, however, and it was not until the invention of the relay, then of the vacuum tube, and then of the transistor, that large-scale, programmed computation became feasible.

This development has given great impetus to areas of mathematics such as numerical analysis and finite mathematics. It has suggested new areas for mathematical investigation, such as the study of algorithms. It has also become a powerful tool in areas as diverse as number theory, differential equations, and abstract algebra.

In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois.

Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.

## Example #6

Mathematical logic is something that has been around for a very long time. Centuries Ago Greek and other logicians tried to make sense out of mathematical proofs. As time went on other people tried to do the same thing but using only symbols and variables. But I will get into detail about that a little later. There is also something called set theory, which is related to this. In mathematical logic, a lot of terms are used such as axioms and proofs. A lot of things in math can be proven, but there are still some things that will probably always remain theories or ideas.

Mathematical Logic is something that has a very long history behind it. It has been debated for many centuries. If someone were to divide mathematical logic into groups they would get two major groups. Both groups are very long.

One is called “The history of formal deduction” and it goes all the way back to Aristotle and Euclid and other people who lived at that time. The other is “the history of mathematical analysis” which goes back to the times of Archimedes, who was in the same era as Aristotle and Euclid. These to groups or streams were separate for a long time until Newton invented Calculus, which brought Math and logic together.

Somebody who studies mathematical logic and gives his or her own concepts about it is called a logician. Some well-known logicians include Boole and Frege. They were trying to give a definite form to what formal deduction really was. Aristotle had already done such a thing but he had done it with language, Boole wanted to do it with only Symbols. Frege came up with “Predicate Calculus”.

As time went on people did not make new theories as much as they used to in the time of Aristotle. They mostly concentrated on expanding on theories that have been said centuries ago, proving those theories or putting them into symbolic form.

Words that have to do with logic like and, or, not are given symbols like &, V, or an upside-down L reversed. The Letters X, Y, Z and so on are commonly used as variables and P, Q, R are used as predicates, properties or relations.

Sometimes there are theories that have to do with machines that do not exist and usually have things in them that are infinite and they usually work with letters and numbers. For example in Chapter 4 which is “Turning Machines and recursive Functions” it talks about a machine that has a tape running through it. (This is not a real machine)

The tape is endless from both sides. It is divided into little squares and in each square, there is a small letter of the alphabet and a number under it. The machine reads this, changes it moves it one to the right ort one to the left. This experiment was conducted in 1936.

Model theory is the study of different formal languages and their relations with each other. They get a normal sentence and they turn it into variables and symbols, then they compare it with other languages. For example, if you take:

If the boss is in charge and Joe is the boss, then Joe is in charge.

If you convert that you get:

P(b)&E(j,b)–*P(j).

This gets very confusing but the way they get the formula is that they do something with the “Predicate Letter”. Then they turn it into variables and symbols.

In one section it talks about how there is an infinite number of fractions between two rational numbers on the number line such as 1 and two. Does it go like 1/3 1/5 … and so on infinitely. It also compares that to a different kind of number line.

G?del was a different logician then the rest. He concentrated more on expanding on other people’s theories than anything else. Although he has said many good theories himself. That is why many people consider him one of the best logicians and a very good mathematician.

To conclude I want to say that I found this book very difficult and I understood about 20% of it so I read the whole thing and wrote about the parts I understood. It is a 77-page book and it is very interesting.

In one part it said something like -I assume you have some sort of knowledge in Predicate calculus. I have always found Mathematical Logic very interesting and I hope to study it much more in the future.