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    THE NATURE OF MATHEMATICS

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    THE NATURE OF MATHEMATICS

    Post by Admin on Wed Feb 11, 2009 12:00 pm

    THE NATURE OF MATHEMATICS

    Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest. For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge. For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work. Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.
    This chapter focuses on mathematics as part of the scientific endeavor and then on mathematics as a process, or way of thinking. Recommendations related to mathematical ideas are presented in Chapter 9, The Mathematical World, and those on mathematical skills are included in Chapter 12, Habits of Mind.

    PATTERNS AND RELATIONSHIPS

    Mathematics is the science of patterns and relationships. As a theoretical discipline, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world. The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In addressing, say, "Does the interval between prime numbers form a pattern?" as a theoretical question, mathematicians are interested only in finding a pattern or proving that there is none, but not in what use such knowledge might have. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world.
    A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians, like other scientists, are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest elaborateness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof. As mathematics has progressed, more and more relationships have been found between parts of it that have been developed separately—for example, between the symbolic representations of algebra and the spatial representations of geometry. These cross-connections enable insights to be developed into the various parts; together, they strengthen belief in the correctness and underlying unity of the whole structure.
    Mathematics is also an applied science. Many mathematicians focus their attention on solving problems that originate in the world of experience. They too search for patterns and relationships, and in the process they use techniques that are similar to those used in doing purely theoretical mathematics. The difference is largely one of intent. In contrast to theoretical mathematicians, applied mathematicians, in the examples given above, might study the interval pattern of prime numbers to develop a new system for coding numerical information, rather than as an abstract problem. Or they might tackle the area/volume problem as a step in producing a model for the study of crystal behavior.
    The results of theoretical and applied mathematics often influence each other. The discoveries of theoretical mathematicians frequently turn out—sometimes decades later—to have unanticipated practical value. Studies on the mathematical properties of random events, for example, led to knowledge that later made it possible to improve the design of experiments in the social and natural sciences. Conversely, in trying to solve the problem of billing long-distance telephone users fairly, mathematicians made fundamental discoveries about the mathematics of complex networks. Theoretical mathematics, unlike the other sciences, is not constrained by the real world, but in the long run it contributes to a better understanding of that world.

    MATHEMATICS, SCIENCE, AND TECHNOLOGY

    Because of its abstractness, mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences. The relationship between mathematics and the other fields of basic and applied science is especially strong. This is so for several reasons, including the following:
    • The alliance between science and mathematics has a long history, dating back many centuries. Science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analyzing data. Often, abstract patterns that have been studied for their own sake by mathematicians have turned out much later to be very useful in science. Science and mathematics are both trying to discover general patterns and relationships, and in this sense they are part of the same endeavor.
    • Mathematics is the chief language of science. The symbolic language of mathematics has turned out to be extremely valuable for expressing scientific ideas unambiguously. The statement that a=F/m is not simply a shorthand way of saying that the acceleration of an object depends on the force applied to it and its mass; rather, it is a precise statement of the quantitative relationship among those variables. More important, mathematics provides the grammar of science—the rules for analyzing scientific ideas and data rigorously.
    • Mathematics and science have many features in common. These include a belief in understandable order; an interplay of imagination and rigorous logic; ideals of honesty and openness; the critical importance of peer criticism; the value placed on being the first to make a key discovery; being international in scope; and even, with the development of powerful electronic computers, being able to use technology to open up new fields of investigation.
    • Mathematics and technology have also developed a fruitful relationship with each other. The mathematics of connections and logical chains, for example, has contributed greatly to the design of computer hardware and programming techniques. Mathematics also contributes more generally to engineering, as in describing complex systems whose behavior can then be simulated by computer. In those simulations, design features and operating conditions can be varied as a means of finding optimum designs. For its part, computer technology has opened up whole new areas in mathematics, even in the very nature of proof, and it also continues to help solve previously daunting problems.

    MATHEMATICAL INQUIRY

    Using mathematics to express ideas or to solve problems involves at least three phases: (1) representing some aspects of things abstractly, (2) manipulating the abstractions by rules of logic to find new relationships between them, and (3) seeing whether the new relationships say something useful about the original things.
    Abstraction and Symbolic Representation
    Mathematical thinking often begins with the process of abstraction—that is, noticing a similarity between two or more objects or events. Aspects that they have in common, whether concrete or hypothetical, can be represented by symbols such as numbers, letters, other marks, diagrams, geometrical constructions, or even words. Whole numbers are abstractions that represent the size of sets of things and events or the order of things within a set. The circle as a concept is an abstraction derived from human faces, flowers, wheels, or spreading ripples; the letter A may be an abstraction for the surface area of objects of any shape, for the acceleration of all moving objects, or for all objects having some specified property; the symbol + represents a process of addition, whether one is adding apples or oranges, hours, or miles per hour. And abstractions are made not only from concrete objects or processes; they can also be made from other abstractions, such as kinds of numbers (the even numbers, for instance).
    Such abstraction enables mathematicians to concentrate on some features of things and relieves them of the need to keep other features continually in mind. As far as mathematics is concerned, it does not matter whether a triangle represents the surface area of a sail or the convergence of two lines of sight on a star; mathematicians can work with either concept in the same way. The resulting economy of effort is very useful—provided that in making an abstraction, care is taken not to ignore features that play a significant role in determining the outcome of the events being studied.
    Manipulating Mathematical Statements
    After abstractions have been made and symbolic representations of them have been selected, those symbols can be combined and recombined in various ways according to precisely defined rules. Sometimes that is done with a fixed goal in mind; at other times it is done in the context of experiment or play to see what happens. Sometimes an appropriate manipulation can be identified easily from the intuitive meaning of the constituent words and symbols; at other times a useful series of manipulations has to be worked out by trial and error.
    Typically, strings of symbols are combined into statements that express ideas or propositions. For example, the symbol A for the area of any square may be used with the symbol s for the length of the square's side to form the proposition A = s2. This equation specifies how the area is related to the side—and also implies that it depends on nothing else. The rules of ordinary algebra can then be used to discover that if the length of the sides of a square is doubled, the square's area becomes four times as great. More generally, this knowledge makes it possible to find out what happens to the area of a square no matter how the length of its sides is changed, and conversely, how any change in the area affects the sides.
    Mathematical insights into abstract relationships have grown over thousands of years, and they are still being extended—and sometimes revised. Although they began in the concrete experience of counting and measuring, they have come through many layers of abstraction and now depend much more on internal logic than on mechanical demonstration. In a sense, then, the manipulation of abstractions is much like a game: Start with some basic rules, then make any moves that fit those rules—which includes inventing additional rules and finding new connections between old rules. The test for the validity of new ideas is whether they are consistent and whether they relate logically to the other rules.
    Application
    Mathematical processes can lead to a kind of model of a thing, from which insights can be gained about the thing itself. Any mathematical relationships arrived at by manipulating abstract statements may or may not convey something truthful about the thing being modeled. For example, if 2 cups of water are added to 3 cups of water and the abstract mathematical operation 2+3 = 5 is used to calculate the total, the correct answer is 5 cups of water. However, if 2 cups of sugar are added to 3 cups of hot tea and the same operation is used, 5 is an incorrect answer, for such an addition actually results in only slightly more than 4 cups of very sweet tea. The simple addition of volumes is appropriate to the first situation but not to the second—something that could have been predicted only by knowing something of the physical differences in the two situations. To be able to use and interpret mathematics well, therefore, it is necessary to be concerned with more than the mathematical validity of abstract operations and to also take into account how well they correspond to the properties of the things represented.
    Sometimes common sense is enough to enable one to decide whether the results of the mathematics are appropriate. For example, to estimate the height 20 years from now of a girl who is 5' 5" tall and growing at the rate of an inch per year, common sense suggests rejecting the simple "rate times time" answer of 7' 1" as highly unlikely, and turning instead to some other mathematical model, such as curves that approach limiting values. Sometimes, however, it may be difficult to know just how appropriate mathematical results are—for example, when trying to predict stock-market prices or earthquakes.
    Often a single round of mathematical reasoning does not produce satisfactory conclusions, and changes are tried in how the representation is made or in the operations themselves. Indeed, jumps are commonly made back and forth between steps, and there are no rules that determine how to proceed. The process typically proceeds in fits and starts, with many wrong turns and dead ends. This process continues until the results are good enough.
    But what degree of accuracy is good enough? The answer depends on how the result will be used, on the consequences of error, and on the likely cost of modeling and computing a more accurate answer. For example, an error of 1 percent in calculating the amount of sugar in a cake recipe could be unimportant, whereas a similar degree of error in computing the trajectory for a space probe could be disastrous. The importance of the "good enough" question has led, however, to the development of mathematical processes for estimating how far off results might be and how much computation would be required to obtain the desired degree of accuracy.[b]
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    Dragspol

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    Re: THE NATURE OF MATHEMATICS

    Post by Dragspol on Mon Mar 16, 2009 6:41 pm

    I think that in the world we need maths to do everything, like in te computer, it needs millions of numbers in the microprocessor to work in order, to do an spacial trip or anything.
    Maths is vey simple when you understand it and some simbols are difficult to understand but if you learn it is going to be very easy. We need maths to do everything, in an exam or in a scientist probe.
    Maths is a science that everyone can learn and know.
    It is used to count apples, cars or to know how fast do you go in the transport you are.he circle as a concept is an abstraction derived from human faces, flowers, wheels, or spreading ripples; the letter A may be an abstraction for the surface area of objects of any shape, for the acceleration of all moving objects, or for all objects having some specified property; the symbol + represents a process of addition, whether one is adding apples or oranges, hours, or miles per hour.
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    Dondi

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    Re: THE NATURE OF MATHEMATICS

    Post by Dondi on Tue May 05, 2009 6:55 pm

    Not only the professional mathematicians or scientists can us the math in their lives, i am a girl, yes a seventh grade girl but i love math and we can use this play for our lives, math is a game with rules, this is MY favorite game, this is our world and we did it together, but how?, i really think that this is our world because without our work to do better our world we cant have this, this is our creation, our math creation, if you think about what i am saying just for a minute, you can see around you and all the things are math are the creation of math. In other words we can say that the buildings and houses are geometry or our problems, like in the supermarket when you need to pay or when you need to count something we can say that you are using algebra or just in your house, this is the moment to use the mathematics in a good action, we always use the math in a good form but i know that we can do it better.
    The technology is one of the most important thing to create something in this time but nothing is going to change what means math, because we can say that without the math can not exist the technology, why? well because mathematics provides the grammar of science.
    If we doesnt take care of our world we can put our future in problems, now we need to take care of it, we need to take care of it because this is not a good time to be contaminating the world, this is the perfect moment to take care of our actions, i know that the technology is very important to our future but if we put everything in order this can be a better world, the world need balance with the different of technology and trash.
    The math is to express ideas, so if the math is the beautiful game that we are saying, why we need to change this game? no, we dont need to, because this is game that everybody play since the people know, so it was and it is a beautiful game, we dont need to change it, we can play this still we know that we wins the game!! prove this, math is our world. Very Happy

    Selezu

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    Re: THE NATURE OF MATHEMATICS

    Post by Selezu on Tue May 05, 2009 8:02 pm

    Hy i am a student of seventh grade,and i thing that match is very important.Because all the time we ned match,for example in investigatigation of:doctors,scientifics ext.I think that match some topics are very difficults.Bud others no for example Z+ and Z- jaja.So my conclucion is that match we use a lot of time because the match are like a tree.Because we have to use all the time. Smile

    Fer2995

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    Re: THE NATURE OF MATHEMATICS

    Post by Fer2995 on Wed May 06, 2009 7:50 pm

    I believe that mathematics is the most important thing in life that is all that is around us because of its shape and size, some people do not like but if I want my practice and I have to study architecture and I love to get to be but I have to know math.
    I think if you take the math because it would help us in school and in the future of one there to learn why and the scientists who created it is created beautiful art, a way of life and you can build the future and also help us ok thanks ... Very Happy

    Pia

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    Re: THE NATURE OF MATHEMATICS

    Post by Pia on Wed May 06, 2009 10:47 pm

    Exclamation i think that math is very important in life because in every stuff we need math if we go to buy something we need to count our money to see if we can buy it.
    we can't life with out math... is very important for our lives..
    we need math eaven if we dont like math so!! lets put our brain to work and study math!!

    Luisk

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    Re: THE NATURE OF MATHEMATICS

    Post by Luisk on Thu May 07, 2009 6:31 pm

    Hy I only think that the matematic is one of the most important thing in this world because if we dont know them, I thing that our live will be a completely desorder, this is important in many uses like one example is that if we want to buy or to sell we in one moment we with use theme it can be uses to give a change or some thign like that.
    it´s beter to love this important subject,it will chage our main if we put our brain to think a little. For me this subject is very easy the only thing is know how to use them

    Gabi

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    Re: THE NATURE OF MATHEMATICS

    Post by Gabi on Thu May 07, 2009 6:38 pm

    I think mathematics is very important to our lives, because we have to use it everyday, since the moment we wake up at the end of the night. I really dont like math but its very functional to our lives, and our future.
    We, all of us think that mathematics is boring, but in the end of our lives we are gonna see how important they it is.

    Toño

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    Re: THE NATURE OF MATHEMATICS

    Post by Toño on Thu May 07, 2009 8:12 pm

    well hi... i think mathematics is very useful in our lives. we just need it to do all things its very important well i do not like so much mathematics but i need to know it because in future i will probably use it, it will help me too much well bye Very Happy

    Downhill

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    Re: THE NATURE OF MATHEMATICS

    Post by Downhill on Thu May 07, 2009 8:41 pm

    I think mathematics is a very impotant thing in our lifes.
    We use mathematics every day when we buy something we need to know the price to count our money.
    Mathematics is something that Ilike very much!!!!

    Mitsunari34

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    Re: THE NATURE OF MATHEMATICS

    Post by Mitsunari34 on Thu May 07, 2009 10:11 pm

    I think that math is very important because, we all need it in most of the normal activities, without math we cant know how much we need to pay in a super market or we cant know the answer of a simple multiplication.Any way without math we cant live.
    I like math because it makes me use my brain a lot and it will help me to resolve a lot of problems in my hole life. afro lol!

    Pana800

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    Re: THE NATURE OF MATHEMATICS

    Post by Pana800 on Fri May 08, 2009 3:27 pm

    hi, I think that math is pretty important for us because every day we have to use them, as when we buy something we have to count the money, without the math all would be more difficult,I like mathematics because it is a subject that helps me to resolve a lot of things Very Happy

    without mathematics our life would be a disaster study Shocked

    chuchis0614

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    Re: THE NATURE OF MATHEMATICS

    Post by chuchis0614 on Fri May 08, 2009 4:25 pm

    Hi!!! Very Happy I am a 8 grade girl, after I read this article about the Nature Of Mathmatics!!! I realize that the math is very important in our lifes... When I was little, I sincerely hate math, and I think I hated Yestreday, before reading this article....jajaja, but know I have a different perception of this interesting subject! Smile As I said before, now math is a very interesting subject for me... This subject has a very important roll in our lifes, because if you start to study all our activities you will find that math is super mega duper important for us!!! Laughing That is What I think about math!!!


    Last edited by chuchis0614 on Fri May 08, 2009 4:31 pm; edited 1 time in total (Reason for editing : wrong words)

    Nana

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    Re: THE NATURE OF MATHEMATICS

    Post by Nana on Fri May 08, 2009 6:44 pm

    I think that math is very important to our lives because we need it to do diferent things and on a future if we need a job they are gonna ask you for some mathematics things that you will know because you al ready had study that; maybe you will forget some things but others you are gonna remember for the rest of your life and that is what I think about math!!!!!!!!!

    Sebas

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    Re: THE NATURE OF MATHEMATICS

    Post by Sebas on Fri May 08, 2009 7:46 pm

    I think that math is very important to our lives because we need it to do diferent things.!as when we buy something we have to count the money, without the math all would be more difficult,I like mathematics because it is a subject that helps me to resolve a lot of things....
    without math we cant know how much we need to pay in a super market or we cant know the answer of a simple multiplication.Any way without math we cant live.
    I like math because it makes me use my brain a lot and it will help me to resolve a lot of problems in my hole life. Very Happy Smile

    Marlon

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    Re: THE NATURE OF MATHEMATICS

    Post by Marlon on Fri May 08, 2009 9:14 pm

    hi pro! well it seems natural that the majority of the population almost does not know everything on the mathematics and that its relation with them is limited the four rules. This spacing contrasts with the importance that the mathematics have today in the society. The mathematics are in center of our culture and its history is confused, often, with the one of the philosophy. Equally that the cosmological theories and of the evolution they have exerted remarkable influence in the conception that the humans we have of we ourself, nonEuclidean geometries have allowed new ideas on the universe and the theorems of the mathematical logic have shown the limitations of the deductive method. Also in the art there are mathematics. Ever since Pitágoras, the most famous mathematician, discovered numerical reasons in the musical harmony until now the relation of the mathematics with the art has been permanent. These aspects of the mathematics turn into bridge between the humanities and sciences of the nature, between the two cultures of which Snow spoke. We used them to the mathematics in the daily life and are necessary to include/understand and to analyze the abundant information that arrives to us. But its use goes much more there: in practically all the branches of the human knowledge one resorts to mathematical models, and not only in the physics, but thanks to the computers the mathematics are applied to all the disciplines, so that they are in the base of engineerings, of the technologies more outposts, like those of the space flights, the modern techniques of medical diagnosis, like the computerised axial tomography, of meteorology, of the financial studies, the genetic engeneering… But the mathematics are a pure science, whose problems by themselves suppose a naked challenge for intelligence; Jacobi thought that mathematical the unique purpose of the era to award honor to the human spirit. Its language less turns universal them into effective tool for the cooperation between countries and developed more, to favor a scope of collaboration that improves the coexistence and to foment La Paz between the towns. The mathematics have, for twenty-five centuries, an excellent paper in the intellectual education of youth. The mathematics are logical, precision, rigor, abstraction, formalization and beauty, and it is hoped that through those qualities they are reached the capacity to discern the essential of the accessory, the esteem by the intellectually beautiful work and the valuation of the potential of science. All the scholastic matters must contribute to the culture and development of intelligence, the feelings and the personality, but to the mathematics an outstanding place in the formation of intelligence corresponds since, as she indicated Aristotle, the young people can become mathematical very capable, but cannot be wise in other sciences

    Nany

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    Re: THE NATURE OF MATHEMATICS

    Post by Nany on Sat May 09, 2009 7:45 pm

    Very Happy I think math is very important because it help us to understand a little bit more about the fantastic and strange things that happens on earth. Like gravity, and some stuffs about technology. Because without math we couldn't have all these techonoly that we have nowdays.And i think that math helps everybody not only on finding things, but also in your life because without it we will be in a continuous economic conflict. Finally i think that math is the best thing that the human being has discovered even though we don't like to study ir or practice it!

    Pau

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    Re: THE NATURE OF MATHEMATICS

    Post by Pau on Sat May 09, 2009 7:47 pm

    Hey!! Well, I think that the mathematics are very important for us, because they help us in our daily life, day after day with our college, when we are out of the college also. Always it is going to help us in our future, for the career that we are going to choose, if we go shopping jajaja, or to do other things day after day!! Smile

    Ignacillo

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    Re: THE NATURE OF MATHEMATICS

    Post by Ignacillo on Sun May 10, 2009 10:12 am

    scratch well I think that math is really useful in our life,because come on we need math in everything.We need it if we are going to buy something,when I get a job,using the computer,etc scratch I know that everybody in all the answers add the same lol! but come on is the truth...
    -I love math is like a game for me,this what I think about math... study

    rocio muñoz

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    Re: THE NATURE OF MATHEMATICS

    Post by rocio muñoz on Sun May 10, 2009 1:33 pm

    yo creo que las matematicas se ocupan a todo momento porque cuando vamos a ser de comer cuando hay una fiesta por ejemplo hay que calcular cuanta comida hacer para 6 o 7 personas o hasta 10 o cuando estamos calculando como tirar una bolita de papel para que caiga justamente en el basurero hay usamos matematicas tambien otros casos como como los mayas hubieran hecho las grandes piramides que podemos conteplar ahora o como los de el apolo pudieron llegar a la luna tuvieron que calcular exactamente las coordenadas.algunos quieren ser arquitectos o archaelogos y doctores ingenieros todos ellos ocupan las matematicas se no sabriamos sin las matematicas como calcular las casas, o las coordenadas para buscar cosas antiguas o organos del cuerpo.

    SIN LAS MATEMATICAS LA VIDA NO TENDRIA SENTIDO

    kamy

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    Re: THE NATURE OF MATHEMATICS

    Post by kamy on Sun May 10, 2009 3:33 pm

    The mathematics are very important in our life because with out them we can`t do anything because all in this life need the math like when we go to the restuarants we have to pay and if we want to buy something we have to count the money to know if we can buy it.
    If we don`t know math we can`t do anything. We use the math all the time not only in the restaurants or when we buy things, we use it in the construction of buildings, houses... , we also use the math in all the activities we do. What a Face Like a Star @ heaven

    Chacha

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    Re: THE NATURE OF MATHEMATICS

    Post by Chacha on Sun May 10, 2009 6:21 pm

    I think that math is important to our live, because we count the money, make stadistic and we have to use it everyday Shocked Shocked

    tcharrisse07

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    Re: THE NATURE OF MATHEMATICS

    Post by tcharrisse07 on Sun May 10, 2009 6:31 pm

    :
    flower The mathematics are very important in the human beings because it helps to tell money us, to pay accounts if we are going to eat to restaurant, I love you also is very favorable to know like conducting an operation mathematical, in music, Laughing also teach history to us on mathematicians who managed to discover many thing like important tongue

    Quique

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    MATH

    Post by Quique on Mon May 11, 2009 5:41 pm

    I THINK MATH IS VERY IMPORTANT, BECAUSE WITHOUT IT WE WOULDNT BE ABLE TO FUNCTION IN LIVE.MATH IS USED IN EVERYTHING: CODES,MONEY,LICENSE PLATES,PHONE NUMBERS,ETC. EVERYTHING USES MATH, NO MATTER HOW SIMPLE OR COMPLICATED IT IS.

    Hector

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    Re: THE NATURE OF MATHEMATICS

    Post by Hector on Mon May 11, 2009 5:51 pm

    Very Happy HI, I THINK THAT MATH IS VERY IMPORTANT BECAUSE IT IS ONE OF THE THINGS MORE NESSESARI IN OUR LIVES. THAT IS IMPORTANT IN MANY SUBJECTS BECAUSE HELP US TO CONSENTRATE. study lol!

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