Arti : MATEMATIKA
Matematika (dari bahasa Yunani: μαθηματικά - mathēmatiká) secara umum ditegaskan sebagai penelitian pola dari struktur, perubahan, dan ruang; tak lebih resmi, seorang mungkin
mengatakan adalah penelitian bilangan dan angka'. Dalam pandangan formalis, matematika adalah pemeriksaan aksioma yang menegaskan struktur abstrak menggunakan logika simbolik dan notasi matematika; pandangan lain tergambar dalam filosofi matematika.
Struktur spesifik yang diselidiki oleh matematikus sering mempunyai berasal dari ilmu pengetahuan alam, sangat umum di fisika, tetapi mathematikus juga menegaskan dan
menyelidiki struktur untuk sebab hanya dalam saja sampai ilmu pasti, karena struktur mungkin menyediakan, untuk kejadian, generalisasi pemersatu bagi beberapa sub-bidang, atau alat membantu untuk perhitungan biasa. Akhirnya, banyak matematikus belajar bidang dilakukan mereka untuk sebab yang hanya estetis saja, melihat ilmu pasti sebagai bentuk seni daripada sebagai ilmu praktis atau terapan.
SEJARAH MATEMATIKA
Cakupan pengkajian yang disebut sebagai sejarah matematika adalah terutama berupa
penyelidikan terhadap asal muasal temuan baru di dalam matematika, di dalam ruang lingkup yang lebih sempit berupa penyelidikan terhadap metode dan notasi matematika baku di masa silam.
Sebelum zaman modern dan pengetahuan yang tersebar global, contoh-contoh tertulis dari pembangunan matematika yang baru telah mencapai kemilaunya hanya di beberapa tempat. Tulisan matematika terkuno yang pernah ditemukan adalah Plimpton 322 (Matematika Babilonia yang berangka tahun 1900 SM), Lembaran Matematika Moskow (Matematika Mesir yang berangka tahun 1850 SM), Lembaran Matematika Rhind (Matematika Mesir yang berangka tahun 1650 SM), dan Shulba Sutra (Matematika India yang berangka tahun 800 SM).
Semua tulisan yang bersangkutan memusatkan perhatian kepada apa yang biasa dikenal sebagai Teorema Pythagoras, yang kelihatannya sebagai hasil pembangunan matematika yang paling kuno dan tersebar luas setelah aritmetika dasar dan geometri.
Desember 14, 2008
RELEKSI TERHADAP FILSAFAT PENDIDIKAN MATEMATIKA
Oleh: Jero Budi DarmayasaDitinjau dari asal katanya. “Matematika” berasal dari bahasa Yunani yaitu dari kata “Mathema” yang berarti “Sains, Ilmu Pengetahuan atau belajar”. Dalam kegiatan sehari-hari terutama dalam bidang akademis, Matematika sering di-identik-kan dengan sesuatu yang “sulit, membosankan, dan bahkan momok” bagi sebagian besar orang (peserta didik). Namun, jika memang benar matematika adalah sesuatu yang sulit dan membosankan, mengapa matematika diberikan sebagai salah satu mata pelajaran wajib di setiap jenjang pendidikan (mulai TK sampai tingkat Sekolah Menengah Atas)?, Apakah arti dan makna matematika yang sebenarnya?
Dalam suatu cerita, sebuah pasangan suami istri yang sedang berselisih paham bisa berdamai hanya dengan mengancungkan jari yang melambangkan bilangan satu, dua, dan tiga. Dalam hal ini, suatu fenomena yang tidak dapat diselesaikan dengan bahasa verbal tenyata dapat
diselesaikan dengan simbul matematika. Melihat kejadian tersebut, dapat dikatakan bahwa matematika diartikan sebagai “bahasa”. Dalam artinya sebagai bahasa, matematika
diartikan sebagai “aktivitas hidup” dan ketika melakukan aktivitas tersebut, sering kali
dihadapkan pada suatu pilihan yang memerlukan spekulasi untuk memperoleh sesuatu yang lebih baik. Dalam menentukan suatu pilihan, hal pertama yang dilakukan adalah melihat berbagai kemungkinan dan mengkajinya dari berbagai ketentuan yang berlaku (norma, agama, dan hukum) sehingga diperoleh suatu keputusan yang terbaik. Barangkali tidak disadari, dengan belajar matematika, pengambilan keputusan akan lebih maskimal karena kebiasaan berpikir secara deduktif (yaitu proses pengambilan keputusan yang didasarkan pada premis-premis yang kebenaranya telah ditentukan).
Setelah memahami makna dan arti matematika, mungkin akan muncul pertanyaan berikutnya: “Bagaimanakah belajar dan mengajar matematika yang efektif, sementara materi yang ditetapkan cukup padat dan waktu tatap muka yang terbatasí?”. Pertanyaan seperti ini
merupakan pertanyaan dan permasalahan klasik yang tidak hanya dihadapi oleh siswa dan guru di Indonesia. Untuk menghadapi permasalahan seperti ini di era sekarang, Lewin2 memberikan pernyataan sebagai berikut”…. My answer to this problem is to make myself available by telephone seven days a week and to teach my students how to send mathematical questions by email. The software that we use makes this process very simple. Thus, I can sometimes give a good answer to a complicated question at 11:00 PM on a Sunday night”. Ini berarti guru/dosen bisa berinteraksi dan menjawab/menanggapi topik yang disampaikan oleh peserta didiknya dengan menggunakan media elektronik berupa “Handphone” atau “Internet”. Salah satu implementasinya pada saat ini yaitu penggunaan “blog” dalam diskusi matematika sekolah maupun perkuliahan, seperti yang diterapkan dalam perkulian “Filsafat Ilmu” pada Program Magister Pendidikan Matematika UNY.
Hal lain yang sebenarnya mempengaruhi proses belajar mengajar matematika, yaitu: bagaimana siswa belajar, bagaimana guru mengajar, apa yang harus dicapai oleh siswa, dan bagaimana guru menilainya. Dalam belajar matematika, setiap siswa memiliki cara belajar yang berbeda dan secara umum dibedakan menjadi empat kategori3: Alegori, integrasi, analisis, dan sintesis. Siswa yang belajar secara alegori menggunakan konsep yang sudah dipelajari sebelumnya untuk memahami materi atau konsep baru yang diajarkan dan menekankan penggunaan metode dalam bentuk yang mirip. Sementara siswa yang belajar secara integrasi berusaha membandingkan materi baru dengan konsep yang tlah dipahami, tetapi terkadang mereka mengalami kesulitan mencari hubungan antara kedua konsep tersebut. Kelompok siswa yang lain adalah siswa yang belajar dengan cara analisis, yaitu siswa yang mengharapkan penjelasan materi baru secara detail dan memikirkan ide baru dengan mengunakan pemikiran yang logis. Kelompok terakhir adalah siswa yang belajar dengan cara yang sangat abstrak dan berusaha mengembangakan strateginya sendiri yang lebih dikenal dengan kelompok siswa yang belajar secara sintesis.
Sementara itu, bagaimana guru mengajar, menentukan sasaran yang harus dicapai siswa, dan bagaimana hasil kerja siswa dinilai biasanya menyesuaikan dengan karakter guru yang mengajar serta metode dan pendekatan yang digunakan dalam proses belajar mengajar. Guru bisa saja menggunakan metode pengajaran langsung, penemuan, penemuan terbimbing, kooperatif, serta penggunaan teknologi yang tepat4. Untuk penialainnya, bisa menggunakan kertas kerja, penilaian dengan pertanyaan terbuka (open-ended), tes tertulis, fortofolio, penugasan, dan bentuk
penialaian lainnya
http://jerobudy.blogspot.com/2008/12/releksi-terhadap-flsaaat-pendidikan.html
The Meaning of Mathematics
H. Harold Hartzler Goshen College Goshen, Indiana
A word of explanation may be in order concerning the inclusion of a paper with the above title on a program of the American Scientific Affiliation. Most of the papers presented at our previous conventions have been apologetic in character and contained only a relatively small amount of scientific information. As individual members of this affiliation we cannot be equally competent in the various areas of science including mathematics, astronomy, physics, chemistry, biology, sociology, geology, archaeology, anthropology, psychology and philosophy. It seems to me that we need a series of papers presented at our annual meetings, each of which will attempt to explain the essential character of that field of science so that those whose special training does not lie in that area might be aided in their understanding of the problems involved.
Since mathematics has been called "the queen of the sciences" and also their handmaiden and since its essential character is so very poorly understood by many, it seems desirable to delve into the mysteries of this subject. In the first place I would like to point out that mathematics is much more than the art of computation. This reminds me of the young mathematics student who had recently returned from Germany after spending three years there earning the doctor's degree. He was greeted by an old acquaintance in his home town who asked him what he had been doing while abroad. "Studying mathematics" was the reply. "Studying nothing but mathematics for three years" exclaimed his friend as he gazed at a brick wall across the street, "why I suppose you could count the bricks in that wall at a glance." This illustrates a popular misconception in regard to the nature of mathematics. To be able to compute with ease and to perform a few simple mathematical tricks is often taken to mean that one is gifted in mathematics. Such, however, is not necessarily true. In fact, some noted mathematicians have been rather slow at numerical computation.
A fair competence in manipulation is admitted to be a necessary prerequisite to understanding a mathematical argument. But no amount of technical facility will of itself teach anyone what mathematics is or what proof means; nor will it suggest what is probably the most important reason why mathematics is today an even more vital human need and social necessity than it was in the past. Manipulative skill may suffice for the average technician in the trades but it is inadequate as an aid to self-respecting citizenship in even a moderately intelligent society.
Now just what constitutes the essential character of mathematics? I think that we may say that the essential element is reasoning and that it is mainly deductive in character, although not exclusively so, since in the formulation of many theorems, we use the inductive approach.
First of all we start with elements, the undefined terms concerning with which all our mathematical reasoning done. These elements have nothing whatsoever to do with the
constituents of matter which are studied quite extensively in chemistry and physics. In fact, as far as the mathematician is concerned, they have no necessary relation to anything in the world of the senses. They constitute the building blocks, quite few in number, out of which is built the entire mathematical structure. Examples are number, point, line, and plane. These elements or objects or concepts are so fundamental in character as to be incapable of definition. In
really, a circular definition not a definition at all. For example, the definition of the word number as found in Webster's new international unabridged dictionary: "The total aggregate or amount of units (whether of things, persons or abstract units)". Here we find the word number defined in terms of aggregate or amount of units. But what meaning have these terms if not in terms of number?
After we have the elements of the subject, we next have definitions, axioms or postulates, propositions and theorems. The axioms are pure assumptions concerning the undefined elements. They may have been suggested by experience or they may have been chosen on the mere whim of some mathematician interested in seeing what he could make. In no sense are the postulates or axioms eternal truths or necessary; nor are they guaranteed by any extra human necessity or supernatural existence. The laying down of postulates is a free act of human beings.
The totality of the axioms of any branch of mathematics provides the implicit definition of all undefined terms in that area. For applications it is important that the concepts or elements and the axioms or postulates of mathematics correspond well with physically verifiable statements about real tangible objects. The physical reality behind the concept of point is that of a very small object such as a pencil dot, while a straight line is an abstraction from a taut thread or of a ray of light. The properties of these physical points and straight lines are found by experience to agree more or less with the formal axioms of geometry. Quite conceivably more precise
experiments might necessitate modification of these axioms if they are adequately to describe physical phenomena. But if the formal axioms did not agree more or less with the properties of physical objects, then geometry would be of little interest. Thus there is an authority, other than the human mind, that decides the direction of mathematical thought.
We usually require that the postulates be simple and not too great in number. Moreover, the postulates must be consistent, in the sense that no two theorems deducible from them can be mutually contradictory, and they must be complete, so that every theorem of the system is deducible from them. For reasons of economy it is also desirable that the postulates be
independent, in the sense that no one of them is deducible from the others. The question of the completeness and of the consistency of a set of axioms has been the subject of much controversy. Different philosophical convictions concerning the ultimate roots of human knowledge have led to apparently irreconcilable views on the foundations of mathematics. If, as in the Kantian philosophy, mathematical entities are considered to exist in a realm of pure intuition,
independent of definitions and of individual acts of the human mind, then of course there can be no contradictions, since mathematical facts are objectively true statements describing relations considered as real in the realm of pure intuition. From this intuitionist point of view there is no problem of consistency. Unfortunately, it has turned out that the intuitionist attitude, if applied without compromise, would exclude a large and important part of mathematics and would hopelessly complicate the rest. Radical intuitionists deny a legitimate place in mathematics to the number continuum. They completely reject all non-constructive proofs, and. admit only the denumerably infinite as a legitimate child of intuition.
by the use of our fingers. In fact, our word digit comes from the Latin, meaning finger. When you say that you have five objects and hold up five fingers to show the number, you are making use of the principle of one-to-one correspondence, which principle is very important in the consideration of infinite classes. In using this principle it is necessary to have two sets or classes of objects and then to add that if for every object in the first set there corresponds but one object in the second and conversely that for every object in the second there corresponds but one object in the first, then the number of objects in the two sets is the same. Now when we speak of the denumerably infinite we mean that we have a set which can be put into one-to-one
correspondence with the set of the natural numbers: l,2,3,4,... This set of the natural numbers is infinite, which simply means that if .you name any number N, as large as you please, then there are still numbers in the set.
Quite different is the view taken by the formalists. They do not attribute an intuitive reality to mathematical objects, nor do they claim that axioms express obvious truths concerning the realities of pure intuition, their concern is only with the formal logical procedure of reasoning on the basis of postulates. This attitude has a definite advantage over intuitionism, since it grants to mathematics all the freedom necessary for theory and applications. But it imposes on the
formalist the necessity of proving that his axioms, now appearing as arbitrary creations of the human mind, cannot possibly lead to a contradiction. Great efforts have been made during the last twenty five years to find such consistency proofs, at least for the axioms of arithmetic and algebra and for the concept of the number continuum. The results are highly significant, but success is still far off. Indeed, recent results indicate that such efforts cannot be completely successful, in the sense that proofs for consistency and completeness are not possible within strictly closed systems of thought. Remarkably enough, all these arguments on the foundations of mathematics proceed by methods that in themselves are thoroughly constructive and directed by intuitive patterns.
Let us consider a case where mathematical reasoning of the purely formalistic type has led to a contradiction. This involves the use of the concept of set without any restrictions being put upon it. This paradox, first shown by Bertrand Russell is as follows: Most sets do not contain
Let us suppose that we have agreed upon some set of postulates or axioms for the undefined elements. One postulate for our points and lines might be, "two points determine a line," another, "two lines determine a point." The latter, by the way, would not usually be admitted in High School geometry, for in that subject are the exceptions introduced by parallels which, by
definition, are lines having no point in common. But if we introduce an ideal point at infinity, all a matter of words without any clutter of mysticism, the postulate becomes intelligible without any exceptions.
Thus far we have the undefined elements and postulates about them. To the postulates we now apply common logic, or the laws of thought, and see what the postulates imply. The three so-called laws of thought of Aristotle are: (1) A is A (the law of identity), (2) nothing is both A and not-A (the law of excluded middle), (3) everything is either A or not A (the law of
contradiction). These postulates of reasoning were once thought to be superhuman necessities arid not, as they are regarded today, mere assumptions which human beings have made and agreed to accept. So let us refer to Aristotle's classical laws as the postulates of deductive reasoning. Deduction proceeds by an application of these postulates to those of the system, it may be geometry or algebra, which may be under investigation.
It is possible to make different kinds of assertions about the undefined elements. The most important of them are the propositions. A proposition is a statement which is either true or false. A true proposition is sometimes called a theorem. If true, we try to prove propositions by
deductive reasoning. If false, an attempted deductive proof will sometimes reveal the falsity by the indirect method. Proof consists in seeing what the postulates of the system imply. Thus if P and Q are propositions and if Q follows from P by the postulates of deductive reasoning; and if further it is known or assumed that P is true, then Q is true. In particular, if P is one of our postulates which we have assumed at the beginning to be a true proposition, Q is true. But if it is not known whether Q is true, we may tentatively assume that it is false. If from this assumption we can deduce that Q is also true, we have a conflict with the postulate of excluded middle. But we agreed to abide by the postulates of deductive reasoning. To avoid the conflict we say that Q is not false, which we tentatively assumed; namely, Q is true, which we wished to prove.
The whole game is exceedingly simple. There are but two rules. First state all the postulates and second see that no other postulate slips into a chain of deductive reasoning. In geometry, for example, it looks as if a straight line which cuts one side of a triangle at a point other than a vertex must also cut another side. This is the sort of assumption which Euclid or some of his modern imitators might easily make, If it cannot be deduced from the remaining postulates it should be put in plain view with them as another postulate.
Although Russell's remark may tend to overemphasize the view of the older British school that mathematics is identical with logic; a view which, outside of Great Britain, is now generally regarded as untenable; it does call attention to a distinction between pure mathematics and applied mathematics. To see this, consider the statement often seen in elementary texts that the a,b,c, ....x,y,z of algebra represent numbers. Rather it should be stated that the letters are mere undefined marks or elements about which certain postulates are made. The very point of
elementary algebra is simply that it is abstract, that is, devoid of any meaning beyond the formal consequences of the postulates laid down for the marks. Some of the elementary algebra is true when interpreted in terms of rational numbers; some of it is false for these same numbers; for example, the statement (which might be taken as a postulate in a first course) that every equation has a root. But we miss the whole point of algebra if we insist on any particular interpretation. Algebra stands on its own feet as a hypothotico-deductive system. An interpretation of the abstract system is an application.
To illustrate what has been said about mathematical systems let us glance at an elegant set of seven postulates for common algebra, from E. V. Huntington (Transactions of the American Mathematical Society, vol.4, 1903, pp. 31-37). The system defined by these postulates is usually called a field, and is identical abstractly, with common, rational algebra. The fundamental concept involved is that of a class in which two rules of combination (or operations), denoted by
a b and a b are uniquely known elements of the class. This is sometimes expressed as "the
class is closed under operations , ." Neither a b nor a b belong to the class unless so stated explicitly. These remarks are merely by way of introduction; the postulates follow.
Postulate Al. If a, b and b a belong to the class, then a b = b a.
Postulate A2. If a, b, c, a b, b c, and a (b c) belong to the class, then (a b) c =a
(b c).
Postulate A3. For every two elements a and b (a = b or a b), there is an element x such as a
x = b.
Postulate Ml. If a, b and b a belong to the class, then a b = b a.
Postulate M2. If a, b, c, a b, b c and a (b c) belong to the class, then (a b) c = a (b c).
Postulate M3. For every two elements a, b (a = b or a b), provided a a a and b b b, there is an element y such that a y = b.
then a (b c) = (a b) (a c).
The unusual , instead of the familiar /, x are used to prevent any possible misconception that we are talking about numbers as in arithmetic. We are not; the marks or undefined elements a, b, c ... are marks and nothing more, end the seven postulates state everything that we are
assuming about these marks and , . It is easy to see, as already suggested, that the postulates define common school algebra (including the ban against attempting to divide by zero) up to the point where radicals are introduced. Perhaps the complete freedom, the arbitrariness of what we are doing will be more obvious when we realize that the seven postulates are independent of one another. That is, it is possible to exhibit a system which does not satisfy any particular one of of the sven postulates, but which does satisfy the remaining six. You may easily verify that the set
of all positive rational numbers with a b and a b now defined to be b and a, b respectively,
that is, a b equals b and a b equals a, b, satisfies all the postulates except Al. In the same
way, a system satisfying all except Ml is the system of all integral numbers with a b = a b
and a b = b. A system which satisfies all integral numbers with a b = a b and a b = a b. Perhaps you are getting bored with this discussion so I shall pass on to two very fine cases of mathematical reasoning which more nearly approach everyday experience.
The first case is that of the proof of the theorem which states that the number of prime numbers is infinite. As you will recall, a prime number is one divisible only by itself and unity. By the number of such numbers being infinite is meant that if one attempts to name any very large number as representing all the primes, that there are still more primes to bc found. The proof is as follows. Let us assume that a largest prime number exists. Call it P. Now let us form a number N as equal to the product of all the primes from the first one which is two to the last one which we have assumed to be P and than let us add one to this product. Therefore N = (a,3,5,7 ... P) 1. Now the question is whether this number is prime numbers starting with two and ending with P. Hence we see that this number is not divisible by any number other than itself and unity. Therefore it is prime and surely larger than P which we had assumed to be the largest prime. Hence we have a contradiction and we conclude that there does not exist a largest prime number and therefore the number of prime numbers is infinite.The next case we wish to prove is that the square root of the number two is irrational, that is, it is a real number which is not the quotient of two integers. To prove this let us assume that the square root of two is rational and denote it by the ratio p/q, where p and q have no common divisor, that is, they have been reduced to lowest terms. Now square both sides of the equality and we have 2 = p2/q2 or p2 = 2q2. Here we have
applied the axiom that when equals are multiplied by equals the results are equal. If, however, p2
= 2q2, then p2 is even and hence p is even since only even numbers when squared result in even
numbers. If is even we can express it in the form p = 2n. From this equation we have p2 = 4n2
again by the application of the axiom that when equals are multiplied by equals the results are equal. Now applying the axiom that things equal to the same thing are equal to each other we have 2q2 = 4n2 or q2 = 2n2. This latter equation states that q2 is even and hence is even. This is a
and the proof is complete. In both of these cases we have used the method of proof known as reductio ad absurdum.
We come now to the question of the application of mathematics to the physical world and to all of God's creation. We have emphasized the fact that pure mathematics is an invention of the human mind. The more nearly that the elements and the postulates of the subject correspond with what we consider as physical realities, the more closely will the results correspond and can be used to predict the manner in which physical phenomena will perform. Hence as was stated in the beginning of this paper, mathematics becomes the handmaiden or servant of the sciences. But just to say that something has been proved mathematically, does not insure that the results will correspond with physical phenomena. For example, it was proved a numher of years ago that it was impossible for a heavier-than-air craft to fly through the air. This is no discredit to
mathematics, but rather it serves to warn us that we need to be very careful in the applications of mathematics. Since all sciences are continually making more and more use of mathematical methods, we need to keep constantly on our guard that the results obtained correspond as closely as possible with physical phenomena.
Finally I would like to say a word concerning the relationship existing between mathematics and the Christian idea of God. Since I believe with Professor Jaarsma in his paper entitled, "Christian Theism and the Empirical Sciences" that "the God of Christianity as the Creator is the
unconditioned Conditioner of all things, including the very facts and conclusions of science," I feel that even the thoughts of mathematicians have their ultimate source in God. However to say, as some have said, "that the Great Architect of the Universe now begins to appear as a pure mathematician," appears to me to belittle the idea of God. The pure mathematician is just a puny little man with a quite finite mind doing a small bit of purely human reasoning. If some of this reasoning does seem to aid us in delving into the mysteries of God's creation, we should give more glory to His name for allowing us this privilege. But to put the infinite God, creator and sustainer of the universe, as well as savior of our souls, into this category seems to me to be quite a serious blunder. May we then, as Christian men of science, make more use of the mathematical method in science, since it has proved so fruitful in leading us into a deeper understanding of God's creation.
Mathematics
Mathematics is commonly defned as the study oa patterns oa structure, change, and space; more inaormally, one might say it is the study oa 'fgures and numbers'. In the aormalist view, it is the investigation oa axiomatically defned abstract
structures using logic and mathematical notation; other views are described in Philosophy oa mathematics.
The specific structures that are investigated by mathematicians often have their origin in the
natural sciences, most commonly in physics, but mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common
calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science.
Mathematics is often abbreviated to math in North America and maths in other English-speaking countries.
Overview and history of mathematics
See the article on the history of mathematics for details.
The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning".
The major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.
The study of structure starts with numbers, firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vector, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.
The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler and compass constructions were finally settled by Galois theory. The modern fields of differential geometry
and algebraic geometry generalize geometry in different directions: differential geometry
emphasizes the concepts of functions, fiber bundles, derivatives, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations.
Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for
quantum mechanics among many other things. Many phenomena in nature can be described by
dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior.
In order to clarify and investigate the foundations of mathematics, the fields of set theory,
mathematical logic and model theory were developed.
When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory,
information theory and algorithmic information theory. Many of these questions are now investigated in theoretical computer science. Discrete mathematics is the common name for those fields of mathematics useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences.
Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.
http://www.knowledgerush.com/kr/encyclopedia/Mathematics/
BAHASA MATEMATIKA
Written by Administrator Thursday, 12 March 2009
Bahasa merupakan suatu sistem yang terdiri dari lambang-lambang, kata-kata, dan kalimat-kalimat yang disusun menurut aturan tertentu dan digunakan sekelompok orang untuk berkomunikasi. Dalam tulisannya, Mudjia Rahardjo mengatakan: "Di mana ada manusia, di sana ada bahasa". Keduanya tidak dapat dipisahkan. Bahasa tumbuh dan berkembang karena manusia. Manusia berkembang juga karena bahasa. Keduanya menyatu dalam segala aktivitas kehidupan. Hubungan manusia dan bahasa meruapakan dua hal yang tidak dapat dinafikan salah satunya. Bahasa pula yang membedakan manusia dengan makhluk ciptaan Tuhan yang lain.
Berkaitan dengan hal ini, dapat dikatakan bahwa syarat terjadinya proses komunikasi harus terdapat dua pelaku, yakni pengirim dan penerima pesan, sehingga yang perlu ditekankan selanjutnya adalah bagaimana cara kita menyampaikan pesan agar dapat berjalan secara efektif.. Dalam hal ini, Badudu (1995), mengemukakan ada beberapa faktor yang harus diperhatikan, yaitu: a). orang yang berbicara; b). orang yang diajak bicara; c). situasi pembicaraan apakah formal atau non-formal; dan d). masalah yang dibicarakan (topik).
Menurut Galileo Galilei (1564-1642), seorang ahli matematika dan astronomi dari Italia,"Alam semesta itu bagaikan sebuah buku raksasa yang hanya dapat dibaca kalau orang mengerti bahasanya dan akrab dengan lambang dan huruf yang digunakan di dalamnya. Dan bahasa alam tersebut tidak lain adalah matematika. Berbicara mengenai matematika sebagai bahasa, maka pertanyaan yang muncul kemudian adalah dalam sudut pandang mana matematika itu disebut sebagai bahasa, dan apa perbedaan antara bahasa matematika dengan bahasa-bahasa lainnya.
Merujuk pada pengertian bahasa di atas, maka matematika dapat dipandang sebagai bahasa karena dalam matematika terdapat
sekumpulan lambang/simbol dan kata (baik kata dalam bentuk lambang, misalnya ">=" yang melambangkan kata "lebih besar atau sama dengan", maupun kata yang diadopsi dari bahasa biasa, misalnya kata "fungsi" yang dalam matematika menyatakan suatu hubungan dengan aturan tertentu antara unsur-unsur dalam dua buah himpunan).
Matematika adalah bahasa yang melambangkan serangkaian makna dari pernyataan yang ingin kita sampaikan. Simbol-simbol matematika bersifat "artifisial" yang baru memiliki arti setelah sebuah makna diberikan kepadanya. Tanpa itu, maka matematika hanya merupakan kumpulan simbol dan rumus yang kering akan makna. Berkaitan dengan hal ini, tidak jarang kita jumpai dalam kehidupan, banyak orang yang berkata bahwa X, Y, Z itu sama sekali tidak memiliki arti.
Sebagai bahasa, matematika memiliki kelebihan jika dibanding dengan bahasa-bahasa lainnya. Bahasa matematika memiliki makna yang tunggal sehingga suatu kalimat matematika tidak dapat ditafsirkan bermacam-macam. Ketunggalan makna dalam bahasa matematika ini, penulis menyebutnya bahasa matematika sebagai bahasa "internasional", karena komunitas pengguna bahasa matematika adalah bercorak global dan universal di semua negara yang tidak dibatasi oleh suku, agama, bangsa, negara, budaya, ataupun bahasa yang mereka gunakan sehari-hari. Bahasa yang dipakai dalam pergaulan sehari-hari sering mengandung keraguan makna di dalamnya. Kerancuan makna itu dapat timbul karena tekanan dalam mengucapkannya ataupun karena kata yang digunakan dapat ditafsirkan dalam berbagai arti.
Bahasa matematika berusaha dan berhasil menghindari kerancuan arti, karena setiap kalimat (istilah/variabel) dalam matematika sudah memiliki arti yang tertentu. Ketunggalan arti itu mungkin karena kesepakatan matematikawan atau ditentukan sendiri oleh penulis di awal tulisannya. Orang lain bebas menggunakan istilah/variabel matematika yang mengandung arti berlainan. Namun, ia harus menjelaskan terlebih dahulu di awal pembicaraannya atau tulisannya bagaimana tafsiran yang ia inginkan tentang istilah matematika tersebut. Selanjutnya, ia harus taat dan tunduk menafsirkannya seperti itu selama pembicaraan atau tulisan tersebut.
Bahasa matematika adalah bahasa yang berusaha untuk menghilangkan sifat kabur, majemuk, dan emosional dari bahasa verbal. Lambang-lambang dari matematika dibuat secara artifisial dan individual yang merupakan perjanjian yang berlaku khusus suatu permalahan yang sedang dikaji. Suatu obyek yang sedang dikaji dapat disimbolkan dengan apa saja sesuai dengan kesepakatan kita (antara pengirim dan penerima pesan). Kelebihan lain matematika dipandang sebagai bahasa adalah matematika mengembangkan bahasa numerik yang memungkinkan untuk melakukan pengukuran secara kuantitatif. Jika kita menggunakan bahasa verbal, maka hanya dapat mengatakan bahwa Si A lebih cantik dari Si B. Apabila kita ingin mengetahui seberapa eksaknya derajat kecantikannya maka dengan bahasa verbal tidak dapat berbuat apa-apa. Terkait dengan kasus ini maka kita harus berpaling ke bahasa matematika, yakni dengan menggunakan bantuan logika fuzzy sehingga dapat diketahui berapa derajat kecantikan seseorang. Bahasa verbal hanya mampu mengemukakan pernyataan yang bersifat kualitatif. Sedangkan matematika memiliki sifat kuantitatif, yakni dapat memberikan jawaban yang lebih bersifat eksak yang memungkinkan penyelesaian masalah secara lebih cepat dan cermat.
Matematika memungkinkan suatu ilmu atau permasalahan dapat mengalami perkembangan dari tahap kualitatif ke kuantitatif.
Perkembangan ini merupakan suatu hal yang imperatif bila kita menghendaki daya prediksi dan kontrol yang lebih tepat dan cermat dari suatu ilmu. Beberapa disiplin keilmuan, terutama ilmu-ilmu sosial, agak mengalami kesukaran dalam perkembangan yang bersumber pada problem teknis dan pengukuran. Kesukaran ini secara bertahap telah mulai dapat diatasi, dan akhir-akhir ini kita melihat perkembangan yang menggermbiarakan, di mana ilmu-ilmu sosial telah mulai memasuki tahap yang bersifat kuantitaif. Pada dasarnya matematika diperlukan oleh semua disiplin keilmuan untuk meningkatkan daya prediksi dan kontrol dari ilmu tersebut.
ekonomis.
Pemodelan matematika merupakan akibat dari penyelesaian permasalahan yang terjadi dalam kehidupan sehari-hari yang diselesaikan menggunakan matematika. Masalah nyata dalam kehidupan biasanya timbul dalam bentuk gejala-gejala yang belum jelas hakikatnya. Kita masih harus membuang faktor-faktor yang tidak/kurang relevan, mencari data-data dan informasi tambahan, lalu kita menemukan hakikat masalah sebenarnya. Lanngkah ini dinamakan sebagai mengidentifikasi masalah. Misalnya seorang pasien datang ke dokter dengan keluhan kepalanya pusing dan perut sakit. Berdasarkan keluhan itu dokter mengadakan beberapa tes dan dengan pengalaman dan dasar ilmunya, ia akan mengadakan analisis, lalu memberikan diagnosis. Diagnosis inilah merupakan identifikasi masalah. Langkah selanjutnya setelah mengidentifikasi masalah, maka melalui beberapa pendefinisian diadakan penerjemahan masalah ke bahasa lambang, yaitu matematika. Penerjemahan ini disebut pemodelan matematika. Setelah model matematika jadi, maka dicari alat yang dapat digunakan untuk menyelesaikannya. Pemodelan inilah yang menjadi kunci dalam penerapan matematika. Memodelkan masalah ke dalam bahasa matematika berarti menirukan atau mewakili objek yang bermasalah dengan relasi-relasi matematis. Istilah faktor dalam masalah menjadi peubah atau variabel dalam matematika. Pada hakikatnya, kerja pemodelan tidak lain adalah abstraksi dari masalah nyata menjadimasalah(model) matematika
Selain sebagai bahasa, matematika juga berfungsi sebagai alat berpikir. Ilmu merupakan pengetahuan yang mendasarkan kepada analisis dalam menarik kesimpulan menurut suatu pola berpikir tertentu. Menurut Wittegenstein, matematika merupakan metode berpikir yang logis. Berdasarkan perkembangannya maka masalah yang dihadapi logika makin lama makin rumit dan membutuhkan struktur analisis yang lebih sempurna. Dalam perspektif inilah maka logika berkembang menjadi matematika, sebagaimana yang disimpulkan oleh Bertrand Russell, "matematika adalah masa kedewasaan logika, sedangkan logika adalah masa kecil matematika". Komunikasi yang terjadi dalam matematika dapat terjadi, di antaranya dalam: 1) Dunia nyata, ukuran dan bentuk lahan dalam dunia pertanian (geometri), banyaknya barang dan nilai uang logam dalam dunia bisnis dan perdagangan (bilangan), ketinggian pohon dan bukit (trigonometri), kecepatan gerak benda angkasa (kalkulus), peluang dalam perjudian (probabilitas), sensus dan data kependudukan (statistika), dan sebagainya.; 2). Struktur abstrak dari suatu sistem, antara lain struktur sistem bilangan (grup, ring), struktur penalaran (logika matematika), struktur berbagai gejala dalam kehidupan manusia (pemodelan matematika), dan sebagainya; dan 3).Matematika sendiri yang merupakan bentuk komunikasi matematika yang digunakan untuk pengembangan diri matematika. Bidang ini disebut
"metamatematika".
Jadi, sejak awal kehidupan manusia matematika itu merupakan alat bantu untuk mengatasi berbagai macam permasalahan yang terjadi dalam kehidupan masyarakat. Baik itu permasalahan yang masih memilki hubungan erat dalam kaitannya dengan ilmu eksak ataupun permasalahan-permasalahan yang bersifat sosial. Peranan matematika terhadap perkembangan sains dan teknologi sudah jelas, bahkan kalu boleh penulis katakan bahwa tanpa matematika, sains dan teknologi tidak akan dapat berkembang..
Oleh: Abdul Halim Fathoni
Penulis adalah Mahasiswa Jurusan Matematika Semester VII, Universitas Islam Negeri (UIN) Malang
http://sigmetris.com/index.php?option=com_content&task=view&id=33&Itemid=28
Mathematics as a Language
Mathematicians need to be clear and concise when they communicate. As a result,
mathematicians are extremely attentive to the foundations of language and logic. Over the years, they developed their own conventions for communications. These conventions have evolved to a point that mathematics can be viewed as a distinct language.
The conventions developed by mathematicians are extremely powerful. The language of
result, scientists have adopted the conventions of mathematics, and you will often hear professors describe mathematics as the language of the sciences.
In this article, I will explore this concept of mathematics as a language. I will begin by briefly skimming over a few of the elements of communication. Then drop a few lines about more complex subjects such as symbolic logic, equations, and computer programming.
For the most part, I see the evolution of mathematics into a language as a positive development. However, there are a few excesses. Far too many people see mathematics as a language. They view it as a foreign language. In many ways the language of mathematics is an elitist language, and as an elitist language it can do more to hamper communications than to enable
communications.
The goal of descriptive mathematics is to find ways to bring this extremely powerful tool of mathematics into our day to day lives. To accomplish this goal, we need to understand how mathematics works as a language, and how we can best make use of this knowledge.
http://descmath.com/desc/language.html
Mathematics as a Language
Blake wrote: "I have heard many People say, 'Give me the Ideas. It is no matter what Words you put them into.'" To this he replies, "Ideas cannot be Given but in their minutely Appropriate Words."
- William Blake (quoted by J. Newman, The World of Mathematics, 1956)
This opinion is seconded by Bertrand Russell. Here is what he says in his Autobiography about meeting G.Peano at an International Congress on Philosophy in 1900:
had been seeking aor years, and that by studying him I was acquiring a new and poweraul technique aor the work that I had long wanted to do.
When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries. Just think of it. Zero gained widespread use only in the last millennium. Systematic introduction of modern algebraic notations began only in the sixteenth century and is most often associated with the French mathematician François Viète (1540-1603). René Descartes (1596-1650) was first to use letters at the end of the alphabet for unknowns. He also introduced the power notations: x2, x3. The sign
of equality (two equal parallel strokes) has been invented by Robert Recorde (c. 1510-1558) in his The whetstone of witte (London, 1557):
I will sette as I doe oaten in worke use, a paire oa paralleles, or Gemowe lines oa one lengthe, thus: =, bicause noe.2. thynges, can be moare equalle.
To help you appreciate the expressive power of the modern mathematical language, and as a tribute to those who achieved so much without it, I collected a few samples of (original but translated) formulation of theorems and their equivalents in modern math language.
Ia a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.4, 300 B.C.)
(a + b)2 = a2 + b2 + 2ab
Ia as many numbers as we please beginning arom a unit be set out continuously in double proportion, until the sum oa all becomes prime, and ia the sum multiplied into the last make some number, the product will be peraect. (Euclid, Elements, IX.36, 300 B.C.)
Ia 1 + 2 + ... + 2n is prime, then
2n(1 + 2 + ... + 2n) is peraect
The area oa any circle is equal to a right-angled triangle in which one oa the sides about the right angle is equal to the radius, and the other to the circumaerence, oa the circle. (Archimedes,
Measurement of a Circle, 225 B.C.)
A = 2r·r/2 = r 2
greatest circle in it. (Archimedes, On the Sphere and the Cylinder, 220B.C.)
Rule to solve x3 + mx = n
Cube one-third the coefcient oa x; add to it the square oa one-hala the constant oa the equation; and take the square root oa the whole. You will duplicate this, and to one oa the two you add one-hala the number you have already squared and arom the other you subtract one-hala the same... Then, subtracting the cube root oa the frst arom the cube root oa the second, the remainder which is leat is the value oa x (Gerolamo Cardano, Ars Magna, 1545).
x =
D in R - D in E aequabitur A quad. (Francois Viete, In
artem analytican isagoge, 1590) DR - DE = A2
Another example shows how different mathematicians might have expressed the modern equation 4x2 + 3x = 10.
However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that
ambiguities exist in plain English (the number's number is just one of them) and in other sciences as well. As a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.
Lest you think that my defense of the mathematical language has no solid basis, I began collecting word usage surprises from non-mathematical fields of activity and sciences. I welcome any examples of language misuse or inherent ambiguity you may want to send me.
