Feb 12, 2013

Mathematical Concept: What are their notions?




By Marsigit
Yogyakarta State University

Bold, T., 2004, notified that the essential components of mathematics covers the concepts of integer numbers, fractions, additions, divisions and equations; in which addition and division are connected with the study of mathematical propositions and the concept of integers and fractions are elements of mathematical concepts. He stated that the concept of equation can fall under mathematical propositions, but it is also related to issues concerning mathematical certainty; and integer numbers are statements about certain properties of bodies; hence, the concept of natural number is explained by study about this particular property of bodies or “notion of unity” or “quantitative property”. He clarified that there are two necessary elements involved with explanation of what a mathematical statement assures: quantifiable bodies and quantitative property of bodies. According to him, “quantifiable bodies” is mathematical statements initially involve only bodies that have capacity to be quantified by the mind; in which, once the concept of quantity is achieved, all bodies are quantifiable and consequently, it will be impossible for an empty mind that only interacts with bodies like sand and air to form the concept of number. While “quantitative property of bodies” is that once quantifiable bodies are present, mathematical statements do not affirm just any properties of those bodies, but only about that particular quantitative property; a mathematical statement is about quantitative properties of bodies, and quantitative bodies are in the bodies. That is how mathematical statements connect with the world.

Bold, T., 2004, further indicated that the second necessary element for interpretation of mathematical concepts is man’s ability of to abstract, that is the mind’s ability to abstract the quantitative property from bodies and use it without the presence of bodies. Due to the fact that all of mathematics is abstract, he believes that one of the motives of intuitionists to think mathematics is a sole product of the mind. He added that a third important element is the concept of infinity; while infinity is based on the concept of possibility. Accordingly, infinity is not a quantity, but a concept based on unrestricted possibility; it is a character of possibility. Next he claimed that the concept of fraction is just based on abstraction and possibleness. According to him, the issue involved with rational and irrational numbers is completely irrelevant for interpretation of concepts of fraction as Arend Heyting is overly concerned. As far as mathematical concepts are concerned, rational numbers as n/p and irrational numbers as q are just a matter of different ways of expression. The difference between them is issue within mathematics to be explained by mathematical terms and language.

On the other hand, Podnieks, K., 1992, claimed that the concept of natural numbers developed from human operations with collections of discrete objects; however, it is impossible to verify such an assertion empirically and the concept of natural number was already stabilized and detached from its real source viz. the quantitative relations of discrete collections in the human practice, and it began to work as a stable self-contained model. According to him, the system of natural numbers is an idealization of these quantitative relations; in which people abstracted it from their experience with small collections and extrapolated their rules onto much greater collections (millions of things) and thus idealized the real situation. He insisted that the process of idealization ended in stable, fixed, self-contained concepts of numbers, points, lines etc and ceased to change. While the stabilization of concepts is an evidence of their detachment from real objects that have led people to these concepts and that are continuing their independent life and contain an immense variety of changing details.

According to Podnieks, K., 1992, when working in geometry, a mathematician does not investigate the relations of things of the human practice directly, he investigates some stable notion of these relations viz. an idealized, fantastic "world" of points, lines etc; and during the investigation this notion is treated subjectively as the "last reality", without any "more fundamental" reality behind it. Further he claimed that if during the process of reasoning mathematicians had to remember permanently the peculiarities of real things, then instead of a science viz. efficient geometrical methods, we would have an art - simple, specific algorithms obtained by means of trial and error or on behalf of some elementary intuition. He summed that Mathematics of Ancient Orient stopped at this level and Greeks went further. According to him Plato treats the end product of the evolution of mathematical concepts that is a stable, self-contained system of idealized objects, as an independent beginning point of the evolution of the "world of things"; Plato tried to explain those aspects of the human knowledge, which remained inaccessible to other philosophers of his time.

Jones, R.B.,1997, elaborated that in the hands of the ancient Greeks mathematics becomes a systematic body of knowledge rather than a collection of practical techniques; mathematics is established as a deductive science in which the standard of rigorous demonstration is deductive proof. According to him, Aristotle provides a codification of logic which remains definitive for two thousand years; while the axiomatic method is established and is systematically applied to the mathematics of the classical period by Euclid, whose Elements becomes one of the most influential books in history. Jones insisted that the next major advances in logic after Aristotle appear in the nineteenth century, in which Boole introduces the propositional (boolean) logic and Frege devises the predicate calculus. This provides the technical basis for the logicisation of mathematics and the transition from informal to formal proof. On the other hand, Russell's paradox shakes the foundational advances of Frege, but is quickly resolved.

Further, Jones, R.B.,1997, noted that the Pythagoreans, were first inclined to regard number theory as more basic than geometry; the discovery of in-commensurable ratios presented them with a foundational crisis not fully resolved until the 19th century. Since Greek number theory, which concerns only whole numbers, cannot adequately deal with the magnitudes found in geometry, geometry comes to be considered more fundamental than arithmetic. Therefore, despite the inadequacies of the available number systems the desire to treat geometry numerically remains. Descartes1 , by inventing co-ordinate geometry advances an understanding of how geometry can be reduced to number. Meanwhile Mathematics continues to develop as Newton and Leibniz invent the calculus despite weakness in the underlying number system and Berkeley is one of the vocal critics of the soundness of the methods used. Jones2 claimed that not until the 19th Century do we see the foundational problems resolved by precise definition of the real number system and elimination of the use of infinitesimals from mathematical proofs. On the other side, Cantor's development of set theory together with Frege's advances in logic pave the way for Zermelo's first order axiomatisation of set theory, which provides the foundations for mathematics in the twentieth century.3

Landry, E., 2004, quoted Bolzano's that mathematical truths can and must be proven from the mere [the analysis of] concepts. Bolzano4 did this by demonstrating how mathematical rigor could be both an epistemological as well as a semantic notion; his demonstration of the dual character of mathematical rigor was the distinction between what he termed subjective and objective representations. According to Bolzano5 , meaning relates not to the subjective representation but rather relates to the inter-subjective content and as such is in no need of assistance from intuitions, either empirical or pure. Meanwhile, Hempel, C.G., 2001, argued that the validity of mathematics rests neither on its alleged self-evidential character nor on any empirical basis, but derives from the stipulations which determine the meaning of the mathematical concepts, and that the propositions of mathematics are therefore essentially "true by definition." He insisted that for the rigorous development of a mathematical theory proceeds not simply from a set of definitions but rather from a set of non-definitional propositions which are not proved within the theory; these are the postulates or axioms of the theory.

Hempel, C.G., 2001, exposed the example that the multiplication of natural numbers may be defined by definition which expresses in a rigorous form the idea that a product nk of two integers may be considered as the sum of k terms each of which equals n, that is (a) n.0 = 0; (b) n.k' = n.k + n. We may prove the laws governing addition and multiplication, such as the commutative, associative, and distributive laws (n + k = k + n; n.k = k.n; n + (k + I) = (n + k) + I; n.(k.l) = (n.k).l;n.(k + l) = (n.k) + (n.l)), as

commutative associative distributive
n + k = k + n
n.k = k.n n + (k + l) = (n + k) + l n.(k.l) = (n.k).l
n.(k + l) = (n.k) + (n.l)

Hempel6 concluded that in terms of addition and multiplication, the inverse operations of subtraction and division can then be defined; but it turns out that these "cannot always be performed"; i.e., in contradistinction to the sum and the product, the difference and the quotient are not defined for every couple of numbers; for example, 7-10 and 7/10 are undefined; and this situation suggests an enlargement of the number system by the introduction of negative and of rational numbers.

Ford & Peat, 1988, insisted that mathematical notation has assimilated symbols from many different alphabets and fonts includes symbols that are specific to mathematics; in mathematics a word has a different and specific meaning such as group, ring, field, category, etc; mathematical statements have their own moderately complex taxonomy, being divided into axioms, conjectures, theorems, lemmas and corollaries; and there are stock phrases in mathematics, used with specific meanings, such as "if and only if", "necessary and sufficient" and "without loss of generality". 7Any series of mathematical statements can be written in a formal language, and a finite state automaton can apply the rules of logic to check that each statement follows from the previous ones. According to them, various mathematicians attempted to achieve this in practice, in order to place the whole of mathematics on a axiomatic basis; while Gödel's incompleteness theorem shows that this ultimate goal is unreachable in which any formal language is powerful enough to capture mathematics will contain un-decidable statements. 8

Ford & Peat, 1988, claimed that the vast majority of statements in mathematics are decidable, and the existence of un-decidable statements is not a serious obstacle to practical mathematics. 9 According to them mathematics is used to communicate information about a wide range of different subjects covering to describe the real world viz. many areas of mathematics originated with attempts to describe and solve real world phenomena that is from measuring farms (geometry) to falling apples (calculus) to gambling (probability); to understand more about the universe around us from its largest scales (cosmology) to its smallest (quantum mechanics); to describe abstract structures which have no known physical counterparts at all; to describe mathematics itself such as category theory in which deals with the structures of mathematics and the relationships between them.

References:

1 In Jones, R.B.,1997, A Short History of Rigour in Mathematics,
2Jones, R.B.,1997, A Short History of Rigour in Mathematics,
3 Jones, R.B.,1997, A Short History of Rigour in Mathematics,
4 In Landry, E., 2004, Semantic Realism: Why Mathematicians Mean What They Say,
5 Ibid.
6 Hempel, C.G., 2001, On the Nature of Mathematical Truth, http://www.ltn.lv/ ~podniek/gt.htm
7 Ford & Peat, 1988, Mathematics as a language, Wikipedia, the free encyclopedia,
8 Ibid.
9 Ibid.

13 comments:

  1. Kunny Kunhertanti
    16709251060
    PPs Pendidikan Matematika kelas C 2016

    Landry, E., 2004, mengutip Bolzano bahwa kebenaran matematis dapat dan harus dibuktikan dari konsep belaka. Bolzano melakukan ini dengan menunjukkan bagaimana ketelitian matematis bisa menjadi gagasan epistemologis maupun semantik; Peragaan karakter ganda dari keteguhan matematis adalah pembedaan antara apa yang disebutnya representasi subyektif dan obyektif. Sehingga memang dalam membuktikan kebenaran matematis juga harus diikuti dengan konsep-konsep yang menyusun dalam proses pembuktian tersebut.

    ReplyDelete
  2. ARNY HADA INDA
    16709251079
    PPS-MAT D 2016
    Masih sedikit siswa yang menyukai pelajaran matematika yang disebabkan berbagai hal. Dari hasil wawancara diantaranya penyebab siswa tidak menyukai matematika adalah anggapan negatif siswa tentang matematika dan pembelajaran konvensional yang dilakukan guru. Pembelajaran konvensional yang dilakukan guru membuat mereka menganggap matematika itu sulit, karena berkaitan dengan menghafal rumus dan berhitung. Selain itu, objek matematika juga abstrak yaitu terdiri atas fakta, konsep, operasi dan prinsip sehingga siswa terkadang sulit membayangkan secara konkret.

    ReplyDelete
  3. Elli Susilawati
    16709251073
    Pmat D pps16

    Pemahaman konsep dalam pembelajaran matematika sangat penting bagi siswa. Dengan pemahaman konsep, siswa dapat mengkonstuksikan konsep-konsep yang sudah ada dengan konsep yang baru sehingga menjadi sebuah skema yang utuh pada materi tertentu. Namun pentingnya pemahaman konsep oleh siswa sering diabaikan beberapa guru, dikarenakan pemahaman konsep membutuhkan waktu yang tidak sedikit. Sehingga langsung saja diberikan rumus, dilengkapi dengan istilah “pokoknya”. Contohnya: pokoknya kalau soalnya begini rumusnya begini dan seterusnya.

    ReplyDelete
  4. Elli Susilawati
    16709251073
    Pmat D pps16

    Pemahaman konsep dalam pembelajaran matematika sangat penting bagi siswa. Dengan pemahaman konsep, siswa dapat mengkonstuksikan konsep-konsep yang sudah ada dengan konsep yang baru sehingga menjadi sebuah skema yang utuh pada materi tertentu. Namun pentingnya pemahaman konsep oleh siswa sering diabaikan beberapa guru, dikarenakan pemahaman konsep membutuhkan waktu yang tidak sedikit. Sehingga langsung saja diberikan rumus, dilengkapi dengan istilah “pokoknya”. Contohnya: pokoknya kalau soalnya begini rumusnya begini dan seterusnya.

    ReplyDelete
  5. Uswatun Hasanah
    17701251022
    S2 PEP B

    Saya menjadi tahu bahwa dalam matematika itu tidak hanya berkaitan dengan bilangan/angka/simbol. Namun, lebih dari itu matematika memiliki definisi yang sangat beragam dari berbagai ahli. Mulai dari yang mengatakan matematika berpusat pada logika/berpikir abstrak, operasional bilangan diskrit dan sebagainya. Saya tertarik dengan matematika di bidang geometri dimana membahas tentang representasi subyektif dan obyektif. Pada dasarnya representasi subyektif dan obyektif tersebut lebih dari sekedar definisi namun sejauhmana ketajaman analisis dari konsep yang ada. Matematika menjadi begitu menarik bagi saya jika pemaknaan dari simbol-simbol dapat mendeskripsikan dan memecahkan fenomena yang ada di dunia nyata.

    ReplyDelete
  6. Charina Ulfa
    17709251039
    PPs Pendidikan Matematika B 2017

    Assalamu'alaikum wr wb.
    Pada awalnya konsep matematika itu hanya berkaitan dengan pengukuran yang dapat diukur dengan pikiran, selalu berkaitan dengan benda-benda yang dapat dihitung. Tapi seiring berjalannya waktu, matematika terus berkembang, sehingga matemtika juga bisa untuk mengukur benda-benda yang abstrak dan konkrit. Matematika bukan hanya memecahkan masalah dengan bahasa matematis(angka) saja, akan tetapi juga dengan bahasa istilah. Bahasa istilah ini bisa di tuliskan dengan bahasa matematika atau simbol-simbol matematika. Banyak bidang matematika berasal dari usaha untuk mendeskripsikan dan memecahkan fenomena dunia nyata.

    ReplyDelete
  7. Gamarina Isti R
    17790251036
    Pendidkan Matematika Kelas B (Pascasarjana)

    Pemahaman konsep merupakan salah kemahiran matematika yang diharapkan dapat tercapai dalam belajar matematik.
    Siswa dapat menjelaskan keterkaitan antar konsep, mengaplikasikan konsep bahkan menemukan konsep dari pemecahan masalah. Contohnya pada notasi matematika telah mengasimilasi simbol dari berbagai alfabet dan font yang berbeda termasuk simbol yang spesifik untuk matematika; Dalam matematika kata memiliki makna yang berbeda dan spesifik seperti kelompok, cincin, lapangan, kategori, dll; pernyataan matematika memiliki taksonomi moderat mereka yang rumit, dibagi menjadi aksioma, dugaan, teorema, lemmas dan konsekuensi; dan ada frase stok dalam matematika, yang digunakan dengan makna tertentuMatematika yang bersifat abstrak, harus dapat diolah menjadi matematika realistik agar dalam memahami konsep tersebut siswa mengalami kemudahan dan lebih merasa dekat dengan matematika. Matematika realistik bukan hanya untuk siswa SD saja, tetapi matematika realistik dapat digunakan sebagai pembelajaran di SMP dan SMA. Pemahaman konsep tersebut akan mudah dipahami apabila siswa mengetahui secara real apa yang akan dan telah dipelajari tersebut.

    ReplyDelete
  8. Angga Kristiyajati
    17709251001
    Pps UNY P.Mat A 2017

    Terima kasih Banyak Pak Prof. Marsigit.

    Sepemahaman kami objek dari matematika adalah objek yang abstrak dan hanya ada dalam pikiran. Seorang bayi yang baru saja lahir dapat dipastikan dia sama sekali tidak memiliki kemampuan matematika. Dalam perkembangannya dia akan belajar segala hal (termasuk juga matematika) melalui orang tua, saudara, dan orang-orang disekitarnya dengan mengamati, memperhatikan dan meniru. Dengan ini lah maka intuisinya akan terbentuk (termasuk juga intuisi matematika).

    ReplyDelete
  9. Alfiramita Hertanti
    17709251008
    S2- Pendidikan Matematika kelas A 2017

    Assalamualaikum wr.wb
    Thank you for your posts, sir. To achieve understanding of the mathematics consepts of students is not an easy thing because the understanding of a mathematical concept is done individually. Each students has different abilities in understanding mathematical concepts. Nevertheless increased understanding of mathematical concepts should be sought for the success of students in learning.

    ReplyDelete
  10. Dimas Candra SAputra, S.Pd.
    PPs PMA 2017
    17709251005

    Assalamualaikum prof,
    Konsep matematika sangat beragam. Selain itu juga muncul berbagai pendapat tentang konsep matematika. Menurut Bold, komponen matematika meliputi konsep bilangan bulat, pecahan, penumlahan, pembagian,dan persamaan. Penjumlahan dan pembagian berhubungan dengan ilmu tentang proposisi matematika. Konsep bilangan bulat dan pecahan merupakan elemen dari konsep matematika. Konsep persamaan ada di bawah proposisi matematika dan juga berhubungan dengan kepastian matematika. Elemen lain yang diperlukan untuk menginterpretasikan konsep matematika adalah kemampuan manusia untuk mengabstraksi sifat-sifat kuantitatif dari bagian-bagian tubuh dan menggunakannya tanpa kehadiran tubuh. Elemen yang selanjutnya ialah konsep tentang infinity, infiniti didasarkan pada konsep peluang.Selanjutnya, konsep pecahan hanyalah berdasarkan pada abstraksi dan peluang.

    ReplyDelete
  11. Nama : Habibullah
    NIM : 17709251030
    Kelas : PM B (S2)

    Assalamualaikum wr.wb

    Pemahaman konsep merupakan aspek yang sangat penting untuk dikuasai siswa, karena dapat menjadi pondasi bagi siswa agar mudah ketika belajar matematika terhadap materi yang lebih tinggi dan lebih dalam. Maka dari itu guru harus berusaha melakukan abstraksi dibenak siswa dengan memberikan contoh dari materi yang dipelajari, kemudian mengkomunikasikan hal tersebut dengan membuat klasifikasi contoh agar dapat diberi nama sehingga dengan begitu guru bisa menjelaskan sebuah definisi matematis.

    ReplyDelete
  12. This comment has been removed by the author.

    ReplyDelete
  13. Maghfirah
    17709251007
    S2 Pendidikan Matematika A 2017

    Assalamualaikum Warohmatullah Wabarokatuh
    Everybody has their own notions. As well as mathematics. Each person may has different notions and different understanding of mathematics. But what I can observe, although they concentrate on different elements but elements of it still has a relationship each other.

    ReplyDelete