Oct 10, 2012

Elegi Menggapai "Structuralism as the Epistemological Foundation of Mathematics"




By Marsigit
Yogyakarta State University


Posy C. sets forth that according to structuralist the basic element of mathematics shouldn't be arbitrarily picked, yet nothing dictates its choice and the basic units are structures, not actually objects.



This 1 leads to structuralism to perceive that to be a natural number is to be a place in the sequence; accordingly, if mathematics is totally abstract, why should it have any applicability?

Structuralists 2 argues that mathematics is not about some particular collection of abstract objects but rather mathematics is the science of patterns of structures, and particular objects are relevant to mathematics only in so far as they instantiate some pattern or structure.

Benacerraf 3, as a structuralist says:
When it comes to learning about numbers, they merely learn new names for familiar sets. They count members of a set by determining the cardinality of the set, and they establish this by demonstrating that a specific relation holds between the set and one of the numbers. To count the elements of some k-membered set b is to establish a one-to-one correspondence between the elements of b and the elements of A less than or equal to k. The relation "pointing-to-each-member-of-b-in-turn-while-saying-the-numbers-up-to-and-including-" esta blishes such a correspondence.


Benacerraf 4 concludes that there is no one account which conclusively establishes which sets are the "real" numbers, and he doesn't believe that there could be such an argument.

According to him, any "object" or referent will do as long as the structural relations are maintained.

Benacerraf 5 argues that Frege's belief of some "objects" for number words to name and with which numbers could be identical, stemmed from his inconsistent logic.

Since all objects of the universe were on par, the question whether two names had the same referent always had a truth value; however, identity conditions make sense only in contexts where there exist individuating conditions.

Benacerraf 6 claims that if an expression of the form "x=y" is to have sense, it can be only in contexts where it is clear that both x and y are of some kind or category C, and that it is the conditions which individuate things as the same C which are operative and determine its truth value.

According to Benacerraf 7, Frege fails to realize this fact. It is a thesis that is supported by the activity of mathematicians, and is essential to the philosophical perspective underlying category theory.

Benacerraf 8 concludes that numbers could not be sets at all on the grounds that there are no good reasons to say that any particular number is some particular set, for any system of objects that forms a recursive progression would be adequate.

Benacerraf 9 concludes that a system of objects exhibits the structure of the integers implies that the elements of that system have some properties which are not dependent on structure.

Accordingly, it must be possible to individuate those objects independently of the role they play in the structure; however, this is precisely what cannot be done with numbers.

Benacerraf 10 argues that numbers possess outside of the properties of the structure are of no consequence to the mathematician, nor should they therefore be of concern for the philosopher of mathematics.

Accordingly, there is an activity which theorizes about the unique properties of individual numbers separated from the progressive structure.

According to Structuralist 11, arithmetic is the science exploring the abstract structure that all progressions have in common merely in virtue of being progressions.

Arithmetic is not concerned with particular numbers, and there are no unique set of objects which are the numbers.

Number does not have a singular reference, because the theory is elaborating an abstract structure and not the properties of individual objects.

In counting, we do not correlate sets with initial segments of the sequence of numbers as extra-linguistic entities, but correlate sets with initial segments of the sequence of number words.

The recursive sequence 12 is a sort of yardstick which we use to measure sets; questions of the identification of the referents of number words should be dismissed as misguided in just the way that a question about the referents of the parts of the ruler would be seen as misguided.

According to Structuralist 13, the mathematical description, model, structure, theory, or whatever, cannot serve as an explanation of a non-mathematical event without an account of the relationship between mathematics per se and scientific reality per se.

A mathematical structure can, perhaps, be similarly construed as the form of a possible system of related objects, ignoring the features of the objects that are not relevant to the interrelations.

A mathematical structure is completely described in terms of the interrelations; a typical beginning of a mathematical text consists of the announcement that certain mathematical objects such real numbers are to be studied.

In some cases, at least, the only thing about these objects is that there are certain relations among them and/or operations on them; and one easily gets the impression that the objects themselves are not the problems.

The relations and operations are what we study.

References:
1 Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
2Posy, C. ,1992, “Philosophy of Mathematics”. Retreived 2004
3Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
4 Ibid.
5 Ibid.
6 Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
7 Ibid.
8 Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
9 Ibid.
10 Ibid.
11In Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics”, Retrieved 2004
12Benacerraf in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004
13Shapiro in Stefanik, R., 1994, “Structuralism, Category Theory and Philosophy of Mathematics” Retrieved 2004

10 comments:

  1. Kartika Pramudita
    17701251021
    PEP S2 B

    Pada elegi ini dijelaskan beberapa pendapat struktualis matematika. Matematika itu memiliki struktur. Dalam struktur matematika pasti ada dasar yang mendasarinya.
    Struktualis juga memandang matematika sebagai struktur dan hubungan. Sesuatu dapat dikatakan sebagai matematika apabila memiliki struktur dan hubungan. Dalam memandang matematika sebagai struktur tentunya tidak bisa melepaskan sesuatu dengan sesuatu yang lain. Karena matematika sebagai struktur adalah matematika tentang hubungannya dengan yang lain.

    ReplyDelete
  2. Tri Wulaningrum
    17701251032
    PEP S2 B

    Strukturalisme sebagai landasan epistemologi matematika ialah bahwa elemen dasar matematika merupakan sebuah struktur. Jadi, tolok ukur nilai kebenaran matematika dilihat dari strukturnya, bukan aspek yang lainnya. Oleh karena matematika strukturalis merupakan pola struktur, maka objek tertentu hanya relevan untuk matematika sejauh mereka memberi contoh beberapa pola atau struktur. Dari pernyataan tersebut saya melihat bahwa pola kerja pada matematika strukturalisme ialah suatu obyek akan dianggap keberadaannya ataupun ditindaklanjuti pemcahannya jika objek tersebut mampu mempertahankan diri pada suatu pola struktur.

    ReplyDelete
  3. Muh Wildanul Firdaus
    17709251047
    Pendidikan matematika S2 kls C

    Posy C. menetapkan bahwa untuk strukturalis elemen dasar matematika tidak boleh sewenang-wenang mengambil, namun tidak ada mendikte pilihan dan unit dasar struktur, tidak benar-benar objek.
    Sebuah struktur matematika dapat, mungkin, akan sama ditafsirkan sebagai bentuk sistem kemungkinan obyek terkait, mengabaikan fitur dari benda-benda yang tidak relevan dengan keterkaitan.
    Struktur matematika benar-benar dijelaskan dalam hal keterkaitan; awal khas dari teks matematika terdiri dari pengumuman bahwa objek matematika tertentu bilangan real seperti untuk dipelajari.
    Dalam beberapa kasus, setidaknya, satu-satunya hal tentang benda-benda ini adalah bahwa ada hubungan tertentu antara mereka dan / atau operasi pada mereka; dan satu dengan mudah mendapat kesan bahwa obyek itu sendiri tidak masalah.

    ReplyDelete
  4. Nama: Dian Andarwati
    NIM: 17709251063
    Kelas: Pendidikan Matematika (S2) Kelas C

    Assalamu’alaikum. strukturalisme dalam matematika dipandang sebagai studi yang membahas bagaimana suatu objek matematika itu memiliki struktur dan dari struktur tersebut memiliki hubungan dengan sebuah sistem eksternal objek matematika itu sendiri. Struktur matematis dapat ditafsirkan serupa sebagai bentuk kemungkinan sistem benda-benda terkait, mengabaikan ciri-ciri benda-benda yang tidak relevan dengan keterkaitannya. Struktur matematika benar-benar dijelaskan dalam kaitannya dengan interelasi.

    ReplyDelete
  5. Dewi Thufaila
    17709251054
    Pendidikan Matematika Pascasarjana C 2017

    Assalamualaikum.wr.wb

    Posy C. menetapkan bahwa untuk strukturalis elemen dasar matematika tidak boleh sewenang-wenang mengambil, namun tidak ada mendikte pilihan dan unit dasar struktur, tidak benar-benar objek.
    Sebuah struktur matematika dapat, mungkin, akan sama ditafsirkan sebagai bentuk sistem kemungkinan obyek terkait, mengabaikan fitur dari benda-benda yang tidak relevan dengan keterkaitan.

    Wassalamualaikum.wr.wb

    ReplyDelete
  6. Firman Indra Pamungkas
    17709251048
    S2 Pendidikan Matematika 2017 Kelas C

    Assalamualaikum Warohmatullah Wabarokatuh
    Landasan strukturalism matematika menyatakan bahwa matematika terdiri dari struktur-struktur di dalamnya. Beberapa struktur dalam matematika seperti objek matematika yaitu fakta, konsep, prinsip, algoritma, dan struktur-struktur matematika yang lain. Matematika secara umum ditegaskan sebagai penelitian pola dari struktur, perubahan, dan ruang. Matematika juga dapat didefinisikan sebagai penelitian bilangan dan angka. Dalam pandangan formalis, matematika adalah tentang aksioma yang menegaskan struktur abstrak menggunakan logika simbolik dan notasi matematika. Struktur spesifik yang diselidiki oleh matematikawan seringkali berasal dari Ilmu Pengetahuan Alam, sangat umum di fisika, tetapi matematikawan juga menegaskan dan menyelidiki struktur karena struktur dapat membuat generalisasi dari berbagai konsep atau prinsip.

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

    ReplyDelete
  8. Ulivia Isnawati Kusuma
    17709251015
    PPs Pend Mat A 2017

    Epistemologi merupakan cabang filsafat yang berkaitan dengan ruang lingkup pengetahuan. Epistemologi matematika berarti yang mencakup ruang lingkup matematika bisa berupa abstraksi, deduktif, universal, rasional, dan lain-lain. Epistemologi berusaha meletakkan dasar pengetahuan matematika dan berusaha membuktikan kebenaran dalam matematika. Dalam kajian landasan epistemologi matematika terdapat kajian yang meliputi kebenaran, kepastian, rasionalitas, dan sebagainya.

    ReplyDelete
  9. Dewi Thufaila
    17709251054
    Pendidikan Matematika Pascasarjana C 2017

    Assalamualaikum.wr.wb

    Struktur matematika benar-benar dijelaskan dalam hal keterkaitan; awal khas dari teks matematika terdiri dari pengumuman bahwa objek matematika tertentu bilangan real seperti untuk dipelajari.
    Dalam beberapa kasus, setidaknya, satu-satunya hal tentang benda-benda ini adalah bahwa ada hubungan tertentu antara mereka dan / atau operasi pada mereka; dan satu dengan mudah mendapat kesan bahwa obyek itu sendiri tidak masalah.

    Wassalamualaikum.wr.wb

    ReplyDelete
  10. Isoka Amanah Kurnia
    17709251051
    PPs Pendidikan Matematika 2017 Kelas C

    The structure of the basic elements of mathematics should not take something in vain but does not dictate the choice and basic unit of structure, because it is not really an object. If mathematics is an abstract thing, why should it be applied? Mathematics is not about some collections of certain abstract objects but rather mathematics is the science of patterned structure. The structure of mathematics is really clearly related to each other, mathematics consists of mathematical objects which one of them is related to real life.

    ReplyDelete