THE PHILOSOPHY OF MATHEMATICS EDUCATION Ditulis Oleh: Paul Ernest

FILSAFAT MATEMATIKA DAN PENDIDIKAN MATEMATIKA

THINKING ABOUT MATHEMATICSDitulis Oleh Steward Shapiro

FILSAFAT MATEMATIKA

The Liang Gie

FILSAFAT DAN MATEMATIKA BERHUBUNGAN ERAT SEJAK DULU

FILSAFAT DAN GEOMETRI SESUNGGUHNYA LAHIR PADA MASA YANG SAMA DAN DI TEMPAT YANG SAMA, YAITU PADA SEKITAR TAHUN 640-546 SM DI MILETUS DARI SESEORANG BERNAMA THALES (AHLI FILSAFAT SEKALIGUS GEOMETRI)

HUBUNGAN TIMBAL BALIK DAN SALING PENGARUH MEMPENGARUHI ANTARA FILSAFAT DAN MATEMATIK SALAH SATUNYA DIPICU OLEH FILSUF ZENO DARI ELEA.ZENO MEMPERBINCANGKAN PARADOK-PARADOK YANG BERTALIAN DENGAN PENGERTIAN GERAK, WAKTU DAN RUANG

PARADOK ACHILLES

Pelari cepat Achilles tidak mungkin mengejar seekor kura-kura yang lambat bila binatang itu telah berjalan mendahului pada suatu jarak tertentu. Menurut Zeno Saat Achilles berada dibelakang kura-kura, binatang tersebut telah menempuh jarak tertentu. Ketika Achilles mencapai titik dimana binatang itu semula berada maka binatang itu telah maju lagi dan seterusnya sehingga tak mungkin pelari tersebut mendahukui kura-kura.

PARADOK ZENO : SUATU BENDA YANG BERGERAK MENCAPAI SUATU JARAK TERTENTU, BENDA TERSEBUT HARUS MENEMPUH ½ JARAK YANG DIMAKSUD, SEBELUM MENEMPUH SETENGAH JARAK HARUS MENEMPUH ½ JARAK TERDAHULU, DEMIKIAN SETERUSNYA SETIAP KALI SELALU ADA JARAK ½ YANG HARUS DILEWATINYA SECARA TERUS MENERUS. BERARTI RUANG YANG DAPAT DIBAGI DALAM DIKOTOMI YANG JUMLAHNYA TAK TERHINGGA TIDAK MUNGKIN DITEMPUH DALAM JANGKA WAKTU TERTENTU.

INTERAKSI ANTARA FILSAFAT DAN MATEMATIKA DAPAT TERLIHAT DENGAN ADANYA PADANAN KONSEP DAN PROBLEMA

FILSUF MERENUNGKAN MASALAH-MASALAH KEABADIAN, KEBETULAN, EVOLUSI , GENUS DAN KUANTITAS

AHLI MATEMATIKA MEMPELAJARI KETAKHINGGAAN, PROBABILITAS, KEKONTINUAN, HIMPUNAN DAN BILANGAN.

KESEJAJARAN KEDUANYA DAPAT DIGAMBARKAN SEBAGAI BERIKUTKEABADIAN-KETAKHINGGAAN (IMMORTALITY-INFINITY)KEBETULAN-PROBABILITAS (CHANCE-PROBABILITY)KUANTITAS-BILANGAN (QUANTITY-NUMBER)

KESAMAAN LAIN :FILSAFAT DAN MATEMATIKA BERGERAK PADA TINGKAT GENERALITAS DAN ABSTRAKSI YANG TINGGI, MEMBAHAS BERBAGAI IDE YANG SANGAT UMUM DAN LAZIMNYA MELAMPAUI TARAF KEKONKRETAN

FILSAFAT MATEMATIKA

METODE

DAPAT MENGGUNAKAN BANYAK METODE DEDUKSI

RUANGLINGKUP

PENGALAMAN UMUM UMAT MANUSIA

TIDAK SEMUA PENGALAMAN UMAT MANUSIA DITELAAHQUANTITY, RELATION, POLA, FORM, STRUCTURE

DENGAN BERPANGKAL DARI AKSIOMA ATAU PREMIS DITURUNKAN KESIMPULAN-KESIMPULAN SANGAT JAUH

BEBERAPA BIDANG YANG MUNCUL SEBAGAI PERWUJUDAN DARI INTERAKSI FILSAFAT DAN MATEMATIKAFILSAFAT MATEMATIKALANDASAN MATEMATIKAMETAMATEMATIKAFILSAFAT KEMATEMATIKAAN

LANDASAN MATEMATIKA LEBIH SEMPIT DARI FILSAFAT MATEMATIKA, KHUSUSNYA BERKAITAN DENGAN KONSEP-KONSEP DAN PRINSIP YANG DIGUNAKAN DALAM MATEMATIKA

SECARA HARFIAH METAMATEMATIKA ADALAH BIDANG PENGETAHUAN DI LUAR ATAU DIATAS MATEMATIKA YANG MENELAAH MATEMATIKA ITU SENDIRI.METAMATEMATIKA : SUATU TEORI PEMBUKTIAN UNTUK MENETAPKAN ADA ATAU TIDAK ADANYA KONSISTENSI DALAM MATEMATIKA DAN MENGANALISIS KELENGKAPAN SUATU SISTEM FORMAL.

FILSAFAT KEMATEMATIKAAN SERING DISEBUT FILSAFAT BERDASARKAN MATEMATIKA, KEMUDIAN MENGGUNAKAN MATEMATIKA SEBAGAI TITIK TOLAK DAN SUMBER IDE UNTUK MELAKUKAN PEMIKIRAN FILSAFATI

The links between philosophy and mathematics are ancient and complex. The two disciplines are in a sense coeval: for both, the Ancient Greeks were the first to introduce systematicity, rigor, and the centrality of justification to their practice. Indeed, Plato (428-348(7) BCE) had it inscribed on the gates of his Academy that no one should enter who knew no mathematics.

Let us turn now to a more positive characterization of a philosophical approach to mathematics. It will be helpful to focus on instances of actual mathematics, so let us consider a few theorems and their proofs, and then survey the kinds of typically philosophical issues they raise. The first two both involve the distinction between rational and irrational numbers. A rational number is one that can be expressed as a fraction; for example, 3/5, —19/12, and 8/1 are all rational numbers. An irrational number - phi, for instance - is one that cannot be expressed as such a fraction. The rationals and irrationals together make up the real numbers. Our first theorem dates from Ancient Greece; it is usually atttributed to a member of the school of Pythagoras, although precisely who first proved it is not known. The English mathematician G. H. Hardy (1877-1947) called it a theorem "of the highest class. [It] is as fresh and significant as when it was discovered - two thousand years have not written a wrinkle" on it.

This demonstrates that not all magnitudes - in particular, not the length of the hypotenuse of a right-angled triangle of unit base and height - can be treated by the theory of numerical proportion upon which the mathematics of Ancient Greece was based. It must have thus constituted something of a revolution.

These examples are in a way paradigmatic mathematics. To be sure, mathematics is filled with proofs that are much longer and more complicated, and with theorems that involve concepts far more intricate than those appearing above. But most philosophical questions about mathematics can already be raised with regard to such simple examples. We shall briefly examine a few of these in turn. To begin with, note that (assuming you had not seen these proofs before) you now know three more truths than you did a few moments ago. How did you acquire this knowledge? One way you did not acquire it is through observation. It is true that you used your eyes to read the sentences of the proofs.

platonism insists that mathematics is mind-independent, in the sense that

whether a mathematical statement holds is quite independent of what we

think.

We can imagine certain realms in which the beliefs of observers in effect

settle what is true and what is not. But mathematics, according to the

platonist, is not like this: the truth or falsity of a mathematical claim is

not determined by what anyone believes about its truth value.

This, too, is a plausible position with regard to the theorems above. For

instance, the square root of 2 is irrational regardless of whether anyone

believes or wants it to be; indeed, its irrationality is not contingent on

anyone's having beliefs about it at all. This result obtains, Hardy insisted,

"not because we think so, or because our minds are shaped in one way

rather than another, but because it is so, because mathematical reality is

built that way."

OBJEK KAJIAN FILSAFAT MATEMATIKA

Filsafat Matematika adalah suatu cabang matematika yang memusatkan pengkajiannya pada dua pertanyaan pokok :

1. Memusatkan kajian terhadap arti dari kalimat matematika

2. Memusatkan kajian bertolak dari pertanyaan apakah objek abstrak matematika itu ada.

Terkait dengan yang pertama, akan muncul pertanyaan2:

Sebenarnya apa arti kalimat-kalimat matematika “3 merupakan bilangan prima”, “2+2=4” atau “Terdapat tak hingga bilangan prima”

Sehingga tugas pokok dari filosuf adalah mengkonstruk teori semantik untuk bahasa matematika

semantik=mempelajari makna kata

Kalimat “Kapuas merupakan nama gunung di Jawa” secara semantik adalah salah, tetapi “Semeru merupakan nama gunung di Jawa” secara semantik benar.

Lalu secara semantik, bagaimana dengan kalimat matematika “3 merupakan bilangan prima”, “2+2=4” atau “Terdapat tak hingga bilangan prima”

Alasan para filosof terkait dengan hal ini adalah:

1. Tentang kebenaran yang tidak dapat serta merta dijelaskan

2. Jawaban yang berbeda akan membawa implikasi filosofis yang berbeda

Misalnya tentang kalimat “3 merupakan bilangan prima”, apakah 3? 3 itu apa?

Antirealis mengatakan bahwa bilangan itu tidak ada, bagaimana kita menilai secara semantik?

Realis mengatakan bahwa bilangan itu ada.

Dalam kelompok realis sendiri ada yang menyebut bilangan sebagai objek mental(something like ideas in people’s head) tetapi adapula yang menganggap bilangan ada di luar pikiran ( numbers exist outside of people’s head), seperti pada dunia nyata.

Pandangan lain yaitu dari penganut Plato (platonisme) yang menganggap bahwa bilangan merupakan objek abstrak yang tidak nyata dan bukan objek mental.

Jadi menurut platonis ojek abstrak itu ada tetapi bukan sesuatu pada dunia nyata atau dalam pikiran manusia.

Karena kenyataannya bilangan (dan objek matematika yang lain) tidak ada pada ruang dan waktu manapun.

Pertanyaan berikutnya bagaimanakah objek abstrak ada?

Mathematical Platonism

Platonisme pada matematika, memandang bahwa

a. Terdapat objek abstrak yang secara keseluruhan non spatial-temporal, non physical, dan non mental

b. Terdapat kebenaran kalimat secara matematik yang melengkapi gambaran suatu objek

Diantara Platonist kontemporer, akhirnya tersepakati bahwa yang dimaksud objek abstrak adalah objek yang nonspatialtemporal.

Platonisme merupakan paham dalam matematika yang eksis selama dua milenium setelah itu stagnan, setelah Gotlob Frege mengembangkan logika matematika modern

Versi Platonisme nontradisional

Dikembangkan pada tahun 1980-an dan 1990-an oleh:

1. Penelope Maddy

2. Mark Balaguer dan Edward Zaita

3. Michael Resnik dan Stewart Shapiro

Konsen atas bagaimana orang mendapatkan pengetahuan dari objek abstrak

Menurut Maddy, matematika adalah pengetahuan tentang objek abstrak dan objek abstrak merupakan sesuatu yang nonphysical dan non mental, meskipun berada pada ruang dan waktu

The period in the foundations of mathematics that started in 1879 with the publication of Frege’s Begriffsschrift [18] and ended in 1931 with Go¨del’s [24] U¨ ber formal unentscheidbare S¨atze der Principia Mathematica und verwandter Systeme can reasonably be called the classical period. It saw the development of three majorfoundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert’s formalist and proof-theoretic programme.

Kant claimed that our knowledge of mathematics is synthetic apriori and based on a faculty of intuition. Frege accepted Kant’s claim in the case of geometry, i.e., he thought that our knowledge of Euclidian geometry is based on pure intuition ofspace. But he could not accept Kant’s explanation of our knowledge of statements about numbers.

Frege thought of numerical statements as being objectively true or false. Moreover, he interpreted these statements as literally being about abstract mathematical objects that do not exist in space or time. Now the question arose: How can we have knowledge about numbers and their properties, if numbers are abstract objects?Clearly we cannot interact causally with abstract entities.

In order to show that apriori knowledge of arithmetic is possible, Frege thoughtit necessary and sufficient to establish the logicist thesis that arithmetic is reducible to logic. More precisely, he wanted to show that:

(i) the concepts of arithmetic can be explicitly defined in terms of logical concepts;(ii) the truths of arithmetic can be derived from logical axioms (and definitions) by purely logical rules of inference.

The following four claims are implicit in Frege’s logicist programme: (a)Logic is (or can be presented as) an interpreted formal system (a

Begriffsschrift); (b) It can be known apriori that the axioms of logic are true and that the

logical rules of inference preserve truth; (c) the concepts of arithmetic are logical concepts; and(d) the truths of arithmetic are provable in logic. From (a) and (b) it follows that the theorems of logic are true. Since a

contradiction cannot be true, it follows that logic is consistent.Moreover, it seems to follow from (b) that we can gain aprioriknowledge of the theorems of logic by proving them. In virtue of (d) then,

arithmetic must be consistent and its truths knowable apriori.

Hume’sprinciple says that two concepts F and G have the same cardinal number iff they are equinumerous, i.e., iff there is a one-to-one correspondence between the objectsfalling under F and the objects falling under G. In symbols:

where F ≈ G means that there exists a one-to-one correspondence between the objects that fall under F and G respectively.According to Hume’s principle, the concept of (cardinal) number is obtained by (Fregean) abstraction from the concept of equinumerosity between concepts (orproperties).

A milestone in mathematics is Hilbert’s Grundlagen der Geometrie [33] from 1899.Its importance for the conceptual development of modern mathematics is difficult to overstate. Here Hilbert gave, for the first time, a fully precise axiomatization of Euclidean geometry. The entities like point, line and plane are defined only implicitly by their mutual relations. Generalising this method of implicit definitions it became possible to work also with complicated mathematical systems characterisedaxiomatically up to structural equivalence or isomorphisms. Hilbert’s structuralist approach, of course, goes back to Dedekind’s characterisation in [12] of the natural number system in terms of simply infinite systems. It was also foreshadowed by Felix Klein’s classification of geometries using group invariants (the Erlangen programme).

Hilbert proposed his finitist consistency programme: consider a formal

system T in which all of classical mathematics can be formalised and

prove by finitistic means the consistency of T .

In this way, Hilbert wanted to prove the consistency of classical

mathematics in a particularly elementary part: “finitistic mathematics”.

When Hilbert formulated his programme, he had two significant facts

available:

(i) Classical mathematics can be represented in formal systems of set

theory or type theory.

(ii) These formal systems can be described in a finitistic manner.