THE PHILOSOPHY OF MATHEMATICS EDUCATION Ditulis Oleh: Paul Ernest
FILSAFAT MATEMATIKA DAN
PENDIDIKAN MATEMATIKA
THINKING ABOUT MATHEMATICS Ditulis Oleh Steward Shapiro
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
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
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 BERIKUT
KEABADIAN-KETAKHINGGAAN (IMMORTALITY-INFINITY) KEBETULAN-PROBABILITAS (CHANCE-PROBABILITY)
KUANTITAS-BILANGAN (QUANTITY-NUMBER)
FILSAFAT MATEMATIKA
TIDAK SEMUA PENGALAMAN UMAT MANUSIA DITELAAH
QUANTITY, RELATION, POLA, FORM, STRUCTURE
BEBERAPA BIDANG YANG MUNCUL SEBAGAI PERWUJUDAN DARI
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
The links between philosophy and mathematics are
ancient and complex.
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.
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
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
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
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”
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
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.
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.
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.
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
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 major
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 of
space. But he could not accept Kant’s explanation of our
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 thought it 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
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 apriori
Hume’s
principle 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 objects
falling 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 (or
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 characterised
axiomatically up to structural equivalence or isomorphisms. Hilbert’s structuralist approach, of course, goes back to Dedekind’s
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.
