FILSAFAT MATEMATIKA DAN
PENDIDIKAN MATEMATIKA
THINKING ABOUT MATHEMATICS
Ditulis Oleh Steward Shapiro
THE PHILOSOPHY OF MATHEMATICS EDUCATION
Ditulis Oleh: Paul Ernest
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 640546 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 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.
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.
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)
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 DITELAAH
QUANTITY, RELATION, POLA, FORM,
STRUCTURE
DENGAN BERPANGKAL DARI AKSIOMA
ATAU PREMIS DITURUNKAN KESIMPULANKESIMPULAN SANGAT JAUH
BEBERAPA BIDANG YANG MUNCUL SEBAGAI PERWUJUDAN DARI
INTERAKSI FILSAFAT DAN MATEMATIKA
FILSAFAT MATEMATIKA
LANDASAN MATEMATIKA
METAMATEMATIKA
FILSAFAT 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 major
foundational 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 of
space. 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 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
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 apriori
knowledge of the theorems of logic by proving them. In virtue of (d) then,
arithmetic must be consistent and its truths knowable 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
properties).
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
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.