Belajar Matematika dan Bisnis Online

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Pengembangan Landasan Matematika

Perjalanan bangunan matematika dimulai dari logika, himpunan, relasi,
dan fungsi sebagai landasan pengembangan bidang matematika lainnya.
Bangunan matematika tersebut disusun secara deduktif berpangkal dari
aksioma dan definisi untuk membuktikan secara sah kebenaran-kebenaran
matematis yang disebut teorema.

Ilmu logika semula merupakan bagian dari ilmu filsafat dan telah mulai
dikembangkan oleh para filsuf Yunani Kuno ratusan tahun sebelum
masehi, seperti Thales, Pythagoras, Aristoteles, Euclides, dll.

¤Thales (624-545 SM) adalah ahli filsafat pertama yang menggunakan
metode ilmiah (dan bukan mitologi) untuk menjelaskan fenomena alam
semesta. Secara khusus ia adalah orang pertama Yunani yang
menggunakan metode deduktif untuk membuktikan kebenaran-kebenaran
pernyataan-pernyataan dalam Geometri. Thales sering kali disebut
"Bapak Ilmu Pengetahuan".

¤Pythagoras (566-497 SM)
Murid Thales mengembangkan Geometri dan ilmu yang semula bersifat
empiris menjadi ilmu eksak yang bersifat deduktif dengan membuktikan
kebenaran-kebenaran pernyataan-pernyataan di dalamnya. Salah satu
dalil yang dibuktikan oleh Pythagoras yaitu dikenal dengan "Teorema
Pythagoras". Metode deduktif oleh Thales dan Pythagoras itulah yang
merupakan awal pengembangan logika.

¤Aristoteles (384-322 SM) adalah ilmuwan pertama yang menggunakan
lambang-lambang dalam logika dan mengembangkannya menjadi ilmu logika
formal. Sumbangan utama Aristoteles adalah Silogisma yaitu suatu
kaidah penarikan kesimpulan. Logika yang dikembangkan Aristoteles
adalah Cikal bakal Logika Matematis yang dikembangkan pada abad ke-19.

¤Euclides (322-275 SM) yang dikenal sebagai "Bapak Geometri" adalah
peletak dasar-dasar Geometri sebagai ilmu deduktif. Dalam karya
monumentalnya yang berjudul "Elementa", ia membangun Geometri
berdasarkan seperangkat definisi, postulat (aksioma), dan
pengertian-pengertian umum.

Logika Matematis disebut juga logika simbolik merupakan bagian dari
ilmu logika yang dikembangkan pada zaman modern oleh George Boole,
Augustus De Morgan, Gottlob Frege, Bertrand Russell, dll. dengan
memanfaatkan konsep dan metode matematis. Pada dasarnya ada dua metode
matematis yang digunakan untuk mengembangkan logika yang tradisional
menjadi logika matematis yaitu metode aljabar dan metode aksiomatik.
Metode aljabar adalah metode yang dengan menggunakan lambang-lambang
matematis, mengoperasikan unsur-unsur matematika untuk menghasilkan
unsur lainnya. Metode aksiomatik yaitu metode untuk membangun
matematika berdasarkan seperangkat lambang, formula, aksioma, dan
kaidah-kaidah penarikan kesimpulan.

Matematika pada dasarnya bekerja dengan himpunan yaitu sekumpulan
objek konkret atau abstrak yang mempunyai ciri-ciri tertentu. Konsep
"Himpunan" semula digunakan secara intuitif saja dan hanya mengenai
himpunan-himpunan berhingga. Konsep himpunan secara formal baru mulai
dikembangkan menjelang abad ke-19 oleh matematikawan Jerman, George
Cantor (1845-1918). Melalui penelitiannya mengenai himpunan tak
hingga, ia meletakkan dasar bagi Teori himpunan yang formal.

Himpunan tidak hanya merupakan objek yang dipelajari oleh Matematika,
tetapi juga merupakan sarana yang dipergunakan untuk membangun dan
mengembangkan konsep-konsep dan kebenaran-kebenaran matematis. Salah
satu konsep dasar dalam matematika yang dibangun dengan menggunakan
himpunan adalah konsep relasi yang merumuskan kaitan antara entitas
dalam matematika. Dengan konsep relasi dapat dirumuskan hubungan
antara elemen-elemen dalam suatu himpunan dan lain-lain. Dari situ
muncul konsep-konsep lainnya seperti Partisi, poset, kisi, dsb.

Salah satu bentuk khusus yang amat penting dari relasi adalah fungsi (
atau pemetaan), yang mendasari konsep penting lainnya dalam
matematika. Dengan menggunakan konsep fungsi dapat dibandingkan dua
buah himpunan termasuk dua buah himpunan dengan Struktur Aljabar
tertentu (seperti Grup, Gelanggang, Medan, Ruang Vektor) dan dapat
dibangun konsep-konsep dasar lainnya dalam berbagai bagian dari
matematika, seperti Determinan, operasi, barisan, transformasi
geometri, dsb

My Referensi:
Susilo, Frans.2012. Landasan Matematika. Yogyaarta:Graha Ilmu

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