Rigor and Proof in Mathematics: A Historical Perspective

Kleiner, Israel

doi:10.2307/2690647

Cited by 27 publications

(15 citation statements)

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“…Tujuan dari melakukan tinjauan sejarah adalah untuk mencoba melihat perubahan pandangan dari proses membuktikan sepanjang sejarahnya. Kleiner (1991) menyatakan bahwa gagasan pembuktian itu bukanlah hal yang absolut. Matematikawan memandang bahwa apa yang mendasari keterterimaan bukti semakin meningkat.…”

Section: Evolusi Bukti Matematikaunclassified

“…Matematikawan memandang bahwa apa yang mendasari keterterimaan bukti semakin meningkat. Masih menurut Kleiner (1991) matematika rigor mirip seperti memakai pakaian, gayanya hendaknya disesuaikan dengan kesempatan tertentu, hal ini akan mengurangi kenyamanan dan menghalangi kebebasan bergerak jika terlalu longgar atau terlalu ketat. Dari dua pernyataan Kleiner tersebut bisa kita katakan bahwa standar dari kerigoran dari bukti matematika dapat berubah-ubah dan tidak harus dari yang kurang rigor menuju ke yang lebih rigor.…”

Section: Evolusi Bukti Matematikaunclassified

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Mengatasi Kesulitan Mahasiswa Ketika Melakukan Pembuktian Matematis Formal

Sentosa¹

2014

JPMIPA

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ABSTRAKMelakukan pembuktian matematis formal merupakan hal yang sentral dalam matematika. Ketika seseorang mempunyai dugaan tentang suatu hal, salah satu cara yang paling tepat untuk meyakinkan bahwa hal tersebut benar adalah dengan melakukan pembuktian matematis yang sah. Proses mendefinisikan pembuktian ini berkembang dari masa ke masa sesuai dengan perkembangan jaman. Walaupun belum ada konsensus yang disepakati oleh matematikawan secara keseluruhan tentang apa itu bukti matematis formal, namun proses membuktikan ini merupakan suatu masalah tersendiri ketika bukti matematis dikenalkan saat pembelajaran berlangsung. Kesulitan ini terjadi tidak hanya pada mahasiswa tingkat pertama perkuliahan, namun ternyata mahasiswa program yang lebih tinggi (pascasarjana) pun mengalami kesulitan dalam membuktikan walaupun dengan porsi yang lebih kecil. Jika ditelusuri proses berpikir pembuktian matematikawan ternyata sangat berbeda dengan alur berpikir yang disajikan pada buku-buku teks matematika saat ini. Sehingga terdapat masalah ketika mahasiswa melakukan proses pembuktian. Tulisan ini mencoba untuk menjelaskan pembuktian matematis secara komprehensif dimulai dari sejarah pembuktian sampai dengan masa kini, dan mengeksplorasi kesulitan-kesulitan pembuktian yang telah diteliti oleh para peneliti. Tulisan ini pun mencoba untuk mendiskusikan suatu cara yang paling baik yang dapat dilakukan untuk mengatasi kesulitan ketika mahasiswa dihadapkan pada proses membuktikan. Dimulai dari aspek kognitif, afektif, dan commognition.Kata Kunci: afektif, aspek kognitif, bukti formal, commognition ABSTRACT Doing formal mathematical proof is central in mathematics. When someone has a conjecture about something, one of the most appropriate ways to ensure that we do the right thing is to do a legitimate mathematical proof. The process of defining formal mathematics proof is changing from time to time in accordance with the changing times. Although there has been no consensus reached by mathematicians as a whole about what the formal proof, but the process of proving is another issue when mathematical proof introduced during the learning. This difficulty occurs not only in the first years student in undergraduate, but it turned out to graduted student have difficulty proving althought with smaller portions. If we try to trace mathematicians' process to do prove, we can see that it turned out very different from the logic presented in mathematics textbooks today. This paper tries to explain in a comprehensive mathematical proof starts from the historical of proofing to the present, and explores the difficulties of proof which has been studied by researchers. This paper attempts to discuss the best way to solve the difficulties when students are faced with proving process. Starting from the cognitive, affective, and commognition

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Section: Evolusi Bukti Matematikaunclassified

Mengatasi Kesulitan Mahasiswa Ketika Melakukan Pembuktian Matematis Formal

Sentosa¹

2014

JPMIPA

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“…Emphasis on rigor should not overshadow a more conceptual overview or a work's physical motivation. Simmons remarked [27] "Mathematical rigor is like clothing: in its style it ought to suit the occasion, and it diminishes comfort and restricts freedom of movement if it is either too loose or too tight." Indeed, students majoring in mathematics learn the concept of compactness without knowing its physical or historical background.…”

Section: Introductionmentioning

confidence: 99%

Compactness and Dirichlet's Principle

Seo¹,

Zorgati²

2014

Journal of the Korea Society for Industrial and Applied Mathema

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ABSTRACT. In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet's principle. We emphasize on the intuition in Riemann's statement on the principle criticized by Weierstrass'requirement of rigor followed by Hilbert's restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion.Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly after Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincaré and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.

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“…This myth holds that because mathematics is a discipline based solely on logic, mathematical knowledge and truth are independent of time and culture. For instance, even though philosophers and historians argue that the standards and styles of mathematical argumentation have evolved over time (Kleiner, 1991), even among contemporary branches of mathematics (Rav, 1999), the mathematicians interviewed by Shanahan et al were the only experts in their study who believed that the utility and interpretation of their domain-specific text was not time dependent. Consistent with this myth is the belief that when mathematicians read arguments, they do not need to consider the source of the arguments, only the contents of the argument itself.…”

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confidence: 99%

The Influence of Sources in the Reading of Mathematical Text

Weber

Mejía-Ramos

2013

Journal of Literacy Research

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In a recent article published in this journal, Shanahan, Shanahan, and Misischia investigated the differences in how chemists, historians, and mathematicians read text specific to their disciplines. Unlike the chemists and historians, the pair of mathematicians in this study did not consider sources when reading and evaluating their text. In this response, we contend that most mathematicians regularly do consider sources when reading mathematical arguments and papers. We summarize the results of four empirical studies, which suggest that this is the case.

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Rigor and Proof in Mathematics: A Historical Perspective

Cited by 27 publications

References 0 publications

Mengatasi Kesulitan Mahasiswa Ketika Melakukan Pembuktian Matematis Formal

Mengatasi Kesulitan Mahasiswa Ketika Melakukan Pembuktian Matematis Formal

Compactness and Dirichlet's Principle

The Influence of Sources in the Reading of Mathematical Text

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