نوع مقاله : مقاله علمی- پژوهشی
نویسنده
دکترای فلسفه، گروه فلسفه علم، واحد علوم و تحقیقات دانشگاه آزاد اسلامی، تهران، ایران.
چکیده
The concept of Mathematical Proof has been controversial for the past few decades. Different philosophers have offered different theories about the nature of Mathematical Proof, among which theories presented by Lakatos and Hersh have had significant similarities and differences with each other. It seems that a comparison and critical review of these two theories will lead to a better understanding of the concept of mathematical proof and will be a big step towards solving many related problems. Lakatos and Hersh argue that, firstly, “mathematical proof” has two different meanings, formal and informal; and, secondly, informal proofs are affected by human factors, such as individual decisions and collective agreements. I call these two theses, respectively, “proof dualism” and “humanism”. But on the other hand, their theories have significant dissimilarities and are by no means equivalent. Lakatos is committed to linear proof dualism and methodological humanism, while Hersh’s theory involves some sort of parallel proof dualism and sociological humanism. According to linear proof dualism, the two main types of proofs are provided in order to achieve a common goal: incarnation of mathematical concepts and methods and truth. However, according to the parallel proof dualism, two main types of proofs are provided in order to achieve two different types of purposes: production of a valid sequence of signs (the goal of the formal proof) and persuasion of the audience (the goal of the informal proof). Hersh’s humanism is informative and indicates pluralism; whereas, Lakatos’ version of humanism is normative and monistic.
کلیدواژهها
عنوان مقاله [English]
Lakatos and Hersh on Mathematical Proof
نویسنده [English]
- Hossein Bayat
PhD in Philosophy, Department of Philosophy of Science, Science and Research Branch of Islamic Azad University, Tehran, Iran.
چکیده [English]
The concept of Mathematical Proof has been controversial for the past few decades. Different philosophers have offered different theories about the nature of Mathematical Proof, among which theories presented by Lakatos and Hersh have had significant similarities and differences with each other. It seems that a comparison and critical review of these two theories will lead to a better understanding of the concept of mathematical proof and will be a big step towards solving many related problems. Lakatos and Hersh argue that, firstly, “mathematical proof” has two different meanings, formal and informal; and, secondly, informal proofs are affected by human factors, such as individual decisions and collective agreements. I call these two theses, respectively, “proof dualism” and “humanism”. But on the other hand, their theories have significant dissimilarities and are by no means equivalent. Lakatos is committed to linear proof dualism and methodological humanism, while Hersh’s theory involves some sort of parallel proof dualism and sociological humanism. According to linear proof dualism, the two main types of proofs are provided in order to achieve a common goal: incarnation of mathematical concepts and methods and truth. However, according to the parallel proof dualism, two main types of proofs are provided in order to achieve two different types of purposes: production of a valid sequence of signs (the goal of the formal proof) and persuasion of the audience (the goal of the informal proof). Hersh’s humanism is informative and indicates pluralism; whereas, Lakatos’ version of humanism is normative and monistic.
کلیدواژهها [English]
- formal proof
- informal proof
- practical proof
- humanism
- proof dualism
- Davis, Phillip J., Hersh, Reuben, (1990), The Mathematical Experience, Houghton Mifflin.
- Detlefsen, M., (2008) “Proof: Its Nature and Significance”, in Proof and Other Dilemmas, Eds: Gold, B., et. al., The Mathematical Association of America, pp 3-33.
- Hersh, R., (1997), What Is Mathematics, Really? Oxford University Press, New York.
- Kitcher, Philip, (1984), The Nature of Mathematical Knowledge, Oxford University Press.
- Lakatos, Imre, (1976), Proofs and Refutations: the logic of mathematical discovery, Eds: John Worrall, Elie Zahar, Cambridge University Press.
- Lakatos, Imre, (1978), “Mathematics, science and epistemology” in Philosophical Papers vol 2, Eds: John Worrall, Gregory Currie, Cambridge University Press, pp 61-69.
- Olsker, T. C., (2011), “What Do We Mean by Mathematical Proof?” in Journal of Humanistic Mathematics 1, No 1, January.
- Popper, Karl R., (1996), “The Myth of The Framework: In defence of science and rationality”, Routledge.
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