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 thesis, 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.
خلاصه ماشینی:
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.
com A formalistic definition of "mathematical proof" which is frequently seen in various related courses and textbooks is something like: A finite sequence of sentences in a formal language, arranged by a certain set of rules (each sentence in the sequence is either an axiom or an assumption or follows from the preceding sentences in the sequence by a rule of inference).
). (Hersh, 1997:49) Olsker adds the following explanation to clarify Hersh’s standpoint on the subject: The practical meaning implies that proof has a subjective side; the goal of a proof is to convince the mathematical community of the truth of a theorem.
Lakatos’ theory About three decades before Hersh, the Hungarian philosopher, Imre Lakatos made similar claims in his book, “Proofs and refutations: the logic of mathematical discovery” (1957) and his article, “what does mathematical proof prove?” (written between 1959 and 1961).
Lakatos holds that pre-formal proofs are an important part of the procedure to make informal mathematical theories, which are the main base and source for construction of formal systems.
The other difference between Lakatos and Hersh is the relation between formal and informal proofs in their theories.