Strategic Maneuvering in Mathematical Proofs

This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs.

Socratic Proofs

Journal of Philosophical Logic - Our aim is to express in exact terms the old idea of solving problems by pure questioning. We consider the problem of derivability: “Is A derivable from...     

An Exploration of the Potential for Mathematical Experiences in Informal Learning Environments

ABSTRACT

As young learners develop understanding of concepts in mathematics, formal learning environments tend to focus on providing isolated instruction, with limited opportunities for learning experiences that help them make connections. Informal learning environments might thus play a significant role in providing mathematical learning experiences that complement formal classroom experiences for young learners. This study was organized to determine the potential for mathematical experiences through an examination of existing informal learning opportunities and to ascertain a general perspective of the value of mathematical thinking in informal learning environments from current informal educators. This study served as exploratory research conducted to provide more information about the current evidence of math learning in informal settings. Data were collected and organized from observations at 3 local informal learning sites, interviews with leading museum educators in the region, and a written survey completed by museum educators across the nation. The findings reveal that parents/families play a critical role and the potential for mathematics in current exhibits is not emphasized.     

Truth,Proofs and Functions

Synthese - There are two different ways to introduce the notion of truthin constructive mathematics. The first one is to use a Tarskian definition of truth in aconstructive (meta)language....     

宇宙大爆炸:有什么证据?

大爆炸理论是关于宇宙形成的最有影响的一种学说,英文说法为Big Bang,也称为大爆炸宇     

Proofs,pictures, and Euclid

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid’s reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received view, this essay provides a contrary analysis by introducing a formal account of Euclid’s proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid’s arguments. It specifies what diagrams Euclid’s diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored.     

Ontological Proofs of Existence and Non-Existence

Caramuels’ proof of non-existence of God is compared with Gödel’s proof of existence.     

Meaning Approached Via Proofs

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence.     

Socratic Proofs for Quantifiers

First-order logic is formalized by means of tools taken from the logic of questions. A calculus of questions which is a counterpart of the Pure Calculus of Quantifiers is presented. A direct proof of completeness of the calculus is given. *Research for this paper was supported by The Foundation for Polish Science (both authors), and indirectly (in the case of the first author) by a bilateral exchange project funded by the Ministry of the Flemish Community (project BIL 01/80) and the State Committee for Scientific Research, Poland.     

Remarks on Independence Proofs and Indirect Reference

In the last two decades, there has been increasing interest in a re-evaluation of Frege's stance towards consistency- and independence proofs. Papers by several authors deal with Frege's views on these topics. In this note, I want to discuss one particular problem, which seems to be a main reason for Frege's reluctant attitude towards his own proposed method of proving the independence of axioms, namely his view that thoughts, that is, intensional entities are the objects of metatheoretical investigations. This stands in contrast to more straightforward interpretations, which claim that Frege's hesitancy is mainly due to worries concerning the logical constants or what counts as a logical inference.     

非形式逻辑与演绎主义

非形式逻辑出现于上个世纪70年代,它首先是一种寻求更好的大学逻辑教学方式的努力。进而,非形式逻辑学者们逐渐开始触及和探讨越来越多的理论议题,即发展不依赖于形式逻辑的关于论证以及好论证的理解方式。在1998年世界哲学大会上,布莱尔和我阐释了非形式逻辑为哲学所带来的理论后果,其中一点我们称之为“演绎主义的终结”。那时我们并未给出演绎主义的定义,而只是把它等同于麦金泰尔的那个精炼的说法：“任何推理,不是演绎的,就是有缺陷的。”但现在看来,我们在当时就作出“演绎主义的终结”这一论断,显然是有点为时过早了。因为演绎主义似乎至今都还很有生命力,甚至在那些倾向于非形式逻辑的学者那里它也能得到支持：恩尼斯长期以来都辩护演绎主义是一种论证重构策略,格罗尔克更是致力于辩护它是一种论证评价理论。在本文中我将论证,非形式逻辑最好被理解为一种不诉诸于演绎技巧和演绎规范的逻辑研究。当我们意识到演绎和演绎主义已经如何深深地扎根于我们的哲学发展史中,并且牢牢地控制了我们关于逻辑研究的理解,我们就能明白非形式逻辑这一理论努力是多么的困难和重要。我将首先澄清“演绎主义”的意思,然后再分别考察那些赞成和反对演绎主义的论证,最后,我将表明非形式逻辑是从演绎主义中挽救逻辑的理论尝试。     

Program Extraction from Normalization Proofs

This paper describes formalizations of Tait's normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.