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首页> 外文期刊>Studia Logica >Confronting Ideals of Proof with the Ways of Proving of the Research Mathematician
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Confronting Ideals of Proof with the Ways of Proving of the Research Mathematician

机译：用证明数学家的证明方式来证明证明的理想

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摘要

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the "axiomatic conception" of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books. Along with Cellucci, Rav, Grattan-Guinness and Grosholz, we deplore this view of mathematical proof, and favour instead the "analytic conception" of mathematical proof, where the axiomatic proof, when it exists at all, is only the core of a proof. An analytic proof solves a problem, by making hypotheses and using a mixture of deductive moves and induction (loosely construed to include diagrams, etc.) to present a solution to the problem. This implies that proofs are not always finite, that it might involve much more than axioms and straight logical inferences from these deductions and a proof can always be questioned. Moreover, this is where a lot of the interesting conceptual work of mathematics takes place. We view proofs as communicative acts made within the mathematical community which ensures correctness through application, context and standards of rigor.
机译：在本文中，我们讨论了哲学家和许多数学家关于数学证明的普遍观点。在切鲁奇之后，我们将主流观点称为证明的“公理概念”。这个概念包括以下思想：证明是有限的，它源自公理，它是结论问题的最终定论。收到的视图可以追溯到Frege，Hilbert和Gentzen等人，并且在数学教科书和逻辑教科书中都很普遍。与Cellucci，Rav，Grattan-Guinness和Grosholz一起，我们对这种数学证明的观点感到遗憾，而代之以数学证明的“解析概念”，其中公理证明（即使存在）仅是证明的核心。解析证明通过做出假设并结合演绎运动和归纳法（松散地解释为包括图表等）来解决问题，从而解决了问题。这意味着证明并不总是有限的，它可能涉及的不仅仅是公理和从这些推论中得出的直接逻辑推论，而且总是可以质疑一个证明。此外，这是发生许多有趣的数学概念性工作的地方。我们将证明视为在数学界内部进行的交流，这通过应用，上下文和严格标准来确保正确性。

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