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- Solomon Feferman, And so on … : Reasoning with Infinite Diagrams.A proof of a theorem in mathematics is what we require to convince ourselves and others of the truth of the statement made by the theorem. Here ‘truth’ is taken in its prima facie sense, i.e. the notions involved in the statement of the theorem are supposed to be meaningful, and if it is to be truth for us, we are supposed to understand the meaning of those notions. In order to be convinced of a proof, one must follow the argument and check the various steps for ourselves, making use not only of what is given in the proof itself but what is required from background knowledge, i.e. previous statements that we have already accepted to be true on some ground or other. And that background knowledge may require understanding other notions not explicitly involved in the statement of the theorem. So both background knowledge and the understanding of meanings is an essential component of what it takes to accept a given proof. Even given that, it is possible to go through the steps of a given proof and not understand the proof itself. That is a different level of understanding, which, when successful, leads one to say, ”Oh, I see!” In other words, this “really understanding the proof” is a special kind of insight into how and why the proof works. And that is necessary if one wants to follow proofs of related theorems and contribute to the subject by creating new proofs oneself. It follows that one cannot truly be a consumer and producer of mathematics without achieving real understanding of the arguments.No categoriesIn my reading list | Discuss this article | Edit | Categorize |My bibliography
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Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams (...)
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