Mathematical Reasoning with Diagrams by Mateja Jamnik

By Mateja Jamnik

Mathematicians at each point use diagrams to turn out theorems. Mathematical Reasoning with Diagrams investigates the probabilities of mechanizing this kind of diagrammatic reasoning in a proper computing device evidence approach, even providing a semi-automatic formal evidence system—called Diamond—which permits clients to end up arithmetical theorems utilizing diagrams.

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To be more accurate, the diagrammatic proof given here proves the following version of the theorem: (1 + 2 + 3 + · · · + n)2 = 1 × 12 + 2× 22 + 3 × 32 + · · · + n × n2 , because we are not appealing to any three dimensional property of a cube. A three dimensional version of this diagrammatic proof is to think of dots as spheres, and take for each thick ell, all of the sectioned squares and join them one on top of another to form a cube. The example just given shows that the degree of the polynomial in a theorem does not uniquely determine the dimension of the space for a diagram representing this theorem.

Furthermore, a whole new area of programming using visual languages has been established (see Burnett and Baker 1994). Yet, we have not come closer to defining any precise distinction between the two. It seems that scientists adopt a distinction which is suitable within the 15 Of course, any reasoning by a computer system could be considered as symbolic, because everything needs to be represented at the lowest level in terms of symbols on a computer. However, we do not support this view, we consider reasoning on different levels of abstraction.

8, we give structured informal schematic proofs for examples of Category 2 proofs given in Chapter 3. 1 Motivation The theorems whose diagrammatic proofs we aim to formalize and automate, are typically proved by mathematical induction in a symbolic logical proof. 4). An alternative way of capturing the generality of a diagrammatic proof of a theorem is to use schematic proofs. 17 A schematic proof is a recursive program with some parameters. By instantiation of these parameters the program generates ground instances of a particular proof.

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