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On Automating Diagrammatic Proofs of Arithmetic Arguments
 Journal of Logic, Language and Information
, 1999
"... . 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 corres ..."
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Cited by 29 (7 self)
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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 are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the "inference steps" of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive !rule provides the mathematical basis for this step from schematic proofs to theoremhood. In ...
Architectural and representational requirements for seeing processes, protoaffordances and affordances. Research paper, for Workshop Proceedings COSYTR0801a
"... Abstract. This paper, combining the standpoints of philosophy and Artificial Intelligence with theoretical psychology, summarises several decades of investigation by the author of the variety of functions of vision in humans and other animals, pointing out that biological evolution has solved many m ..."
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Cited by 12 (9 self)
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Abstract. This paper, combining the standpoints of philosophy and Artificial Intelligence with theoretical psychology, summarises several decades of investigation by the author of the variety of functions of vision in humans and other animals, pointing out that biological evolution has solved many more problems than are normally noticed. For example, the biological functions of human and animal vision are closely related to the ability of humans to do mathematics, including discovering and proving theorems in geometry, topology and arithmetic. Many of the phenomena discovered by psychologists and neuroscientists require sophisticated controlled laboratory settings and specialised measuring equipment, whereas the functions of vision reported here mostly require only careful attention to a wide range of everyday competences that easily go unnoticed. Currently available computer models and neural theories are very far from explaining those functions, so progress in explaining how vision works is more in need of new proposals for explanatory mechanisms than new laboratory data. Systematically formulating the requirements for such mechanisms is not easy. If we start by analysing familiar competences, that can suggest new experiments to clarify precise forms of these competences, how they develop within individuals, which other species have them, and how performance varies according to conditions. This will help to constrain requirements for models purporting to explain how the competences work. For example, Gibson’s theory of affordances needs a number of extensions, including allowing affordances to be composed in several ways from lower level protoaffordances. The paper ends with speculations regarding the need for new kinds of informationprocessing machinery to account for the phenomena.
The hierarchical product of graphs
 DISCRETE APPL. MATH., SUBMITTED. (AVAILABLE AT HTTP://HDL.HANDLER.NET/2117/672
, 2007
"... A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the correspondin ..."
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Cited by 1 (1 self)
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A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some wellknown properties of the cartesian product, such as a reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree Tm (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.
Automation of Diagrammatic Proofs in Mathematics
 New Bulgarian University
, 1996
"... Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than ..."
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Cited by 1 (1 self)
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Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding logical proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. The proposed research project is to identify and ultimately automate this diagrammatic reasoning on mathematical theorems. The system that we are in the process of implementing will be given a theorem and will (initially) interactively prove it by the use of geometric manipulations on the diagram that the user chooses to be the appropriate ones. These operations will be the inference steps of the proof. The constructive !rule will be used as a tool to capture the generality of diagrammatic proofs. In this way, we hope to verify and to show that the diagra...
A Proposal for Automating Diagrammatic Reasoning in Continuous Domains
"... . This paper presents one approach to the formalisation of diagrammatic proofs as an alternative to algebraic logic. An idea of `generic diagrams' is developed whereby one diagram (or rather, one sequence of diagrams) can be used to prove many instances of a theorem. This allows the extension o ..."
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. This paper presents one approach to the formalisation of diagrammatic proofs as an alternative to algebraic logic. An idea of `generic diagrams' is developed whereby one diagram (or rather, one sequence of diagrams) can be used to prove many instances of a theorem. This allows the extension of Jamnik's ideas in the Diamond system to continuous domains. The domain is restricted to nonrecursive proofs in real analysis whose statement and proof have a strong geometric component. The aim is to develop a system of diagrams and redraw rules to allow a mechanised construction of sequences of diagrams constituting a proof. This approach involves creating a diagrammatic theory. The method is justi ed formally by (a) a diagrammatic axiomatisation, and (b) an appeal to analysis, viewing the diagram as an object in R 2 . The idea is to then establish an isomorphism between diagrams acted on by redraw rules and instances of a theorem acted on by rewrite rules. We aim to implement these ideas in an interactive prover entitled Rap (the Real Analysis Prover). 1
DOI: 10.1017/S000000000000000 Printed in the United Kingdom Towards a diagrammatic classification
"... In this article I present and discuss some criteria to provide a diagrammatic classification. Such a classification is of use for exploring in detail the domain of diagrammatic reasoning. Diagrams can be classified in terms of the use we make of them static or dynamic and of the correspondence bet ..."
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In this article I present and discuss some criteria to provide a diagrammatic classification. Such a classification is of use for exploring in detail the domain of diagrammatic reasoning. Diagrams can be classified in terms of the use we make of them static or dynamic and of the correspondence between their space and the space of the data they are intended to represent. The investigation is not guided by the opposition visual vs. non visual, but by the idea that there is a continuous interaction between diagrams and language. Diagrammatic reasoning is characterized by a duality, since it refers both to an object, the diagram, having its spatial characteristics, and to a subject, the user, who interprets them. A particular place in the classification is occupied by constructional diagrams, which exhibit for the user instructions for the application of some procedures. 1 Introduction: recentering the discussion on visual thinking In the process of describing the world around us, not only language and verbal thought matter. In recent years, an interest has grown around the phenomenon of non verbal thought, such as the use of visualization and visual thinking in reasoning in general and in scientific argumentation in particular. One motivation to pursue this study is the observation that, as Ferguson has suggested,
Architectural and Representational Requirements for Seeing Processes and Affordances.
, 2008
"... Original longer version here ..."
VISUALIZATION OF ORDINALS
, 2007
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. ..."
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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics.
iv Promotor: prof.dr. W.R. de Jong
"... Proofs, intuitions and diagrams Kant and the mathematical method of proof ..."
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Proofs, intuitions and diagrams Kant and the mathematical method of proof
Diagrammatic Inference and Graphical Proof
, 2001
"... In this paper we present a diagrammatic inference scheme that cn be used in the proof and discovery of diagrammatic theorems. First, we present the theory of abstraction moxkers nd notationa! keys for explaining how abstraction can be incorporated in the interpretation of graphics through both s ..."
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In this paper we present a diagrammatic inference scheme that cn be used in the proof and discovery of diagrammatic theorems. First, we present the theory of abstraction moxkers nd notationa! keys for explaining how abstraction can be incorporated in the interpretation of graphics through both syntactic and semantic means. We aso explore how the process of reinterpretation of graphics is essenrio] for learning and proving graphical theorems. Then, we present the diagrammatic inference scheme; it is illustrated with the proof of the theorem of the sum of the odd numbers. The paper concludes with a discussion on the relation between abstraction, visualization, interpretation change and leoxning, applied to understand a purely diagrammatic proof of the Theorem of Pythagoras.