Local axioms in disguise: Hilbert on Minkowski diagrams

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski??s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating ??rigorously?? with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as ??drawn formulas??, and formulas as ??written diagrams??, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski??s diagrammatic methods in number theory prompted Hilbert??s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert??s assessment of Minkowski??s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics.     

Counterexample Search in Diagram-Based Geometric Reasoning

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.     

Scientific Discovery With Law-Encoding Diagrams

This article introduces the concept of law-encoding diagrams (LEDs) and presents the argument that they have had a role in scientific discovery that has not been previously recognized. An LED is a representation that correctly encodes the underlying relations of a law, or a system of simultaneous laws, in the structure of a diagram by the means of geometric, topological, and spatial constraints, such that the instantiation of a particular diagram represents a single instance of the phenomena or a particular case of the law(s). Examples of LEDs in the history of science are discussed, and the benefits of using LEDs in discovery are considered. LEDs are distinguished from other forms of diagrammatic representation. Previous work on the computational modeling of diagrammatic law induction is reinterpreted in terms of the search for diagrammatic constraints of LEDs. A general characterization of the role of LEDs in discovery is considered, and a framework for classifying processes of discovery based on LEDs is proposed.     

Kant on geometry and spatial intuition

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant??s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant??s theory of geometrical intuition can, at best, capture only part of what Kant??s conception involves and that, for example, they cannot explain why Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant??s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories.     

Evidence for abstract, schematic knowledge of three spatial diagram representations

Spatial diagram representations such as hierarchies, matrices, and networks are important tools for thinking. Our data suggest that college students possess abstract schemas for these representations that include at least rudimentary information about their applicability conditions. In Experiment 1, subjects were better able to select the appropriate spatial diagram representation for a problem when cued to use general category information in memory about those representations than when cued to use specific example problems given during the experiment. The results of Experiment 2 showed that the superior performance in the general category condition was not based on a comparison of the test problems with examples in memory. The results of Experiment 3 showed that the superior performance was not due to learning that occurred during the experiment or to transfer appropriate processing. The General Discussion section considers the nature of students' representation schemas and the question of why college students have only rudimentary schemas for common and widely applicable diagrammatic representations.     

HOW DIAGRAMS CAN IMPROVE REASONING

Abstract— We report an experimental study on the effects of diagrams on deductive reasoning with double disjunctions, for example:
Raphael is in Tacoma or Julia is in Atlanta, or both. Julia is in Atlanta or Paul is in Philadelphia, or both. What follows?
We confirmed that subjects find it difficult to deduce a valid conclusion, such as
Julia is in Atlanta, or both Raphael is in Tacoma and Paul is in Philadelphia.
In a preliminary study, the formal of the premises was either verbal or diagrammatic, and the diagrams used icons to distinguish between inclusive and exclusive disjunctions. The diagrams had no effect on performance. In the main experiment, the diagrams made the alternative possibilities more explicit. The subjects responded faster (about 35 s) and drew many more valid conclusions (nearly 30%) from the diagrams than from the verbal premises. These results corroborate the theory of mental models and have implications for the role of diagrams in reasoning.     

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer??s calculus of ??conditional density operators??. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.     

Electrifying diagrams for learning: principles for complex representational systems

Six characteristics of effective representational systems for conceptual learning in complex domains have been identified. Such representations should: (1) integrate levels of abstraction; (2) combine globally homogeneous with locally heterogeneous representation of concepts; (3) integrate alternative perspectives of the domain; (4) support malleable manipulation of expressions; (5) possess compact procedures; and (6) have uniform procedures. The characteristics were discovered by analysing and evaluating a novel diagrammatic representation that has been invented to support students' comprehension of electricity—AVOW diagrams (Amps, Volts, Ohms, Watts). A task analysis is presented that demonstrates that problem solving using a conventional algebraic approach demands more effort than AVOW diagrams. In an experiment comparing two groups of learners using the alternative approaches, the group using AVOW diagrams learned more than the group using equations and were better able to solve complex transfer problems and questions involving multiple constraints. Analysis of verbal protocols and work scratchings showed that the AVOW diagram group, in contrast to the equations group, acquired a coherently organised network of concepts, learnt effective problem solving procedures, and experienced more positive learning events. The six principles of effective representations were proposed on the basis of these findings. AVOW diagrams are Law Encoding Diagrams, a general class of representations that have been shown to support learning in other scientific domains.     

From Design to Self-organization,or: A Proper Structure for a Proper Function

It is suggested that Charles Sanders Peirce's triadic semiotics provides a framework for a diagrammatic representation of a sign's proper structure. The action of signs is described at the logical and psychological levels. The role of (unconscious) abductive inference is analyzed, and a diagram of reasoning is offered. A series of interpretants transform brute facts into interpretable signs thereby providing human experience with value or meaning. The triadic structure helps in de-mystifying the relations between Penrose's three worlds when the latter are considered as constituting a semiotic triangle. Conference “Dynamic Ontology: An Inquiry into Systems, Emergence, Levels of Reality, and Forms of Causality” University of Trento, Italy, September 7–11, 2004.     

The twofold role of diagrams in Euclid’s plane geometry

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid??s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid??s plane geometry (EPG). Euclid??s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: (i) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; (ii) EPG objects inherit some properties and relations from these diagrams.     

Translating Toulmin Diagrams: Theory Neutrality in Argument Representation

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two.     

Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs.     

The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton??s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition??the putting-together in space??of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton??s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler??s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.     

Falsifying mental models: Testing the predictions of theories of syllogistic reasoning

Four experiments are reported that tested the claim, drawn from mental models theory, that reasoners attempt to construct alternative representations of problems that might falsify preliminary conclusions they have drawn. In Experiment 1, participants were asked to indicate which alternative conclusion(s) they had considered in a syllogistic reasoning task. In Experiments 2-4, participants were asked to draw diagrams consistent with the premises, on the assumption that these diagrams would provide insights into the mental representation being used. In none of the experiments was there any evidence that people constructed more models for multiple-model than for single-model syllogisms, nor was there any correlation between number of models constructed and overall accuracy. The results are interpreted as showing that falsification of the kind proposed by mental models theory may not routinely occur in reasoning.     

The effect of emergent features on judgments of quantity in configural and separable displays

Two experiments investigated effects of emergent features on perceptual judgments of comparative magnitude in three diagrammatic representations: kiviat charts, bar graphs, and line graphs. Experiment 1 required participants to compare individual values; whereas in Experiment 2 participants had to integrate several values to produce a global comparison. In Experiment 1, emergent features of the diagrams resulted in significant distortions of magnitude judgments, each related to a common geometric illusion. Emergent features are also widely believed to underlie the general superiority of configural displays, such as kiviat charts, for tasks requiring the integration of information. Experiment 2 tested the extent of this benefit using diagrams with a wide range of values. Contrary to the results of previous studies, the configural display produced the poorest performance compared to the more separable displays. Moreover, the pattern of responses suggests that kiviat users switched from an integration strategy to a sequential one depending on the shape of the diagram. The experiments demonstrate the powerful interaction between emergent visual properties and cognition and reveal limits to the benefits of configural displays for integration tasks.     

The forgotten individual: diagrammatic reasoning in mathematics

Parallelism has been drawn between modes of representation and problem-sloving processes: Diagrams are more useful for brainstorming while symbolic representation is more welcomed in a formal proof. The paper gets to the root of this clear-cut dualistic picture and argues that the strength of diagrammatic reasoning in the brainstorming process does not have to be abandoned at the stage of proof, but instead should be appreciated and could be preserved in mathematical proofs.     

论开放域的表征(英文)

本文通过引入开放域的概念来拓展文恩-i图系统(带个体的文恩图)。"隐无(absence)"的概念在本论文中被当做独立的范畴加以讨论。"隐无"和开放域的概念共同不相容于集合论中绝对补的概念。     

工作记忆成分在传递性推理中的作用

采用双任务(dual-task)实验范式,探讨工作记忆成分在五项系列问题(five-term task)这种传递性推理中的作用。被试为80名大学生,实验材料为32个空间和时间内容的传递性推理题目。结果发现:(1)空间位置和时间关系的推理结果都支持心理模型理论,而不支持形式规则理论。(2)在推理前提的加工阶段,中央执行系统和视空间模板发挥着关键作用,语音环路没有参与;而在结论的推理阶段,工作记忆的三个成分均发挥了作用。(3)传递性推理在工作记忆系统中采用视空间编码进行表征。