The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (shrink)

It is common to distinguish two great families of representation. Symbolic representations include logical and mathematical symbols, words, and complex linguistic expressions. Iconic representations include dials, diagrams, maps, pictures, 3-dimensional models, and depictive gestures. This essay describes and motivates a new way of distinguishing iconic from symbolic representation. It locates the difference not in the signs themselves, nor in the contents they express, but in the semantic rules by which signs are associated with contents. The two kinds of rule have (...) divergent forms, occupying opposite poles on a spectrum of naturalness. Symbolic rules are composed entirely of primitive juxtapositions of sign types with contents, while iconic rules determine contents entirely by uniform natural relations with sign types. This distinction is marked explicitly in the formal semantics of familiar sign systems, both for atomic first-order representations, like words, pixel colors, and dials, and for complex second-order representations, like sentences, diagrams, and pictures. (shrink)

An important part of the explanatory role of concepts is that they enable us to combine a wide variety of objects, properties and relations in thought, with contents spanning diverse domains. I discuss an argument that appears to show that paradigmatic non-linguistic representational formats are unsuited to play this role, and thus conceptual representation could not occur in these formats. I show that this argument fails, because it overlooks the possibility of individual concepts being shared between a number of special (...) purpose representational systems. Demonstrating this requires defending the possibility of cross-format redeployment of concepts. (shrink)

Mental capacities for perceiving, remembering, thinking, and planning involve the processing of structured mental representations. A compositional semantics of such representations would explain how the content of any given representation is determined by the contents of its constituents and their mode of combination. While many have argued that semantic theories of mental representations would have broad value for understanding the mind, there have been few attempts to develop such theories in a systematic and empirically constrained way. This paper contributes to (...) that end by developing a semantics for a ‘fragment’ of our mental representational system: the visual system’s representations of the bounding contours of objects. At least three distinct kinds of composition are involved in such representations: ‘concatenation’, ‘feature composition’, and ‘contour composition’. I sketch the constraints on and semantics of each of these. This account has three principal payoffs. First, it models a working framework for compositionally ascribing structure and content to perceptual representations, while highlighting core kinds of evidence that bear on such ascriptions. Second, it shows how a compositional semantics of perception can be compatible with holistic, or Gestalt, phenomena, which are often taken to show that the whole percept is ‘other than the sum of its parts’. Finally, the account illuminates the format of a key type of perceptual representation, bringing out the ways in which contour representations exhibit domain-specific form of the sort that is typical of structured icons such as diagrams and maps, in contrast to typical discursive representations of logic and language. (shrink)

Since Tolman’s paper in 1948, psychologists and neuroscientists have argued that cartographic representations play an important role in cognition. These empirical findings align with some theoretical works developed by philosophers who promote a pluralist view of representational vehicles, stating that cognitive processes involve representations with different formats. However, the inferential relations between maps and representations with different formats have not been sufficiently explored. Thus, this paper is focused on the inferential relations between cartographic and linguistic representations. To that effect, we (...) appeal to heterogeneous inference with ordinary maps and sentences. In doing so, we aim to build a model to bridge the gap between cartographic and linguistic thought. (shrink)

An ongoing philosophical discussion concerns how various types of mental states fall within broad representational genera—for example, whether perceptual states are “iconic” or “sentential,” “analog” or “digital,” and so on. Here, I examine the grounds for making much more specific claims about how mental states are structured from constituent parts. For example, the state I am in when I perceive the shape of a mountain ridge may have as constituent parts my representations of the shapes of each peak and saddle (...) of the ridge. More specific structural claims of this sort are a guide to how mental states fall within broader representational kinds. Moreover, these claims have significant implications of their own about semantic, functional, and epistemic features of our mental lives. But what are the conditions on a mental state's having one type of constituent structure rather than another? Drawing on explanatory strategies in vision science, I argue that, other things being equal, the constituent structure of a mental state determines what I call its distributional properties—namely, how mental states of that type can, cannot, or must co‐occur with other mental states in a given system. Distributional properties depend critically on and are informative about the underlying structures of mental states, they abstract in important ways from aspects of how mental states are processed, and they can yield significant insights into the variegation of psychological capacities. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from (...) computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

Concrete Causation centers about theories of causation, their interpretation, and their embedding in metaphysical-ontological questions, as well as the application of such theories in the context of science and decision theory. The dissertation is divided into four chapters, that firstly undertake the historical-systematic localization of central problems (chapter 1) to then give a rendition of the concepts and the formalisms underlying David Lewis' and Judea Pearl's theories (chapter 2). After philosophically motivated conceptual deliberations Pearl's mathematical-technical framework is drawn on for (...) an epistemic interpretation and for emphasizing the knowledge-organizing aspect of causality in an extension of the interventionist Bayes net account of causation (chapter 3). Integrating causal and non-causal knowledge in unified structures ultimately leads to an approach towards solving problems of (causal) decision theory and at the same time facilitates the representation of logical-mathematical, synonymical, as well as reductive relationships in efficiently structured, operational nets of belief propagation (chapter 4). (shrink)

Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images (...) in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences. (shrink)

Abstract In this essay, I will argue that images can play a substantial role in argumentation: exploiting information from the context, they can contribute directly and substantially to the communication of the propositions that play the roles of premises and conclusion. Furthermore, they can achieve this directly, i.e. without the need of verbalization. I will ground this claim by presenting and analyzing some arguments where images are essential to the argumentation process. Content Type Journal Article Pages 1-14 DOI 10.1007/s10503-011-9259-y Authors (...) Axel Arturo Barceló Aspeitia, Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México, Cto. Mario de la Cueva s/n, Cd. Universitaria, Coyoacán, 04500 DF, Mexico Journal Argumentation Online ISSN 1572-8374 Print ISSN 0920-427X. (shrink)

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ?external? negation not-R; and the assertion of R in the pair of propositions ?it is (un)true that R? belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended. (shrink)

Visual arguments can seem to require unique, autonomous evaluative norms, since their content seems irreducible to, and incommensurable with, that of verbal arguments. Yet, assertions of the ineffability of the visual, or of visual-verbal incommensurability, seem to preclude counting putatively irreducible visual content as functioning argumentatively. By distinguishing two notions of content, informational and argumentative, I contend that arguments differing in informational content can have equivalent argumentative content, allowing the same argumentative norms to be rightly applied in their evaluation.

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the (...) effectiveness of mathematics in natural science. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

In contrast to Constructivist Views, which construe perceptual cognition as an essentially reconstructive process, this article recommends the Deictic View, which grounds perception in perceptual-demonstrative reference and the use of deictic tracking strategies for acquiring and updating knowledge about individuals. The view raises the problem of how sensory-motor tracking connects to epistemic and integrated forms of tracking. To study the strategies used to solve this problem, we report a study of the ability to track distal individuals when only their directions (...) can be perceived and not their locations. We introduce a new experimental paradigm named the 'Modified Traveling Salesman Problem' (MTSP), which requires subjects to visit n invisible targets in a 2D display once each. Surprisingly, subjects are competent at this task for up to 10 targets. We consider two types of tracking strategies that subjects might use: 'location-based' strategies and 'deictic direction-based' strategies. A number of observations suggest that subjects used the latter, at least for larger numbers of targets. We hypothesize that subjects used perceptual-demonstrative reference and deictic strategies (i) to perform the sensory-motor tracking of directional segments, (ii) to bind the segments with their updated status in the task, and (iii) to perform the epistemic tracking of invisible targets by means of perception-based inferences. (shrink)

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

6. Seeing With the Mind’s Eye 1: The Puzzle of Mental Imagery .................................................6-1 6.1 What is the puzzle about mental imagery?..............................................................................6-1 6.2 Content, form and substance of representations ......................................................................6-6 6.3 What is responsible for the pattern of results obtained in imagery studies?.................................6-8..

The years 1880-1914 were a time of intense experimentation in the visual arts. Representative conventions became variable, and artists deliberately departed from a concept of depiction considered as physical resemblance or photographic similarity. Visual representations progressed toward a conceptualization of figures and objects that transcended perceptual data, and the rendering of pictorial objects turned into an experiment involving complex visualization processes. This paper explores the interplay between artistic and scientific representative practices between 1880 and 1914. I argue that science and (...) technology acted as substantial challenges upon the concept of resemblance in art and that the rhythm of scientific and technological discoveries at the turn of the 20th century paralleled a shift from a notion of similarity to one of homomorphism in the conceptualization of pictorial representation. Homomorphism denotes representations which dispense with a point-to-point correspondence between depicted objects and perceptual data. I developed the concept from a scholarly study of Charles S. Peirce’s pragmatic account of representation, and in particular his theory of iconicity. Peirce defined iconic signs as “partaking in the character of the object” (CP 4.531), that is, as preserving the same relational structure as their object. A theory of iconicity as homomorphism successfully exemplifies representative relations based on structure preservation. Applied to 20th century representative practice, homomorphism offers a plausible explanation for representations in which a considerable conceptual effort is required, independently of a point-to-point correspondence between depicted objects and perceptual data. Using four case studies - the photographer Alfred Stieglitz and the painters André Derain, Max Weber and Pablo Picasso - I argue that representative practice between 1880 and 1914 was strongly informed by experimental scientific practices and that the shift from figurative to conceptual representation in art was triggered by a more significant theoretical shift involving representation as a general philosophical notion. Rather than a normative quest for the necessary and sufficient conditions for representation, I will propose a pragmatic evaluation of the means and strategies through which artists and scientists devise perspicuous and useful representations of the world. My analysis of the correlations between artistic and scientific representations at the turn of the 20th century aims to fulfill a twofold purpose. From a historical viewpoint, it draws significant parallels between the experimental aspects of representative practices in art and science considered as ways of exploring natural phenomena and intervening upon them. From a philosophical viewpoint, my goal is to propose a novel epistemological framework to assess how the shift in the conceptualization of representations affected subsequent styles of knowing and experimental practices in art and science. Ultimately, by combining the relative merits of historical and philosophical accounts of representation, I will argue for the advantages and desirability of a philosophically informed history of representative practice. (shrink)

Last year at VSS, Bullot, Droulez & Pylyshyn reported studies using a Modified Traveling Salesman Paradigm in which a virtual vehicle had to visit up to 10 targets once and only once, and in which the invisible targets were identified only by line segments pointing from the vehicle toward each target. We hypothesized that subjects used two distinct strategies: A “location-based strategy”, which kept track of where targets were located in screen coordinates, and a “segment-based strategy” that kept track of (...) which segments corresponded to visited targets. We report new studies that further explore these two strategies. Subjects passively observed a computer-controlled virtual vehicle that visited a number of targets. Two forms of display were used: an “Allocentric” display, in which the vehicle moved and the targets remained fixed in screen coordinates, and an “Egocentric” display, in which the vehicle's position on the screen remained fixed while the targets moved -- as if the environment were being viewed by an observer on the vehicle. At the end of each trial, the directional segments were extended to the edge of the screen and subjects were asked to perform two tasks by referring to these segments. In the “status task” observers had to indicate for each segment whether the corresponding target had been visited or not. In the “locating task” they had to locate each target along its directional segment. Performance on these two tasks measures the use of the two hypothesized strategies. Results showed that observers do well on the status task with 4 or 6 targets in both display conditions, but do poorly on the locating task, especially in the egocentric condition when there are more targets. These results are consistent with the hypothesis that in the egocentric condition the MTSP task is carried out primarily by segment-tracking, which can be viewed as a deictic strategy. (shrink)