Learning the Structure of Abstract Groups Dirk Schlimm (dirk.schlimm@mcgill.ca) Department of Philosophy and School of Computer Science, McGill University 855 Sherbrooke St. W., Montreal, QC H3A 2T7, Canada Thomas R. Shultz (thomas.shultz@mcgill.ca) Department of Psychology and School of Computer Science, McGill University 1205 Penfield Avenue, Montreal, QC H3A 1B1, Canada Abstract It has recently been shown that neural networks can learn particular mathematical groups, for example, the Klein 4- group (Jamrozik & Shultz, 2007). However, there are groups with any number of elements, all of which are said to instantiate the abstract group structure. Learning to differentiate groups from other structures that are not groups is a very difficult task. Contrary to some views, we show that neural networks can learn to recognize finite groups consisting of up to 4 elements. We present this problem as a case study that exhibits the advantages of knowledge-based learning over knowledge-free learning. In addition, we also show the surprising result that the way in which the KBCC algorithm recruits previous knowledge reflects some deep structural properties of the patterns that are learned, namely, the structure of the subgroups of a given group. Keywords: Mathematical groups; neural networks; KBCC; CC. Introduction Mathematical groups are remarkable for two reasons: On the one hand, they pervade many areas of mathematics and also everyday life, and on the other hand, their abstract structure is quite difficult to learn from instances. In this paper we use the task of learning the abstract structure of small finite groups as a case study to make the following three points: (a) We provide an argument against purported limitations of neural network approaches. (b) We show how neural networks can learn from previously existing knowledge and that this increases the speed of their learning. (c) Our results reveal that the neural networks show some ‘deep understanding’ of the abstract structure of the problems they are learning. Some authors have argued that ordinary artificial neural networks are inherently unable to learn systematically structured representations such as mathematical groups (Phillips, & Halford, 1997; Marcus, 1998). However, a first step has recently been made to invalidate this claim, because it has been demonstrated that neural networks can learn particular mathematical groups, like the Klein 4-group, in a fashion that simulates learning by humans (Jamrozik & Shultz, 2007). We now extend the task to learning the abstract structure of small finite groups with three and four elements and we compare knowledge-based learning with knowledge-free learning. Artificial neural networks most always learn from scratch, unlike people who almost always try to build new learning on top of their existing knowledge. Our results show how neural networks can also learn from existing knowledge. In particular, we are curious to know whether it helps the learning process to have previously learned the structure of smaller groups. This is especially interesting, because similarities between two different groups and differences between groups and other domains cannot in general be characterized by structure-preserving mappings, but only by formulating their underlying laws (Schlimm, 2008). Learning of group structure based on previous knowledge of smaller groups is a natural area of application for constructive algorithms that build their learning on top of existing knowledge. One such algorithm is knowledge- based cascade-correlation (KBCC), which constructs a neural-network topology by recruiting previously learned networks and single hidden units (Shultz & Rivest, 2001). Its performance can be readily compared to ordinary cascade-correlation (CC), which builds a network only by recruiting single hidden units (Fahlman & Lebiere, 1990). Past comparisons have revealed that KBCC tends to recruit relevant knowledge whenever it can and that this speeds learning (Shultz & Rivest, 2001) and in some cases makes learning possible (Egri & Shultz, 2006). Finally, we investigate the relation between the group to be learned and the kinds of networks that are recruited from the knowledge base: Does only the size of the recruits matter, or does the quality of their knowledge also play a role? Our results provide a surprising answer to this question that points to promising new lines of work on knowledge and learning. The Abstract Group Structure A mathematical group consists of a set of elements together with an associative operation * (i.e., where a*(b*c) and (a*b)*c yield the same result), for which the following three properties hold: (i) the objects are closed under the group operation, i.e., for elements a and b, a*b is also an element of the group; (ii) there exists a neutral element n, such that a*n=a for all elements a of the group; and (iii) that for every element a there exists an inverse element a', such that a*a'=n.
Abstract Tables are widely used for storing, retrieving, communicating, and processing information, but in the literature on the study of representations they are still somewhat neglected. The strong structural constraints on tables allow for a clear identification of their characteristic features and the roles these play in the use of tables as representational and cognitive tools. After introducing syntactic, spatial, and semantic features of tables, we give an account of how these affect our perception and cognition on the basis of fundamental principles of Gestalt psychology. Next are discussed the ways in which these features of tables support their uses in providing a global access to information, retrieving information, and visualizing relational structure and patterns. The latter is particularly important, because it shows how tables can contribute to the generation of new knowledge. In addition, tables also provide efficient means for manipulating information in general and in structured notations. In sum, tables are powerful and efficient representational tools.
Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other.
Moritz Pasch (1843–1930) gave the first rigorous axiomatization of projective geometry in his Vorlesungen über neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span of Pasch’s career, the latter decades of which he devoted primarily to careful reflections on the nature of mathematics and of mathematical knowledge, Pasch’s highly original, but virtually unknown, philosophy of mathematics is presented.
Abstract This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of historical developments of mathematics has been primarily on the former, even if they claim to be about the latter.
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