Papers by Jorge Soto-Andrade
III Congreso Internacional Virtual de Educacion Estadistica (CIVEEST), 21-24 febrero de 2019. [ww... more III Congreso Internacional Virtual de Educacion Estadistica (CIVEEST), 21-24 febrero de 2019. [www.ugr.es/local/fqm126/civeest.html]
Metaphors in Mathematics Education
Encyclopedia of Mathematics Education
The role of metaphors and the switch in cognitive modes in relation to visualization in learning ... more The role of metaphors and the switch in cognitive modes in relation to visualization in learning and teaching mathematics is discussed, based on examples and case studies with students and teachers. We present some preliminary evidence supporting our claims that visualization requires the activation of various metaphors, that it is rather hampered than facilitated by traditional teaching in mathematics, but it is however a trainable capacity in teachers and students.
Journal fĂĽr die reine und angewandte Mathematik (Crelles Journal), 1983
PROTEOMICS, 2007
The definition of life has excited little interest among molecular biologists during the past hal... more The definition of life has excited little interest among molecular biologists during the past half-century, and the enormous development in biology during that time has been largely based on an analytical approach in which all biological entities are studied in terms of their components, the process being extended to greater and greater detail without limit. The benefits of this reductionism are so obvious that they need no discussion, but there have been costs as well, and future advances, for example for creating artificial life or for taking biotechnology beyond the level of tinkering, will need more serious attention to be given to the question of what makes a living organism living. According to Robert Rosen's theory of (M, R)-systems (metabolism-replacement systems), the central idea missing from molecular biology is that of metabolic circularity, most evident from the obvious but commonly ignored fact that proteins are not given from outside but are products of metabolism, and thus metabolites. Among other consequences this implies that the usual distinction between proteome and metabolome is conceptually artificial-however useful it may be in practice-as the proteome is part of the metabolome.
Journal of Theoretical Biology, 2006
This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he... more This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called Ă°M; RĂž systems. His main contribution was an attempt to prove that metabolic closure (or metabolic circularity) could be explained in purely formal terms, but his work remains very obscure and we try to clarify his line of thought. In particular, we clarify the algebraic formulation of Ă°M; RĂž systems in terms of mappings and sets of mappings, which is grounded in the metaphor of metabolism as a mathematical mapping. We define Rosen's central result as the mathematical expression in which metabolism appears as a mapping f that is the solution to a fixed-point functional equation. Crucially, our analysis reveals the nature of the mapping, and shows that to have a solution the set of admissible functions representing a metabolism must be drastically smaller than Rosen's own analysis suggested that it needed to be. For the first time, we provide a mathematical example of an Ă°M; RĂž system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of Ă°M; RĂž systems. In addition, by extending Rosen's construction, we show how one might generate self-referential objects f with the remarkable property f Ă°f Ăž ÂĽ f , where f acts in turn as function, argument and result. We conclude that Rosen's insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks.
Journal of Theoretical Biology, 2010
This is a preprint of an article accepted for publication in the Journal of Theoretical Biology, ... more This is a preprint of an article accepted for publication in the Journal of Theoretical Biology, to be published in 2010. The major insight in Robert Rosen's view of a living organism as an (M, R)system was the realization that an organism must be "closed to efficient causation", which means that the catalysts needed for its operation must be generated internally. This aspect is not controversial, but there has been confusion and misunderstanding about the logic Rosen used to achieve this closure. In addition, his corollary that an organism is not a mechanism and cannot have simulable models has led to much argument, most of it mathematical in nature and difficult to appreciate. Here we examine some of the mathematical arguments and clarify the conditions for closure.
There are deep underlying similarities between Rosen's (M,R) systems as a definition of life and ... more There are deep underlying similarities between Rosen's (M,R) systems as a definition of life and the RAF sets (Reflexive Autocatalytic systems generated by a Food source) introduced by Hordijk and Steel as a way of analyzing autocatalytic sets of reactions. Using RAF concepts we have systematically explored the set of possible small idealized metabolic networks, searching for instances of (M,R) systems. This exhaustive search has shown that the central requirement of Rosen's framework, unicity of , becomes harder and harder to obtain as the network grows in size. In addition, we give an expression for operators , and in terms of RAF sets.
One of the most important characteristics observed in metabolic networks is that they produce the... more One of the most important characteristics observed in metabolic networks is that they produce themselves. This intuition, already advanced by the theories of Autopoiesis and (M, R)-systems, can be mathematically framed in a weird looking equation, full of implications and potentialities: f (f)= f. This equation (here referred as Ouroboros equation), arises in apparently dissimilar contexts, like Robert Rosen's synthetic view of metabolism, hyperset theory and, importantly, untyped lambda calculus. In this paper we survey how ...
Metaphors in Mathematics Education
Encyclopedia of Mathematics Education, 2014
Mathematical Transgressions 4 Proceedings , 2021
We argue that metaphorising and enacting are natural means of cognitive transgression, usually th... more We argue that metaphorising and enacting are natural means of cognitive transgression, usually thwarted by traditional teaching of mathematics. In the latter, the prevailing didactic contract sets stringent boundary conditions for the learner's acting and reacting. We explore here different ways in which this didactic contract may be transgressed by the learner, particularly through idiosyncratic metaphorisation triggered by dislike of a posed problem, or through enaction of a problematic mathematical situation. We report on the implementation of our transgressive metaphoric and enactivist approach to the learning of mathematics, including the construction of a therapeutic framework based on the operative group technique to enable the learners to transgress themselves mathematically, when facing anxiety triggered by our approach. This implementations involves various cohorts of learners consisting of first year university students majoring in the humanities, prospective mathematics and physics secondary school teachers, prospective mathematicians, in service primary school teachers and primary and secondary school students.
Tensor products as induced representations: The case of finite GL(3)
Mathematical Notes, 2012
We describe the tensor products of two irreducible linear complex representations of the group G ... more We describe the tensor products of two irreducible linear complex representations of the group G = GL(3, $\mathbb{F}_q $) in terms of induced representations by linear characters of maximal tori and also in terms of Gelfand-Graev representations. Our results includeMacDonald’s conjectures for G and are extensions to G of finite counterparts to classical results on tensor products of principal series as well as holomorphic and antiholomorphic representations of the group SL(2,ℝ); besides, they provide an easy way to decompose these tensor products with the help of Frobenius reciprocity. We also state some conjectures for the general case of GL(n, $\mathbb{F}_q $).
arXiv: Representation Theory, 2009
The first and second authors were partially supported by Pontificia Universidad Cat´olica de Valp... more The first and second authors were partially supported by Pontificia Universidad Cat´olica de Valpara ´iso. The second and third authors were partially supported by FONDECYT Grants 1040444, 1070246 and by PICS CNRS 1413
Problematizing as an Avatar of Mathematical Activity: Replications and Prospects
Représentations de Weil de SL*(2,A) et SL(n,q)
Nous construisons, par contraction d'un G-fibre hilbertien adequat [6], une representation de... more Nous construisons, par contraction d'un G-fibre hilbertien adequat [6], une representation de Weil du groupe G = SL * (2,A) sur un anneau involutif fini (A,*) Si A = M m (k), k corps fini, nous retrouvons la representation de Weil classique de Sp(2m,k), dont nous calculons le cocycle en termes de sommes de Gauss geometriques. Enfin, par restriction a un plongement de SL(n,k), nous recuperons les representations de Weil generalisees de [4] et [5].
Journal of theoretical biology, Mar 7, 2010
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Proceedings of The American Mathematical Society, 1988
We consider the higher-order local field analogue of the real euclidean plane afforded by a Galoi... more We consider the higher-order local field analogue of the real euclidean plane afforded by a Galois field extension of degree n, and we describe the structure of its corresponding "rigid motion" group G.
Biological Cybernetics
Carlton (1988) has proposed an attractive hypothesis to link perceived visual images to brain ele... more Carlton (1988) has proposed an attractive hypothesis to link perceived visual images to brain electrical patterns via a linear representation of the Euclidean group onto an appropriate functional space. We show that the construction she proposes is (I) biologically restrictive, and (2) cannot be completed in the desired way. We conclude by presenting other possible means to pursue Carlton's approach. biologically questionable way. Second, we argue that this does not correspond to extending Bargman's representation as claimed, and furthermore such an extension seems demonstrably impossible in the desired way. We conclude with some suggestions as to how Cariton's elegant idea could be nevertheless carried through with different means, but basic problems remain essentially open. (For the rest of this note we assume that the reader is familiar with the paper being discussed).
Axiomathes, 2011
We review and discuss A. H. Louie's book ''More than Life Itself: A Reflexion on Formal Systems a... more We review and discuss A. H. Louie's book ''More than Life Itself: A Reflexion on Formal Systems and Biology'' from an interdisciplinary viewpoint, involving both biology and mathematics, taking into account new developments and related theories.
Biological Research, 2007
This paper describes a notable convergence between biological organization and programming langua... more This paper describes a notable convergence between biological organization and programming language abstractions. Our aim is to explore possibilities of cross-fertilization, at both conceptual and empirical levels, towards the understanding of what cognition and cognitive systems might be.
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Papers by Jorge Soto-Andrade