Beyond Representation

Epistemological Analysis of Multiple Representations in Secondary Mathematics: The Case of the Function Concept

Mathematical concepts are inherently inaccessible to learners except through their representations. Yet what does it mean to "represent" a mathematical object, and why does the movement between different forms of representation constitute such a persistent site of difficulty? This paper presents a sustained theoretical analysis that integrates Duval's theory of semiotic representation registers with Bachelard's notion of epistemological obstacles, using Sfard's reification framework as a bridging construct. Taking the function concept in secondary mathematics as a case study, we argue that representation conversion is not a technical skill superimposed upon conceptual understanding but is constitutive of mathematical thinking itself. We analyze how the function is expressed across five semiotic registers (natural language, algebraic-symbolic, graphical, tabular, and dynamic-technological) and identify four epistemological obstacles that emerge during conversion: the variable obstacle, the graph-as-picture obstacle, the formula-function conflation, and the correspondence-dependence tension. We further ground this analysis in specific examples drawn from the Chinese People's Education Press (PEP) textbook series and the 2022 Chinese mathematics curriculum standards, examining how curricular choices generate particular epistemological conditions. The paper contributes an integrative theoretical framework and a series of philosophical reflections on the nature of mathematical representation, concept formation, and the irreducibility of semiotic mediation to mathematical knowing.

AMBIGUITY FROM THE PERSPECTIVE OF MENTAL SPACE THEORY AND FRAME SEMANTICS IN WHATSAPP MESSAGES

This paper has as its research problem the fact that although many scholars present different approaches to the concept of ambiguity in their works, among those that were found, few – if any – studies are based on more current theoretical frameworks such as those of Cognitive Linguistics; a fact we consider to be relevant, as research in this field has made significant advances. Within this context, the general objective of this study is to refine the concept of ambiguity by considering, in addition to grammatical aspects, the mental spaces theory (MST). To this end, we investigate how meaning is constructed in WhatsApp messages that contain ambiguous stickers, based on the theories of mental spaces by Fauconnier (1994 and 1997), conceptual blending by Fauconnier and Turner (2002), and frame semantics by Fillmore (1976, 1982, and 1985). The analysis, based on WhatsApp messages, explains how double meanings are constructed, highlighting the specific blending processes involved in building the ambiguous meaning of the messages.

Correlationism: the dogma that was, is and will be

Doabilism: A Critical framework to understand Perceived Feasibility of Action

Every decision in human life is not merely the result of external circumstances; rather, a complex psychological, ethical, and intellectual process operates behind it. Often, a person chooses a particular course of action not because it is inherently righteous or ethical, but because they feel capable of executing it. It is this very ‘doability’ that gradually begins to serve as the basis for the action’s propriety, rationale, and moral legitimacy. Thus, the sense of ‘I can do this.’ gradually transforms into the conviction that ‘I should do this.’ The present study attempts to understand this psychological and philosophical process through a theoretical framework termed Doabilism. This theory analyzes the interrelationships among the sense of capability, feasibility, self-interest, the construction of justification, and rational argumentation within the human decision-making process. It suggests that in many situations, rather than serving as the initial source of a decision, reason becomes a means to justify pre-existing inclinations. The Indian philosophical tradition, especially the Srimad Bhagavad Gita, offers an alternative ethical paradigm to this tendency. According to the Gita, the basis of action should not be power or selfishness, but svadharma, buddhiyoga, selflessness and dharmabodh. The principle of unselfish action elevates man above the tendency of self-justification arising from the accomplishment of action and orients him towards duty, self-restraint and collection of the people. In this light, ‘Doabilism’ offers a descriptive account of human decisions, whereas the Gita’s principle of Nishkama Karma (selfless action) establishes their normative direction. The dialogue between the two not only provides an opportunity to reinterpret decision-making, ethics, and action within the Indian intellectual tradition but also creates a novel conceptual foundation for contemporary literary criticism, moral philosophy, and the study of human behavior.

Methodological Asymmetry of the Projects by G. Deleuze and M. DeLanda: From Metaphysics to Operationalization

The article examines the epistemic asymmetry between G. Deleuze’s metaphysical project and its operational reconstruction undertaken by M. DeLanda. Based on an analysis of categorical transformations (plane of im-manence, “assemblage”, the virtual, diagram), the author demonstrates a fundamental difference between the epistemic strategies: while Deleuze designs the transcendental conditions of thought relying on heuristic meta-phors in science, then DeLanda translates his categories into the parametric language of nonlinear dynamics and complexity theory asserting materialist realism and the applied status of philosophical analysis. Methodo-logical asymmetry is qualified as the institutionalization of a dichotomy between a conceptual apparatus consti-tution and its subsequent operationalization. DeLanda’s adaptation, as shown in the article, enables the inte-gration of Deleuzian ontology into the discourse of philosophy of science offering an alternative both to classi-cal reductionism and to uncritical poststructuralism, yet it is conjugated with a narrowing of the original meta-physics critical potential. The conclusions of the article are relevant for the development of instrumentarium for the analysis of complex, emergent, and large-scale variable systems.

Quatro considerações sobre fazer Realismo filosófico

The purpose of this article is to develop four considerations that will allow us to approach philosophical Realism from a complex point of view. It begins by 1) seeking an operational definition of Realism, contrasting it with unrealist, anti-realist, idealist, and materialist positions, with the aim of considering critiques from these positions in order to reject or appropriate some of their guidelines into our own position. Next, we move into 2) a consideration of the term "Complexity," contrasting it with literalist assertions, differentiating between reality and givenness, and providing some considerations into categorical intuition. Then, 3) we explore the connection between Complex Realism and two forms of pluralism (material and formal/categorical) through the principle of irreductions and the conception of the subdisciplines of Ontography and Categorical Analysis. This leads to the notion of pluralism(s) as the milieu and/or dhamma of Complex Realism. Finally, 4) through the notion of interobjectivity, the category of Ontological Work is explored as the ontological factory floor that operates by creating ontognoseological conditions for the production of knowledge, theory, practice, and pragmatism.