Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to (...) avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism. (shrink)

There has been much discussion recently about the scope and limits of purely symbolic models of the mind and about the proper role of connectionism in cognitive modeling. This paper describes the symbol grounding problem : How can the semantic interpretation of a formal symbol system be made intrinsic to the system, rather than just parasitic on the meanings in our heads? How can the meanings of the meaningless symbol tokens, manipulated solely on the basis of their shapes, be grounded (...) in anything but other meaningless symbols? The problem is analogous to trying to learn Chinese from a Chinese/Chinese dictionary alone. A candidate solution is sketched: Symbolic representations must be grounded bottom-up in nonsymbolic representations of two kinds: iconic representations, which are analogs of the proximal sensory projections of distal objects and events, and categorical representations, which are learned and innate feature-detectors that pick out the invariant features of object and event categories from their sensory projections. Elementary symbols are the names of these object and event categories, assigned on the basis of their categorical representations. Higher-order symbolic representations, grounded in these elementary symbols, consist of symbol strings describing category membership relations. Connectionism is one natural candidate for the mechanism that learns the invariant features underlying categorical representations, thereby connecting names to the proximal projections of the distal objects they stand for. In this way connectionism can be seen as a complementary component in a hybrid nonsymbolic/symbolic model of the mind, rather than a rival to purely symbolic modeling. Such a hybrid model would not have an autonomous symbolic module, however; the symbolic functions would emerge as an intrinsically dedicated symbol system as a consequence of the bottom-up grounding of categories ' names in their sensory representations. Symbol manipulation would be governed not just by the arbitrary shapes of the symbol tokens, but by the nonarbitrary shapes of the icons and category invariants in which they are grounded. (shrink)

John Searle began to discuss his recently published book `The Construction of Social Reality' with Anthony Freeman, and they ended up talking about God. The book itself and part of their conversation are introduced and briefly reflected upon by Anthony Freeman. Many familiar social facts -- like money and marriage and monarchy -- are only facts by human agreement. They exist only because we believe them to exist. That is the thesis, at once startling yet obvious, that philosopher John Searle (...) explores in The Construction of Social Reality. (shrink)

Drawing on the work of Scheler, Heidegger, Merleau-Ponty, Husserl and Sartre, this article presents an overview of some of the diverse approaches to intersubjectivity that can be found in the phenomenological tradition. Starting with a brief description of Scheler’s criticism of the argument from analogy, the article continues by showing that the phenomenological analyses of intersubjectivity involve much more than a ‘solution’ to the ‘traditional’ problem of other minds. Intersubjectivity doesn’t merely concern concrete faceto-face encounters between individuals. It is also (...) something that is at play in simple perception, in tool-use, in emotions, drives and different types of self-awareness. Ultimately, the phenomenologists would argue that a treatment of intersubjectivity requires a simultaneous analysis of the relationship between subjectivity and world. It is not possible simply to insert intersubjectivity somewhere within an already established ontology; rather, the three regions ‘self’, ‘others’, and ‘world’ belong together; they reciprocally illuminate one another, and can only be understood in their interconnection. (shrink)