Critical Foundations of the Contextual Theory of Mind

The contextual mind is found attested in various usages of the term complement, in the background of Kant. The difficulties of Kant's intuitionism are taken up through Quine, but referential opacity is resolved as semantic presence in lived context. A further critique of rationalist linguistics is developed from Jakobson, showing generic functions in thought supporting abstraction, binding and thereby semantic categories. Thus Bolzano's influential philosophy of mathematics and science gives way to a critical view of the ancient heritage acknowledged by Plato.\ud \u

Proof phenomenon as a function of the phenomenology of proving

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings

Learning to Understand: Mathematical Preparation of Prospective Teachers

This paper describes the development of a two-course sequence in mathematics content for prospective elementary teachers. Community college and university personnel collaborated to develop a course sequence that would prepare prospective elementary teachers to teach mathematics with an understanding of concepts to support their abstract mathematical knowledge. The strategy was to begin with a broad vision and then focus on the smaller pieces which would achieve that vision. The course changes are validated by documents published by various educational and mathematical groups advocating an increased emphasis on teaching for understanding rather than rote learning. Significant change is difficult without support from colleagues and sufficient time, both necessary to the change process. The noteworthy components of Austin Community College’s revised course are a safe environment in which students become independent learners and written communication as an integral part of the course resulting in students who have increased their conceptual understanding. As a result of taking the course, students accept responsibility for their own learning, have increased self-confidence, and show enthusiasm for mathematics. While requiring a major commitment from faculty, the results are well worth the effort

Parikh and Wittgenstein

A survey of Parikh’s philosophical appropriations of Wittgensteinian themes, placed into historical context against the backdrop of Turing’s famous paper, “On computable numbers, with an application to the Entscheidungsproblem” (Turing in Proc Lond Math Soc 2(42): 230–265, 1936/1937) and its connections with Wittgenstein and the foundations of mathematics. Characterizing Parikh’s contributions to the interaction between logic and philosophy at its foundations, we argue that his work gives the lie to recent presentations of Wittgenstein’s so-called metaphilosophy (e.g., Horwich in Wittgenstein’s metaphilosophy. Oxford University Press, Oxford, 2012) as a kind of “dead end” quietism. From early work on the idea of a feasibility in arithmetic (Parikh in J Symb Log 36(3):494–508, 1971) and vagueness (Parikh in Logic, language and method. Reidel, Boston, pp 241–261, 1983) to his more recent program in social software (Parikh in Advances in modal logic, vol 2. CSLI Publications, Stanford, pp 381–400, 2001a), Parikh’s work encompasses and touches upon many foundational issues in epistemology, philosophy of logic, philosophy of language, and value theory. But it expresses a unified philosophical point of view. In his most recent work, questions about public and private languages, opportunity spaces, strategic voting, non-monotonic inference and knowledge in literature provide a remarkable series of suggestions about how to present issues of fundamental importance in theoretical computer science as serious philosophical issues

What is Time? A New Mathematico- Physical and Information Theoretic Approach

A New Mathematico-Physical and Information Theoretic Approach Examination of the available hard core information to firm up the process of unification of quantum and gravitational physics leads to the conclusion that for achieving this synthesis, major paradigm shifts are needed as also the answering of `What is Time?' The object of this submission is to point out the means of achieving such a grand synthesis. Currently the main pillars supporting the edifice of physics are: (i) The geometrical concepts of space- time-gravitation, (ii) The dynamic concepts involving quantum of action, (iii) Statistical thermodynamic concepts, heat and entropy, (iv) Mathematical concepts, tools and techniques serving both as a grand plan and the means of calculation and last but not least v)Controlled observation, pertinent experimentation as the final arbiter. In making major changes the author is following Dirac's dictum "....make changes without sacrificing the existing superstructure". It is shown that time can be treated as a parameter rather than an additional dimension. A new entity called "Ekon" having the properties of both space and momentum is introduced along with a space called "Chalachala". The requisite connection with Einstein's formulation and mathematical aperatus required have been formulated which is highly suited for the purpose. The primacy of the Plancks quantum of action and its representation geometrically as a twist is introduced. The practical and numerical estimates have been made and applied to evaluation of the gravitational constant in a a seperate submission "Estimations of gravitational constant from CMBR data".Comment: 29 pages, pdf fil

The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity - II: Dirac versus Bergmann observables and the Objectivity of Space-Time

(abridged)The achievements of the present work include: a) A clarification of the multiple definition given by Bergmann of the concept of {\it (Bergmann) observable. This clarification leads to the proposal of a {\it main conjecture} asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). b) The analysis of the so-called {\it Hole} phenomenology in strict connection with the Hamiltonian treatment of the initial value problem in metric gravity for the class of Christoudoulou -Klainermann space-times, in which the temporal evolution is ruled by the {\it weak} ADM energy. It is crucial the re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (nearly unknown) connection to gauge transformations on-shell; this is expounded in the first paper (gr-qc/0403081). The use of the Bergmann-Komar {\it intrinsic pseudo-coordinates} allows to construct a {\it physical atlas} of 4-coordinate systems for the 4-dimensional {\it mathematical} manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent {\it Dirac observables}), and to realize the {\it physical individuation} of the points of space-time as {\it point-events} as a gauge-fixing problem, also associating a non-commutative structure to each 4-coordinate system.Comment: 41 pages, Revtex

On the Relevance of the Bayesian Approach to Statistics

In this essay, I argue about the relevance and the ultimate unity of the Bayesian approach in a neutral and agnostic manner. My main theme is that Bayesian data analysis is an effective tool for handling complex models, as proven by the increasing proportion of Bayesian studies in the applied sciences. I thus disregard the philosophical debates on the meaning of probability and on the random nature of parameters as things of the past that ultimately do a disservice to the approach and are irrelevant to most bystanders.Bayesian inference, Bayes model choice, foundations, testing, non-informative prior, Bayes factor, computational statistics