47,721 research outputs found
Commentary on Robert Riley's article "A personal account of the discovery of hyperbolic structures on some knot complements"
We give some background and biographical commentary on the postumous article
that appears in this [journal issue | ArXiv] by Robert Riley on his part of the
early history of hyperbolic structures on some compact 3-manifolds. A complete
list of Riley's publications appears at the end of the article.Comment: 5 page
In the Beginning Was the Verb: The Emergence and Evolution of Language Problem in the Light of the Big Bang Epistemological Paradigm.
The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution is perceived today as one of the most important scientific problems. \ud
The purpose of the present study is actually to outline such a solution to our problem which is epistemologically consonant with the Big Bang solution of the problem of the Emergence of the Universe}. Such an outline, however, becomes articulable, understandable, and workable only in a drastically extended epistemic and scientific oecumene, where known and habitual approaches to the problem, both theoretical and experimental, become distant, isolated, even if to some degree still hospitable conceptual and methodological islands. \ud
The guiding light of our inquiry will be Eugene Paul Wigner's metaphor of ``the unreasonable effectiveness of mathematics in natural sciences'', i.e., the steadily evolving before our eyes, since at least XVIIth century, \ud
``the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics''. Kurt Goedel's incompleteness and undecidability theory will be our guardian discerner against logical fallacies of otherwise apparently plausible explanations. \ud
John Bell's ``unspeakableness'' and the commonplace counterintuitive character of quantum phenomena will be our encouragers. And the radical novelty of the introduced here and adapted to our purposes Big Bang epistemological paradigm will be an appropriate, even if probably shocking response to our equally shocking discovery in the oldest among well preserved linguistic fossils of perfect mathematical structures outdoing the best artifactual Assemblers
Machines, Logic and Quantum Physics
Though the truths of logic and pure mathematics are objective and independent
of any contingent facts or laws of nature, our knowledge of these truths
depends entirely on our knowledge of the laws of physics. Recent progress in
the quantum theory of computation has provided practical instances of this, and
forces us to abandon the classical view that computation, and hence
mathematical proof, are purely logical notions independent of that of
computation as a physical process. Henceforward, a proof must be regarded not
as an abstract object or process but as a physical process, a species of
computation, whose scope and reliability depend on our knowledge of the physics
of the computer concerned.Comment: 19 pages, 8 figure
Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better
Joel Cohen offers a historical and prospective analysis of the relationship between mathematics and biolog
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