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Numerical Uncertainty and Its Implications

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DOI: 10.4236/jamp.2014.23004    4,352 Downloads   7,095 Views   Citations

ABSTRACT

A scrutiny of the contributions of key mathematicians and scientists shows that there has been much controversy (throughout the development of mathematics and science) concerning the use of mathematics and the nature of mathematics too. In this work, we try to show that arithmetical operations of approximation lead to the existence of a numerical uncertainty, which is quantic, path dependent and also dependent on the number system used, with mathematical and physical implications. When we explore the algebraic equations for the fine structure constant, the conditions exposed in this work generate paradoxical physical conditions, where the solution to the paradox may be in the fact that the fine-structure constant is calculated through different ways in order to obtain the same value, but there is no relationship between the fundamental physical processes which underlie the calculations, since we are merely dealing with algebraic relations, despite the expressions having the same physical dimensions.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Rodrigues, A. and Martins, N. (2014) Numerical Uncertainty and Its Implications. Journal of Applied Mathematics and Physics, 2, 33-44. doi: 10.4236/jamp.2014.23004.

References

[1] M. Atiyah, “Collected Works,” Oxford University Press, Oxford, 2005.
[2] N. Guicciardini, “Isaac Newton on Mathematical Certainty and Method,” MIT Press, Cambridge, 2009.
[3] A. Einstein, “The Meaning of Relativity,” Princeton University Press, Princeton, 1922.
http://dx.doi.org/10.4324/9780203449530
[4] M. Eckert, “Arnold Sommerfeld: Science, Life and Turbulent Times, 1868-1951,” Springer, New York, 2013.
http://dx.doi.org/10.1007/978-1-4614-7461-6
[5] R. P. Feynman, “QED: The Strange Theory of Light and Matter,” Princeton University Press, Princeton, 1985.
[6] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio and B. Odom, “New Determination of the Fine Structure Constant from the Electron g Value and QED,” Physical Review Letters, Vol. 97, No. 3, 2006, Article ID: 030802.
[7] R. Nair, P. Blake, A. Grigorenko, K. Novoselov, T. Booth, T. Stauber, N. Peres and A. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science, Vol. 320, No. 5881, 2006, p. 1308.
[8] J. Reed, B. Uchoa, Y. Joe, Y. Gan, D. Casa, E. Fradkin and P. Abbamonte, “The Effective Fine-Structure Constant of Freestanding Graphene Measured in Graphite,” Science, Vol. 330, No. 6605, 2010, pp. 805-808.
http://dx.doi.org/10.1126/science.1190920
[9] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell and M. B. Bainbridge, “Indications of a Spatial Variation of the Fine Structure Constant,” Physical Review Letters, Vol. 107, No. 19, 2011, Article ID: 19110.
[10] F. Nasseri, “Corrections to the Fine Structure Constant in D-Dimensional Space from the Generalized Uncertainty Principle,” Physical Letters B, Vol. 618, No. 1-4, 2005, pp. 229-232. http://dx.doi.org/10.1016/j.physletb.2005.05.050
[11] G. Marques, “Comment to: ‘Corrections to the Fine Structure Constant in the Spacetime of a Cosmic String from the Generalized Uncertainty Principle’,” Physical Letters B, Vol. 638, No. 5-6, 2006, pp. 552-553.
http://dx.doi.org/10.1016/j.physletb.2006.06.012
[12] F. Nasseri, “Reply to: ‘Comment to: Corrections to the Fine Structure Constant in the Spacetime of a Cosmic String from the Generalized Uncertainty Principle’,” Physical Letters B, Vol. 645, No. 5-6, 2007, pp. 470-471.
http://dx.doi.org/10.1016/j.physletb.2006.12.033
[13] P. Davies, “Is the Universe Transparent or Opaque?” Journal of Physics A: General Physics, Vol. 5, No. 12, 1972, pp. 1722-1737. http://dx.doi.org/10.1088/0305-4470/5/12/012
[14] U. Segerstrale, “Good to the Last Drop? Millikan Stories as ‘Canned’ Pedagogy,” Science and Engineering Ethics, Vol. 1, No. 3, 1995, pp. 197-214. http://dx.doi.org/10.1007/BF02628797
[15] P. Mohr, T. Barry and D. Newell, “CODATA Recommended Values of the Fundamental Physical Constants: 2010,” Reviews of Modern Physics, Vol. 84, No. 4, 2012, pp. 1527-1605. http://dx.doi.org/10.1103/RevModPhys.84.1527
[16] W. Bertozzi, “Speed and Kinetic Energy of Relativistic Electrons,” American Journal of Physics, Vol. 32, No. 7, 1964, pp. 551-555. http://dx.doi.org/10.1119/1.1970770
[17] S. Livingston and P. Blewett, “Particle Accelerators,” Mcgraw Hill Book Co. Inc., New York, 1962.
[18] D. Landau, M. Liftshitz and P. Pitaevskii, “Electrodynamics of Continuous Media,” Pergamon Press, New York, 1984.
[19] D. Saltzberg, P. Gorham, D. Walz, C. Field, R. Iverson, A. Odian, G. Resch, P. Schoessow and D. Williams, “Observation of the Askaryan Effect: Coherent Microwave Cherenkov Emission from Charge Asymmetry in High Energy Particle Cascades,” Physical Review Letters, Vol. 86, No. 13, 2001, pp. 2802-2805. http://dx.doi.org/10.1103/PhysRevLett.86.2802
[20] G. Askaryan, “Excess Negative Charge of an Electron-Photon Shower and Its Coherent Radio Emission,” Soviet Physics JETP-USSR, Vol. 14, No. 2, 1962, pp. 441-443.
[21] P. Gorham, D. Saltzberg, R. Field, E. Guillian, R. Milincic, D. Walz and D. Williams, “Accelerator Measurements of the Askaryan effect in Rock Salt: A Roadmap Toward Teraton Underground Neutrino Detectors,” Physical Review, Vol. D72, 2005, 16 p.
[22] P. Gorham, S. Barwick, J. Beatty, D. Besson, W. Binns, C. Chen, P. Chen, J. Clem, A. Connolly, P. Dowkontt, M. DuVernois, R. Field, D. Goldstein, A. Goodhue, C. Hast, C. Hebert, S. Hoover, M. Israel, J. Kowalski, J. Learned, K. Liewer, J. Link, E. Lusczek, S. Matsuno, B. Mercurio, C. Miki, P. Miocinovic, J. Nam, C. Naudet, J. Ng, R. Nichol, K. Palladino, K. Reil, A. Romero-Wolf, M. Rosen, D. Saltzberg, D. Seckel, G. Varner, D. Walz and F. Wu, “Observations of the Askaryan Effect in Ice,” Physical Review Letters, Vol. 99, No. 17, 2007, 4 p.
[23] A. N. Whitehead, “The Principle of Relativity,” Cambridge University Press, Cambridge, 1922.
[24] A. N. Whitehead, “Process and Reality: An Essay on Cosmology, Macmillan, London, 1929.
[25] A. N. Whitehead and B. Russell, “Principia Mathematica,” Cambridge University Press, Cambridge, 1910, 1912, 1913.

  
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