The paper reviews the most consequential defects and
rectification of traditional mathematics and its foundations. While this work
is only the tip of the iceberg, so to speak, it gives us a totally different
picture of mathematics from what we have known for a long time. This journey
started with two teasers posted in SciMath in 1997: 1) The equation 1 = 0.99…
does not make sense. 2) The concept ？does not
exist. The first statement sparked a debate that raged over a decade. Both
statements generated a series of publications that continues to grow to this day. Among the new findings are: 3) There does not exist
nondenumerable set. 4) There does not exist non-measurable set. 5) Cantor’s
diagonal method is flawed. 6) The real numbers are discrete and countable. 7)
Formal logic does not apply to mathematics. The unfinished debate between
logicism, intuitionism-constructivism and formalism is resolved. The resolution
is the constructivist foundations of mathematics with a summary of all the
rectification undertaken in 2015, 2016 and in this paper. The extensions of the
constructivist real number system include the complex vector plane and
transcendental functions. Two important results in the 2015 are noted: The
solution and resolution of Hilbert’s 23 problems that includes the resolution of Fermat’s last theorem and proof Goldbach’s conjecture.
References
[1]
Escultura, E.E. (2015) The Resolution of the Great 20th Century Debate in the Foundations of Mathematics. Advances in Pure Mathematics, 6, 144-158. http://file.scirp.org/Html/3-5301056_63915.htm https://doi.org/10.4236/apm.2016.63012
[2]
Escultura, E.E. (2009) The New Real Number System and Discrete Computation Calculus. Neural, Parallel and Scientific Computation, 17, 59-84.
[3]
Escultura, E.E., Bhaskar, T.G., Leela, S. and Laksmikantham, V. (2009) Revisiting the Hybrid Real Number System. Nonlinear Analysis, C-Series: Hybrid Systems, 3, 101-107. https://doi.org/10.1016/j.nahs.2008.11.001
[4]
Escultura, E.E. (2009) Extending the Reach of Computation. Applied Mathematics Letters, 21, 1074-1081. https://doi.org/10.1016/j.aml.2007.10.027
[5]
Escultura, E.E. (2016) The Constructivist Real Number System. Advances in Pure Mathematics, 6, 597-607. http://www.scirp.org/journal/apm/
[6]
Andrew David Irvine, Bertrand Russell; andrew.irvine@ubc.ca.
[7]
Intuitionism in the Philosophy of Mathematics. https://plato.stanford.edu/entries/intuitionism/
[8]
Zach, R. (2005) Hilbert’s Program Then and Now. arXiv:math/0508572 [math.LO]
[9]
Russell’s Letter to Frege. http://www.logicmuseum.com/reference/fregerussell1904.htm
Weisstein, E.W. Brouwer Fixed Point Theorem. MathWorld. http://mathworld.wolfram.com/BrouwerFixedPointTheorem.html
[12]
Weisstein, E.W. Zermelo-Fraenkel Set Theory. MathWorld. http://mathworld.wolfram.com/Zermelo-FraenkelSetTheory.html
[13]
Royden, A. (1983) Real Analysis. 3rd Edition, MacMillan, New York.
[14]
Bell, J.L. The Axiom of Choice. https://plato.stanford.edu/entries/axiom-choice/
[15]
Escultura, E.E. (2015) Rational Thought, Cognition and Knowledge. International Education Research, 3, 21-37. http://www.todayscience.org/IER/article/ier.v3i1p21.pdf https://doi.org/10.12735/ier.v3i1p21
[16]
Escultura, E.E. (2007) The Pillars of the New Physics and Some Updates. Nonlinear Studies, 14, 241-260.
[17]
Young, L.C. (1980) Mathematicians and Their Times. North-Holland, Amsterdam.
[18]
Escultura, E.E. (1998) Exact Solutions of Fermat’s Equation (Definitive Resolution of Fermat’s Last Theorem. Nonlinear Studies, 5, 227-254.
[19]
Ito, K. (1993) Encyclopedic Dictionary of Mathematics. 2nd Edition, Corporate Mathematical Society of Japan, MIT Press, Cambridge, MA.
[20]
Escultura, E.E. (2003) The New Mathematics and Physics. Applied Mathematics and Computation, 138, 127-149. https://doi.org/10.1016/S0096-3003(02)00126-1
[21]
Escultura, E.E. (2002) The Mathematics of the New Physics. Applied Mathematics and Computations, 130, 45-169. https://doi.org/10.1016/S0096-3003(01)00088-1
[22]
Horgan, H. (1993) The Death of Proof. Scientific American, 269, 92-103. https://doi.org/10.1038/scientificamerican1093-92
[23]
Escultura, E.E. (2008) The Grand Unified Theory. Nonlinear Analysis: A-Series: Theory, Methods and Applications, 69, 823-831.
[24]
Young, L.C. (1969) Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders, Philadelphia.
[25]
Escultura, E.E. (2001) Vortex Interactions. Problems of Nonlinear Analysis in Engineering Systems, 7, 30-44.
[26]
Escultura, E.E. (1997) The Solution of the Gravitational n-Body Problem. Nonlinear Analysis, Series Ai Theory, Methods and Applications, 30, 5021-5032. https://doi.org/10.1016/S0362-546X(96)00133-2
[27]
Zeigler, B.P. (2005) An Introduction to Calculus Course Based on DEVS: Implications of a Discrete Reformulation of Mathematical Continuity. International Conference on Simulation in Education ICSiE’05, New Orleans, 23-25 January 2005.
[28]
Escultura, E.E. (1993) Diophantus: Introduction to Natural Philosophy (With Counterexamples to Fermat’s Last Theorem. Kalikasan Press, Manila.
[29]
Zeigler, B.P. (2004) Continuity and Change (Activity) Are Fundamentally Related in DEVS Simulation of Continuous Systems. Keynote Talk at AI, Simulation and Planning AIS’04, Kor, 4-6 October 2004.
Davies, P.J. and Hersch, R. (1981) The Mathematical Experience. Birkhauser, Boston.
[32]
Smolin, L. (2008) The Trouble with Physics. Penguin Books, New York.
[33]
CERN. https://en.wikipedia.org/wiki/CERN
[34]
Escultura, E.E. (2013) The Logic and Fundamental Concepts of the Grand Unified Theory. Journal of Modern Physics, 4, 213-222. http://file.scirp.org/Html/21-7501396_36280.htm https://doi.org/10.4236/jmp.2013.48A021
[35]
Escultura, E.E. (2007) Dynamic Modeling and the New Mathematics and Physics. Neural, Parallel and Scientific Computations, 15, 527-538.
[36]
Escultura, E.E. (2000) Set-Valued Differential Equations and Applications to Quantum Gravity. Mathematical Research, 6, 58-69.
[37]
Escultura, E.E. (2009) Qualitative Model of the Atom, Its Components and Origin in the Early Universe. Nonlinear Analysis: Real World Applications, 11, 29-38. https://doi.org/10.1016/j.nonrwa.2008.10.035
[38]
Escultura, E.E. (2013) Chaos, Turbulence and Fractal: Theory and Applications. International Journal of Modern Nonlinear Theory and Application, 2, 176-185. https://doi.org/10.4236/ijmnta.2013.23025
[39]
Escultura, E.E. (2001) From Macro to Quantum Gravity. Problems of Nonlinear Analysis in Engineering Systems, 7, 56-78.
[40]
Escultura, E.E. (2010) The Leap from Traditional to the New Science. Nonlinear Studies, 21, 283-291.
[41]
Escultura, E.E. (2009) The Mathematics of the Grand Unified Theory. Nonlinear Analysis, A-Series: Theory, Methods, 71, e420-e431. https://doi.org/10.1016/j.na.2008.11.003
[42]
Escultura, E.E. (1999) Superstring Loop Dynamics and Applications to Astronomy and Biology. Nonlinear Analysis, A-Series: Theory, Methods and Applications, 35, 259-268.
[43]
Escultura, E.E. (2005) Dynamic Modeling of Chaos and Turbulence. Nonlinear Analysis, A-Series: Theory, Methods and Applications, 63, e519-e532. https://doi.org/10.1016/j.na.2005.02.052
[44]
Escultura, E.E. (1999) Recent Verification and Applications. 2rd International Conference Tools for Mathematical Modeling, St. Petersburg, 4, 74-89.
[45]
Escultura, E.E. (2010) Our Fractal Universe and Applications. International Conference on Dynamic Systems and Applications, Morehouse College, Atlanta.
[46]
Escultura, E.E. (2012) The Big Bang and What It Was. In: O’Connell, J.R. and Hale, A.L., Eds., The Big Bang: Theory, Assumptions and Problems, Nova Science Publishers, 61-102.
[47]
Escultura, E.E. (2012) The Physics of Intelligence. Journal of Education and Learning, 1, 51-64. http://www.ccsenet.org/journal/index.php/jel/article/view/20095/13268 https://doi.org/10.5539/jel.v1n2p51
[48]
Escultura, E.E. (2014) The Physics of the Mind. The Journal of the Science of Healing Outcome, 6, 6-15.
![\"\"](\"Edit_8e29f17a-92df-4ec7-a230-89770633671d.bmp\")
