Archive
Special Issues

Volume 3, Issue 2, April 2014, Page: 26-39
On the System Analysis of the Foundations of Trigonometry
Temur Z. Kalanov, Home of Physical Problems, Pisatelskaya 6a, Tashkent, Uzbekistan
Received: Mar. 28, 2014;       Accepted: Apr. 18, 2014;       Published: Apr. 30, 2014
Abstract
Analysis of the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Keywords
Mathematics, Physics, Mathematical Physics, Geometry, Engineering, Formal Logic, Philosophy of Science
Temur Z. Kalanov, On the System Analysis of the Foundations of Trigonometry, Pure and Applied Mathematics Journal. Vol. 3, No. 2, 2014, pp. 26-39. doi: 10.11648/j.pamj.20140302.12
Reference
[1]
T.Z. Kalanov. “The critical analysis of the foundations of theoretical physics. Crisis in theoretical physics: The problem of scientific truth”. Lambert Academic Publishing. ISBN 978-3-8433-6367-9, (2010).
[2]
T.Z. Kalanov. “Analysis of the problem of relation between geometry and natural sciences”. Prespacetime Journal, Vol. 1, No 5, (2010), pp. 75-87.
[3]
T.Z. Kalanov. “Logical analysis of the foundations of differential and integral calculus”. Bulletin of Pure and Applied Sciences, V. 30 E (Math.& Stat.), No. 2, (2011), pp. 327-334.
[4]
T.Z. Kalanov. “Critical analysis of the foundations of differential and integral calculus”. International Journal of Science and Technology, V. 1, No. 2, (2012), pp. 80-84.
[5]
T.Z. Kalanov. “On rationalization of the foundations of differential calculus”. Bulletin of Pure and Applied Sciences, V. 31E (Math.& Stat.), No. 1, (2012), pp. 1-7.
[6]
T.Z. Kalanov. “Critical analysis of the mathematical formalism of theoretical physics. I. Foundations of differential and integral calculus”. Bulletin of the Amer. Phys. Soc., (APS April Meeting, 2013), Vol. 58, No. 4.
[7]
T.Z. Kalanov. “The logical analysis of the Pythagorean theorem and of the problem of irrational numbers”. Asian Journal of Mathematics and Physics. ISSN: 2308-3131. Vol. 2013, pp. 1-12.
[8]
T.Z. Kalanov. “The critical analysis of the Pythagorean theorem and of the problem of irrational numbers”. Global Journal of Advanced Research on Classical and Modern Geometries. ISSN: 2284-5569. Vol. 2, No 2, (2013), pp. 59-68.
[9]
T.Z. Kalanov. “On the logical analysis of the foundations of vector calculus”. Research Desk. ISSN: 2319-7315. Vol. 2, No. 3, (2013), pp. 249-259.
[10]
C.B. Boyer. “A history of mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. (1991).
[11]
E.S. Kennedy. “The History of Trigonometry”. 31st Yearbook (National Council of Teachers of Mathematics, Washington DC). (1969).
[12]
R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002).
[13]
W.B. Ewald. “From Kant to Hilbert: a source book in the foundations of mathematics”. Oxford University Press US. ISBN 0-19-850535-3. (2008).
[14]
M. Hazewinkel (ed.), “Trigonometric functions”. Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4. (2001).
[15]
A. Einstein. “Geometrie und Erfahrung”. Sitzungsber. Preuss. Akad. Wiss., (1921), V. 1, pp. 123-130.
[16]
A. Grünbaum. “Philosophical Problems of Space and Time” (Second edition), (1973).
[17]
N.I. Lobachevski. Selected works on geometry. Moscow, 1956.
[18]
D. Hilbert. “Grundlagen der Geometrie”. Siebente Auflage, Lpz. – Berl., 1930.