On the System Analysis of the Foundations of Trigonometry

Temur Z. Kalanov

doi:doi:10.11648/j.pamj.20140302.12

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On the System Analysis of the Foundations of Trigonometry

Temur Z. Kalanov

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 2)

Received: 28 March 2014 Accepted: 18 April 2014 Published: 30 April 2014

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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.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 2)
DOI	10.11648/j.pamj.20140302.12
Page(s)	26-39
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Mathematics, Physics, Mathematical Physics, Geometry, Engineering, Formal Logic, Philosophy of Science

References

[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.

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Temur Z. Kalanov. (2014). On the System Analysis of the Foundations of Trigonometry. Pure and Applied Mathematics Journal, 3(2), 26-39. https://doi.org/10.11648/j.pamj.20140302.12

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Temur Z. Kalanov. On the System Analysis of the Foundations of Trigonometry. Pure Appl. Math. J. 2014, 3(2), 26-39. doi: 10.11648/j.pamj.20140302.12

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AMA Style

Temur Z. Kalanov. On the System Analysis of the Foundations of Trigonometry. Pure Appl Math J. 2014;3(2):26-39. doi: 10.11648/j.pamj.20140302.12

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@article{10.11648/j.pamj.20140302.12,
  author = {Temur Z. Kalanov},
  title = {On the System Analysis of the Foundations of Trigonometry},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {2},
  pages = {26-39},
  doi = {10.11648/j.pamj.20140302.12},
  url = {https://doi.org/10.11648/j.pamj.20140302.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140302.12},
  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.},
 year = {2014}
}

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TY  - JOUR
T1  - On the System Analysis of the Foundations of Trigonometry
AU  - Temur Z. Kalanov
Y1  - 2014/04/30
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140302.12
DO  - 10.11648/j.pamj.20140302.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 39
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140302.12
AB  - 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.
VL  - 3
IS  - 2
ER  -

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Author Information

Temur Z. Kalanov

Home of Physical Problems, Pisatelskaya 6a, Tashkent, Uzbekistan

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APA Style

Temur Z. Kalanov. (2014). On the System Analysis of the Foundations of Trigonometry. Pure and Applied Mathematics Journal, 3(2), 26-39. https://doi.org/10.11648/j.pamj.20140302.12

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ACS Style

Temur Z. Kalanov. On the System Analysis of the Foundations of Trigonometry. Pure Appl. Math. J. 2014, 3(2), 26-39. doi: 10.11648/j.pamj.20140302.12

Copy | Download

AMA Style

Temur Z. Kalanov. On the System Analysis of the Foundations of Trigonometry. Pure Appl Math J. 2014;3(2):26-39. doi: 10.11648/j.pamj.20140302.12

Copy | Download

@article{10.11648/j.pamj.20140302.12,
  author = {Temur Z. Kalanov},
  title = {On the System Analysis of the Foundations of Trigonometry},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {2},
  pages = {26-39},
  doi = {10.11648/j.pamj.20140302.12},
  url = {https://doi.org/10.11648/j.pamj.20140302.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140302.12},
  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.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - On the System Analysis of the Foundations of Trigonometry
AU  - Temur Z. Kalanov
Y1  - 2014/04/30
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140302.12
DO  - 10.11648/j.pamj.20140302.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 26
EP  - 39
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140302.12
AB  - 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.
VL  - 3
IS  - 2
ER  -

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