Differential Calculus Based on the Double Contradiction

Download Download as PDF (Size:493KB)  HTML   XML  PP. 420-427  
DOI: 10.4236/ojpp.2016.64039    443 Downloads   601 Views  
Author(s)    Leave a comment

ABSTRACT

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction.

Cite this paper

Kotani, K. (2016) Differential Calculus Based on the Double Contradiction. Open Journal of Philosophy, 6, 420-427. doi: 10.4236/ojpp.2016.64039.

References

[1] Archimedes (1897). Measurement of a Circle. In T. L. Heath, Translated, The Works of Archimedes. London: Cambridge University Press.
[2] Aristotle (1996). Book 6. In R. Waterfield, Translated, Physics. New York: Oxford University Press.
[3] Dedekind, R. (1963). Continuity and Irrational Numbers. In W. W. Beman, Translated, Essays on the Theory of Numbers. New York: Dover Publications.
[4] Dehaene, S. (2011). The Number Sense: How the Mind Create Mathematics (2nd ed.). New York: Oxford University Press.
[5] Euclid (1956). Book 5. In T. L. Heath, Translated, The Thirteen Books of Euclid’s Elements (2nd ed.). New York: Dover Publications.
[6] Heath, T. L. (1921). A History of Greek Mathematics (Vol. 1). London: Oxford University Press.
[7] High-Speed Camera (n.d.). Wikipedia.
https://en.wikipedia.org/wiki/High-speed_camera
[8] Hoffman, D. D. (2000). Visual Intelligence: How We Create What We See. New York: W.W. Norton & Company.
[9] Hubel, D. H. (1995). Eye, Brain and Vision (2nd ed.). New York: Henry Holt & Company.
http://hubel.med.harvard.edu/book/b1.htm
[10] Klein, M. (1972). Mathematical Thought from Ancient to Modern Times (Vol. 1). New York: Oxford University Press.
[11] Klein, M. (1998). Calculus: An Intuitive and Physical Approach (2nd ed.). New York: Dover Publications.

  
comments powered by Disqus

Copyright © 2017 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.