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.

Highlights

  • When the position of an object is given as a function of time, the average velocity from time t to time t + Δt is represented by v =

  • We provide the following as an example of the differential calculus based on the double contradiction

  • When the double contradiction is applied to inequality (13), s(C) can be only one value, which is the slope of the tangent

Read more

Summary

Introduction

Why was differential calculus developed? The consideration of this problem inevitably. We can only observe the position of the object at a point in time. We can only measure an average velocity for an interval of time between observations. Formal differential calculus is based on the concept of the limit. As Δt decreases, the average velocity approaches the instantaneous velocity at time t. The instantaneous velocity at time t is denoted as v(t). The average velocity never reaches the instantaneous velocity. A limit of the average velocity is defined as the instantaneous velocity. The novel method of differential calculus based on the double contradiction is introduced in this paper. The method is easier to accept intuitively than the traditional method of differential calculus. The meaning of the double contradiction in the foundation of mathematics is considered

Zeno’s Arrow Paradox
The Geometrical Meaning of the Double Contradiction
Discussion
Conclusion
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call