Fathering the Calculus

Mathematics is the study of abstract bodies and objects that allows us to explain the universe and its abstractions work. Essentially an exercise in first order and second order logic Mathematics is a conservative discipline which since it’s foundation in the time of Euclid and Archimedes has only gone through 2 or 3 paradigmatic shifts. One of such shifts was the development of the Calculus. The Calculus, from the Latin “to count with small stones,” was developed in the late 1600’s by two men independently, Isaac Newton and Gottfried Leibniz. However, this new field has also spawned some of the greatest debate with in mathematics, and asked the question ‘who developed the Calculus?’ The debate still rages on today and is an important question in the metaphysics of mathematics. While the calculus is responsible, in large part, for most of the modern era’s scientific and technological breakthroughs it is the mind set behind it is also responsible for the progression of mathematics as a discipline. In terms of how the Calculus has advanced mathematics is what needs to be addressed, after all the Calculus is a field in mathematics. In developing the Calculus it is clear (or soon will be) the Leibniz and Newton had very different philosophies on what the Calculus was. However, regardless of who published first or which man was more popular at the time, the man with the correct thinking of what that Calculus is must then be the proverbial father of the Calculus in that this mind set would lead to the progression of mathematics.

In a nut shell the Calculus is the study of the relationship between any function, a relationship between two sets a domain and an image (or range), and how the domain related to the image. For example if one were to plot the distance traveled of a motorcycle over some time domain the Calculus looks at how changed in the time domain leads to some change in the distance image through some function. The earliest research in this question of how the domain and the image of a function related goes back to the time of Archimedes when he proved that the area under a simple parabola and a linear line cutting the parabola was proportional to the inscribed triangle between the parabola and the linear line. In the centuries that followed mathematicians learned of relationships such as the derivative (the change of the image with respect to the domain, or the set of tangent functions) and the anti- derivative (given a function g, the anti-derivative if the function f such that the derivative of f is g) but found no relationship between the two. Now enters Newton and Leibniz in the late 1600’s who both found the relationship between the derivative and the anti-derivative. Know as the Fundamental Theorem of Calculus:
First part
This part of the theorem is sometimes referred to as the First Fundamental Theorem of Calculus. A real-valued function F is defined on a closed interval [a, b] by setting, for all x in [a, b],

where ƒ is a real-valued function continuous on [a, b]. Then, F is continuous on [a, b], differentiable on the open interval (a, b), and

for all x in (a, b).
Second part
This part is sometimes referred to as the Second Fundamental Theorem of Calculus or the Newton-Leibniz Axiom. Let ƒ be a real-valued function defined on a closed interval [a, b] that admits an anti-derivative F on [a, b]. That is, ƒ and F are functions such that for all x in [a, b],

If ƒ is integrable on [a, b] then

Because it does not assume that ƒ is continuous, the Second Part is slightly stronger than the Corollary. When an anti-derivative F exists, there are infinitely many anti-derivatives for ƒ, obtained by adding an arbitrary constant to F. Also, by the first part of the theorem, anti-derivatives of ƒ always exist when ƒ is continuous.

Both Newton and Leibniz both came to the same realization, however through different understandings of the problem. Newton views functions in terms of one domain, the time domain and the functions as taking a time to a distance or a velocity or acceleration. The Calculus to Newton was no more than a means to do physics and much of his findings were done using geometric approximations. In mathematics what Newton saw did not matter, he viewed functions and the areas under the functions to have a physical meaning, i.e. that the derivative of the distance function is the velocity and the area under the velocity function over some time is the distance. We see this physical understanding of Newton’s in his notation where there is no mathematical meaning to it. He symbolized functions with the letter ‘y’ and then symbolized the velocity of ‘y’ as ‘y’ with one dot over it and two dots for the acceleration. These notations add nothing to how we in mathematics expand our understanding of mathematics, but allows for the progress of physics. These notations and equating this line of thought to moving bodies led to classical or Newtonian Mechanics. Newton was very parochial in his describing the calculus and left very vague notations which at the end of the day is not more than applied mathematics.

Leibniz, on the other hand, made much more general realizations. Instead of looking at just the time domain and the distance or the other images he looked at any two sets, one and image and one a domain. What is interesting then in the philosophy which further distinguishes Newton and Leibniz is the idea of the infinitesimal. The infinitesimal is an infinitely small area under a curve which Leibniz saw could be added up using sums. The infinite sums that Leibniz developed led to his developing the Calculus. The idea of having extremely small quantities led to the progression of mathematics which in turn has led to the modern era of mathematics (the Hilbert Paradigm) in which every statement in mathematics must be proven. Having extremely small quantities allows mathematicians to build a system in which the calculus can be proven, this system is called real analysis and is the corner stone of modern day mathematics.

Clearly both men are important individuals in mathematics and physics respectively. But in terms of the advancement of mathematics Leibniz did more to lay the foundation ground work for how mathematics is thought of then Newton did. Leibniz began mathematics on a new path where the deductive power of mathematics would separate the discipline from science where mathematics was applied. Leibniz moved mathematics to a more independent state, with his generalization of functions and domains and images the Calculus was a much more important tool than Newton could have believed. Even the notation that Leibniz used ,’dy/dx’ (meaning the change in ‘y’ over the change in ‘x’), created applications and understanding of the Calculus ranging from Political Science to Economics and of course in all three of the natural sciences. Leibniz is rightfully the father of the Calculus, he saw the generalization of the derivative and the anti-derivative in way Newton did not, he saw functions as abstract entities which Newton did not and he did indeed view the Calculus as counting by small, very small, stones.