Calculus: The Logical Extension of Arithmetic

In order to understand not only “how”, but also “why”, calculus evolved into that major subdivision of mathematics that it presently occupies, the binary arithmetic of three special numbers is selected as the starting point. When these three, herewith designated by the adjective “foundational”, numbers are combined with the six familiar arithmetic operations of addition, subtraction, multiplication, division, raising to a power and extracting a root, the basis for a new, enlarged perspective of ‘what is mathematics?’ is created. Note that the selected term, foundational number, was chosen over others who have referred to this same set of three numbers in a meta-mathematical context. Notwithstanding that this same set of three numbers was designated in [1] as “boundary numbers”, such a name is herewith eschewed as the traditional concept of “boundary” has a very different denotation in mathematical physics.

Rather than rigidly following the traditional approach, based on function theory, which has been taught in high schools and colleges ever since it was independently developed by Leibnitz and Newton a third of a millennium ago, this treatise focuses attention on the arithmetic taught in elementary school, but from a more advanced standpoint in much the same manner as Felix Klein’s Erlanger Program and its Elementary Mathematics From an Advanced Standpoint [2]. The single-most important difference in this formulation is that there is a new perspective which incorporates that long familiar, but not fully exploited, recognition that there does not exist a last number in the counting sequence. In other words, for any given number, call it “n“, there is another unique number n+1. Similarly, for n+1 there is an n+2; etc.

Because of the above, instead of the traditional side-stepping of the question, both this author and the reader have been left to fixate on this set of foundational numbers –which respectively quantify the heuristic concepts of “none”, “some” and “all”, along with the respective names of “zero”, “one” and “infinity”. Note that the alignment of “some” in symbolic logic with “one” in mathematics arises because in logical systems the term “some” denotes “1 or more”, while in mathematical systems. (as will be shown in the formulation of a co-ordinate system in Chapter 2, Section 2) the arbitrary choice of a measuring reference is usually standardized, irrespective of what is being quantified, by focusing on some pre-selected entity being assigned a magnitude equal to 1.

Furthermore, in the historical evolution of mathematical thought, these three numbers have produced paradigm shifts in understanding “what”, “how”, and “why” in mathematics. An example of this, which is one of the main compasses for this book, is a treatise written in 1941 by Richard Courant and Herbert Robbins which set out to answer the question: “What is mathematics?” Such a choice was made not only because this classical treatise [3] is still one of the standards of excellence today, which has remained in wide circulation for three-quarters of a century since its original publication, but also the observation that the intended audience of that book is the one that this author hopes to reach with his treatise. Unlike these two mentoring references, which cut a wide swath of the domain of mathematics, the monograph which follows is limited to that subdivision of mathematics subsumed by the term “calculus”, along with emphasizing its place in the larger picture of intellectual inquiry. The following remarks, extracted from the preface of [3], expresses precisely the audience to whom this monograph is directed:

“For more than two thousand years some familiarity with mathematics has been regarded as an indispensible part of the intellectual equipment of every cultured person. . . . Teachers, students, and the educated public demand constructive reform . . . It is possible to proceed on a straight road from the very elements to vantage points from which the substance and driving forces of modern mathematics can be surveyed. . . . The present book is an attempt in this direction. . . . It requires a certain degree of intellectual maturity and a willingness to do some thinking on one’s own. This book is written for beginners and scholars, for students and teachers, for philosophers and engineers, for class rooms and libraries”.

As a prerequisite to appreciating that domain of mathematics referred to as “calculus”, this chapter re-examines important ideas supposedly (or maybe one should say “hopefully”), learned in previous studies. The author’s objective in including this chapter is to emphasize (and thus help to understand) WHY, in contradistinction to merely HOW, algebraic operations are performed. Notwithstanding that this set of topics had been developed in previous encounters with mathematics, they are now viewed from an advanced viewpoint. One begins by reiterating that over a millennium ago arithmetic was simplified by assigning a number (zero) to “nothing”; thereby causing a paradigm shift that brought mathematics into the mainstream. A similar new paradigm shift, focused on a number that represents the concept of “all” (in that philosophical trichotomy of none, some and all) is herewith proposed. This role will be filled by a new number, denoted as “infinity”, which includes the infinitely large, the infinitely small, and the infinitely dense. Having made such an introduction, the rest of this treatise, starting with Chapter 3, examines the relationship between the set of three “foundational” numbers (zero, one, and infinity) upon which, we assert, the development of calculus should be formulated, and the familiar arithmetic operations of compounding and undoing previous operations.

This chapter begins with a brief historical introduction, wherein the seeming paradox of Achilles and the turtle is examined. Although this treatise does follow part of the tradition and progresses to the concepts of limits and continuity, including the more formal perspective of using epsilon and delta type proofs that have been the touchstone of calculus’s foundation for over three centuries, no further development of such a protocol is undertaken. In its place a very different “Weltanschauung” (world philosophy) that focuses on what this author asserts is the appropriate underlying foundation of calculus is promulgated; namely, the relation of the concepts of none, some and all (algebraically expressed as 0, 1 and ∞) to the six fundamental operations of numbers (addition, subtraction, multiplication, division, raising to a power and extracting a root). From the 54 potential binary combinations of these sets, the seven traditional indeterminate l’Hôpital forms, as well as three additional related forms that mathematicians have missed for over three centuries are distilled. In the process, attention is focused on combinations deliberately disallowed in previous mathematics courses; especially those that arise with respect to infinity and division by zero. One particular combination, which has as its objective the determination of those extreme values that the given function can reach both globally (over all of space), and locally (in a given interval), is postulated to be the foundation upon which, provided the appropriate constraints are included, the first of the major techniques of calculus is to be built. The philosophy espoused herein views a specific related function, derived from the given function and thus named as “the derivative of that function”, as the division of two, considered to be even more elementary, functions, called “differentials”. Each of these differentials, which are primarily algebraic constructs, is equivalent to having a limit value of 0. Consequently, the derivative may be viewed as giving meaning to the indeterminate form 0\0 , under a set of constraints to be designated at a later time. Meanwhile, selected other entities, which had been historically defined, such as the concept traditionally expressed as “concavity”, are viewed as having been relegated to the status of insignificance. This is, in contradistinction to many traditional calculus textbooks which belabor concavity as being nearly equal in importance with the extreme values of maxima and minima. The topological subtleties, often forming the basis of theoretically biased courses, are included only when they add to an intuitive understanding of the subject matter, and thus become of interest to applied scientists and engineers.

Two other l’Hôpitalian combinations, which are similarly depicted as forming the foundation for the other two significant terms that comprise the principal domain associated with calculus will be introduced and developed in Chapters 4 and 6 respectively.

In this chapter, the second of the processes in the traditional sequence of courses that comprise the calculus curriculum, “integral calculus”, is examined in terms of its algebraic foundation. A new paradigm is proposed that views “integration”, also often designated as “anti- (or inverse) differentiation”, as a form of infinity multiplied by zero. Such a protocol is demonstrated as being the inverse operation to the previously developed process of differentiation in Chapter 3. However, unlike the function produced by the process of differentiation, this inverse is either not unique and needs to be supplemented with an arbitrary unspecified constant (which is addended to the generated function) or a set of limits. Along with “sloughing through” several of the techniques associated with such anti-differentiation, the mathematical underpinning of this inverse operation is introduced. This is then supplemented by an expansion of the horizon of “what is mathematics?” to introduce (1) a special (more advanced) function, called the Dirac delta function, which, in an altogether different manner is also subsumed by the over-arching concept of infinity multiplied by zero, and (2) the theoretical base from one limited to continuous functions (called “Riemann integration”) to a larger set that includes selected discontinuous functions (called “Lebesgue integration”).

In the now traditional subdivision of mathematics into algebra, geometry and analysis popularized by Felix Klein over a century ago, calculus was the first subset of analysis taught in the educational system. Meanwhile, precisely because many of the uses of calculus were associated with a geometrical perspective, a special subset of mathematics, called “analytic geometry”, was formulated and taught (sometimes interdigited and sometimes as a separate course) concurrent with the underlying foundations of calculus. In particular, as well as the basic algebra discussed up to this point, an entire chapter is herewith interjected before continuing on this author’s path to understanding “what is calculus”. Because one specific geometric figure, the hyperbola, will be shown to play an important role in that third fundamental level of complexity, exponentiation and root extraction, this treatise diverts its focus from the l’Hôpitalian aspects associated with extending algebra beyond the real finite domain in order to probe what may be considered to be a “tangential” path. (Here the word “tangential” has the layman’s connotation of moving away from a central idea that is in the process of being developed. This is in contradistinction to any role as a term in trigonometry).

In an attempt to remove the blinders that have been firmly fastened onto the student by the traditional presentation, this treatise eschews always remaining focused on a single dimension (be it either a line, a plane or a three dimensional embedding space) and assumes the perspective of multi-dimensionality. Additionally, even though most of the presentation is in the domain of real numbers, when appropriate, the place of complex numbers in the overall scheme of “analysis” is not overlooked. This will be especially useful in understanding the correlation between traditional (circular) trigonometry of Chapter 2 and a differently focused “hyperbolic” trigonometry, which will be the focus of Chapter 6.

This chapter returns the overall focus of this treatise to the development of those unique attributes of calculus begun in Chapters 3 and 4; namely differentiation and integration. Having diverted attention in order to follow what was a seemingly tangential path of analytic geometry in Chapter 5, the main stream of calculus is returned to with a third indeterminate combination manifesting some unique attributes of major scientific significance. Much as the number pi was discovered four millennia ago, another constant irrational number, discovered much more recently, is associated with that particular integral for which the formula for integration of polynomials could not be extended; namely, when the exponent n had the value of −1.

The physically important constant number discovered by John Napier and designated as e in honor of Leonhart Euler [23], along with its associated function ex, are next examined. Instead of the traditional definitions given in modern calculus textbooks, such as [14] through [21], this treatise proposes as the source of definition that binary indeterminate form 1∞. This constant will be associated with properties common to logarithms in algebra. For such a development one reiterates the denotation and connotation of the terms “exponent” and “logarithm”; two terms which are essentially synonymous in denotation but whose connotations are that “exponent” is usually associated with an integer or a common fraction, while “logarithm” is traditionally a decimal. In Chapter 2, the properties of logarithms were developed for any base, but primarily when the base of the numbering system was the number 10. Such logarithms were designated with the adjective “common” in conformity with the basis of our number system being the biological “accident” that the human species has evolved to have ten fingers.

The definition of e will be in terms of a definite integral with the argument of the function being one, or both, of its limits and the integrand being a “dummy variable”. The added perspective that will accompany this fundamental constant of both nature and of advanced mathematics, e, will be a third member of the set of indeterminate forms, cataloged by l'Hôpital in 1696, which was discussed, but not then assigned a name other than L7 in Section 3.2; namely one raised to the infinite power (1∞).

Next, employing algebraic properties associated with exponents, a pragmatic technique, which is applicable to functions that are combinations of multiplication, division, exponents and roots, while simultaneously being limited with respect to addition and subtraction will be described. This technique, called logarithmic differentiation, has been devised so as to decrease the tedium of selected traditional differentiations in many instances by employing properties of algebra.

In a similar manner, this number e, as the base of the function ex, has the important functional identity property that its derivative is equal to the function itself. Moreover, every higher derivative (and integral, when the constants of integration are set equal to zero) is also equal to this function. It is now further observed that the sum and difference of this function and its reciprocal bear a seemingly serendipitous relation to the respective cosine and sine functions of trigonometry. It is precisely one-half of each of these two relations which have been the traditional nearly-universal definition of that group of functions referred to as the hyperbolic trigonometric functions; i.e., cosh x = (ex + e-x)/2 and sinh x = (ex − e-x)/2. To the contrary, this treatise defines such a set of functions, as their names imply, starting from the geometry of a selected reference hyperbola. By the proposed geometric definitions, all of the identity properties of these functions are derivable without reference to their relation to exponential functions. In other word, the exponential relationships are downgraded to being secondary properties, while the trigonometry of the reference hyperbola is elevated to being the basis for definition. That the exponential relations are, in fact, valid is an intrinsic property of the confluence of algebra and geometry. This will be shown in Chapter 7 by expanding each of these functions using infinite series. Consequently, each is a different path to the same mathematical description in much the same manner as the set of six fundamental functions in circular trigonometry was defined by starting from either angles in a right triangle or lengths in a unit circle.

Up to this point in the proposed new perspective for understanding “what is calculus?” the domain associated with both integration and differentiation has mostly been confined to continuous functions. As a concluding chapter of this opus, the focus is directed to a discussion of discrete variables with an examination of the domain of sequences and series; then a re-definition of important functions, in particular trigonometric and exponential functions, in term of infinite series, and a broad look at the concept of infinity as both a cardinal and an ordinal number.

This chapter begins by defining the concept of sequences and both the mathematical limitations and the heuristic expectations that are fundamental to a quantitative, as well as a qualitative, development of the question “is the sequence of counting numbers unending?” and the related question “if there is such a “last” number, to which the name “infinity” has been given, what are its properties?” In the preceding chapters one observed that infinite concepts applied not only to being “infinitely large”, but also to being “infinitely small”. To this latter category the term “infinitesimal” was applied. In this chapter, the further concept, referred to as different “orders” of infinity, will be encountered. Emphasis will be placed on a concept that this author prefers to associate with the heuristic of being “infinitely dense”, in contradistinction to one of being “infinitely large”.