Variable-centric logical foundation of calculus

Question

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For example, in the assertion "If $y=x^2$ then $\frac{dy}{dx}=2x$," the functions $f$ and $f'$ remain unnamed while the variables $x$ and $y$ take center stage. We interpret this as notational finesse, but there seems to be an important philosophical difference between what we say and what we mean.

I have sometimes wondered: Is there an alternate logical foundation of Calculus in which variables, expressions, and equations are the central ideas, and functions per se are implicit?

Answer 1

I would like to argue in the opposite direction. The 17th century notation that is still in use today is a syntactic hodgepodge which does great disservice to students and their teachers alike, despite claims of its usefulness (by people who never tried an alternative). It is partially responsible for the fact that the average mathematician in the street cannot coherently describe the notion of a bound variable, thinks there isn't much difference between $f$ and $f(x)$, and is willing to believe that $\frac{\partial L}{\partial \dot q}$ is a sensible notation.

Functions as mathematical objects (as opposed to symbolic expressions) are fundamental to differential calculus. Moreover, important concepts such as derivative, definite integral, differential operator, gradient, etc., are themselves functions of higher order (they take functions as arguments).

Let me mention two modern foundational explorations in analysis.

First there is Synthetic differential geometry (introductory reading material here/archive and here/archive) whose distinguishing features are that it calculates with nilpotent infinitesimals, and that arbitrary function spaces can be formed at will (whereas in classical analysis forming a function space is always a Big Thing). This makes certain definitions very easy. For example, the tangent bundle of $M$ is simply the space of function $\Delta \to M$ where $\Delta = \lbrace x \in R \mid x^2 = 0\rbrace$ is the space of infinitesimals (of order 2). And it does not even matter what $M$ is here, the definition just makes sense, both intuitively and technically. The classical approach to analysis requires a whole edifice just so that the tangent bundle can be defined. It is too complicated for the average undergraduate.

A foundation for calculus which is most directly based on functions is the differential $\lambda$-calculus/archive (introduction here/archive). The $\lambda$-calculus is the theory of functions. For example, functional programming languages are based on it. The differential $\lambda$-calculus is an enrichment of $\lambda$-calculus with (abstract) differential operators.

So, while I am sure somebody has cooked up a foundation of calculus based on avoiding functions, the arrow of progress points in the opposite direction.

Answer 2

Here is another approach, which I believe I first learned from Toby Bartels. Suppose $X$ is an arbitrary differentiable manifold (think of the state space of some physical system), and define a variable (one might also say "observable") to be a smooth real-valued function on $X$. If $x:X\to \mathbb{R}$ is such a "variable", then its differential is, as usual in differential geometry, a smooth function ${\rm d}x:T X \to \mathbb{R}$ on the tangent bundle of $X$. We also have the tangent map $T x : T X \to T\mathbb{R} \cong \mathbb{R}\times\mathbb{R}$, with $T x = (x, {\rm d}x)$.

If $y:X\to \mathbb{R}$ is another such "variable", then it might be related to $x$ by an equation such as $y = x^2$ or $x^2 + y^2 = 4$. Being equalities of real-valued functions, these are pointwise equalities. If $y= x^2$, then we can say that "$y$ is a function of $x$" in the sense that there is a function $f:\mathbb{R}\to\mathbb{R}$ such that $y = f\circ x$, namely $f = \lambda u. u^2$ (see this question). In this case, the chain rule of differential geometry tells us that $T y:T X \to T \mathbb{R}$ is the composite $T X \xrightarrow{T x} T \mathbb{R} \xrightarrow{T f} T \mathbb{R}$. Since $T f (u,v) = (f(u), f'(u) \cdot v)$, this means that (in addition to $y = f\circ x$) we have ${\rm d}y = f'(x) \cdot {\rm d}x$. This is a simple pointwise equality of functions $T X \to \mathbb{R}$, so we can divide by ${\rm d} x$ (at least assuming it is never zero) to get $f'(x) = \frac{{\rm d}y}{{\rm d}x}$, or in this case $\frac{{\rm d}y}{{\rm d}x} = 2x$.

Similarly, if $x^2+y^2=4$, then $y$ is not a function of $x$ in this sense, but $x^2+y^2$ and $4$ are two smooth functions $X\to \mathbb{R}$, where the first is expressed as a composite $$X\xrightarrow{(x,y)} \mathbb{R}\times\mathbb{R} \xrightarrow{\lambda u v. u^2+v^2} \mathbb{R}.$$ Thus the chain rule of differential geometry again gives us $2 x \,{\rm d}x + 2 y \,{\rm d}y = 0$ as a pointwise equality of functions $T X \to \mathbb{R}$, so that we can solve it as usual in elementary calculus to get $\frac{{\rm d}y}{{\rm d}x} = -\frac{x}{y}$.

Answer 3

All the various computer algebra system approaches to calculus, which are increasingly prevalent and powerful, although imperfect, seem to me to be prime instances of the kind of foundations you mention. These systems, at least the kind I have in mind, are completely syntactic, working by necessity explicitly on the syntactic strings used to represent the various functions, that is, with the variables, expressions and equations as syntactic objects, rather than with the abstract mathematical objects that these strings represent for us. In particular, the systems have to deal with all kinds of issues such as free and bound variables and variable types and all sorts of irritating syntactic issues involving substitution and composition and whatnot, which we mathematicians usually prefer to glide smoothly past without pause. At bottom, however, the development of these computer algebra systems has required the developers in effect to formulate a syntactic foundation of exactly the kind you seek. The actual meaning of the expressions, the actual abstract mathematical objects, are merely implicit in the operation of the systems.

Answer 4

In one of my comments over at the other thread (this other thread), I had mentioned some discussion at the $n$-Category Café about a differential $\lambda$-calculus (from the looks of it, different to the one alluded to in Andrej Bauer's answer, although I looked only briefly). I'll try to sketch out some of it here, as it seems relevant.

Categorical models of typed $\lambda$-calculus are cartesian closed categories, and the idea was to contemplate the universal (i.e., initial) cartesian closed category that comes equipped with

• A commutative ring object $R$ (finite products suffice to describe what is meant by a commutative ring object),

• A differentiation operator $D\colon R^R \to R^R$ (here $R^R$ is the "function space object" whose existence is given by cartesian closure; the elements of $R^R$ correspond to morphisms $R \to R$), satisfying all the formal expected properties of differentiation (product rule, chain rule, etc.).

This universal cartesian closed category can be constructed syntactically and could be called "the $\lambda$-theory of high school calculus"; I'll call it $\mathit{Diff}$. A model of this theory is by definition a cartesian closed category $C$ together with a functor $S\colon \mathit{Diff} \to C$ which preserves the cartesian closed structure up to isomorphism (meaning the canonical comparison maps $S(a \times b) \to S(a) \times S(b)$ and $S(a^b) \to S(a)^{S(b)}$ are required to be isomorphisms). If the receiving category $C$ consists of concrete structures and structure-preserving maps, then you could call such a model a "denotational semantics" of $\mathit{Diff}$.

Now, there exists no denotational semantics $S\colon \mathit{Diff} \to Set$ which takes the object $R$ to the reals $\mathbb{R}$ as commutative ring. In other words, there is no operator $D\colon \mathbb{R}^{\mathbb{R}} \to \mathbb{R}^{\mathbb{R}}$ that satisfies all the formal properties of differentiation. But, there are other toposes $C$ of interest besides $Set$ which do admit such a semantics, where one can arrange the commutative ring object $R$ in $C$ so that its elements $1 \to R$ correspond exactly to real numbers. There is some flexibility in what one can arrange general morphisms $R \to R$ to be -- certainly they won't correspond to all functions $\mathbb{R} \to \mathbb{R}$, but you can get various interesting subclasses of functions (for which the modeled differentiation operator $D$ coincides with the usual one). For example:

• If $C$ is the topos of functors $CAlg_{fp} \to Set$ where $CAlg_{fp}$ is the category of finitely presented commutative $\mathbb{R}$-algebras, then $\hom(R, R) \cong \mathbb{R}[x]$ is the set of polynomial functions on $\mathbb{R}$.

• If $C$ is the topos of functors $C^{\infty}Alg_{fp} \to Set$ where $C^{\infty}Alg_{fp}$ is the category of finitely presented $C^{\infty}$-algebras (see here for definitions), then $\hom(R, R)$ is isomorphic to the ring of $C^{\infty}$-functions $\mathbb{R} \to \mathbb{R}$.

There were various further ruminations on this, coming under the heading of "snowglobe models" as models "sitting inside $Set$ as a miniature universe", as discussed here.

Answer 5

As mentioned in one comment, you might be interested in Calculus – A modern approach (1955) by Karl Menger:

What Is A Variable?

The conceptual and semantic clarification of calculus is centered on the analysis of the hitherto obscure general term "variable," which is resolved into an extensive spectrum of well-defined meanings.

The only clear (if one-sided) definition heretofore formulated goes back to Weierstrass who, in his celebrated lectures in the 1880's, defined it as a symbol that stands for any number or any element belonging to a certain class of numbers. Bertrand Russell, who at the turn of the century investigated the various aspects of variables probably more thoroughly than anyone before him, said: "Variable is perhaps the most distinctly mathematical of all notions; it is certainly also one of the most difficult to understand […] and in the present work [The Principles of Mathematics, 1903] a satisfactory theory as to its nature, in spite of much discussion, will hardly be found." In fifty years this situation has not been improved.

In this book a solution of the problem is attempted by distinguishing and making precise various equally important uses of the term "variable" in pure and applied calculus. Some of these variables differ from each other as profoundly as do trigonometric and geometric tangents. But whereas no one has, on account of a flimsy equivocation, confused tangents of angles and tangents of curves, this book seems to be the first to maintain clear distinctions between the following three concepts:

I. Variables according to Weierstrass, herein called numerical variables, as $\mathrm{x}$ and $\mathrm{y}$ in $$ \mathrm{x}^2 - 9\mathrm{y}^2 = (\mathrm{x} + 3\mathrm{y})\cdot(\mathrm{x} - 3\mathrm{y})\quad \text{for any two numbers $\mathrm{x}$ and $\mathrm{y}$.} \tag{1}\label{473800_1}$$ Here, as throughout this book, the numerical variables are printed in roman type. Without any change of the meaning, $\mathrm{x}$ and $\mathrm{y}$ may be replaced by any two non-identical letters, e.g., by $\mathrm{a}$ and $\mathrm{b}$ or by $\mathrm{y}$ and $\mathrm{x}$; that is to say, two numerical variables may be interchanged: $$ \mathrm{y}^2 - 9\mathrm{x}^2 = (\mathrm{y} + 3\mathrm{x})\cdot(\mathrm{y} - 3\mathrm{x})\quad \text{for any two numbers $\mathrm{y}$ and $\mathrm{x}$}$$ is tantamount to \eqref{473800_1}. In calculus, numerical variables may be used or they may, as will be shown herein, be dispensed with.

II. Variables or variable quantities in the sense in which scientists use these terms; for instance, $t$, the time; $s$, the distance traveled (in chosen units); $x$ and $y$, the abscissa and ordinate in a physical or postulational plane (relative to a chosen frame of reference); etc. These "variables" are defined and thoroughly discussed in Chapter VII under the names of consistent classes of quantities and — reviving Newton's terminology — of fluents. They are herein consistently denoted by letters in italic type. Fluents cannot be dispensed with in formulas expressing scientific laws, such as Galileo's $$s = 16t^2.\tag{2}\label{473800_2}$$ Nor can they be interchanged: $$2x + 3y = 5\quad\text{and}\quad 2y + 3x = 5$$ are different lines. (If, on the other hand, in pure analytic geometry, the first of these two lines is defined as

the class of all pairs $(\mathrm{x},\mathrm{y})$ of numbers such that $2\mathrm{x} + 3\mathrm{y} = 5$,

where $\mathrm{x}$ and $\mathrm{y}$ are numerical variables, then

the class of all pairs $(\mathrm{y},\mathrm{x})$ of numbers such that $2\mathrm{y} + 3\mathrm{x} = 5$

is an equivalent definition.)

III. Variables in the sense of $u$ and $w$ in statements such as $$\text{If }w = 16u^2\text{, then }\frac{\mathbf{d}\ w}{\mathbf{d}\ u} = 32u\text{ for any two fluents, }u\text{ and }w.\tag{3}\label{473800_3}$$ These "variables" belong to a third type, first explicitly introduced by the author in 1952 (see Bibliography). They are herein referred to as fluent variables, since they partake in characteristics of numerical variables as well as of fluents. In \eqref{473800_3}, $u$ and $w$ may be replaced by any two elements of a certain class — but not by two numbers. If $u$ were replaced by $3$, and $w$ by $144$, then the antecedent $144 = 16\cdot 3^2$ would be valid, and yet the consequent $\dfrac{\mathbf{d}\ 144}{\mathbf{d}\ 3} = 32\cdot 3$, utterly nonsensical. What $u$ and $w$ in \eqref{473800_3} may be replaced by are fluents, such as $t$ and $s$ regarding a motion, or $x$ and $y$ along a plane curve: $$\text{If }s = 16t^2\text{, then }\frac{\mathbf{d}\ s}{\mathbf{d}\ t} = 32t;\tag{$3\ '$}\label{473800_3'}$$ $$\text{If }y = 16x^2\text{, then }\frac{\mathbf{d}\ y}{\mathbf{d}\ x} = 32x.\tag{$3\ ''$}\label{473800_3''}$$

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Disclaimer. I have not read the book yet, and at first glance the suggested formalism feels like an overkill. I would imagine that instead of formalizing variable quantities as mathematical objects in themselves, it should suffice to clarify their use in the spirit of 17th–18th century mathematics, as commonly used today by normal people. This would be all about relations imposed among variable, like in a group presentation by generators and defining relations.

Answer 6

The following papers present a "relational" rather than "functional" approach to differential calculus:

1. Extending the algebraic manipulability of differentials

2. Simplifying and refactoring introductory calculus

3. Total and partial differentials as algebraically manipulable entities

From [1] § "Relationship to historic Leibnizian thought":

The view of differentials presented by Leibniz and those following in his footsteps differed significantly from the modern-day view of calculus. The modern view of calculus focuses on functions, which have defined independent and dependent variables. The Leibniz view, however, according to [5], is a much more geometric view. There is no preferred independent or dependent variable.

The modern concept of the derivative generally implies a dependent and in independent variable. The numerator is the dependent variable and the denominator is the independent variable. In the geometric view, however, there are only relationships, and these relationships do not necessarily have an implied dependency relationship.

Therefore, Leibnizian differentiation doesn't occur with respect to any independent variable. There is no preferred independent variable. Likewise, as we have seen in Sections 6 and 7, the version of the differential presented here allows for the reversal of variable dependency relationships. Similarly, the procedure of differentiation given in Section 3 which allows us to formulate the new notation for the second derivative given in (3) follows the Leibnizian methodology, where the differentiation is done mechanically without considering variable dependencies.