# Variable-centric logical foundation of calculus

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?

• I imagine that by this point, every conceivable logical foundation for Calculus has been explored to within an inch of its life. Don't take what you see in the textbooks as indicative of what's in the journals and monographs. Dec 24, 2011 at 16:24
• @Gerry, maybe not yet. See ncatlab.org/schreiber/show/… Dec 6, 2012 at 0:37
• I have never seen a calculus book that treats the variables properly, e.g. by giving a proper definition of which variables are bound in each expression. Dec 6, 2012 at 4:04
• I vaguely recall from browsing Karl Menger's Calculus in a bygone century that it had a lot to say about variables. Don't recall exactly what.
– bof
Aug 13, 2018 at 7:10
• @LSpice ... and a lost train of thought. ;) Aug 13, 2018 at 18:25

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 $\frac{{\rm d}y}{{\rm d}x} = -\frac{x}{y}$.

• In some sense this reminds one of the foundations for random variables, considered as measurable functions $x : \Omega \to \mathbb{R}$, where $\Omega$ is an unspecified probability space. The parallel isn't exact since most operations on random variables leave the domain $\Omega$ intact, whereas operations on “differential variables” change the domain. Aug 14, 2018 at 7:57
• @UlrikBuchholtz Yes, although one might consider some kind of "universal" collection of "differential variables" by identifying functions $x:X\to \mathbb R$ with the composite $T X \to X \to \mathbb R$ and so on and taking some kind of colimit. Aug 14, 2018 at 14:24
• You probably did learn this from me, Mike, but I want to publicly credit William Lawvere for this way of thinking. For example, the definition of ‘is a function of’ that you use here may be found on page 45 (according to Google) of Lawvere & Schanuel, Conceptual Mathematics. (This is an excellent book, much more than just the introduction to category theory that its subtitle implies, which any mathematics undergraduate can read and which every mathematician should read.) Aug 19, 2018 at 11:02

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.

• +1 to this; that dL/d{q dot} always drove me crazy. Observe also the train-wreck that results when one writes the substitution rule for integrals without being explicit about the functions involved.
– user5117
Dec 6, 2012 at 17:54
• For some reason the link you give to John L. Bell's invitation doesn't work for me. This one does work: publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf Dec 6, 2012 at 23:05
• In what sense is "forming a function space a Big Thing" in classical analysis? Jan 10, 2021 at 21:47
• @LSpice: in the sense that, every time we form a function space we have to have a discussion about what topology to put on it, do we take continuous or uniformly continuous maps, or maye just the linear ones (because otherwise we would have no clue how to topologize) etc. The technical problem is that we're not operating inside a cartesian-closed category. In a world where this wouldn't be a Big Thing, the students of analysis would be told once and for all how to form function spaces, end of discussion. Jan 10, 2021 at 22:02

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.

• True, except that this syntactic foundation is (still) full of holes big enough to drive a truck through. A number of the 'bugs' in computer algebra systems are due to the fact that there is no adequate denotational semantics for their language. The only work that I know of which in this direction is Cezary Kaliszyk's PhD thesis, in which he wrote a mini-CAS in HOL. Dec 6, 2012 at 0:40
• I agree, as I said, that the systems are imperfect, and there is certainly more foundational work to be done. This will be an on-going project. My point, howevever, is that it is these kinds of foundations that the OP is inquiring about. Dec 6, 2012 at 1:16

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.

• IT's the same differential calculus, I think. Dec 7, 2012 at 2:03
• Oh! That's good to know. Dec 7, 2012 at 2:43
• 'This universal cartesian closed category can be constructed syntactically' Do you know if this has been written down somewhere? I just stumbled upon this, hoping to find an answer to my related question. Aug 12, 2018 at 11:06
• @MichaelBächtold I'm trying to think of a good reference to give you. Just generally speaking, I regard this as an application of a general theory of "doctrines" which are, roughly speaking, 2-monads on $Cat$, where the task is to give a suitable 2-coequalizer presentation $F(R) \rightrightarrows F(C) \to \mathit{Diff}$ in a 2-category of cartesian-closed categories, analogous to how we present algebras by generators and relations. This type of thing was explored in "Australian category theory" (e.g., SLNM 420) during the 70's. I've asked at the nLab if anyone knows a good reference. Aug 12, 2018 at 14:32
• Courtesy of Mike Shulman: see D.4.2 in Johnstone's "Elephant": ncatlab.org/nlab/show/Elephant Aug 12, 2018 at 23:11