Hyperreal differentiation with an idempotent ultrafilter

There is a long tradition [13, 2, 16, 4, 14, 11, 5, 12, 6] of attempting to differentiate numbers in various ways. Much attention was focused on derivatives of numbers when Jeffries’ paper on the subject appeared in the Notices of the AMS late last year [10]; almost simultaneously (and apparently independently), Tossavainen et al’s survey on the subject appeared in the College Math Journal [15].

Why should the reader care about differentiating numbers? In general, any time a new theory is introduced, it is natural to seek numerical structures satisfying that theory: thus, when the theory of groups is introduced, it is natural to introduce examples like $(\mathbb{Q},+)$ and $(\mathbb{R}^{+},\cdot)$. Theories about elementary calculus functions, in languages including the unary function symbol $\bullet^{\prime}$, were originally modeled by structures whose universes consisted of elementary calculus functions, not numbers. We can at least try to find numerical models for these theories. The act of interpreting $\bullet^{\prime}$ in a structure whose universe is a number system is the act of “differentiating numbers”. We are hopeful that generalizing models of elementary calculus function theories could eventually be fruitful just like generalizing models of permutation sets led to the abstract theory of groups.

When it comes to numerically modeling theories, different theories might require different number systems: both $(\mathbb{Z},+,\cdot)$ and $(\mathbb{Q},+,\cdot)$ are rings, but only the latter is a field. By gaining knowledge about which number systems are needed for which theories, we gain insight into those theories. We hope that a greater knowledge about which number systems are needed to model various subtheories of elementary calculus functions, will eventually give us insight into those subtheories.

If the only axiom we care about is the Leibniz rule (and the nontriviality axiom $\exists x\,\mbox{s.t.}\,x^{\prime}\not=0$), we can interpret $\bullet^{\prime}$ on $\mathbb{N}$ so as to satisfy that. That is the approach of [2, 16]. But their $(\mathbb{N},\bullet^{\prime})$ does not even satisfy the linearity axiom. Our interpretation (in Section 3) of $\bullet^{\prime}$ on a subset of the hyperreals will satisfy far more axioms of the theory of elementary calculus functions. And in Section 3.2, we will introduce a stricter subset of the hyperreals where not only $\bullet^{\prime}$ can be elegantly interpreted, but $\circ$ as well (in other words, there is an elegant way to define the “composition” of two numbers there), in such a way as to numerically satisfy even the chain rule.

The key idea behind our numerical interpretation of $\bullet^{\prime}$ is to commute the derivative operation with the operation of taking a function’s equivalence class in the hyperreals, in other words, define $[f]^{\prime}=[f^{\prime}]$ (we will spell out the details below). Unfortunately, this is not well-defined in general. However, the idea can be salvaged in several different ways, by making use of certain idempotent ultrafilters. In Section 4 we will use the same approach to well-define a hyperreal variation of the derivative from finite calculus, and as an application of that, we will give a new proof of Hindman’s theorem and also strengthen said theorem.

Throughout the paper, we write $\beta\mathbb{R}$ for the set of ultrafilters on $\mathbb{R}$.

For every $f:\mathbb{R}\to\mathbb{R}$, there are two ways of viewing $f$ in nonstandard analysis. It can be viewed as the number $[f]$ or as the function ${}^{*}f:{}^{*}\mathbb{R}\to{}^{*}\mathbb{R}$. The two are related via $\Omega$. Namely: $[f]={}^{*}f(\Omega)$.

Unfortunately, the following proposition shows that the idea of defining $[f]^{\prime}=[f^{\prime}]$ does not work in general.

In light of Proposition 4, we cannot expect the definition $[f]^{\prime}_{p}=[f^{\prime}]_{p}$ to work for every free $p\in\beta\mathbb{R}$ even if we restrict our attention to everywhere-differentiable $f$; and if we do not so restrict our attention, then we can expect the definition $[f]^{\prime}_{p}=[f^{\prime}]_{p}$ to fail for every $p$. We will show that if we restrict attention to those $f$ such that $[f^{\prime}]$ exists, then the definition $[f]^{\prime}_{p}=[f^{\prime}]_{p}$ does work provided $p$ is idempotent and contains $(0,\epsilon)$ for every $\epsilon>0$.

Clearly an ultrafilter in $0^{+}$ is free. The following lemma illustrates the power of ultrafilters in $0^{+}$.

For the rest of this section, we fix an idempotent $p\in 0^{+}$. The following theorem shows that this suffices to make the definition $[f]^{\prime}=[f^{\prime}]$ well-defined if we restrict it to functions such that $[f^{\prime}]$ exists. Note that since $p\in 0^{+}$, the existence of $[f^{\prime}]$ is equivalent to the statement that for all real $\epsilon>0$, there exists real $\delta\in(0,\epsilon)$ such that $f^{\prime}(\delta)$ exists.

We do not yet know whether $\mathcal{D}$ (from Corollary 10) is a proper subset of ${}^{*}\mathbb{R}$. In other words: can every hyperreal be written in the form $[f]$ where $[f^{\prime}]$ exists? Does this depend on $p$?

If $\mathcal{D}$ is as in Corollary 10 then it follows that the structure $(\mathcal{D},1,\Omega,+,\cdot,\bullet^{\prime})$ satisfies every positive formula in the theory of elementary calculus functions in the language $(1,\mathrm{id},+,\cdot,\bullet^{\prime})$, where, by positive formula, we mean a formula that can be built up without using $\neg$ or $\not=$. In particular this includes the Leibniz rule axiom, linearity, and the power rule schema. The structure also satisfies the nontriviality axiom $\exists x\,\mbox{s.t.}\,x^{\prime}\not=0$. All this remains true if constant symbols for other individual functions (such as $\sin$ and $\cos$), besides just the identity function, are added to the language, interpreted in $\mathcal{D}$ by the hyperreals represented thereby (such as $[\sin]$ and $[\cos]$), provided those hyperreals’ derivatives exist.

In terms of nonstandard extensions, Theorem 9 says that there is a well-defined map which sends every ${}^{*}f(\Omega)$ to ${}^{*}f^{\prime}(\Omega)$. For example, this map sends $e^{\Omega}+\Omega^{3}+\cos 2\Omega$ to $e^{\Omega}+3\Omega^{2}-2\sin 2\Omega$.

One might intuitively wonder whether $[f]^{\prime}=0$ implies $f$ is constant (at least ultrafilter often). The following proposition provides a counterexample.

We have proven Corollary 10 under the assumption that $p$ is idempotent and in $0^{+}$. Similar reasoning would hold if $p$ were idempotent and in $0^{-}$ (i.e., if $p$ were required to contain $(-\epsilon,0)$ for every positive real $\epsilon$). We currently do not know whether Corollary 10 holds for any other type of ultrafilter.

In so-called finite calculus, one considers the “derivative” $f(x+1)-f(x)$ of $f$, see Section 2.6 of [8]. In this section, we will investigate a variant of this derivative, namely $\Delta f(x)={}^{*}f(x+\Omega)-f(x)$ where $\Omega=[n\mapsto n]$ is the canonical hyperreal in the hyperreals constructed using an idempotent ultrafilter on $\mathbb{N}=\{1,2,\ldots\}$ (note that we omit $0$ from $\mathbb{N}$). We will show that this finite derivative $\Delta$ has unexpected connections to the standard derivative, and that the equivalence class (in an iterated ultrapower construction) of $(\Delta f)|_{\mathbb{N}}$ is well-defined as a function of $[f|_{\mathbb{N}}]$ (we elaborate what we mean by “in an iterated ultrapower construction” in Remark 28 below).

The following Definitions 20 and 22, and Lemma 21 are $\mathbb{N}$-focused analogies of the $\mathbb{R}$-focused Definitions 5 and 1, and Lemma 7 above, respectively. We prefer this slight redundancy (instead of defining everything in higher generality) for the sake of concreteness.

For the remainder of the paper, we fix an idempotent ultrafilter $q$ on $\mathbb{N}$. Lemma 24 below will replace $0^{+}$.

Note that the following lemma depends on $q$ being idempotent.

For the rest of the section, let $\Omega=[n\in\mathbb{N}\mapsto n]$ (the canonical element of ${}^{*}\mathbb{R}\backslash\mathbb{R}$).

The following theorem shows that, when restricted to $\gamma$-periodic functions for a fixed irrational real number $\gamma>0$, $\Delta$ is a constant multiple of $\bullet^{\prime}$ (up to an infinitesimal error), at least where $f^{\prime}$ exists. And even when $f^{\prime}$ does not exist, $\Delta$ (times the same constant multiple) sometimes still provides information about the slope in question.

In particular, $\Delta(f\circ g)(x)/[\mathrm{rm}_{\gamma}]\approx(\Delta f(g(x))/[\mathrm{rm}_{\gamma}])(\Delta g(x)/[\mathrm{rm}_{\gamma}])$ (with infinitesimal error) provided $f^{\prime}(g(x))$ and $g^{\prime}(x)$ exist and $g$ is $\gamma$-periodic; thus $\Delta/[\mathrm{rm}_{\gamma}]$ satisfies a chain rule. Contrast this with Graham, Knuth and Patashnik’s claim that “there’s no corresponding chain rule of finite calculus” [8].

Without the idempotency requirement, it would be possible to find free $q\in\beta\mathbb{N}$ so as to falsify Theorem 26. For example, one could choose $q$ such that $\{n\in\mathbb{N}\,:\,\mathrm{rm}_{\gamma}(n)\in(\frac{\gamma}{3},\frac{2\gamma}{3})\}\in q$. Then for any $\gamma$-periodic $f:\mathbb{R}\to\mathbb{R}$ such that $f(0)=f^{\prime}(0)=0$ and $f(x)=1$ for all $x\in(\frac{\gamma}{3},\frac{2\gamma}{3})$, it would follow that $\Delta f(0)=1$, implying $\frac{3}{2\gamma}<\Delta f(0)/[\mathrm{rm}_{\gamma}]<\frac{3}{\gamma}$, making the conclusion of Theorem 26 (part 1) impossible at $x=0$.

As in Section 3, we would like to transform the derivative operation $\Delta:\mathbb{R}^{\mathbb{R}}\to({}^{*}\mathbb{R})^{\mathbb{R}}$ into a well-defined derivative on hyperreals. But since $\Delta f(x)$ is itself hyperreal, we will need to be careful.

The following theorem will allow us to well-define $\Delta[f]=\llbracket\Delta f\rrbracket$.

One might intuitively expect that $\Delta[f]=0$ should imply that $f$ is constant (at least ultrafilter often); we present a counterexample disproving this intuition and replacing it with a characterization related to periodicity.