1.1 Einstein on Purity in Geometry

We begin with a case study discussed by Luchins and Luchins (Reference Luchins and Luchins1990), from which we take the translations in this section. In 1937 Albert Einstein corresponded with his friend Max Wertheimer, the Gestalt psychologist, about “the problem of axioms.” Seeking to understand Wertheimer’s problem better, Einstein asked if Wertheimer wanted “to compare the value of two proofs which are themselves based on the same system of (concepts and) axioms.” He added that “in that case, surely, we are completely satisfied only if we feel of each intermediate concept that it has to do with the proposition to be proved.”

To clarify this, Einstein then presents two proofs of the theorem of Menelaus as “a pretty example of two proofs of different degrees of perspicuity.” The theorem of Menelaus says that if $△ABC$ is a triangle, and $\ell$ is a line that crosses the sides BC, CA, AB at three distinct points P, Q, R, then

$\frac{AR}{RB}\cdot \frac{BP}{PC}\cdot \frac{CQ}{QA}=1.$

Figure 1 shows the basic configuration of the theorem.

Figure 1 Theorem of Menelaus.

Einstein then gives two proofs of Menelaus, one of which he calls “ugly”, the other “elegant”. The first proof brings in additional, seemingly unrelated constructions, while the second proof restricts itself to what is mentioned in Meneleus’ theorem.

The “ugly” proof relies on what he calls the “principle of similarity”, otherwise known as the theorem of Thales (Euclid VI.2; see Figure 2). It says: let $△ABC$ be a triangle and let DE be parallel to BC, cutting the other two sides at D and E. Then $\frac{AD}{DB}=\frac{AE}{EC}$.

Figure 2 Theorem of Thales.

The “ugly” proof then goes as follows. Augment the Menelaus configuration by a line OA parallel to the transversal PR (see Figure 3). By one application of Thales, we get that $\frac{CQ}{QA}=\frac{PC}{PO}$. By a second application of Thales, we get that $\frac{RB}{AR}=\frac{BP}{PO}$. We have that

Figure 3 “Ugly” proof of Menelaus.

$\frac{CQ}{QA}=\frac{PC}{PO}\text{and}\frac{RB}{AR}=\frac{BP}{PO}.$

Solving both for PO and then setting both equal, we have

$\frac{QA\cdot PC}{CQ}=\frac{AR\cdot BP}{RB}.$

Rearranging terms, we get

$\frac{AR}{RB}\cdot \frac{BP}{PC}\cdot \frac{CQ}{QA}=1.$

This is the theorem of Menelaus.

For the “elegant proof,” Einstein uses the “principle that two triangles with a common (or supplementary) angle are related as the products of the adjacent sides,” and writes it in an equation involving a proportion of the triangles themselves and the adjacent sides. See again Figure 1. For the triangles $△ARQ$ and $△BRP$, Einstein uses this principle to get

$\frac{△ARQ}{△BRP}=\frac{AR\cdot QR}{BR\cdot RP}.$

But what is an equation involving a triangle? A triangle is not a quantity. If we read the “principle” as involving the areas of triangles, however, then it can be understood as an application of the trigonometric formula for the area of a triangle. This formula says that for a triangle $△ABC$ with angle $\angle ABC=\theta$, we have that

$area\left(\Delta ABC\right)=\frac{1}{2}\cdot AB\cdot BC\sin \theta .$

For $△ABC$ and $△DEF$ with a common angle $\angle ABC=\angle DEF=\theta$, we then have

$\frac{area\left(△ABC\right)}{area\left(△DEF\right)}=\frac{\frac{1}{2}\cdot AB\cdot BCsin\theta }{\frac{1}{2}\cdot DE\cdot EFsin\theta }=\frac{AB\cdot BC}{DE\cdot EF}.$

If $\angle ABC$ and $\angle DEF$ are supplementary, then $sin\left(\angle ABC\right)=sin\left(\angle DEF\right)$.

The “elegant proof ” applies the formula

$\frac{area\left(△ABC\right)}{area\left(△DEF\right)}=\frac{AB\cdot BC}{DE\cdot EF}$

three times, on the triangles formed by the vertices of the given triangle with the corresponding segments of the transversal. It then multiplies the resulting equations together, giving the conclusion of Menelaus.

First, apply the formula to $△ARQ$ and $△BRP$, where $\angle ARQ$ and $\angle BRP$ are supplementary, obtaining

$\frac{area\left(△ARQ\right)}{area\left(△BRP\right)}=\frac{AR\cdot QR}{RB\cdot PR}.$

Next, apply the formula to $△BRP$ and $△QPC$, with common angle $\angle QPC$, obtaining

$\frac{area\left(△BRP\right)}{area\left(△QPC\right)}=\frac{PR\cdot BP}{QP\cdot PC}.$

Finally, apply the formula to $△QPC$ and $△ARQ$, with common = opposite angle $\angle AQR$, obtaining

$\frac{area\left(△QPC\right)}{area\left(△ARQ\right)}=\frac{QP\cdot CQ}{QA\cdot QR}.$

We then have

$\begin{array}{c}\frac{area\left(△ARQ\right)}{area\left(△BRP\right)}=\frac{AR\cdot QR}{RB\cdot PR},\\ \frac{area\left(△BRP\right)}{area\left(△QPC\right)}=\frac{PR\cdot BP}{QP\cdot PC},\\ \frac{area\left(△QPC\right)}{area\left(△ARQ\right)}=\frac{QP\cdot CQ}{QA\cdot QR}.\end{array}$

Now multiply both sides of these three equations together. The left-hand side reduces to 1 because in the three terms being multiplied, the area of the three triangles occurs once in a numerator and once in a denominator. We thus obtain

$\begin{array}{cc}1& =\frac{AR\cdot QR}{RB\cdot PR}\cdot \frac{PR\cdot BP}{QP\cdot PC}\cdot \frac{QP\cdot CQ}{QA\cdot QR}\\ & =\frac{AR\cdot \overline{)QR}}{RB\cdot \overline{)PR}}\cdot \frac{\overline{)PR}\cdot BP}{\overline{)QP}\cdot PC}\cdot \frac{\overline{)QP}\cdot CQ}{QA\cdot \overline{)QR}}\\ & =\frac{AR}{RB}\cdot \frac{BP}{PC}\cdot \frac{CQ}{QA},\end{array}$

cancelling the segments QR, PR, QP of the transversal that appear in both numerators and denominators of the expressions. We have thus obtained the conclusion of Menelaus.

Why does Einstein call the first proof “ugly” and the second proof “elegant”? Einstein replies: “Although the first proof is somewhat simpler, it is not satisfying. For it uses an auxiliary line which has nothing to do with the content of the proposition to be proved.” By contrast, “the second proof … can be read off directly from the figure.”

The thought is that the auxiliary line in the first proof is unrelated to the “content” of the theorem of Menelaus, and this renders it ugly. He clarifies this in his comment about the second proof, saying that it can be “read off directly” from the triangle given in the theorem of Menelaus. Every element of the second proof is part of the given triangle and thus contains nothing extraneous to the content of the theorem. The first proof augments the Menelaus configuration by a line parallel to the given transversal, whereas the second adds nothing to the configuration. Einstein judges that this use of an additional element, not part of the original configuration, is to be avoided if we seek “‘elegant” proofs. He thus deems his second proof “better” because it avoids what is extraneous to what is being proved, and is more elegant because of its relative minimality.

Einstein’s remarks are the occasion to introduce the central concern of this Element, purity of methods. Roughly, a proof of a theorem, or a solution to a problem, is pure if it draws only on what is “close” or “intrinsic” to that theorem or problem. Since this rough reckoning of purity invokes a notion of distance, purity can come in degrees; and if a threshold is set, proofs involving what is more remote than that threshold can be said to be impure tout court. In Einstein’s case, the distance between proof and theorem is measured by what belongs to the “content” of the theorem of Menelaus, and the purer proof is better because it is more elegant than the less pure proof. These remarks frame two focal questions for investigating purity constraints. Firstly, how is purity to be measured? Secondly, why is purity, measured this way, valuable? As we will see, purity constraints of many different kinds arise in mathematical practice. For each such constraint, the nature of its measure of purity should be determined, and the reasons for preferring such proofs should be examined.

Let us look more closely at the first question in Einstein’s case. In a follow-up letter to Wertheimer, Einstein seeks to specify his measure of purity more fully. He expresses discomfort even with the second, “better” proof. “A certain uneasiness remains, however, even in the better proof,” he writes, “since we don’t see a priori what the segments on the transversal are supposed to have to do with the matter. We use them in the proof, and then they are cancelled out.” These segments of the transversal aren’t mentioned in the theorem (even though the transversal itself is). Moreover, though they are used in the proof, they cancel each other out during a calculation, furthering the sense that they are unnecessarily brought into the proof. Einstein may be suggesting that since the segments on the transversal are not explicitly mentioned in the statement of Menelaus, they are not permissible in a pure proof of it. While until now the notion of content used in his purity has remained vague, here Einstein moves away from a potential semantic understanding of it toward a syntactic understanding. On this measure, only what is explicitly mentioned in a statement can be used in a pure proof.

Many other proofs of the theorem of Menelaus are known (compare Chemla Reference Chemla1998), and many fail Einstein’s purity constraint. For instance, we can add auxiliary perpendiculars to the original configuration and apply principles of similarity to arrive at the conclusion. We can also use coordinate methods as in Cartesian geometry. Both of these methods go beyond the content of the theorem in Einstein’s sense, as neither these perpendiculars nor coordinates are part of the formulation of Menelaus as considered here.

1.2 Purity in Number Theory: Jacobi’s Four Squares Theorem

At the end of the nineteenth century, Friedrich Engel (Reference Engel1890) published a short book on taste in mathematics. Engel is today best known as the coauthor with Sophus Lie of the monumental three-volume Theorie der Transformationsgruppen (1888, 1890, 1893), which gave a detailed introduction and treatment of what would soon be known as Lie groups. But Engel was also an astute commentator on mathematics, as Engel (Reference Engel1890) reveals. Near the end, Engel considers the state of number theory at the end of the nineteenth century:

Engel here describes the status of purity in number theory, pointing to elementary arithmetic theorems proved by transcendental means. In this section we will describe one such theorem, though as we will see, it is not one that eluded purely arithmetic proof. The case, Jacobi’s four squares theorem, will rather show how this impurity came to be brought to a celebrated result of number theory and how the actors involved in this case thought of it.

In 1621 Claude Gaspar Bachet de Méziriac published a Latin edition of Diophantus’ Arithmetic, in which, in addition to his translation, Bachet added his own commentaries. In Book IV, Problem 31, he comments that every positive integer can be written as the sum of four squares, verifying the result explicitly up to 120 and saying that he verified it up to 325. He adds that it “can easily be extended to any number of squares” (compare Diophantus 1621, p. 242).

Bachet’s conjecture almost immediately captured the imagination of mathematicians. Fermat (Reference Fermat, Tannery and Henry1894) agreed with Bachet that Diophantus had known the theorem, writing in a letter to Mersenne in 1636 that Bachet had verified it experimentally but had no proof (pp. 65–66). In 1638 Descartes (Reference Descartes, Adam and Tannery1898) too wrote of the conjecture to Mersenne, saying that it was “doubtlessly one of the most beautiful that one could find concerning numbers” but that he knew no proof and that he judged it so difficult that he did not dare to start looking for one (p. 256). Fermat learned of Descartes’ remarks and seems to have taken that as further motivation to solve it himself. Noting Descartes’ difficulties with the problem in a letter to Carcavi in 1659, Fermat (Reference Fermat, Tannery and Henry1894), claims to have found a proof by his method of infinite descent, but does not give the proof (p. 433). (Incidentally, it was in a copy of Bachet’s edition of Diophantus that Fermat wrote his infamous marginal comment on what would become known as his Last Theorem.) Euler then worked on the problem but it was only in 1770 that Lagrange, building on Euler’s work, published the first proof (see Lagrange Reference Lagrange1772).

After seeing that every positive integer can be written as the sum of four squares, we can ask in how many different ways it can be done. For instance, Bachet observed that 39 can be written as $1+1+1+36=1^2+1^2+1^2+6^2$ and as $1+4+9+25=1^2+2^2+3^2+5^2$. Are there any other combinations that work? Jacobi (Reference Jacobi1829) turned to this question and solved it (p. 106). The result he proved is now known as Jacobi’s four squares theorem: the number of representations of n as a sum of four squares is 8 times the sum of the positive divisors of n that are not divisible by 4. It follows that 39 can be represented in $8\cdot 56=448$ ways, since the positive divisors of 39 are 1, 3, 13, 39 and their sum is 56.

To see a little more clearly how this works, let’s consider a simpler example, 8. The positive divisors of 8 are 1, 2, 4, and 8, and of these, 1 and 2 are not divisible by 4. Jacobi’s four squares theorem says that 8 is representable in $(1+2)\cdot 8=24$ ways as the sum of four squares. Trivially 0 is a square, and 4 is a square in two ways, as $2^2$ and as $(-2{)}^2$. So, 8 can be written as the sum of four squares in the following four ways: $0+0+2^2+2^2$, $0+0+(-2{)}^2+(-2{)}^2$, $0+0+2^2+(-2{)}^2$, $0+0+(-2{)}^2+2^2$. Order matters here, so these cases need to be distinguished. Similarly, there are six total places where the 4s can go: the last two places as we have seen, the second and fourth places, and so on. Since for each of these placements of the 4s there are four ways that the numbers can be summed, as we saw earlier, we obtain 6 times 4 equals 24 ways, as Jacobi’s result says.

While this purely arithmetic result is simple to state, the proof that Jacobi (Reference Jacobi1829) presents in Fundamenta nova theoriae functionum ellipticarum uses, as the name indicates, the theory of elliptic functions. Elliptic functions in Jacobi’s hands were obtained by inverting elliptic integrals, which were introduced in attempts to find the arc lengths of ellipses. In this development, elliptic functions were thought of as functions of complex variables, though Jacobi’s treatment of complex variables in this work was purely formal rather than a development of complex analysis (compare Gray Reference Gray2015, pp. 88–89). Nevertheless, Jacobi’s proof of his four squares theorem was breathtakingly transcendental. He defines an elliptic function known today as a theta function, a function of a complex variable, and shows that it is periodic. As this is a periodic function, he can then find a Fourier series representing this function. From this series he can read off a power series whose coefficients, following the work of Euler, have a combinatorial interpretation: they give the number of ways a positive integer can be represented as the sum of four squares (compare Hardy and Wright Reference Hardy and Wright1979, chapters 19 and 20).

In so doing, Jacobi brings methods that seem to be quite remote from elementary arithmetic to bear on a simply stated arithmetic theorem. In his announcement of the theorem in Jacobi (Reference Jacobi1828), he writes that this result “seems to be difficult to prove by the known methods of number theory” and that his proof “by the theory of elliptic functions is entirely analytic” (p. 191). He lauded his proof as having bridged two domains previously thought distant. In Jacobi (Reference Jacobi1848), he wrote: “Between analysis and number theory, which were long thought to be completely separate disciplines, more and more frequent and often unexpected connections and transitions have recently been discovered. A rich source of mutual relationships between the two, which will remain unexhausted for a long time, is the analysis of elliptic functions” (p. 61). He saw his work on the four squares theorem in this light.

At the same time, he recognized that not everyone would be satisfied with an analytic proof of this elementary result. In Jacobi (Reference Jacobi1834), he observes:

If he is not himself troubled by the distance between the theorem and the proof, he acknowledges that others might be, the “virorum arithmeticorum,” and accordingly gives a new, purely arithmetic proof. This new proof applies a result about the representation of positive integers by two squares that was known to Fermat and proved by Euler by purely arithmetic means (see Euler Reference Euler1758). Moreover, Jacobi observes that his new proof was “concealed” by the analysis, but can be revealed without difficulty, and that this revelation might help mathematicians skilled in arithmetic make further progress on purely arithmetic problems.

Jacobi continued to apply his transcendental methods to number-theoretic problems and also continued to try to translate these proofs into purely arithmetic terms. He commented on these parallel aims again in Jacobi (Reference Jacobi1848). Firstly, he argued for the value of his transcendental proofs:

By grounding an arithmetic result in a transcendental context, the result gains in generality, as it is now seen to be part of a wider conceptual network. Again, such impurity is not necessary, since purely arithmetic proofs are also available by “translating” the transcendental proof into arithmetic terms:

Thus, while purely arithmetic proofs can be found without difficulty, they can be more complex than the transcendental proofs, though Jacobi does not specify what kind of complexity he has in mind.

Jacobi thus reveals a certain ambivalence about the apparent impurity of his transcendental proof: on the one hand, it bridges what were earlier thought to be distinct branches of mathematics and is valuable for doing so; on the other hand, this “bridge” can be translated into purely arithmetic terms without too much difficulty, using the aforementioned combinatorial interpretations of power series by Euler. However, these translations can be more complex than the original impure proofs; but still, the pure proofs can be useful in helping experts in arithmetic advance their arithmetic studies.

Mathematicians after Jacobi continued to refine his new methods. Eisenstein, who was involved in a controversy with Jacobi over their proofs of certain reciprocity laws (see Collison Reference Collison1977), found another purely arithmetic proof of Jacobi’s four squares theorem, but judged his results to not be mere translations of transcendental methods: “In my investigations these propositions are proved by purely arithmetic considerations, and appear as special cases of more general propositions” (see Eisenstein Reference Eisenstein1847, p. 135). For similar reasons Dirichlet too sought a new purely arithmetic proof of Jacobi’s theorem. In 1856 an extract of a letter from Dirichlet to Liouville was published (Dirichlet Reference Dirichlet1856) in which Dirichlet recalls a recent conversation they had had about Jacobi’s “beau” theorem. He observes that Jacobi “first proved by his elliptic series and has since given an arithmetic demonstration … [but] which Jacobi himself warned was only a translation of the first proof ” (p. 210). Dirichlet says he has already looked for a long time for other principles on which to found the theorem, but that is not the subject of his letter today. Instead, it is to simplify Jacobi’s purely arithmetic proof by “exposing the arithmetic or rather algebraic fact that forms its principal foundation.” While there may exist straightforward ways to render the transcendental methods arithmetic, the pure proof resulting from this translation is not particularly clear. Here he echoes Jacobi himself. There is reason, says Dirichlet, to search for new proofs that are more clearly situated within arithmetic, rather than awkwardly replicating the analytic structure in arithmetic language.

1.3 Summing up the Case Studies

In these two case studies two type of purity seem to be involved. Einstein looks for a proof that does not draw on resources beyond what is needed to determine the content of the theorem he is proving. Jacobi, by contrast, looks for a proof that stays within the domain or branch of the theorem he is proving. We will go on to call the former type of purity “topical” and the latter, “geographical.” We will come to distinguish other types of purity as well, and discuss how these types of purity are related.

1.4 Purity of Proof vs. Purity of Definition

In this Element we will focus on treating purity of proof. But purity concerns arise for other mathematical practices as well, such as definition. For instance, Samuel Eilenberg and Norman Samuel Eilenberg and Norman Steenrod (Reference Eilenberg and Steenrod1945) sought an axiomatic definition of the concept of homology group in algebraic topology, invoking purity as a constraint on their search:

The purely algebraico-topological concept of homology group should not be defined using analytic geometry, they write, alluding to its development by Henri Poincaré with its attachment to particular topological spaces. Nevertheless, their interest in a topologically pure definition of homology group was linked to their interest in topologically pure proofs. In their textbook (Eilenberg & Steenrod Reference Eilenberg and Steenrod1952, p. x), they add that “Proofs based directly on the axioms are usually simple and conceptual. It is no longer necessary for a proof to be burdened with the heavy machinery used to define the homology groups.” Their pure definition of homology group thus leads to topologically pure proofs. The quest for “good” definitions in mathematics is frequently linked to the values realized by proofs from such definitions (see Tappenden Reference Tappenden and Mancosu2008), and so we will continue to focus principally on purity of proof.

1.5 Looking Ahead

In this Element we will investigate the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. In Section 2, we’ll give a brief history of purity in mathematics. We’ll then, in Section 3, discuss several different types of purity, based on several different measures of distance between theorem and proof. In Section 4 we’ll discuss reasons for preferring pure proofs, for the varieties of purity constraints presented in Section 3. In Section 5 we conclude by reflecting briefly on purity as a preference for the local and how issues of translation intersect with the considerations we have raised throughout this Element.

3.1 Geographical Purity

In Section 2, we saw how purity can be formulated in terms of branches of mathematics. The classic formation of this kind is Aristotle’s, writing that “we cannot, in demonstrating, pass from one genus to another. We cannot, for instance, prove geometrical truths by arithmetic.” Accordingly, we can define a proof of a statement to be geographically pure if it draws only on what belongs to the branch of mathematics to which the statement belongs.

Several of the examples of purity we have already surveyed are of this type, like Jacobi’s and Bolzano’s. Ferraro and Panza (Reference Ferraro and Panza2012) have studied a case of what we call geographic purity in Lagrange’s theory of analytic functions. Lagrange sought to prove theorems of the infinitesimal calculus using only “an algebraic, purely formal theory centered on the manipulation of (finite or infinite) polynomials through the method of indeterminate coefficients” (p. 96). He sought algebraic solutions for what he understood to be algebraic problems, rather than those using principles of infinitesimals “foreign to the spirit of analysis, which should have no metaphysics but that which consists in the first principles and in the fundamental operations of calculation” (see Lagrange Reference Lagrange1799, p. 233, translation from Ferraro and Panza Reference Ferraro and Panza2012, p. 97).

Talk of purity in number theory is often geographical. We can illustrate this with an example that we will have reason to revisit. Legendre and Gauss hypothesized that the number of primes up to an integer n, $\pi \left(n\right)$, is approximately $\frac{n}{logn}$, but they did not prove it. The path to a proof opened with Euler’s clever proof of the infinitude of primes, observing that when s=1, $\sum _{n\ge 1}\frac{1}{n^s}=\prod _p\frac{1}{1-p^{-s}}$. Suppose there are finitely many primes; then the product is finite but the sum, the harmonic series, diverges. The infinitude of primes follows by contradiction. Riemann built on Euler’s work, using the identity just described to define his “zeta” function, $\zeta \left(s\right):=\sum _{n\ge 1}\frac{1}{n^s}$. This sum converges absolutely for real-valued $s>1$, and Riemann showed, by powerful new methods, how to extend it to complex-valued $s\ne 1$ (that is, to numbers $s=a+bi$, where $i=\sqrt{-1}$, and a and b are real numbers). Riemann’s work entailed, again by complex analysis, that Legendre and Gauss’ hypothesis regarding the number of primes less than an integer n is equivalent to the nonexistence of zeros of $\zeta \left(s\right)$ on the line $\text{Re}\left(s\right)=1$. Hadamard and de la Vallée Poussin proved the latter, independently, in 1896, using Riemann’s powerful complex-analytic methods. Thus Legendre and Gauss’ hypothesis is now called the prime number theorem.

These complex-analytic proofs irked some number theorists. For instance, A. E. Ingham (Reference Ingham1932) remarked that “the solution just outlined may be held to be unsatisfactory in that it introduces ideas very remote from the original problem, and it is natural to ask for a proof of the prime number theorem not depending on the theory of a complex variable” (pp. 5–6). Bram Pel (Reference Pel2023) has shown how concern about the use of imaginary numbers in solving real-valued integrals marked a dispute between Pierre-Simon Laplace and Siméon-Denis Poisson in the early nineteenth century. We will comment a little later on the “elementary” proofs of the prime number theorem found by Atle Selberg and Paul Erdős, independently in 1949 – a discovery deemed important enough that it contributed to Selberg’s winning a Fields Medal in 1950. For now, we remark that the search for a pure proof of the prime number theorem is sometimes expressed geographically. For instance, Harold Davenport (Reference Davenport2008) writes:

Geographicity emerges in the talk of what “belongs properly to” and what is “foreign to” the discipline of number theory.

We have adapted the term “geographical” from Jean-Michel Salanskis’ work on the “geographicity of mathematics,” already introduced in the previous section. Salanskis (Reference Salanskis2008) observes that mathematics has always been and continues to be divided into branches, but also that this division is “as shifting and relative as the map of Europe or of the world” (p. 175). Even as the ancient division of mathematics into arithmetic and geometry is reflected today in the distinction between discrete and continuous mathematics, mathematics today admits more fine-grained divisions. According to the Mathematics Subject Classification 2020, designed by the American Mathematical Society and zbMATH Open (formerly Zentralblatt MATH), mathematics has more than sixty branches, including number theory, algebra, analysis, and geometry, but also branches that mix other branches, for example algebraic geometry and analytic number theory.

The landscape of branches of contemporary mathematics sets limits on the usefulness of geographical purity. Firstly, results belonging to several branches that mark movement toward new fused branches do not by themselves pose obstacles to the search for purity. As we saw in the previous section, algebraic geometers can seek purity, avoiding analytic or topological means, for example. Poincaré’s uniformization theorem – that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere – belongs at once to algebraic topology, differential geometry, and complex analysis. A geographically pure proof of uniformization would draw on these branches but not others. But these branches comprise a great deal of mathematics, and one might wonder about the use of such a broad and tolerant notion of purity. An advocate for the unity of mathematics in which the boundaries between branches have been forgotten might say that they too advocate purity: mathematical proofs for mathematical theorems, nothing more. This wide conception of purity is far from the fine-grained distinctions characteristic of historical purity discriminations. Yet, as we have seen, mathematicians continue to make those fine-grained distinctions.

Another concern for the local shows further limits of geographical purity. A researcher might want to use only part of a particular discipline to prove a result. For example, consider the infinitude of primes theorem: that for all natural numbers, there exists a greater prime number. The classical Euclidean proof from Elements IX.20 is purely arithmetic and is thus geographically pure. Let us consider, though, an investigator who wants to prove the theorem without addition, seeing it as extraneous. Neither primality nor the infinitude of natural numbers seems to make use of addition. A prime number is a number divisible only by 1 and itself, and divisibility is not an additive notion. As for infinitude, this rests on an ordering, which is also not an additive notion; simply thinking of the numbers as following in discrete succession from 1 generates the ordering, but the unary successor operation is not the full binary operation of addition. This is a purity project, but not a geographical one, because the geographic conception does not allow for such fine-grained distinctions within mathematical branches. The infinitude of primes is an arithmetic theorem, and arithmetic includes addition as a basic operation. To make sense of this purity project, we will need another conception of purity.

One might riposte that the multiplicative fragment of arithmetic is its own discipline, studied by logicians as Skolem arithmetic. This purity project could thus be framed geographically as the search for a proof of the infinitude of primes in Skolem arithmetic. But one would be hard-pressed to find the study of this fragment of arithmetic outside of mathematical logic. From the point of view of school children and number theorists alike, Skolem arithmetic is a curio rather than a branch of mathematics. Framing this purity project in terms of geometrical purity seems to require taking a stance on the question of what it takes to count as a branch of mathematics, a question that may strike us as merely terminological.

3.2 Topical Purity

An example from geometry will point the way toward another conception of purity better suited to these cases that geographical purity does not handle well. The Sylvester–Gallai theorem says “that in any configuration of n points in the plane, not all on a line, there is a line which contains exactly two of the points” (Aigner and Ziegler Reference Aigner and Ziegler2010, p. 63). H. S. M. Coxeter (Reference Coxeter1948) sought a proof of this result that avoided metric notions, remarking that “it seems to me that parallelism and distance are essentially foreign to this problem, which is concerned only with incidence and order” (p. 27). Coxeter (Reference Coxeter1989) acknowledged that metric notions belong to geometry, observing that “etymologically, ‘geometry without measurement’ looks like a contradiction in terms” (p. 176). On a geographic conception of purity, Coxeter’s purity project looks badly framed, like our arithmetician looking for a nonadditive proof of the infinitude of primes. But Coxeter (Reference Coxeter1989) insisted that the concept of straight line is not a metrical notion as some would understand it, as the shortest distance between two points, but rather as an ordinal notion. He notes that what he calls Euclid’s “famous definition,” that “a line (segment) is that which lies evenly between its ends … suggests the possibility of regarding intermediacy as a primitive concept and using it to define a line segment as the set of all points between two given points” (p. 176). Thus did Coxeter justify his view that the Sylvester–Gallai theorem “is concerned only with incidence and order,” and his concomitant successful search for a purely incidence and order-theoretic proof of it (see Arana Reference Arana, Preyer and Peter2008, p. 39, and Arana Reference Arana2009, pp. 4–5).

As with the previous case, we can try to frame Coxeter’s quest geographically. Indeed, Coxeter called his fragment of geometry “ordered geometry,” following earlier work of Pasch, Veblen, and Hilbert; in the latter’s Grundlagen der Geometrie, ordered geometry corresponds to the first two groups of axioms taken together. As with Skolem arithmetic, though, ordered geometry does not attract much interest besides that of logically-inclined geometers (see Pambuccian Reference Pambuccian2009 and Pambuccian Reference Pambuccian2011).

We can avoid a decision on whether ordered geometry is genuinely a branch of mathematics by following a methodological suggestion of Hilbert and adopting another conception of purity. In his 1900 address in Paris to the International Congress of Mathematicians, Hilbert (Reference Hilbert1901) claimed that in solving a mathematical problem we ought to use only the conceptual resources used in stating that problem:

In lectures given shortly before the Paris address, Hilbert (Reference Hilbert, Hallett and Majer2004) emphasized his view of purity in explaining his search for a purely planar projective proof of the planar Desargues theorem (compare Arana and Mancosu Reference Arana and Mancosu2012):

Thus Hilbert judged a proof to be pure just in case it draws only upon what is “suggested by the content of the theorem” being proved, or on what “lies in” the presentation of the problem being solved; or, perhaps better, inasmuch as it stays as close as possible to the conceptual resources used in understanding that theorem or problem. Since he judged that the “content of Desargues’ theorem belongs completely to planar geometry,” he maintained that a three-dimensional proof of it is impure in his sense. (See also Hilbert Reference Hilbert1971, pp. 106–107, where Hilbert calls purity a form of the “ground rule” of his method in this work.)

From Hilbert’s views we may extract the conception of purity we may call “topical.” Following Hilbert’s fin de siècle stress on problems, we focus on topically pure solutions to problems rather than solely on proofs of theorems. We reprise here from Detlefsen and Arana (Reference Detlefsen and Arana2011) the representation of problems by ordered triples $P=({?}_{y/n},P,\varphi )$, where ‘ ${?}_{y/n}$’ stands for a “yes-no” interrogative attitude; ‘P’ stands for a propositional content; and ‘ $\varphi$’ stands for a formulation of P. A problem is the subject of an investigation by what we will refer to as an investigator, though an investigator need not be an individual human person, but could be a community of some sort. A formulation is the means by which an investigator represents a content to herself. For example, a community of English-speaking number theorists may ask whether every even number greater than 3 is equal to the sum of two prime numbers – that is, they may seek a resolution to the Goldbach Conjecture. In so doing, they take a yes-no interrogative attitude to the content of this problem, formulated in mathematical English.

We call the topic of a problem the family of commitments that together determine what its content is for a given investigator. These are the definitions, axioms, and inferences such that if the investigator were to stop accepting any one of them, then the content of the problem would not be what it is for her. That is to say, the problem whose solution she had been seeking would no longer be the object of her investigation. To put the point epistemically, their understanding of the problem would be thereby changed. For example, if the aforementioned number theory community were to rescind their commitment that every natural number has a successor, then they would no longer understand the natural numbers as an indefinitely extended sequence. Were this the case, the Goldbach Conjecture would no longer have the same content as it did for this community, since it would concern instead a finite sequence. By contrast, if this community stopped accepting that every rectangle has four right angles, the content of the Goldbach Conjecture would remain unchanged for them. Thus the axiom that every natural number has a successor belongs to the topic of the Goldbach Conjecture for this investigator, but the definition of a rectangle as a four-sided polygon with four right angles does not.

Our talk of rescinding commitments in mathematics may strike some as odd. After all, we are talking about commitments to axioms, definitions, and rules of inference, matters which seem to be “foundational.” Yet such rescinding is a feature of mathematical life, even if in times of stability it is infrequent. Minor changes in definitions are common, particularly as a concept is being isolated; think of the changes in the definition of group surveyed in Wussing (Reference Wussing2007). With the rise of non-Euclidean geometry, the geometry community rescinded their uniform commitment to the parallel postulate, admitting it thereafter as marking a special domain of geometry, that of Euclidean geometry. Rules of inference too can be rescinded, as the case of Hermann Weyl’s time as an intuitionist shows (compare van Dalen Reference van Dalen1995). Indeed, for a commitment to earn its name, it must be something that is held but whose holding, at least implicitly, requires agency; and such agency can be employed in differing ways.

The upshot of this analytic work is that we can now introduce another conception of purity. A solution of a problem is topically pure if it draws only on what belongs to the topic of the problem. This makes sharper Hilbert’s view that purity concerns solutions that use only “means that are suggested by the content of the theorem,” the theorem whose proof constitutes a solution to a problem under investigation.

Topical purity differs from geographical purity in that it limits what is pure to what is contained within the problem’s topic, rather than to the entire branch to which the problem (or theorem) belongs. In that way, topical purity is a “local” purity constraint. We can then make sense of purity attributions where the object of the investigation seems itself mixed. For instance, Poincaré’s uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. Uniformization belongs to each of algebraic topology, differential geometry, and complex analysis, at minimum, because Riemann surfaces are at once topological, geometrical, analytic, and algebraic. No sense can be made of a geographical purity attribution for uniformization, since it would require some single branch of mathematics containing all these subjects that does not simply flatten their differences into some indistinct unity. But the topic of uniformization entails no such difficulty, since a topic need not be identified with any existing branch. On the topical conception, then, it becomes intelligible to talk of a pure proof of uniformization – the search for which, we would claim, helps makes sense of Poincaré’s work on the problem during the almost thirty-year period that the problem occupied him (see Saint-Gervais (Reference Saint-Gervais2010) for a detailed telling of this result’s story).

Instances of topical purity abound. We saw it in Section 1 in Einstein’s example from elementary geometry, where he remarked that “we are completely satisfied only if we feel of each intermediate concept that it has to do with the proposition to be proved.” Here Einstein localized his search for purity to the proposition being proved, adding that impurity arises in his first proof when it “uses an auxiliary line which has nothing to do with the content of the proposition to be proved.”

We can also look to number theory for examples of topical purity. Let’s look again at the infinitude of primes, or for short, IP (recapitulating a longer discussion in Arana Reference Arana and Link2014). Its topic includes definitions and axioms for the natural numbers with an ordering, and primality. The wider public, as well as number theorists, understands the natural numbers to begin with a first number 1, followed by other numbers that are understood to be the successors of the numbers that came before. These suppositions can be codified by axioms for successor and an induction axiom, following Poincaré’s argument that induction is essential to understanding the natural numbers (see Poincaré Reference Poincaré1902). The ordering induced by successor is normally understood as linear and discrete, and axioms specifying these too can be added. Finally, a natural number is normally understood as prime if it is greater than 1 and only divisible by 1 and itself, so in addition to adding this definition to the topic of IP, we must add axioms defining divisibility.

At this point we could stop, taking the topic as determined for an investigator for whom divisibility is basic. We can understand such an investigator as capable of assessing whether one natural number is divisible by another, for instance by a combinatorial understanding of whether particular finite collections can be partitioned into a certain number of partitions. Or we could continue, from the perspective of an investigator who understands divisibility in terms of multiplication: A number a is divisible by another b if there is a number c such that c multiplied by b is a. In the latter case, the topic of IP would also include axioms for multiplication. These are two different topics and we need not choose between them, because each corresponds to a different investigator. There is no single answer to the question of what a problem’s or theorem’s topic is.

A first candidate for a topically pure proof of IP is the classical Euclidean proof. If $a=1$, then since $2=S\left(1\right)$ is prime, where S is the successor function, we know that there is a prime greater than $a=1$. So, suppose that $a>1$. Let $p_1,p_2,\dots ,p_n$ be all the primes less than or equal to a, and let $Q=S(p_1\cdot p_2\cdots p_n)$. Then Q has a prime divisor b that is not equal to any of the $p_i$, and so $b>a$. For our investigator who takes multiplication to be part of IP’s topic, this proof is, at first glance, topically pure. It makes no use of addition and so makes sense of the purity project we discussed in the last section as falling outside the scope of geographical purity.

For our investigator who does not include multiplication, however, the matter is more complex. The classical proof uses multiplication to compute Q, so this proof is not topically pure for our nonmultiplicative investigator. We are aware of no simple purely divisibility-theoretic proof of IP. Cegielski has worked on axiomatic theories of the arithmetic of divisibility (see Cegielski Reference Cegielski1984, Cegielski, Matijasevich, and Richard Reference Cegielski, Matijasevich and Richard1996), but these theories include IP as an axiom. A more promising approach was developed by Julia Robinson in Robinson (Reference Robinson1949). She defines multiplication in terms of just successor and divisibility; using this definition, we can translate the classical proof of IP into a topically pure proof (according to our multiplico-centric investigator). Robinson noted that this “mechanical” translation will yield a proof from arithmetic axioms that are “complicated and artificial” (pp. 102–103), notably longer than typical arithmetic axioms. She found a “simple and elegant axiom system” for successor and divisibility arithmetic, with a second-order induction axiom; this theory has the same provable consequences as the usual arithmetic theory as well as her “artificial” translated system. She was unable to prove that the “simple and elegant” system with first-order induction has the same provable consequences, however, and to our knowledge this remains open.

This mention of induction turns us to a second matter concerning the topicality of proofs of IP. Our definition of a topic also included the inference rules that determine the identity of a problem or theorem. Logicians differentiate between several different versions of arithmetic induction; each of these will determine a different topic, for an investigator who accepts that version of induction. Classical first-order induction suffices for the (Robinsonian) Euclidean proof as does intuitionistic first-order induction and second-order induction; thus, the Euclidean proof is topically pure for each of these three topics. Turning to finitary arithmetic as a topic of IP, where the induction axiom is limited to primitive recursive arithmetic (following Tait Reference Tait1981), the Euclidean proof can again be seen to be topically pure. This is because it inducts on no formula of arithmetic complexity higher than ${\Sigma }_1^0$, and $I{\Sigma }_1$, the arithmetic theory with induction limited to ${\Sigma }_1^0$, is typically thought equivalent to primitive recursive arithmetic (see Hájek and Pudlák Reference Hájek and Pudlák1998, pp. 44–47). Finally, taking feasible arithmetic as a topic of IP, matters are cloudier. On Parikh’s identification of feasible arithmetic with $I{\Delta }_0$, where induction is limited to formulas with bounded quantifiers, it is open whether there is a topically pure proof of IP. We do know that the Euclidean proof fails in this setting, since $I{\Delta }_0$ does not prove that every product of primes exists (see D’Aquino Reference D’Aquino1992, p. 13).

In this example, we have seen how we need not settle on any particular topic as being the “right” one, but can instead identify many different topics, each of which license different proofs as being topically pure or impure. This is typical. As we saw earlier, Coxeter thought that distance did not belong to the topic of the Sylvester–Gallai theorem, because straight lines should be defined in terms of betweenness. By contrast, Legendre (Reference Legendre1794) held that “the line is the shortest path from one point to another” (p. 1). This would give rise to a metrical topic for the Sylvester–Gallai theorem, on which the metrical solution by L. M. Kelly presented by Coxeter would in fact be topically pure (see Coxeter Reference Coxeter1948, p. 28; also Arana and Mancosu Reference Arana and Mancosu2012, section 4.6).

The question of what exactly is contained in a problem’s topic is difficult. We can see some of this difficulty in Alexander Paseau’s study of proofs of the compactness theorem in mathematical logic (see Paseau (Reference Paseau2010); also see section 2 of Arana Reference Arana2009). The compactness theorem for first-order logic says that if every finite subset of a set of first-order sentences has a model, then the set itself has a model. Students of logic learn Leon Henkin’s proof of this result, one step of which hinges on adding elements to the model being constructed that “witness” the truth of existential sentences in this model. Paseau asks whether Henkin’s use of witness completion is intrinsic to compactness. One could argue “that the notion of a witness is intrinsic to understanding the existential quantifier, part of the language of which the first-order compactness theorem is about” or that “witness-completeness is intrinsically connected to the concepts in the statement of compactness, because the latter says that some set of sentences has a model, and to exhibit a model one needs to be able to refer to its elements” (p. 77). It is not clear, however, on what terms such arguments are to be made. Paseau concludes that “the extent to which a proof uses notions foreign to the theorem it proves can be unclear, even indeterminate, so that questions of intrinsicness can be correspondingly difficult to answer or unanswerable.”

Indeed, one might resist the notion of topic, that there is anything definite that is “suggested by the content of the theorem.” Penelope Maddy (Reference Maddy2000) derides Solomon Feferman’s interest in “teasing out ‘what is implicit in the concepts and principles’ in a given theory” (p. 417, quoting Feferman Reference Feferman1998, p. 122), as well as what is “part of the very concept of set,” as “philosophical niceties.” Instead, she argues that we should take what she calls a “naturalistic point of view” in making sense of mathematical practices, drawing instead on “specifically mathematical considerations, considerations directly linked to the goals of the particular practice in question” (p. 415). In fact, our discussion of topics here does exactly that. Topical purity is one goal, among others, of particular mathematical practices, and we can observe how the actors of these practices make judgments about what is and is not topically pure by how they carry out their purity-seeking practices. From this we can see how the actors “tease out” the topics of their problems.

We can attempt to get a foothold on these worries by turning to a plausibly topically impure proof of IP, Hillel Furstenberg’s topological proof. The proof (presented and discussed in some detail in Detlefsen and Arana Reference Detlefsen and Arana2011, §§ 4–5) proceeds by putting a topology on the integers with arithmetic progressions as the basic open sets. After showing that these open sets are in fact also closed, one supposes that there are finitely many primes, looking for a contradiction. The union of all sets of products of primes is then a finite union of closed sets and thus closed, and contains all integers except $\pm 1$. Then $\{-1,1\}$, as the complement of a closed set, is open, a contradiction since the basic open sets are all infinite. This proof is topically impure for any investigator for whom commitments about topological notions like open set and basis set do not play a role in determining the content of IP.

In response, one might observe that Furstenberg’s proof employs only point-set topology, formulable in a weak subsystem of set theory. Reinhard Kahle and Gabriele Pulcini have highlighted Idris Mercer’s recasting of Furstenberg’s proof into a proof where “no topology on $Z$ ever comes into play” and in which its topological notions are “replaced by easy results involving the basic set-theoretic operations of union and intersection” (see Mercer Reference Mercer2009; and Kahle and Pulcini Reference Kahle, Pulcini, Arazim and Lávička2018, p. 132). Kahle and Pulcini conclude that Furstenberg’s proof is only a case of “apparent impurity,” whose “resort to such extraneous notions is nothing else but an avoidable roundabout.” They judge set theory, at least this basic part, to belong to IP’s topic, and thus hold that Mercer’s proof is topically pure.

By adding set theory to the topic of an arithmetic theorem, Kahle and Pulcini are in sync with what we may call “set-theoretic descent”: that all mathematics, deep down, is about sets. For IP, one might think that a result about the natural numbers is really a result about the set of natural numbers; or that mathematical induction is a rule of inference only for inductively defined sets. More generally, one might hold that set theory is the “basis” of mathematics, in that the content of every mathematical problem or theorem is fully determined by its set-theoretic articulation. As John Burgess (Reference Burgess2022) puts it:

On this view, all of mathematics is “about” sets, no matter how things may appear, and so naturally set theory will belong to every problem’s topic.

This kind of view, that formulations of ordinary mathematics occlude its real content, is not exclusive to set-theoretic descent. Let us return to Fermat’s Last Theorem. Briefly, a nontrivial integral solution to the equation of Fermat’s Last Theorem that is greater than 2 can be understood as giving rise to an elliptic curve, whose “modularity” – “a kind of two-dimensional generalization of the familiar sine and cosine functions from trigonometry” (see Harris Reference Harris2019) – is established by a result known as the modularity theorem (formerly known as the Shimura–Taniyama–Weil conjecture). Before Wiles, number theorists knew that the elliptic curve in question, if modular, would have certain properties that it in fact does not have. Wiles and Richard Taylor were able to prove the modularity theorem in the case needed for Fermat. Thus no nontrivial integral solution greater than 2 to the equation of Fermat’s Last Theorem is possible. Writing about the modularity theorem, the number theorist Barry Mazur (Reference Mazur1991) observes:

Mazur asserts that the modularity theorem is doubtlessly an arithmetic proposition, but one that can be reformulated in other terms, involving quite different parts of mathematics.

Mazur’s recognition that theorems admit multiple comprehensions lets us accommodate the urge behind set-theoretic descent and other views about the clash between “face value” and what we might call “deeper” contents. We need not concede that the deeper content is the “truer” content, what a problem or theorem is “really” about. Instead, we can simply conclude that different topics for whatever problem is in question have been found: a basic topic, composed of what determines the problem’s content taken at face value, and a deep topic, including the set theory that can be used to define this content. Indeed, there might be many different levels of set-theoretic analysis that can define this content, some of which use intermediate definitions like that of union or intersection, as in Mercer’s set-theoretic reformulation of Furstenberg’s proof, others using fewer and fewer auxiliary definitions in favor of just the primitive notions of set and membership. These different degrees of what we might call “analytic depth” will determine different topics. Rather than the exclusivist view that one of the deeper topics is the “right” one for every investigator, that an investigator who does not pursue a higher degree of analytic depth has misunderstood her own problem, we want to keep a place for the investigator who wants to solve a problem with means limited to its basic topic.

We might wonder if the urge to identify the “true” content of problems or theorems with deep contents are driven by inferentialism, the view that contents are determined by their roles within inferential practices. If one identifies the content of a mathematical claim in terms of its inferential connections, then one is forced to identify the content of a mathematical claim with what we have called the “deep” content. However, if a mathematical claim has only this deep content, then, as we will argue, there is no way to make sense of the importance of topical purity in mathematical practice. Given that we must make sense of the importance of topical purity in mathematical practice if we are to take that practice seriously, we must reject inferentialism about content.

We begin this argument by looking at how some philosophers have taken contents to be deep contents. Daniel Isaacson has argued that some sentences that appear to be purely arithmetic on face value in fact have hidden or tacit “higher-order,” i.e. infinitary, nonarithmetical content. Gödel sentences and the Paris–Harrington sentence are arithmetically equivalent to sentences expressing metamathematical properties of arithmetic by means of Gödel coding (see Paris and Harrington Reference Paris, Harrington and Barwise1977). Moreover, these sentences are

As a result, Isaacson maintains that “the understanding of these sentences rests crucially on understanding this coding and our grasp of the situation being coded” (p. 214). The key is the provable equivalence between the Gödel sentences and the coded metamathematical statements, since this shows that these sentences all have the same inferential role in whatever theory their equivalence can be demonstrated. (John Baldwin has added that this can be treated model-theoretically by replacing the notion of inferential role of a given statement with the collection of its models: see Baldwin Reference Baldwin2018, p. 33, chapter 12.)

Michael Hallett has echoed Isaacson’s argument for a different theorem, the planar Desargues theorem. This theorem says that if two triangles lying in the same plane are such that the lines connecting their corresponding vertices intersect at a point, then the intersections of their corresponding sides are collinear (see Figure 4). As documented at length in Arana and Mancosu (Reference Arana and Mancosu2012), a projective proof of this planar result was known in solid (that is, three-dimensional) geometry since Desargues, by proving a solid version of Desargues’ theorem and then projecting that result into the plane, but no purely planar proof had been found by the end of the nineteenth century. As we chronicled in Arana and Mancosu (Reference Arana and Mancosu2012), the search for a purely planar and projective proof of this result occupied many, including Karl Georg Christian von Staudt, Felix Klein, Giuseppe Peano, and David Hilbert. It also gave rise to a fascinating pedagogical debate in Italy about whether planar and solid geometry should be taught together – “fusionism” – or separately. Hilbert clarified the matter in the last years of the nineteenth century, bringing the methods developed for proving the independence of the parallel postulate from the other Euclidean axioms to bear on planar Desargues. This work was the context for his remarks on topical purity that we quoted earlier in this section. We will narrate this work now in order to give and respond to Hallett’s argument.

Figure 4 The planar Desargues theorem.

Say that a plane is Desarguesian if it satisfies the planar Desargues theorem. Hilbert showed that if a planar geometry satisfies the planar incidence axioms, the order axioms, and the parallel axiom, in his axiomatization of geometry, then that plane is not necessarily Desarguesian. But if all the axioms of incidence are used, not only the planar axioms but also the spatial axioms (like that there are at least four points not lying in the same plane), then that plane is necessarily Desarguesian. Hence planar Desargues is a necessary condition for a plane’s being an element of a spatial geometry; or to put it another way, for a plane’s being “embeddable” into space. Hilbert also showed that planar Desargues is a sufficient condition for a plane’s being embeddable into space. That is, he showed that a plane satisfying the planar incidence axioms, with order and parallels, and also satisfying planar Desargues, will also satisfy the spatial incidence axioms. Hilbert proved this by showing firstly, given a planar geometry satisfying the planar incidence, order, and parallel axioms, how to construct its “algebra of segments,” and that if this plane satisfies planar Desargues, multiplication in this algebra is associative, so that this algebra is an ordered division ring. Thus, if a plane is Desarguesian, then that plane is “coordinatized” by a division ring. He secondly showed how this ordered division ring can be used to construct a model of the spatial incidence, order, and parallel axioms, that is, a model of spatial geometry. (In fact, order is inessential for these results.) As a consequence, Hilbert (Reference Hilbert, Hallett and Majer2004) said, “[planar] Desargues is a necessary condition for the plane’s being regarded as a plane in space … the only thing that distinguishes the plane from space, and we could say that everything that can be proven in space can now be proven with Desargues in the plane” (p. 240). That is, Desargues’ theorem can be used as a replacement for Hilbert’s spatial axioms; it has the same provable consequences as those axioms in Hilbert’s axiomatic system. Or, to put it another way, a plane’s being Desarguesian is a necessary and sufficient condition for that plane’s being embeddable in a spatial geometry.

While spatial considerations would seem “at first sight” to be impure for proving Desargues’ theorem, Hallett (Reference Hallett and Mancosu2008) infers from Hilbert’s work that they are not, for Desargues’ theorem is in fact a theorem with spatial content, albeit hidden or tacit spatial content. As he puts it:

In contrast to Hilbert’s view that “the content of Desargues’ theorem belongs completely to planar geometry,” Hallett maintains that Hilbert’s work shows that planar Desargues has spatial content. From this, Hallett (Reference Hallett and Mancosu2008) concludes:

Hallett does not further explain his theory of tacit content, but in a footnote to this last sentence he notes that “[t]here is surely here more than an analogy with the ‘hidden higher-order content’ stressed by Isaacson in connection with the Gödel incompleteness phenomena for arithmetic,” citing Isaacson’s paper.

Isaacson holds that Gödel sentences have higher-order content, despite being expressed in purely arithmetical terms, in virtue of having the same inferential role as sentences expressed in higher-order terms. Hallett takes the same position with Desargues’ theorem, replacing “higher-order” with “spatial.” They both echo Carnap (Reference Carnap1937) in Logical Syntax of Language, who wrote, “[t]he question whether two sentences have the same logical sense is concerned only with the agreement of the two sentences in all their consequence-relations” (p. 42). Carnap contrasts this “logical” construal of meaning with “psychological” construals of meaning for which the “images” associated with a sentence are determinative. On our reading, Isaacson and Hallett are adopting Carnap’s notion of logical content as the “hidden” content of the theorems they are studying. As Hallett (Reference Hallett and Mancosu2008) puts it, “Hilbert’s axiomatic method abandons the direct concern with the kind of knowledge the individual propositions represent because they are about the primitives they are, and concentrates instead on what he calls ‘the logical relationships’ between the propositions in a theory” (p. 212). Thus Hallett sees Hilbert’s “organizational” practice as yielding a way to determine the content of theorems of axiomatic geometry: their content is determined by their inferential role within axiomatic theories. He reads Hilbert’s results as showing that planar Desargues plays the inferential role of a spatial sentence, and concludes that it has (tacit) spatial content. As a result, the classical spatial proof of planar Desargues draws only on what is “suggested by the content of the theorem.” and is thus topically pure, in Hallett’s judgment. A similar conclusion could be drawn for Gödel sentences and the Paris–Harrington statement (though Isaacson does not do so).

In response, we again invoke the distinction between basic and deep topics. These equivalencies that we have been discussing do not force this distinction to collapse. If the content of the planar Desargues theorem were spatial, even if only tacitly, it would seem to follow that an investigator with no beliefs or commitments concerning space (such as a character of Edwin Abbott’s (Reference Abbott1884) novel Flatland) could not understand Desargues’ theorem. But this is implausible, since Desargues configurations are purely planar phenomena, and thus the sort of thing that Flatlanders normally understand. This is why efforts to conflate basic and deep content fail: they take the results of lengthy enquiries for starting points. The mathematician who said that Fermat’s Last Theorem is really about elliptical modular functions did not mean that a student encountering number theory for the first time must first master complex analysis and algebraic geometry before trying to understand a problem about Diophantine equations. Rather, knowledge of this identification is the result of years of effort.

Furthermore, slippage from basic to deep topics threatens to obliterate topical purity as a genuine constraint. Indeed, Hallett says as much in his introduction to Hilbert’s 1898–1899 lectures, declaring that Hilbert’s work “reveals that Desargues’ planar Theorem has hidden spatial content, perhaps showing that the spatial proof of the Planar Theorem does not violate ‘Reinheit’ after all” (see Hilbert Reference Hilbert, Hallett and Majer2004, p. 197). If we were to replace basic topics with deeper ones systematically in this manner, seeing that topical purity is in these cases assured, then the continuing importance of purity as an epistemic objective would cease to be intelligible. That would be a significant loss on our part as philosophers of mathematics; it would be to withdraw our claim to make sense of the practice we purport to study in order to promote the philosophical agenda of inferential role semantics for mathematics.

None of that is to say that deep topics are not topics, or that theorems cannot have hidden or tacit content. Rather, it is to say that basic topics make sense of the commitments that must be engaged by an investigator to work with a problem, to begin an inquiry with it, but not necessarily the commitments that will allow her to understand the problem fully. What those commitments are can only be found a posteriori, by means of investigation into problems.

Our point of view is that each problem can have many topics, and some can generally be said to be more basic or deeper than others. There is, as far as we can see, no precise way to measure this kind of depth (though we will address one such candidate, reverse mathematics, in our discussion of elemental purity later). The question is then, which topic should be used to evaluate topical purity claims? Our methodology is that this is a question of philosophical modeling. We can consider particular purity attributions and try to identify the topic or topics that best match the actors’ practices. This is what we have tried to do here in our examples. For instance, it would be hard to understand why someone who accepted a spatial topic for planar Desargues would have struggled so much with the question of whether topical purity is possible in that case. We should then resist the claim that the best topic to study in making sense of purity attributions about planar Desargues is a solid one. We can then study what makes topical purity valuable in a more general way, treating the topic of a problem as a parameter. Then any solution determined to be topically pure for a given topic can be said to apport this particular value. This will be our approach in Section 4.4.

All this being said, there has been some work in recent years trying to show how the topics of problems can be more precisely determined. As we discussed earlier, Baldwin has taken a model-theoretic approach to topics, as the collection of models of a statement. Kahle and Pulcini (Reference Kahle, Pulcini, Arazim and Lávička2018) say that the topic of a problem should include not only the objects that the problem is about, but also the objects contained in the closure of the operations occurring in that problem (p. 134). They call this collection of objects the “ontology” of the problem, the smallest numerical domain closed under its operations, and similarly define the ontology of a proof. Robin Martinot (Reference Martinot2023) follows Kahle and Pulcini in focusing on the objects that proofs and theorems are about, calling her precisification of topical purity, “ontological purity.” Like Baldwin, and Kahle and Pulcini, she reads the ontology off the vocabulary and axioms of the statements and proofs in question.

Kahle and Pulcini call a proof “operationally pure” if the ontology of the proof is a subset of the ontology of the problem. For example, the operational closure of addition and multiplication is the set of natural numbers; adding the operations of subtraction and division, the operational closure becomes the rational numbers. An operationally pure proof of IP can draw on the rationals, then, even if the theorem ranges only over the natural numbers. In general, topics construed operationally are more inclusive, with regard to objects, than our basic topics.

These “ontological’ readings of topical purity are ways to try to make more precise what are the topics of problems. As the authors acknowledge, they are set-theoretically dependent ways of doing so, and this will distort topics in ways we have already explained. It is unsurprising that Mercer’s recasting of Furstenberg’s proof of IP is pure on Kahle and Pulcini’s account, for they have built the closures of union and intersection into IP’s topic. Their topics are accordingly deeper than our basic ones in general.

3.3 Syntactic Purity

Another way to clarify purity is to treat it syntactically. In this section we will discuss one attempt to do so, emerging from proof theory (and discussed more fully in Arana Reference Arana2009).

The setting is Gerhard Gentzen’s sequent calculus formulation of the first-order predicate calculus without equality, where each step of a proof consists of sequents $A_1,\dots ,A_m⊢B_1,\dots ,B_n$ where the $A_i$ and $B_j$ are formulas. Proofs in sequent calculus have axioms of the form $A⊢A$. The inference rules consist of logical rules, the introduction and elimination rules for logical constants, plus a few structural rules, mostly for switching formulas around. One structural rule needs singling out, the “cut” rule, which has the form

$\frac{\Gamma \Rightarrow \Delta ,AA,\Gamma \Rightarrow \Delta }{\phantom{\Gamma \Rightarrow \Delta ,A}\Gamma \Rightarrow \Delta \phantom{A,\Gamma \Rightarrow \Delta }},$

where A is called the “cut formula.” Cuts are commonly compared to lemmas in informal proofs. To prove $\Gamma ⊢\Delta$ (say, concerning circles and lines), a cut uses something about $\Gamma$, $\Delta$, and something new, represented by A (say, concerning right angles). More precisely, just as a lemma may draw on resources that are not used elsewhere in the proof, in a cut inference the cut formula occurs in the upper sequent but not in the lower sequent and hence is not a “subformula” of the conclusion: the subformulas of a given statement are all the substrings of the formulas comprising that statement that are themselves formulas.

In this framework we can measure the distance between a theorem and a proof by the number of formulas in a proof that are not subformulas of the theorem. When this number is zero, we may call this proof Gentzenian pure, and have in any case a measure of the degree of impurity of the proof, in this syntactic sense. This purity measure is appealing because it is sharply determinate when a proof is pure; indeed, it is decidable in the sense of the theory of computation.

Gentzen’s interest in what we are calling Gentzenian purity was related to his mathematical work. Gentzen showed that in the sequent calculus, every proof of a statement can be transformed into a cut-free proof of that statement. This result, known as cut-elimination, implies that every provable statement in the sequent calculus has a proof such that all of the proof’s formulas are subformulas of the statement proved. Gentzen (Reference Gentzen and Szabo1934–1935) described the importance of this subformula property as follows:

In Gentzen’s sequent calculus, every provable statement has a Gentzenian pure proof. As the proof theorist Gaisi Takeuti (Reference Takeuti1987) added, “This means that any theorem in the predicate calculus can be proved without detours, so to speak” (pp. 21–22).

Being syntactic, this measure is sensitive to formulations. Consider again the infinitude of primes (IP), that for all natural numbers a, there exists a natural number $b>a$ such that b is prime. This result can be formulated in the language of first-order Peano arithmetic:

$\forall a\exists b[b>a\wedge \forall x[\exists y(x\cdot y=b)\to (x=1\vee x=b)\left]\right].$(3.1)

The familiar Euclidean solution can be formulated straightforwardly in the language of first-order Peano arithmetic, but a sticking point, as we have seen already for its informal analogue, will be its use of the successor function. No such function symbol occurs in (3.1), and so no formula involving the successor function is a subformula of (3.1). Thus a formalization of the Euclidean proof that uses successor will be Gentzenian impure. We have already seen that the Euclidean proof is geographically pure and that the Robinsonian reformulation of this proof using successor is topically pure. Gentzenian purity thus differs from the other purity measures we have considered.

Another limitation of Gentzenian purity concerns its scope. Gentzen and Takeuti’s enthusiasm stemmed from the fact that every provable statement in their formal systems has a Gentzenian pure proof, since cut-elimination implies the subformula property. But as Jean-Yves Girard (Reference Girard1987) has observed, “The cut-elimination theorem holds for predicate calculus, but fails for first-order theories, as soon as they contain proper axioms” (p. 104). By “proper axioms” Girard means nonlogical axioms like $\Rightarrow A\to B$ and $\Rightarrow A$ , rather than the axioms of the form $A⊢A$ of Gentzen’s sequent calculus. He shows that there is no cut-free proof of $\Rightarrow B$ from $\Rightarrow A\to B$ and $\Rightarrow A$ in sequent calculus, by checking every way of proving $\Rightarrow B$ from these axioms. As a result, there is no Gentzenian pure proof of $\Rightarrow B$ from $\Rightarrow A\to B$ and $\Rightarrow A$.

In general, then, the guarantee of Gentzenian purity assured for purely logical systems does not hold for systems with mathematical axioms. Sara Negri and John von Plato (Reference Negri and von Plato2001) have attempted to overcome this limitation by developing a means of adding nonlogical axioms to sequent calculi while preserving a subformula property (in chapter 6). Their strategy is to add axioms not as initial sequents of proofs, but rather as new inference rules, in such a way that cut-elimination is more or less preserved. They give a general strategy for converting axioms into inference rules and prove that every (classical) quantifier-free theory can be converted to such an extension of sequent calculus, where the theory’s axioms have been replaced with inference rules following this general schema. Using this strategy, they obtain inference rule extensions of sequent calculus for the first-order predicate calculus with equality, partial orders, and plane affine geometry, among others. They next prove a cut-elimination theorem for such axiomatic extensions of sequent calculus and obtain a version of the subformula property for these extensions. They show that all the formulas occurring in proofs in the resulting systems are either subformulas of the conclusion, or atomic formulas.

This might be thought to salvage something like Gentzen’s observation for sequent calculi with nonlogical axioms and thus to provide for a syntactic measure of purity. However, Negri and von Plato’s methods at present only apply to quantifier-free theories, and there is reason to think these methods cannot be extended to all nonlogical theories. That is not to say that this strategy cannot be carried out further or that other syntactic measures of purity cannot be developed. But we have seen how purity is frequently construed in terms of content, as in Hilbert.

3.4 Logical Purity

At the start of this section we saw Dieudonné characterize purity as a search for economy of means, means that are maximally “close” to what is being proved. We observed that this characterization can be untangled into one sense of purity in which means of proof of a theorem are close to that theorem if they are contained in the theorem or to the branch of mathematics to which it belongs. This was the goal of our discussions of geographical, topical, and syntactic purity. Dieudonné’s characterization can also be read as saying the means of proof are close to the theorem if they are no stronger than what is being proved. We now want to show how purity in this sense functions.

How we should understand “stronger” here is a chief concern. One such sense has been evoked by Anand Pillay (Reference Pillay2021), in a discussion of pure proofs of the Sylvester–Gallai theorem:

Purity in this sense limits a proof to what is logically necessary for proving it.

In Arana (Reference Arana, Preyer and Peter2008) we tried to make this purity constraint more precise, and John Baldwin (Reference Baldwin2018) later improved upon our formulation (pp. 267–268). He calls a collection of assumptions S “fully logically minimal” for a statement P if there is a proof of P from S and there is no set of assumptions $S^{\prime }$ such that there are proofs of all the elements of $S^{\prime }$ from S and a proof of P from $S^{\prime }$, but no proof of the elements of S from $S^{\prime }$. He then defines a proof of a statement P as logically pure if it draws only on assumptions from a collection of assumptions that are fully logically minimal for P. While this shows one way how talk of logical minimality can be made precise, Baldwin observes that his notion of logical purity is a stronger version of the notion we called “strong logical purity” in Arana (Reference Arana, Preyer and Peter2008): a proof of a statement P from a set of assumptions S, with background assumptions T that we call a “base theory,” is strongly logically pure over T if there is a proof of (the elements of) S using P and (the elements of) T as assumptions. That is, over the base theory T, P and S are logically equivalent.

Baldwin then observes that “strong logical purity has a long history including Sierpinski’s equivalents of the continuum hypothesis in the 1920s, Rubin’s 101 equivalents of the axiom of choice,” as well as Victor Pambuccian’s “reverse geometry” (Pambuccian Reference Pambuccian2001, p. 393; Pambuccian Reference Pambuccian2005, p. 19) and Harvey Friedman and Stephen Simpson’s “reverse mathematics” (on which more soon; see Friedman Reference Friedman1976, Friedman Reference Friedman and James1975, and Simpson Reference Simpson2009). “These are,” Baldwin says, “searches for the weakest hypotheses in terms of proof theoretic strength” (p. 268). Similarly, Dieudonné’s view of purity as the search for proofs no stronger than what is being proved can thus be seen as an instance of logical purity.

3.5 Elemental Purity

There is another reading of strength of means of proof giving rise to a related but distinct purity measure. When Coxeter (Reference Coxeter1989) decries Kelly’s solution to the Sylvester–Gallai as impure, he adds that “it is like using a sledge hammer to crack an almond” (p. 181). Strength here is not merely logical strength. What it might be can be gleaned by reflecting on number theorists’ use of the term “elementary.”

The term “elementary” in number theory goes back at least to Edmund Landau. In his textbook on prime number theory, Landau (Reference Landau1909) says he will show how far one can get in proving the prime number theorem by “elementary methods” [elementaren Methoden], by which he means, in addition to ordinary arithmetic considerations, some real analytic means, namely finite sums and properties of the logarithm, but must avoid the integral calculus and complex analysis (p. viii). As already noted, this quest for an “elementary” proof of the prime number theorem is widely acknowledged to have ended in success with proofs by Atle Selberg and Paul Erdős (Selberg Reference Selberg1949 and Erdős Reference Erdős1949). Selberg (Reference Selberg1949) characterized his proof as “elementary” because “it uses practically no analysis, except the simplest properties of the logarithm” (p. 305). This usage has now solidified in number-theoretic practice, so that in contemporary number-theoretic textbooks like Nathanson (Reference Nathanson2000), elementary proofs are defined as those that “do not use contour integrals, Cauchy’s theorem, or other results from analytic function theory, but only basic facts about arithmetic functions and the distribution of prime numbers” provable by the means Landau and Selberg singled out (p. 280).

What is elementary about these proofs? A senior number theorist told me, in a personal communication, that an elementary proof is one that does not use analytic continuation or the Fourier inversion theorem, which are considered “mysterious” and “deep” results of complex analysis. Elementary proofs may use other less “deep” properties of analytic functions, however. This epistemic reading of elementarity dates at least to Landau (Reference Landau1909), who stressed that problems concerning the distribution of primes, including the prime number theorem, are “also understandable by the layman” [auch dem Laien verständlich] (p. v).

Here, though, there is ambiguity. Some proofs are hard to follow, on account of their intricacy or length, although using concepts that are easy to understand, while others are easy to follow although using concepts that are difficult to understand. Harold Diamond (Reference Diamond1982) shows how this distinction colors the language used to evaluate such proofs:

On this view, a proof is not elementary on account of its length or its surveyability (compare Floyd Reference Floyd2021, section 4.4). Rather, its elementarity is a function of the difficulty of the components of the proof to comprehend.

Difficulty to comprend is a matter for the cognitive science of mathematics (see Gilmore, Göbel, and Inglis Reference Gilmore, Göbel and Inglis2018), but its research focuses on rudimentary mathematics rather than the sort that occupies us here. We could instead focus on interpretability strength, a perspective that emerged from work in computability theory, proof theory, and descriptive set theory (see Dean and Walsh Reference Dean and Walsh2017). As a most basic level we can follow Hilbert’s (Reference Hilbert1931) attention to finitary reasoning, “the fundamental way of thinking that I hold as necessary for mathematics and in general for all scientific thought, understanding and communication, and without which mental activity is not at all possible” (p. 486). There is a connection here with “elementary” signifying what is understandable by the layman as in Landau. Bill Tait (Reference Tait1981) called this “a minimal kind of reasoning presupposed by all non-trivial mathematical reasoning about numbers” (p. 525) and identified the formal theory PRA (for primitive recursive arithmetic) as formalizing this body of reasoning. To this theory, we can add more axioms and arrive at proof-theoretically stronger systems, as is done in reverse mathematics. Stephen Simpson has observed that the “Big Five” theories studied by reverse mathematics correspond to well-established epistemic foundational programs in the philosophy of mathematics: for instance, that $RC{A}_0$ corresponds to the constructivism of Errett Bishop Bishop (Reference Bishop1967), and $AC{A}_0$ to the predicativism of Weyl (Reference Weyl1918) and Feferman (Reference Feferman and Shapiro2005) (see Eastaugh Reference Eastaugh2019 for a discussion of these correspondences). Thus we have a series of formal theories graded by their interpretability strength, and a story about how these degrees of strength can be assessed epistemically.

Reverse mathematics seeks to answer what Simpson (Reference Simpson2009) calls the “Main Question”: “which set existence axioms are needed to prove the theorems of ordinary, non-set-theoretic mathematics?” (p. 2). It works by identifying proving logical equivalences between theorems and antecedently interesting set-theoretic theories like the aforementioned Big Five, over a logically weaker base theory. Put thusly, reverse mathematics is a search for strongly logically pure proofs. But it is more than that. The epistemic calibration by interpretability strength mentioned earlier means that reverse mathematics identifies the epistemically weakest assumptions capable of proving given results. It does this by identifying what Denis Hirschfeldt calls the “combinatorial core,” and what Beno $\hat{i}$ t Monin and Ludovic Patey call the “computational content,” of mathematical theorems (Hirschfeldt Reference Hirschfeldt2014, p. 2; Monin and Patey Reference Monin and Patey2022, p. 475).

So far we have looked at elementarity measured epistemically or cognitively, that can be measured comprehensionally as with Landau and the number theorists, or computationally as in reverse mathematics. Another way to measure elementarity, which we can call “ontic,” would order statements in terms of some objective order of priority. Such a measure is characteristic of classical rationalist views held for instance by Aristotle, Leibniz, Bolzano, and Frege, as we saw in Section 2, on which it is held that there is an objective hierarchy of truths whose recapitulation and exhibition by “scientific demonstrations” or “groundings” provide for understanding of the truth in question (see Detlefsen Reference Detlefsen1988). There is a wide-ranging literature on grounding today (see Correia and Schnieder Reference Correia and Schnieder2012), and the notion has been applied by contemporary philosophers of mathematics, sometimes in the context of purity (see Lange Reference Lange2019 and Poggiolesi and Genco Reference Poggiolesi and Genco2023).

We can now bring all this together to define another sense of purity. We say that a proof of a theorem that only draws on what is more elementary than the theorem is elementally pure. Having distinguished two measures of elementarity, we can distinguish two notions of elemental purity. In the first sense, an elementally pure proof of a theorem would involve only assumptions that are cognitively prior to the theorem. We can understand or grasp the proof before understanding or grasping the theorem. Dieudonné’s characterization of purity as a search for economy of means is of this type. So is Hilbert’s consistency program, as Kreisel (Reference Kreisel1969) suggested: “Hilbert’s idea was very much the same as the widely current idea that an arithmetic theorem must have an arithmetic proof, or even that a ‘simple’ statement, if it can be proved at all, must have a simple proof. … the autonomy of elementary mathematics” (p. 60).

In the second sense, an elementally pure proof would involve only assumptions that are metaphysically prior to the theorem. That is, in the order of being (if not our understanding), the proof proceeds from the more basic to the less basic. This second kind of elemental purity can be seen in Bolzano and Frege, for example, in Section 2.

4.1 Purity and Rigor

Kreisel continues:

In addition to raising the parallel question of the value of impurity, which we will address alongside the value of purity, Kreisel describes purity as an “ideal of rigour.” A rigorous proof can be characterized as a proof free of errors or one with no gaps (compare Stanley Tanswell Reference Stanley Tanswell2024, p. 5; Hamami Reference Hamami2014, p. 9). On this view, an impure “proof ” would not be a proof at all. In Section 2 we saw Aristotle say that “we cannot, for instance, prove geometrical truths by arithmetic.” We also saw Scaliger’s Aristotelian critique of Archimedes’ solution to the quadrature problem by exhaustion as “worthless,” as summarized by van Roomen. Such Aristotelian views hinge on the necessity of purity for demonstrations capable of providing the best kind of knowledge, epistēmē. Impure proofs can play a different, inferior epistemic role, giving what Aristotle called in Posterior Analytics A13 “knowledge of the fact” rather than “knowledge of the reason why,” justifying belief in the conclusion without revealing the reason why the conclusion necessarily holds (see Allen (Reference Allen2001) for a discussion of what Aristotle calls “inferences from signs”).

Elemental purity in particular has been lauded for its contribution to rigor. A chief reason we seek proofs is to obtain conviction in their conclusions. A proof whose concepts are more mysterious than those of the conclusion, or whose steps are less evident than the conclusion, will be deficient in providing such conviction. This is the context in which we should consider Ingham’s remarks concerning a “Tauberian” theorem of G. H. Hardy and J. E. Littlewood proved in Hardy and Littlewood (Reference Hardy and Littlewood1919): “This theorem and its relation to the theory of numbers were first investigated by Hardy and Littlewood, but their proof does not provide a proof of the prime number theorem, since it depends explicitly on a theorem a little deeper than the prime number theorem itself” (compare Ingham Reference Ingham1932, p. 39). Does not provide a proof of the prime number theorem: Ingham suggests that an elementally impure proof fails in one key task of a proof, namely in providing conviction of its conclusion, on account of what Ingham judges to be a kind of circularity. Evidently there is an elementary “proof ” of the prime number theorem from Hardy and Littlewood’s theorem, but since the proof of Hardy and Littlewood’s theorem draws on “deeper” (on our reading, harder to comprehend and hence less evident) means than the prime number theorem, this elementary “proof ” fails to be a proof at all. Whatever doubt one has in the prime number theorem, this alleged proof does not provide sufficient evidence for assuaging that doubt. In this regard, an elementally impure proof is circular. It instead enmeshes its reader in further mystery. Elemental purity thus has the epistemic virtue of avoiding this kind of circularity and thus contributing toward conviction in the statement proved.

4.2 Purity, Understanding and Explanation

We now turn away from rigor and toward proofs giving the reason why. In Section 3 we saw Dieudonné promote purity on the basis that “one must always try to understand what one is doing as well as one can.” Understanding and explanation are commonly linked by philosophers of mathematics, with explanatory proofs yielding understanding (see Tappenden Reference Tappenden, Mancosu, Jørgensen and Pedersen2005). As we discussed in Section 3.5, philosophers have recently linked mathematical explanations and proofs that “ground” their conclusions. Such work naturally turns to Bolzano, whose interest in purity we discussed in Section 3 in the context of his project of organizing domains of mathematical knowledge (see Rusnock Reference Rusnock2022).

Several recent works, however, have shown how purity and explanation come apart (see McCarthy Reference McCarthy2021, Pincock Reference Pincock2023, pp. 37–38). We can focus, for instance, on Mark Steiner’s account of explanation, and give examples of proofs that are pure but not explanatory, and explanatory but impure. Steiner argues that inductive proofs that the sum $S\left(n\right)$ of the first n positive integers equals $\frac{n(n+1)}{2}$ are not explanatory, but a similar kind of analysis to the one in Section 3.2 for the infinitude of primes gives that bounded inductive proofs are topically pure (see Arana Reference Arana, Posy and Ben-Menahem2023, section 4). Marc Lange (Reference Lange2019) gives other such examples with proofs by “brute force” that he argues can be pure but generally not explanatory (section 2). Next, Steiner (Reference Steiner1978) argues that complex-analytic proofs of theorems about real-valued functions can be explanatory, but these will generally not be (topically) pure (pp. 18–19).

Indeed, Patrick Ryan (Reference Ryan2021) has argued that impurity, rather than purity, makes for better explanations in mathematics. He argues that Hillel Furstenberg’s topically impure ergodic proof of Szemerédi’s theorem in additive combinatorics is explanatory by revealing “the dichotomy between structure and randomness” at the heart of the result (p. 34). Endre Szemerédi’s original, purely combinatorial proof, by contrast, seems to be topically impure but, according to Ryan, not explanatory (also see Arana Reference Arana2015, pp. 168–170, for a discussion of purity in the context of Szemerédi’s theorem). Ryan appeals to the “simplicity and unification” afforded by Furstenberg’s proof and thus aligns himself with the unification model of mathematical explanation exemplified by Philip Kitcher’s work (see Kitcher Reference Kitcher1981 and Kitcher Reference Kitcher, Kitcher and Salmon1989). Similarly, Ellen Lehet (Reference Lehet2021) has argued that impurity can promote explanation, on account of the “generality and unification that accompany these impure methods” (p. 79). Ryan and Lehet thus echo Suzanne Bachelard (Reference Bachelard1967), who acknowledged purity as a regulative principle in contemporary mathematics, but maintained that a too strict preference for purity over impurity would sterilize the development of a unified mathematics (p. 32).

Lehet’s praise for the generality of impure proofs in contemporary mathematics, particularly by means of category-theoretic proofs, should be put into perspective. A topically pure proof will use only those resources required for understanding the content of the theorem, and will thus draw on a minimum of epistemic resources. A logically pure proof uses the logically weakest hypotheses possible for its conclusion. These proofs will accordingly have wide range, capable of application to a variety of settings with stronger hypotheses. The case that impurity better provides for generality than purity is thus equivocal.

That pure proofs draw on minimal resources ensures other benefits. Topically pure solutions to problems require minimal epistemic resources for their comprehension and transmission. Since such solutions draw only on those definitions, axioms, and inference rules that determine the content of the problem, an agent who understands a problem has understood everything necessary to understand and communicate a topically pure solution of it. Elementally pure proofs minimize comprehensional resources in a different way, in that they require nothing more difficult to comprehend than what is needed to comprehend the conclusion. Recall Kreisel’s characterization of Hilbert’s program, quoted at the start of Section 2, that “finitist theorems should have finitist proofs.” Since finitary proofs were taken by the Hilbert school to be more secure than infinitary proofs, the minimization of comprehensional resources in finitism illustrates the value of elemental purity (as would a related point about predicativist proofs of predicative theorems).

The geometer Corrado Segre (Reference Segre1891) wrote that in addition to the scientific value of this minimality afforded by purity, it has “didactic” value (pp. 396–397). Augustus De Morgan (Reference De Morgan1849) observed similarly that the mixing of geometry and algebra poses difficulties for students. As he wrote:

To these pedagogical accounts of the value of purity, we can add Dieudonné’s remark, in the now oft-mentioned quote at the start of Section 3, that purity is “good discipline for the mind”. In Detlefsen and Arana (Reference Detlefsen and Arana2011, pp. 5–6), we called this an “intervenient” value of purity, in that it contributes to the training of the capacity to reason rather than to the distinctively epistemic value of purity.

4.3 Geographical Purity and Local Knowledge

Until now we have said nothing specifically about the value of geographical purity. Detlefsen (Reference Detlefsen1990a) has written about Poincaré’s concern for “subject-specific insight… the easy, loping stride of one familiar with the twists and turns of a given local terrain” (pp. 502–503). Taking a branch of mathematics as a “local terrain,” geographical purity is thus a kind of localism about mathematical knowledge, whose value we will now investigate.

Penelope Maddy (Reference Maddy2001) described this localism as follows:

She stresses the locality of knowledge within particular branches, noting that branches come with their own ways of thinking and working. Rather than thinking of branches as divided by subject matters – algebra by groups, rings, modules, and fields; geometry by manifolds; set theory by sets – she says that the essence of each branch consists in the “special sensitivity” practitioners of that branch will come to acquire.

Gaston Bachelard (Reference Bachelard1966) drew attention to such localisms with his notion of “regional rationalisms,” organizations of particular scientific practices. Bachelard calls such regional developments “cultures” that are socially determined by the fact of internal consensus about the rational values within each culture’s practices, about what that scientific culture’s goals are and how best they can be accomplished (p. 133). He focuses on the regionality of cultures of physics, leaving such a discussion for mathematics for a time when more mathematicians are occupied with its foundations (p. 119) – a project that his daughter, Suzanne Bachelard (Reference Bachelard1967), would begin in her study of purity and impurity in representations of imaginary numbers by algebraic, geometrical, and topological considerations.

Karine Chemla and Evelyn Fox Keller have brought renewed attention to the notion of cultures in scientific practices, including mathematics, in their book Chemla and Keller (Reference Chemla and Keller2017). Adopting the term “epistemic cultures,” from Karin Knorr Cetina’s work (see Knorr Cetina Reference Knorr Cetina1999) and from Keller’s own earlier work, they identify antecedent work in that of Gilles-Gaston Granger (Reference Granger1968), Alistair Crombie (Reference Crombie1994) and Ian Hacking (Reference Hacking1992) on styles of scientific thought and practice (compare Mancosu (Reference Mancosu and Zalta2021) and Rabouin (Reference Rabouin, Chemla and Keller2017) for recent work on mathematical styles). Chemla and Keller, even in the title of their book, Cultures Without Culturalism, warn against cultural essentialisms, such as identifying epistemic cultures with national or ethnic identities, as in Orientalist explanations of Japanese science (see Ito Reference Ito, Chemla and Keller2017), or the Nazi Ludwig Bieberbach’s identification of different mathematical “styles” as racially determined (see Segal Reference Segal2003, pp. 361–368; we noted the practical consequences of this latter identification in Section 1). Instead, an epistemic culture is, like for Bachelard, a particular way of knowing and working.

We have spoken of “ways of knowing.” This matter deserves a full treatment of its own, but we can give at least a hint of what we mean. Mathematicians frequently observe that different parts of mathematics support different ways of thinking. Timothy Gowers (Reference Gowers2008a) has observed, for instance, that “there is a definite difference between algebraic and geometric methods of thinking – one more symbolic and one more pictorial – and this can have a profound influence on the subjects that mathematicians choose to pursue” (p. 2). While this sort of thinking was widespread into the nineteenth century, and reinforced by Kant’s dichotomy of the forms of intuition, it was then tainted by its associations with the scientific racism of the Nazis (as discussed in the introduction). More recently this type of view has been cast in cognitive terms (see Lakoff and Núñez Reference Lakoff and Núñez2000). The significance of this for the unity of mathematics has been noted by A. R. D. Mathias (Reference Mathias1992), speculating “that the physiological separation by the brain of the processing of spatial from the processing temporal thought supports the thesis that a complete unification of mathematics is not possible” (p. 11).

Mathematicians commonly formulate projects in ways coherent with there being different mathematical cultures: algebraic and geometric cultures, for instance. Consider firstly Colin Rourke and Dennis Sullivan’s article, Rourke and Sullivan (Reference Rourke and Sullivan1971). They describe their purpose as finding a “geometrical definition” of the Kervaire obstruction, a notion in topology, and state their workings at a couple of points as finding a “more geometrical proof ” of particular statements. Secondly, consider Stéphane Sabourau’s article, Sabourau (Reference Sabourau2010), in differential geometry. Responding to earlier work by Florent Balacheff, Sabourau aims “to present an alternative (more geometrical) proof of the local extremality of the Calabi-Croke sphere, which does not rely on the uniformization theorem” of Poincaré (p. 549). We could similarly give examples of mathematicians seeking more algebraic proofs, more arithmetical proofs, more topological proofs, more combinatorial proofs, and so on. The point is that the disciplines of mathematics pertinent to geographical purity are understood by many practitioners as cultures in the sense of Chemla and Keller. Geographical purity is a localism, a preference for what is local to a particular mathematical culture in proving results deemed to belong to that culture. A mathematician might hold that a geometric approach to a result affords different knowledge than an algebraic approach, and so favor a geometric proof of a geometric theorem for the particular kind of knowledge that proof promises.

4.4 Topical Purity and the Stability of Knowledge

We turn now to an epistemic value of topical purity, one we have called in Detlefsen and Arana (Reference Detlefsen and Arana2011) the stability account. In brief, we have argued that pure proofs provide for enduring, stable knowledge of their conclusions. In the following we will synthesize and amplify this view.

We have seen that the partisans of deep content often change the theorems being proved. One seeks to prove a theorem about parabolas, for instance, but after having grasped its connections with algebra, one arrives instead at a theorem about equations – a theorem with a good solution, perhaps, susceptible to being generalized to other cases. All the same, we may have reasons to prefer the original theorem and desire a solution to it. This is to take a vectorial conception of problem solving, wherein our investigations are directed toward a particular problem. On this conception, a solution is a solution to a particular problem to the extent that it is directed at that problem and not some other. A solution may succeed as a solution to some different problem while failing as a solution to the problem toward which it was directed.

Problem solving (of which theorem proving is a chief means) aims to relieve the “specific” ignorance represented by that problem. On this view, a problem is an irritation, to be removed by solving the problem. By calling problem-solving a “relief,” we stress the connection between the search for knowledge and the satisfaction of desire that Aristotle identified in writing that all men by nature desire to know. Just as the satisfaction of a desire is a relief, so is the resolution of a problem.

Every solution to a problem, pure or impure, provides such relief. But how stable is this relief? To describe stability as we understand it, we draw on the following observation of Plato (Reference Grube1997b), from the Meno:

In Plato’s metaphor, the difference between mere true opinion and knowledge is that true opinion is transient while knowledge perdures. I cannot know a proposition while also believing that at some time in the future that that proposition will be false or no longer justified. Similarly, to believe that a solution solves a problem, I must also believe that it will continue to be a solution in the future. We call this epistemic state stability and hold that the knowledge engendered by a solution is best when it is stable. Solutions should thus be stable with respect to changes in our epistemic position; they should perdure as solutions as long as the problem being solved perdures. We will make the case that topically pure solutions engender knowledge that is more stable than the knowledge engendered by topically impure solutions in general, and thus better realize this ideal of knowledge.

The key to our case is the observation that there are two different ways to solve a problem. Firstly, we may provide an answer to that problem. Secondly, we may rationally dissolve that problem. This second type of solution arises from inspection of the etymology of the verb “to solve,” from the Latin solvere: to loosen, release, unbind. Both types of solutions eliminate the specific ignorance represented by a problem, either by providing the desired knowledge, or by eliminating that ignorance as a source of desire for knowledge. When the ignorance represented by a problem has been eliminated for an agent, by some rational means, then that problem ceases to be a problem for her and thus has been solved.

We have spoken of rational means of dissolving problems. More precisely, we say that a problem has been dissolved when, on account of a change in the investigator’s beliefs or attitudes concerning the problem’s content, that problem no longer represents an ignorance for that investigator. For instance, if I rescind my commitment that every natural number has a successor, then I will no longer understand the natural numbers as a discrete sequence, and thus I will have dissolved (for me) basic arithmetic problems like the infinitude of primes. Then those problems are no longer problems for me, in that they no longer represent specific ignorances that I seek to relieve. When I dissolve a problem in this fashion, then, I have solved that problem in our second sense. By contrast, if I rescind my belief that every holomorphic function is analytic, my understanding of the infinitude of primes does not change, and so the problem is not dissolved.

We may now relink this discussion with the notion of a problem’s topic. Any rescission of an element of a problem’s topic dissolves that problem for that investigator, and hence that problem is no longer an ignorance for that investigator. This is the case even though a different problem may have been opened by this rescission, maybe even a better problem by some measures; but the original problem has been closed by being dissolved.

To explain how topically pure solutions more stably solve the problems toward which they are directed, we need to discuss the rescission of solutions. An investigator rescinds a solution to a problem when that investigator no longer accepts one of the premises or inferences of that solution, and what remains of that solution no longer determines the propositional content of that problem. A recent example of rescission comes from symplectic geometry, where Dusa McDuff and Katrin Wehrheim brought attention to errors in work by Kenji Fukaya (see Hartnett Reference Hartnett2017). In order to solve the Arnold conjecture, a key problem in symplectic geometry about a kind of fixed point of symplectic manifolds, Fukaya and his collaborator Kaoru Ono introduced the notion of Kuranishi structures in Fukaya and Ono (Reference Fukaya and Ono1999). Their work on this notion, however, was not sufficiently intelligible to the community working in symplectic geometry to be widely accepted. In 2012 McDuff and Wehrheim made public errors they had found in Fukaya and Ono’s work and began publishing work that fixed the errors they had found (see McDuff and Wehrheim Reference McDuff and Wehrheim2015). These errors were not unique to Fukaya; indeed, Wehrheim pointed out that some of McDuff’s own work contained errors as well, rooted in unclarity in the foundations of symplectic geometry as developed until then. While Fukaya did not think that McDuff and Wehrheim’s work contributed any new ideas, calling them a “mere technicality,” the community acknowledged that their work was necessary to make precise what Fukaya had not. As a result, Fukaya’s methods, suitably corrected, have been accepted by the symplectic geometry community, though new methods for the problems arising from the Arnold conjecture have also been developed.

In this episode we can see several moments of acceptance and rescission by different communities. At first, imprecise ideas about symplectic manifolds seem to have been widely accepted by symplectic geometers at large. Fukaya and Ono’s work advanced new ideas, but their imprecision blocked their wider acceptance. McDuff and Wehrheim’s “whistleblowing” (adopting their term) made a part of the community rescind their acceptance of many of the methods they had earlier taken up, though Fukaya did not do so. After McDuff and Wehrheim’s work, the wider community accepted the modified Fukaya methods. We thus see how the messiness of mathematical practice can involve rescission, change, and acceptance of modified methods, though we make no claims about the purity of any of this work. (Thanks to Moon Duchin for bringing this example to my attention.)

With the notion of solution rescission clarified, we may return to the question of how topically pure solutions more stably solve the problems toward which they are directed than topical impure solutions. This is because rescission of a topically pure solution by an investigator will generally dissolve that problem for that investigator. Since each premise and inference of a topically pure solution belongs to the problem’s topic, their rescission will dissolve that problem. That is not necessarily so for topically impure proofs.

We have identified stability as an ideal of knowledge: knowledge that perdures through changes in our epistemic situation – an invariance condition on knowledge. As we have seen, the knowledge of a theorem engendered by a topically pure proof of that theorem perdures through changes in our epistemic attitude toward that theorem, but that is not necessarily so for topically impure proofs. Thus, other things being equal, topically pure proofs are better at providing stable knowledge of their conclusions than are topically impure proofs.

4.5 The Simplicity of Purity and Impurity

Impurity, like purity, is valued by many mathematicians, as we have noted throughout this work. Douglas Marshall has shown how thinking about impurity, which he calls “internal applications of mathematics,” contributes to work on traditional questions in metaphysics and philosophy of science (cf. Marshall Reference Marshall2023). We have been taking the view that purity and impurity can coexist as values, since we can give multiple proofs of theorems. But whatever advantages pure proof may have over impure proof would be countered by disadvantages if impure proof were systematically simpler than pure proof. We will now address this possibility.

Such claims are common in the literature. Carlo Cellucci (Reference Cellucci1985) has claimed that “the use of ‘impure’ methods leads to a marked improvement in efficiency” (p. 173). In Section 2 we saw Lagrange’s call for the unity of algebra and geometry on efficiency grounds. Jacques Hadamard (Reference Hadamard1945) famously observed that “the shortest and best way between two truths of the real domain often passes through the imaginary one” (p. 123).

We can distinguish two different questions concerning the simplicity of pure and impure proof. Firstly, we can investigate whether impure proofs are generally simpler to verify than pure proofs of the same statement; that is, to check the validity of a given proof candidate (see Detlefsen Reference Detlefsen1990b, p. 376f 24; also Detlefsen Reference Detlefsen and Shanker1996, p. 87). Secondly, we can investigate whether impure proofs are generally simpler to discover than pure proofs of the same statement.

Both questions can be approached by proof theory, though there is a considerable literature pointing out the limitations of proof theory for philosophical reflection on proofs in informal mathematical practice. In Arana and Stafford (Reference Arana and Stafford2023), section 5, we both discuss this literature and defend the significance of proof theory for such questions. Briefly, at whatever level of granularity we consider a proof, it will have some logical structure, and proof theory can be applied to this structure. This is so even if there are other epistemic features of proof to which proof theory cannot be made sensitive.

In Arana (Reference Arana, Kossak and Ording2017) we addressed the first question, whether impure proofs are generally simpler to verify than pure proofs of the same statement. In this work we considered conservative extensions of the formal theory Primitive Recursive Arithmetic (PRA) mentioned in Section 3.5, in which proofs of theorems of PRA can be given that are, arguably, impure. Specifically, we looked at arithmetic extensions adding induction schemas for more inclusive classes of arithmetical formulas, and set-theoretic extensions adding comprehension principles for certain kinds of sets. The former make for cases of elemental impurity, while the latter make for cases of topical impurity. To compare the simplicity of proofs of theorems of PRA with proofs of these same theorems in extensions of PRA, we consider the “speed-up” of proofs in extensions of PRA. We say that theory $T_1$ is at most a polynomial speed-up of $T_2\subset T_1$ when for every $\phi$ provable in $T_2$, the length of the shortest proof (measured in terms of total number of symbol occurrences) of $\phi$ in $T_2$ is less than some fixed polynomial multiple of the length of the shortest proof of $\phi$ in $T_1$. Next, $T_1$ is said to have a roughly superexponential speedup over $T_2\subset T_1$ when for every $\phi$ provable in $T_2$, the length of the shortest proof in $T_2$ of $\phi$ is a superexponential multiple of the length of the shortest proof of $\phi$ in $T_1$. Superexponential speedups are considered to be significant, while polynomial speedups are not (following the tradition in complexity theory; see Dean Reference Dean and Zalta2016, § 2.2). Using this notion of speedups, we showed that there is no evidence of a general pattern of improvement in simplicity in moving from pure to impure proof, thus answering the first question negatively.

In Arana and Stafford (Reference Arana and Stafford2023) we addressed the second question, whether impure proofs are generally simpler to discover than pure proofs of the same statement. We discussed Alessandra Carbone’s topological measure of proof simplicity, building on work of Richard Statman, for proofs in propositional sequent calculus with rules permitting inferences with computations of binary functions (see Carbone Reference Carbone2009; Statman Reference Statman1974). Associating each proof with a graph, this measure evaluates the simplicity of discovering a proof by the topological genus of its associated graph. Carbone’s idea is that topological genus is a measure of “discovermental complexity” because it is a measure of what Statman calls the “global structural complexity” of a proof, since genus is a property of a graph as a whole; and a high global structural complexity means that the proof has many highly interconnected ideas of which the discoverer or reader must keep track. Intuitively, then, the higher the genus of a proof, the more difficult it is to discover that proof, on account of its convolutedness.

Recalling the identification of pure proofs with cut-free proofs that we saw in Section 3.3, Carbone proves that there are pure (that is, cut-free) proofs of arbitrarily high genus. This can be taken to say that pure proofs may have arbitrarily high “discovermental complexity.” Since no such result is known for impure proofs, one can conclude that impure proofs may not have arbitrarily high discovermental complexity, and thus, that impure proofs are generally simpler to discover than pure proofs.

We nevertheless have good reasons to reject this conclusion. Firstly, it is possible that Carbone’s main result is also obtainable for proofs with cut. In that case, the epistemic asymmetry between pure and impure proofs would be broken. Secondly, her present result applies only to propositional sequent calculus, but an adequate formalization of ordinary mathematics would need more expressive and inferential resources, including quantifiers and the ability to begin proofs with nonlogical axioms. As we saw in Section 3.3, extending the result to these wider contexts is likely to be difficult. Thirdly, the proofs with large genus that Carbone builds have graphs with many edges, and this is what makes them hard to comprehend. But this complexity is not a special feature of proofs with high genus. It will also be true of many fully formalized proofs (in PA, say) that are not typically judged to be particularly complex. Informal proofs that “correspond” to these formal proofs suppress a lot of the intermediate logical steps that are generating complexity here, suggesting that this complexity may be in part an artifact of formalization. Fourthly, Carbone’s measure assigns a special kind of graph to proofs, what Sam Buss (Reference Buss1991) calls a “logical flow graph”. But, we can argue, the genus of the logical flow graph of a proof does not capture the global structural complexity of that proof, because it ignores the structure introduced by the logical connectives, by definition. Instead, the logical flow graph tracks where a formula is within the sequents of a proof, its “flow” through the proof. But the application of logical rules have structural effects on proofs. For instance, right $\wedge$ binds together two proofs, one for each conjunct, but this structure is entirely missing from logical flow graphs. Thus Carbone’s genus measure misses structural features of proofs that should be tracked by a discovermental simplicity measure.

For both of the two questions about the relative simplicity of pure and impure proofs, then, the evidence from proof theory is not definitive. That should not mean that we dismiss the evocations of the efficiency of impure proof that we have seen from Lagrange, Hadamard, and Cellucci. It is to say that attempts to make their claims precise, by the best formal means we have today, are not successful.