In Sects. 3 and 4, we argued that, according to the CTE, mathematical explanations involve counterpossibles. Our argument might give rise to an interesting objection: the CTE diverges from actual mathematical theorising, because mathematicians do not use—or do not need to use—anything quite so exotic as counterfactuals and counterpossibles. Perhaps all they are doing is looking at deductive inferences from various assumptions. Some of the assumptions turn out to be true and some turn out to be false. According to this objection, there is no need for putting counterpossibles or, indeed, any conditionals in the mouths of mathematicians.Footnote 19
This objection includes two related worries that should be distinguished here. The first is a claim about mathematical practice: mathematicians do not, in fact, use such counterpossible language. The second is the modal claim that there is no need for mathematicians to use such counterfactuals. As we will show in this section, mathematicians do seem happy to use conditionals, including counterfactuals and counterpossibles. This, of course, does not show that they are right in using such language. There may be some reconstruction of their practice along the lines of the suggested objection, appealing only to assumptions and deductive consequences without appeal to conditionals. Be that as it may, the fact that mathematicians do use counterfactuals in such circumstances (as we will show) gives us prima facie reason to take such counterfactuals seriously and we are reluctant to engage in too much reconstruction or reinterpretation of mathematical practice—at least not without good reason.
To avoid misunderstandings, it is not the purpose of this section to provide more examples of explanation in mathematics. What we are going to demonstrate instead is something more general, namely, that mathematicians do use counterfactual conditionals, and counterpossibles in particular, in their writings, and that their choices regarding the grammatical form of their statements do not seem to be accidental. In other words, we want to show that counterfactual and counterpossible conditionals are not just idiosyncrasies of philosophical reconstructions (such as the CTE).
One way to empirically determine if a certain linguistic community speaks in a particular way is to search the corpus of their language, that is, a collection of texts (typically records of both written and spoken word) produced by that community. Since the goal of finding out if mathematicians use counterfactual language is relatively modest, our pilot corpus study did not need to employ any sophisticated, computational methods developed in the field of corpus linguistics. Instead, we searched through a sample of texts written by mathematicians to find examples of the kind of language use we are interested in.
The first step in our pilot study was to assemble a corpus consisting of 20 mathematical texts published between 1984 and 2018. We collected three different types of texts:
-
1.
A selection of research papers exploring, among other things, the consequences of as yet unproven hypotheses, such as Riemann hypothesis, or of assuming the existence of impossible mathematical objects, such as the field with one element.
-
2.
Essays collected in an anthology of “survey papers presenting the status of some essential open problems in pure and applied mathematics, including old and new results as well as methods and techniques used toward their solution.” (Nash and Rassias 2016, p. v.)
-
3.
Undergraduate and graduate textbooks and lecture notes introducing students to various fields of mathematics, such as: Analysis, Calculus, Geometry, or Number Theory.
The selection of the texts was dictated partly by their availability in digital form via open access resources such as arXiv.org, researchers’ personal webpages, or resources accessible through the university libraries’ subscriptions, such as, SpringerLink. In these texts, we searched for the occurrences of subjunctive conditionals. Since the purpose of the study is to argue against the claim that counterfactual language plays no role in mathematical practice, a single instance, in principle, makes the point. For this reason, we will only present a number of examples showing that mathematicians do not only use conditional and counterfactual language, but also, that they use it purposefully, leaving any quantitative analyses for future studies.
To facilitate the search, we focused on the paradigmatic surface structure of a subjunctive conditional, that is, we looked for the sentences consisting of an if-clause and a main clause involving the auxiliary ‘would.’ We ended up with a list of 42 conditionals of the form ‘if it had been the case / if it were the case that \(\varphi\), then it would be / it would have been the case that \(\psi\).’ More specifically, we found instances of counterfactual language in 11 out of 17 essays collected in Nash and Rassias (2016), summing up to total 21 conditionals, in 4 out of 12 research papers, total 7 conditionals, and in 6 out of 7 textbooks, total 14 conditionals. It is then an empirical fact that mathematicians use counterfactual conditionals in their writing. In fact, mathematicians use conditionals both in indicative and subjunctive moods, depending on what they are writing about.
For instance, in a paper on the consequences of the Generalised Riemann Hypothesis by Deshouillers et al. (1997), we can find the following statements:
-
(1)
“If the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers.” (p. 99)
-
(2)
“If the primes up to \(10^8\) were uniformly distributed, which they are not, a proportion of about \(0.885^2\) of the even numbers would not be covered by [the set of even numbers] \(\mathcal {F}_2\).” (p. 102)
The paper is devoted to “The 3-Primes Problem,” that is, the question whether every odd number greater than 5 can be written as a sum of three prime numbers. The sentence (1) is the main theorem of the paper, while (2) occurs in the context of a presentation of a computer search method for verification of the Goldbach conjecture on a given interval [a, b] (which in the authors’ own experiments was an interval of the length of \(10^8\)), involved in the proof of (1). Since the Riemann Hypothesis has not been proven one way or another, the truth value of the antecedent of (1) is unknown, hence the use of the indicative conditional is a natural choice.Footnote 20 By contrast, when the authors entertain an antecedent which is not only false, but also known to be false, such as “the primes up to \(10^8\) are uniformly distributed” in (2), they choose to phrase the dependency between this assumption and whatever follows from it as the subjunctive conditional. Note that (2) is not only a counterfactual but also a counterpossible.
A subjunctive form can also be used when the antecedent is not known to be false, that is, when its truth value is itself an open question, though the choice of subjunctive tends to reveal the author’s belief in its falsehood. Many instances of subjunctive conditionals can be found when mathematicians explore the consequences of not-yet-proven conjectures such as the Riemann Hypothesis mentioned above, The Chromatic Number of the Plane Problem, or, to consider an example more familiar to philosophers, \(\textsf {P} \ne \textsf {NP}\).Footnote 21 For instance, in his essay on the \(\textsf {P} \ne \textsf {NP}\) hypothesis, Scott Aaronson writes:
-
(3)
“If just one of these problems [i.e. problems that have been shown to be in \(\textsf {P}\)] had turned out to be both \(\textsf {NP}\)-complete and in \(\textsf {P}\), that would have immediately implied \(\textsf {P} = \textsf {NP}\).” (Aaronson 2016: 25)
The counterfactual conditional (3) occurs in a context of an empirical argument for the inequality of two classes of computational complexity, \(\textsf {P}\) and \(\textsf {NP}\).Footnote 22 This argument rests on an observation that while thousands of problems have been shown to be either in \(\textsf {P}\) or to be \(\textsf {NP}\)-complete, there is not a single one that has been shown to be both. The antecedent of (3) has not been proven to be false—in principle, it might still happen that an \(\textsf {NP}\)-complete problem will turn out to be in \(\textsf {P}\). Yet the use of the subjunctive is appropriate as it corresponds to a belief that is empirically justified and widely shared in the computer science community.
Counterfactual language can also be found in texts that are primarily of a didactic nature, such as undergraduate textbooks to mathematics or lecture notes. Authors use conditionals to explain basic notions, e.g., the notion of logical equivalence:
-
(4)
“If X and Y are logically equivalent, and X is false, then Y has to be false also (because if Y were true, then X would also have to be true).” (Tao 2016a: 311)
More interestingly, conditionals can be used to explain consequences of certain assumptions such as, for instance, the infamous axiom of Universal Specification, that is, an assumption that every property corresponds to a set. Let us define P(x) as the following property: “x is a set and \(x \notin x\),” and the set \(\Omega\) as a set of all such x of which P(x) is true, that is, a set of all sets that do not contain themselves. In the following passage from a textbook to Analysis, Terrence Tao explains why these assumptions lead to the Russell’s paradox:
-
(5)
“If \(\Omega\) did contain itself, then by definition this means that \(P(\Omega )\) is true, i.e., \(\Omega\) is a set and \(\Omega \not \in \Omega\). On the other hand, if \(\Omega\) did not contain itself, then \(P(\Omega )\) would be true, and hence \(\Omega \in \Omega\). Thus in either case we have both \(\Omega \in \Omega\) and \(\Omega \not \in \Omega\), which is absurd.” (Tao 2016a: 47)
As we emphasised above, it was not the aim of our pilot corpus study to provide more examples of explanations, but to demonstrate that mathematicians use counterfactual language. Nevertheless, textbooks and lecture notes may be considered particularly valuable sources of data on the language used by mathematicians in the context of explanation, given that the primary goal of proofs found in such texts is arguably to explain key ideas to students. In the literature on mathematical education, it has been emphasised that the role of a proof is not restricted to showing that a theorem holds, but first and foremost to provide an explanation of why a theorem is true (Hanna 1990; Hersh 1993). Unsurprisingly, then, in teaching materials, we can find multiple examples of conditional and counterfactual language used in the context of proofs, particularly the reductio ad absurdum proofs.
For instance, Clark (2002) in his lecture notes on number theory, in the contexts of a discussion of primality tests, presents a proof of a theorem (a converse of Fermat’s Little Theorem) that states that if \(m \ge 2\) and for all a such that \(1 \le a \le m - 1\) it holds that \(a^{m - 1}\) is congruent to 1 modulo m, then m must be prime. The first step of the proof is phrased as an indicative conditional: “if the hypothesis holds, then for all a with \(1 \le a \le m - 1\), we know that a has an inverse modulo m, namely, \(a^{m-2}\) is an inverse for m modulo m,” The next step makes use of a theorem proven earlier (p. 72), which is also an indicative conditional, namely: if the product of two integers a and b is congruent to 1 modulo \(m > 0\) then both a and b are relatively prime to m, that is, the greatest common divisor of a and m, written gcd(a, m), equals 1, and so does gcd(b, m). In virtue of this fact, the first step amounts to an observation that for \(1 \le a \le m - 1\), the greatest common divisor of a and m is 1. Now, to show that m must be prime, one can consider the consequences of the assumption that it is not. Such an assumption leads to a contradiction, which is naturally phrased as a subjunctive conditional:
-
(6)
“...if m were not prime, then we would have \(m = ab\) with \(1< a < m\), \(1< b < m\). Then \(\gcd (a, m) = a > 1\), a contradiction. So m must be prime.” (2002: 97)Footnote 23
Again, this conditional is not only a counterfactual, but also a counterpossible: its antecedent is necessarily false.
Although this research does not show that mathematicians need to use counterfactuals and counterpossibles, we do have sufficient empirical evidence to support the claim that counterfactual language is used in mathematical writing, including, importantly, didactic texts such as textbooks and lecture notes, which can be said to have, broadly speaking, explanatory goals.Footnote 24 This finding lends additional support to our proposal of extending the CTE from scientific to mathematical explanations.