How to make (mathematical) assertions with directives

Abstract

It is prima facie uncontroversial that the justification of an assertion amounts to a collection of other (inferentially related) assertions. In this paper, we point at a class of assertions, i.e. mathematical assertions, that appear to systematically flout this principle. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs are sequences of directives. The claim is backed up by linguistic data on the use of imperatives in proofs, and by a pragmatic analysis of theorems and their proofs. Proofs, we argue, are sequences of instructions whose performance inevitably gets one to truth. It follows that a felicitous theorem, i.e. a theorem that has been correctly proven, is a persuasive theorem. When it comes to mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success.

1 Introduction

Assertions generally abide by the following principle:

Assertions Justify Assertions (AJA)

The justification of an assertion amounts to a collection of other assertions.

A standard justification for the assertion “It’s raining outside the car” is a collection of other (inferentially related) assertions: “There are raindrops on the windshield”, “There is an intermittent noise that sounds like rain falling on the car’s roof”, etc. The principle finds clear expression in Robert Brandom’s work (1983, p. 642):

Assertions are treated as in order, that is, as warranted, until challenged. Responding to such a challenge consists in producing further assertions whose contents are appropriately inferentially related to the original one. Each justifying consists of further assertings, which may themselves be challenged and stand in need of further justification.

In this paper, we discuss a class of assertions that appear to systematically flout AJA, i.e. mathematical assertions. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs, we claim, are sequences of instructions, i.e. directives. Mathematical assertions, therefore, abide by

Directives Justify Assertions (DJA)

The justification of a mathematical assertion amounts to a sequence of directives.

We begin by examining the pragmatics of mathematical assertions and how they differ from ordinary assertions (Sect. 2). We then move on to argue for the directive nature of proofs (Sect. 3), and conclude by discussing the theoretical significance of DJA (Sect. 4). We maintain that, when it comes to theorems and other mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success. A felicitous theorem (i.e. a theorem that has been correctly proven) is ipso facto a persuasive theorem. The directive character of proofs, we hypothesize, serves the goal of persuasion.

2 Ordinary assertions and mathematical assertions

Assertion is a species of illocutionary act. Illocutionary acts—things such as promising, ordering, marrying, and so on—comply with certain norms and engender certain normative effects. For example, a pronouncement of marriage can only be performed by an authorized minister and gives the newly married couple new rights and duties. Assertions are no exception. They are defined by their normative effects,¹ and bring those effects about by adhering to a set of norms.² Marco Ruffino, Luca San Mauro, and Giorgio Venturi have recently claimed that mathematical assertions form a proper subclass of the class of assertions inasmuch as they comply with peculiar norms or felicity conditions (Ruffino et al. 2021). This Section expands upon their work by exploring the normative effects that mathematical assertions bring about.

When one claims that something is the case (i.e. when one makes an assertion), one takes on distinctive responsibilities. A well-established view understands assertoric responsibility as a commitment to the proposition asserted.³ As we will see shortly, a ‘commitment to a proposition’ can be cashed out in more familiar terms as a ‘commitment to do something’. Ordinary assertions and mathematical assertions alike engender a commitment to a proposition. However, what such a commitment involves—what the speaker is required to do, in virtue of asserting—varies from one subclass of assertions to the other.

2.1 Ordinary assertions

Utterances such as “It’s raining outside the car”, “Paris is the capital of France”, and “The US economy is in poor shape” are all instances of ordinary assertions. A speaker who makes an ordinary assertion presents its propositional content as true, and commits themself to it. But what does a ‘commitment to a proposition’ amount to? What kind of ‘ought’ does a speaker acquire in virtue of making an assertion?

One prominent answer is that, in asserting that p, a speaker commits themself to providing reasons (evidence, grounds) in favor of p, if queried.⁴ As Brandom (1983, pp. 641–642) nicely puts it, assertoric commitment amounts to

the undertaking of justificatory responsibility for what is claimed. In asserting a sentence, one [...] commits oneself to justifying the original claim. [...] Specifically, one undertakes the conditional task responsibility to justify the claim if challenged.

In a similar spirit, MacFarlane (2003, p. 14) writes,

To assert a sentence S (at a context U) is (inter alia) to commit oneself to providing adequate grounds for the truth of S (relative to U), in response to any appropriate challenge.

So, ordinary assertions change the speaker’s normative status by imposing a certain (conditional) commitment upon them. It is easy to see how this translates into a felicity condition, requiring the speaker to have reasons in support of the asserted proposition. Committing oneself to p is to take on the conditional task responsibility to provide reasons in support of p if queried. One can fulfill this responsibility only if one has reasons in support of p. Therefore, asserting requires having reasons in support of what is asserted.

Ordinary assertions also affect the hearer’s normative status by giving them a reason for believing the proposition asserted, as well as an entitlement to re-assert it on the speaker’s epistemic authority (Brandom 1994, pp. 174–175). A speaker who takes on a commitment to defending p if challenged presupposes that they have some evidence in favor of p—that p is defensible. This supplies the hearer with a reason for believing that p: had the speaker not asserted that p, and everything else being equal, the hearer would have had one less reason to believe that p.⁵ The strength of the reason for belief the hearer acquires will vary with the speaker’s perceived credibility. For example, my assertion that Paris is the capital of France gives you a reason to believe that Paris is the capital of France, which may add up to (or clash with) reasons for belief you had prior to my utterance, and whose strength will vary with how credible you take me to be when it comes to basic political geography. It also entitles you to re-assert that Paris is the capital of France, and to adduce that I asserted it in lieu of a justification.

To sum up, an ordinary assertion commits the speaker to justifying their claim if challenged. This reverberates on the felicity conditions for asserting. In particular, for an ordinary assertion to be felicitous, the speaker must have some evidence for what they assert.⁶ In addition, ordinary assertions give the hearer a reason to believe that things are as the speaker claims they are, and an entitlement to re-assert that things are that way on the speaker’s authority.⁷

2.2 Mathematical assertions

Utterances such as “There are infinitely many prime numbers”, “For any finite group G, the order of every subgroup H of G divides the order of G”, and “The sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse” are all instances of mathematical assertions. A mathematical assertion, as we conceive of it, is an illocutionary act whose propositional content presents something as being the case, only includes purely mathematical terms (be they primitive or previously defined terms), and the truth of which is proven.⁸Theorems are the paradigm of mathematical assertions; lemmas and corollaries provide further instances.⁹ While this may do as a first approximation, a more accurate characterization would rely on, and elucidate, the essential normative effects of a mathematical assertion.

Qua assertions, theorems and their ilk commit the speaker to the proposition expressed. Specifically, they commit the speaker to proving that proposition. In asserting that there are infinitely many prime numbers, for example, one undertakes a non-conditional task responsibility to prove one’s claim.¹⁰ An assertion cannot count as a theorem without a proof. Thus, no challenges have to be raised for a mathematician to be required to prove what they put forth as a theorem. Mathematical assertions are not ‘standalone’ speech acts: a felicitous mathematical assertion is necessarily tied to a set of further proof-constituting speech acts.

Clearly, a proof is such only if it is correct: mistaken proofs aren’t proofs after all. When we say that a theorem commits the speaker to providing a proof, we are thus saying that it commits them to providing a correct proof. Just as an unproven assertion cannot count as a mathematical assertion, so too a purported theorem that comes with an incorrect proof is not a theorem. Any mathematical assertion must be accompanied by a correct proof. This implies that any mathematical assertion expresses a true proposition. Ordinary assertions are liable to error: an ordinary assertion can be felicitous (i.e. backed up by good reasons) and yet things may turn out to be different from how the speaker claimed they were.¹¹ Mathematical assertions are not liable to error in the same way: if felicitous (i.e. correctly proven), a mathematical assertion cannot turn out to be false.

Note also that ordinary assertoric commitment is a gradable notion. One can take on more or less responsibility toward the asserted proposition and modulate the degree of commitment undertaken via mitigation devices. I can felicitously assert that, maybe, it’s raining outside the car, or that, as far as I know, it’s raining outside the car. These are less committal assertions than “It’s raining outside the car”: their speakers take on less responsibility toward the asserted content, and can provide weaker reasons in its support if challenged.¹² Mathematical assertions generate a commitment that does not come in degrees, and hence they do not admit of mitigation. “Maybe, there are infinitely many prime numbers” or “As far as I know, there are infinitely many prime numbers” would not be mathematical assertions at all. To this, one might object that explicit foundational assumptions prefaced to a theorem (e.g. “Assuming the Axiom of Choice, p”) are mitigation operators downgrading the degree of assertoric commitment that the speaker undertakes.¹³ We don’t think this is the case, though. “Assuming the Axiom of Choice” does not put the speaker in a position to provide weaker reasons in favor of p: any justification short of a proof would not do. The opening parenthetical does not mitigate the degree of strength of the speaker’s assertion—it does not make it less than a full-fledged assertion. Rather, it makes its propositional content explicitly conditional (“P, if the Axiom of Choice holds”).¹⁴ Our point therefore stands: a speaker cannot felicitously and simultaneously advance an assertion as a theorem and downgrade their commitment to the proposition it expresses via mitigation devices like ‘maybe’ or ‘as far as I know’—for doing so would defeat their speech act.

So, mathematical assertions impact on the speaker’s normative status by assigning a certain (non-conditional) commitment to them. This translates into a felicity condition, requiring that the speaker have and provide a correct proof. Interestingly, this often comes as a future felicity condition relative to the time of utterance. Within mathematical texts, mathematical assertions typically occur in graphically separated blocks (Ruffino et al. 2021, p. 10075). A block has a two-part structure: the assertion of a (purported) theorem, which comes at the top, is followed by a (purported) proof. Since the correctness of a proof can be checked only once the proof has been entirely given, the assertion labeled as a ‘theorem’ in the top block counts as a mathematical assertion only retroactively. It is only once the proof is given that the top-block mathematical assertion may be deemed felicitous and retroactively count as a genuine theorem.

Mathematical assertions also affect the hearer’s normative status in a way that sets them apart from ordinary assertions. In particular, while ordinary assertions only give the hearer a reason to believe that p, mathematical assertions make the hearer (or reader)¹⁵believe that p. When it comes to mathematical assertions, the distinction between illocutionary and perlocutionary success becomes somewhat blurred. I can assert that it’s raining outside the car without convincing you that it is. I can give you a number of good reasons to believe that it’s raining outside the car, and you can still rationally doubt it (perhaps, it is the automatic sprinkler of the field we are skirting that explains the water droplets on the windshield). But I cannot assert that there are infinitely many prime numbers, and prove that there are, without convincing a sufficiently competent hearer that there are—without making them ‘see’ the truth of what I say. (We will argue below that this can be accounted for by conceiving of proofs as sequences of instructions, rather than of assertions: see Sect. 4.)

Much like ordinary assertions, mathematical assertions also entitle the hearer to re-assert their content. Both ordinary and mathematical re-assertions allow for ‘justification by deference’. An ordinary re-asserter, if challenged, can defer the responsibility to provide supportive evidence to the original speaker. Similarly, a mathematician who re-asserts a theorem can avoid providing a proof themself and defer to the work of someone else. However, while an ordinary re-assertion can be based only on the original speaker’s epistemic authority (p—because S, who’s an expert on the relevant subject matter, says that p), a mathematician who re-asserts a theorem cannot merely rely on the authority of whomever first advanced it, but must invoke the correctness of an already known proof (p—because it has been proven).¹⁶

Taking stock, a mathematical assertion commits the speaker to proving their claim. This reverberates on the felicity conditions for asserting in mathematics. In particular, for a mathematical assertion to be felicitous, the speaker must have and provide a proof for what they assert. In addition, a mathematical assertion entitles the hearer to re-assert its content on the basis of the correctness of the proof given. Also, and importantly, mathematical assertions do not only give a sufficiently competent hearer a reason for believing the asserted proposition but make them believe it. The gap between giving a reason for belief and convincing is bridged, we suggest, by the directive character of proofs.

3 The directive nature of proofs

Before disentangling the illocutionary nature of proofs, let us have a look at how proofs are standardly defined.

3.1 Proofs and derivations

Here is the standard formalization of a proof.

Definition 3.1

A proof of $\varphi$, where $\varphi$ is a formula in a theory T equipped with a deduction operator $\Delta$, is a finite sequence of formulas $\varphi _0,\varphi _1,\ldots ,\varphi _n$ such that

• (1) $\varphi _n=\varphi$,

• (2) each $\varphi _i$ is either an axiom of T or is derivable from the previous $\varphi _j$’s by applying the deduction rules of $\Delta$.

Almost any logic textbook has a definition like this. As is often the case in logic, Definition 3.1 is an abstract idealization, which fails to apply to many concrete proofs. Consider the proof¹⁷ of Theorem 3.2.

Theorem 3.2

There are infinitely many prime numbers.

Proof

Suppose otherwise and let P be the set of all (finitely many) primes. Denote by z the following number

$$\begin{aligned} z := \left( \prod _{p \in P} p\right) +1. \end{aligned}$$

Observe that, for all $p\in P$, p cannot divide z, since $z \equiv 1 \; \hbox {(mod}\,p)$. But then z is either prime or there is a prime $n \notin P$ which divides z, a contradiction. $\square$

As one can see, the proof of Theorem 3.2 does not meet the formal standard expressed by Definition 3.1. For one, it is not written in formal language, but in a version of English augmented with mathematical symbols. Furthermore, it has gaps that have to be filled by practitioners (e.g. it tacitly relies on the Fundamental Theorem of Arithmetic, which guarantees that every integer greater than 1 can be uniquely represented as a product of primes).¹⁸

Definition 3.1 captures what we may refer to as derivations. In opposition to a longstanding philosophical tradition, Lakatos (1976a, 1976b) has famously argued that proofs cannot be fully reduced to derivations. Or, put the other way round, derivations give an oversimplified image of proofs.¹⁹

In the past few decades, scholars have increasingly recognized the theoretical significance of the discrepancy between proofs and derivations. Such a recognition has often been enabled by the adoption of interdisciplinary angles blending logic and philosophy of mathematics with history, sociology, and computer science.²⁰ We here follow suit with this growing trend. We adopt the angle of speech act theory, as developed within philosophy of language and linguistics. Proofs, we claim, are complex illocutionary acts, i.e. illocutionary acts whose overall performance comprises the performance of many illocutionary passages.

3.2 Proofs as illocutionary complexes

Some illocutionary acts are simple: their performance requires one single illocutionary step. A promise that this is my last cigarette only requires an utterance of “I promise this is my last cigarette”. And an utterance of “Would you do the dishes, please?” is all it takes to request that you wash the dishes.²¹ Other illocutionary acts are complex: their performance requires multiple illocutionary steps. A contract, for instance, includes a number of preliminary definitions, a series of clauses, and the signatures of all contracting parties. Preliminary definitions are (Searlean) declarations by which the parties are assigned a functional reference (e.g. “Seller”, “Buyer”). Most contract clauses are commissive (e.g. “After acceptance by all Parties, the Buyer agrees to make a payment in the amount of 5000 euros by 10 July 2022”), but contracts can also include assertives (e.g. “No loan or financing of any kind is required in order to purchase the Property”) as well as directives (e.g. “If the Seller does not remedy any defects in the title of the Property, the Buyer shall have the option of canceling this Agreement”²²). The signatures of the parties count as the conclusive commissive step, by which the parties commit to respecting the contract clauses.

Like contracts, proofs are illocutionary complexes: their performance requires a number of illocutionary passages, which typically display an array of illocutionary forces. To those who move from a logical take on proofs, as distilled by Definition 3.1, this may come as a surprise. After all, each of the formulas $\varphi _i$’s in that definition is naturally interpreted as an assertion within the background theory T. In abstracting from real-world mathematical proving, the logical formalization of proofs conflates the different types of speech acts that make up concrete proofs to one. To see this, consider the following proof of the Bolzano-Weierstrass Theorem.

Theorem 3.3

Every bounded sequence of real numbers has a convergent subsequence.

Proof

Let ${x_n}$ be a bounded sequence and without loss of generality assume that every term of the sequence lies in the interval [0, 1]. Divide [0, 1] into two sub-intervals, [0, 1/2] and [1/2, 1]. At least one of the halves contains infinitely many terms of ${x_n}$; denote that interval by $I_1$, which has length 1/2, and let ${x_{n'}}$ be the subsequence of ${x_n}$ consisting of every term that lies in $I_1$. Now divide $I_1$ into two halves, each of length 1/4, at least one of which contains infinitely many terms of the subsequence ${x_{n'}}$, and denote that half by $I_2$. Let ${x_{n''}}$ be the subsequence of ${x_{n'}}$ consisting of all of the terms that lie in $I_2$. Continuing in this way, we construct a sequence of nested intervals $I_1 \supseteq I_2 \supseteq \cdots$, where the length of $I_n$ is $1/2^n$, and each interval contains an infinite number of terms of the original sequence ${x_n}$. Finally, we construct a subsequence ${z_n}$ of ${x_n}$ made up of one term from each interval $I_n$. This subsequence is clearly Cauchy:

$$\begin{aligned} (\forall s)(m, n > s \Rightarrow |z_m - z_n| < {1/2^s}). \end{aligned}$$

Therefore the subsequence ${z_n}$ converges. $\square$

The proof of Theorem 3.3 includes a number of illocutionary passages, displaying various illocutionary forces. For example, “Divide [0, 1] into two sub-intervals, [0, 1/2] and [1/2, 1]” is a directive. “This subsequence is clearly Cauchy” is an assertive. And “Let ${x_n}$ be a bounded sequence” is (what we call) a directive declaration. Qua auxiliary definition, the utterance constitutes a declaration—i.e. a speech act that “change[s] the world solely by saying so” (Searle and Vanderveken 1985, p. 37).²³ The mathematician says that ${x_n}$ is a bounded sequence, and ‘${x_n}$’ therewith comes to pick out a bounded sequence. However, unlike standard declarations (e.g. “I name this ship the Queen Elizabeth”), the definition is given, as is typical of within-proof definitions, in the imperative mood—as an instruction for the reader to follow. This is why we classify it as a hybrid speech act, having the force both of a declaration and a directive.

In spite of consisting of a variety of (simple) illocutionary acts, illocutionary complexes are illocutionary acts themselves, whose overall nature can be pinned down. Contracts, for instance, are overall commissive speech acts, for their essential (definitional) normative effect is to bind the parties to certain terms and conditions. In contrast with the logical take on proofs, we argue that proofs are overall directive speech acts – their essential normative effect being to give the reader binding reasons to carry out a number of truth-conducive steps. There are at least three orders of reasons to classify proofs as directives.

First, a linguistic datum: mathematical proofs make a massive use of the imperative mood.²⁴ ‘Let’, ‘suppose’, ‘observe’, ‘divide’, ‘denote’, ‘assume’, ‘apply’, ‘substitute’, ‘solve’, ‘order’ are but a few instances of the typical imperatives in proofs.²⁵ Some direct the reader’s (mental) gaze toward something (e.g. ‘observe’), others require that they carry out a (mental) operation (e.g. ‘substitute’, ‘solve’, ‘order’, ‘apply’), and still others instruct them as to the functional reference of certain terms (e.g. ‘let’, ‘denote’). Even though the imperative mood systematically correlates with directive force, the correlation is not necessary. “Get well soon!”, for example, is more of an expressive than a directive speech act.²⁶ In principle, then, within-proof imperatives could constitute speech acts other than directives. After all, one might maintain that imperatives in proofs can be replaced with ‘we-indicatives’ without loss of meaning, and that this suggests that, beneath the surface, they constitute assertives. “Divide [0, 1] into two intervals, [0, 1/2] and [1/2, 1]” can be translated into “We divide [0, 1] into two intervals, [0, 1/2] and [1/2, 1]”; “Denote that interval by $I_1$” can be rendered as “We denote that interval by $I_1$”; etc. While we acknowledge that the pervasiveness of imperatives in proofs is not per se a decisive reason for their directive nature, it is important to note that we-indicatives may serve as instructions, and thus constitute directive speech acts notwithstanding their grammatical mood. Suppose a gang leader is walking their men through the steps of a robbery. “We break into the bank from the basement tunnel”, “We unlock the vault by the cloned keycard”, etc. These are instructions, despite their ‘we-indicative’ form. And indeed, it would not make much sense to ask whether “We break into the bank from the basement tunnel” is true. And should one of the gang members question it (“Wouldn’t it be more cautious to break into the bank from the adjacent garage?”), they would question the appropriateness of a command—not the truthfulness of a statement. “We break into the bank from the basement tunnel” does not generate a commitment to truth; rather, it imposes an obligation to act in a certain way upon the gang members and its leader. Insofar as the leader will take part in the robbery, they will be bound to follow the plan as much as the others. In a somewhat similar vein, insofar as, in providing the proof of a theorem, the mathematician is (re-)executing the steps they instruct the reader to take, the use of we-indicatives may well serve a directive function.²⁷ This is further supported by the fact that, while the truth of “This subsequence is clearly Cauchy" (from the proof of Theorem 3.3) can appropriately be questioned (perhaps, the subsequence is not Cauchy), an appropriate challenge to “We divide [0, 1] into two intervals, [0, 1/2] and [1/2, 1]” would question its efficiency as an instruction (perhaps, dividing [0, 1] into two intervals is an unnecessary step).

Second, the Curry–Howard isomorphism states a correspondence between (natural deduction) proofs and ($\lambda$-calculus) programs: hence, formal derivations can be transformed into suitable sequences of commands (that is, directives).²⁸

Third, mathematical proofs are inescapably persuasive: any rational and sufficiently competent reader will judge a proven theorem true. Mathematical assertions, if felicitous, leave no room for rational disagreement. Granted, one may disagree as to whether the provided proof is the easiest way to prove its theorem, or as to whether its phrasing can be streamlined further. And, to be sure, disagreement may arise about foundational issues—consider the acceptability (or lack thereof) of using the Axiom of Choice within the constructivist framework; or think of large cardinals, beyond measurable, from the perspective of Gödel’s constructible universe. But no one can rationally disagree as to whether the conclusion of a correct proof is true. Any competent mathematician—even a strong advocate of intuitionism—would acknowledge that, if the Axiom of Choice is assumed, then a ball in the ordinary Euclidean space can be doubled by only partitioning it into subsets, replacing a set with a congruent set, and reassembling it; or that the existence of a supercompact cardinal implies that $V \ne L$. In the next Section, we suggest that ‘inescapable persuasion’ cannot be the product of a series of assertions, but is granted by proofs functioning as sequences of directives.

4 Mathematical assertions, proofs, and persuasion

A proof is “a sequence of thoughts convincing a sound mind”, says Gödel (1953, p. 341). Gödel’s remark has often been understood as highlighting the difficulty (or perhaps the impossibility) of delimiting, once and for all, what counts as a proof. But it could also be construed as an attempt to offer a minimal characterization of what mathematical proofs are for. Stripped away of its internal intricacies, a proof may be a linguistic tool to convince or persuade a reader that a given theorem holds.

To an extent, this isn’t striking: just like giving reasons in favor of the proposition expressed by an ordinary assertion may be a means to persuade your addressee that it faithfully represents the world, so too providing a proof of a mathematical assertion is a means to persuade your reader that its content is true. But the analogy does not go much further. For, no matter how good your reasons in support of an ordinary assertion are, your addressee can rationally doubt it, and thus fail to be persuaded.²⁹ You assert that the US economy is in poor shape because inflation has hit its highest level since the 1980s. Your addressee rebuts that the US is worlds away from where things stood when the pandemic began—which suggests that the US economy isn’t in poor shape after all. This space for disagreement is precluded by mathematical proving. If a proof is correct—if what a mathematician presents as a proof is in fact a proof, then the reader cannot rationally doubt that its theorem holds. Ordinary assertions illocutionarily invite the hearer to believe that p, and may perlocutionarily make them believe that p—but illocutionary felicity is no guarantee of perlocutionary success. Mathematical assertions, by contrast, are felicitous—i.e. they count as theorems, lemmas, etc.—only if they are proven. And this is to say that they cannot be felicitous without being persuasive. When it comes to mathematical assertions, illocutionary and perlocutionary success are one.

Let us now get back to the question as to how to characterize the illocutionary nature of proofs. Bracket momentarily their massive use of imperatives and the transformability of derivations into suitables sequences of commands granted by the Curry–Howard isomorphism, and suppose that a proof were a series of assertions. If a proof were a series of ordinary assertions, each and every step could in principle be put in question. The reader could doubt one or more of the proof-constituting assertions, and fail to be persuaded by the theorem. If a proof were a series of ordinary assertions and mathematical (re-)assertions, some of its constitutive assertions (i.e. those pertaining to the ordinary domain) could be challenged, and the proof could overall fail to persuade the reader. We rule out the possibility that a proof is a series of mathematical assertions only, for clearly this is not the case. Even if “Divide [0, 1] into two sub-intervals, [0, 1/2] and [1/2, 1]” (from the proof of Theorem 3.3) could be fully translated into an assertion, this clearly would not be a mathematical assertion or re-assertion.

The hypothesis we put forth is that proofs overall have a directive nature. At its core, a proof is a sequence of instructions for actions to be carried out in order to apprehend a mathematical truth. This does not imply that proofs do not include any mathematical (re-)assertions. They do, but in the economy of a proof those (re-)assertions play the role of assumptions that the reader must make in order to move on through the proof. An imperative such as “Apply Lemma 3”, for example, can be translated into an assertion stating the content of Lemma 3. This mathematical (re-)assertion would play the role of an assumption that the reader must make—and thus constitute an indirect instruction³⁰ itself as to what is to be assumed—, or else they would be unable to carry out the subsequent passages of the proof.

Construing the passages of a proof as instructions or binding directives gives us an elegant way to explain why mathematical assertions are inescapably persuasive. Recall that any felicitous mathematical assertion comes with a proof. Our hypothesis is that a proof is a sequence of instructions, giving the reader binding reasons for action.³¹ Each and every step of a proof requires the reader to do what they are told, on the implicit promise that, if they follow the instructions, they will apprehend a truth. The language game of mathematical proving is one in which the mathematician, who has already taken the steps that they are now instructing (and accompanying) the reader to take, implicitly vouches for the correctness of the instructions they give. As Tanswell (forthcoming) suggests, proofs are, to a certain extent, akin to IKEA or LEGO instructions: if you follow them step by step, you will get the piece of furniture or the LEGO model that the author of the user manual has implicitly promised you. In an analogous fashion, if you follow the instructions constituting a proof, you will get what the mathematician has implicitly promised you—you will apprehend a mathematical truth.

Of course, executing the instructions of a proof would generally be prohibitive for a non-working mathematician (and in fact the same is often true for mathematicians working in distant fields), but whoever is able to execute them all will be rewarded with a certain end truth-product.³² The distinctive feature of proofs (and of deductive reasoning more broadly) is, in a slogan, that no agent—or “sound mind”, in Gödel’s words—can follow a proof and fail to be persuaded.³³

Justifying a mathematical assertion by a sequence of instructions is what bridges the gap between giving a reason for belief and persuading—a gap that remains open with ordinary assertions, whose justification goes through further assertings. Within the broader class of assertions, theorems (lemmas, etc.) can be isolated as those assertions in which illocutionary felicity is one with perlocutionary success: if felicitous, a theorem necessarily makes whomever can follow its proof believe that its content is true. This unity of illocution and perlocution is granted by the directive character of proofs: the steps of a proof oblige the reader to carry out a number of (mental or non-mental) actions whose overall performance leads them to truth, and to see it as such.

5 Conclusion

We stated at the outset that assertions, as Brandom explicitly argues for and speech act theorists tend to take for granted, comply with

Assertions Justify Assertions (AJA)

The justification of an assertion amounts to a collection of other assertions.

If our hypothesis as to the directive nature of proofs is correct, then there is at least one subclass of assertions, i.e. mathematical assertions, whose justification does not amount to a collection of other assertions. In particular, mathematical assertions (theorems and their kin) follow

Directives Justify Assertions (DJA)

The justification of a mathematical assertion amounts to a sequence of directives.

This is a significant result both for speech act theory and for our collective image of mathematics. It is theoretically surprising, and it has so far been largely neglected, that certain assertions can be supported by reasons for action, rather than reasons for belief. Assertions express beliefs. As such, in order to convince the hearer about their truth, it is prima facie obvious that one should give them reasons for belief in the form of further assertions. But mathematical assertions seem to systematically flout this principle. They express and prove truths—and once a mathematical truth has been correctly proven, it is inescapably persuasive. Mathematical assertions reach this level of persuasiveness by providing the reader with a series of instructions giving them binding reasons to act in such a way as to eventually see that the theorem holds.

Summarizing common sense and the logical take on mathematics, Ganesalingam (2013, p. 11) writes: “the language of mathematics consists purely of assertions about mathematical objects”. Our analysis challenges this apparent truism. The language of mathematics does not consist purely of assertions. As we have highlighted throughout, mathematics makes use of speech acts other than assertions and assertives, and gives directives pride of place in its central practice, i.e. the practice of proving.