Abstract
We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics (not just assertions), and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics (which tends to see mathematics as a realm in which no pragmatic features of ordinary language are present) and in speech act theory (which tends to pay attention solely to communication in ordinary language but not to formal languages). As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts (silently) present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion.
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Notes
Although the speech act dimension is a neglected chapter in the philosophy of mathematics, we can find contributions in the near literature. To name just a few examples: the role of assertions is a main topic in mathematical fictionalism, and in particular the so-called force fictionalism is at the center of debate (Balaguer 2009, 2011; Armour-Garb 2011) (for our part, we will assume that when a mathematician is asserting something she is really doing an assertion); Williamson (2000) investigates the constitutive rules of assertion and devotes a whole section to assertions within mathematics; finally, there are contributions that equate mathematical objects and social institutions in Searle’s sense (Cole 2013; Hersh 1997). We defer to a continuation of this work a thourough discussion of the relations between the illocutionary dimension of mathematical language and the classical problems of the philosophy of mathematics.
“Even such a conscientious and rigorous writer as Euclid often makes unmentioned use of premises to be found neither among his axioms and postulates nor among the premises of the particular theorem being proved [...] The reader, though, would have to pay particular attention to notice these sentences being skipped over, the more so because they seem almost as fundamental as the laws of thought themselves, to be used, as it were, with the same freedom as those laws themselves.” (Frege 1972a, p. 85).
Whitehead and Russell (1912) adopt the same symbol for assertion in their formal language, with the same interpretation (i.e., assertive force indicator).
From “My Basic Logical Insights”, Frege (1981 p. 252).
Wittgenstein expresses in the Tractatus (Wittgenstein 1961, 4.063, 4.442) a somewhat critical view about the use of assertion sign in Frege, Russell and Whitehead. The exact interpretation of Wittgenstein’s point is a matter of some controversy. However, as some scholars point out (e.g., Proops 1997), it seems that Wittgenstein projects on Frege something that was actually Russell’s view (around 1903) on thoughts and propositions, namely, that a thought is composed of a purely propositional element combined with a non-propositional element (the assertive part). This is not exactly Freges view, however; as he says many times, the assertoric force does not contribute at all for the content. Wittgenstein probably takes Frege’s remark in Begriffsschrift §3 (that his formal language can be seen as having “only a single predicate for all judgements, namely, ‘is a fact”’) as representing Frege’s view. But the remark must be taken as simply metaphorical. (For more on Wittgenstein’s critical view on Frege’s assertion sign in logic, see Dummett 1981, pp. 295–363).
We here follow Searle (1975, p. 14), who treats permissions as directive acts, along with orders, suggestions, invitations, etc. A permission that P can be formally understood as the negation of a prohibition that P; a prohibition that P on its turn, can be seen as ordering (not- P). So, a permission that P can be formally understood as the negation of an order that not-P, i.e., as a composition of a directive act with negation (See Searle and Vanderveken 1985, pp. 201–2).
Cook in his Appendix to The Basic Laws of Arithmetic (2013) also notices that
Concept-script expressions containing instances of the Roman letter generality device are only provided with a “semantics” when such expressions are prefixed with an occurrence of the judgement-stroke.[...] [T]he Roman letter generality device cannot be understood as a simple function-name within the taxonomy of objects and function-name types that Frege mobilizes elsewhere in Grundgesetze. (Cook 2013, A-14)
For a different view, see Heck (2012, §3.2). Heck sees the use of Roman letters as an anticipation of Tarski’s later notion of assignments for free variables. He places no importance to the fact that Roman letters only have meaning if preceded by the judgement stroke.
A remarkable exception is Vanderveken:
Modern logicians and philosophers of language did not recognize fully the philosophical importance of Frege’s idea of admitting illocutionary force markers in the ideal object-languages of logic. Rapidly Frege’s assertion sign was eliminated from the object-language of logic, and became a meta-linguistic sign used only to identify the sentences of object-language which are provable in axiomatic systems. I believe that this failure to recognize the indispensability of illocutionary force markers in language is responsible for the failure of contemporary logical semantics to interpret adequately performatives and non-declarative sentences (Vanderveken 1990, p. 68).
Something essential behind these dimensions is the notion of direction of fit between word and world, as Searle and Vanderveken call it.
There are few exceptions, e.g., some expressive acts like greeting someone might occur without a propositional content.
Mathematical assertions, as we will see, impose several restrictions on the propositional content, differently from ordinary assertions.
In Searle’s original formulation (Searle 1975), there is no sincerity condition for this kind of act. But in Searle and Vanderveken (1985), the condition might include belief in some cases. In Searle and Vanderveken’s words,
By definition, a declarative illocution is successful only if the speaker brings about the state of affairs represented by its propositional content in the world of utterance [...] All successful declarations have a true propositional content and in this respect declarations are peculiar among speech acts in that they are the only speech acts whose successful performance is by itself sufficient to bring about a word-world fit. In such cases, “saying makes it so”. (Searle and Vanderveken 1985, p. 57)
Although Searle and Vandervekens is the most widely used taxonomy, there are other (more recent) alternatives such as Roberts’ (2018), which divides speech acts only in three big categories based solely on the direction of fit. We leave for a future occasion the question of whether anything said in this paper has to be changed if we switch to the new taxonomy.
Our emphasis.
Williamson (2000) argues for something a little stronger, but compatible with our perspective: he claims that a constitutive rule of any assertion is that one must assert that P only if one knows that P, while knowing that P might allow for different standards in different contexts. But knowing that P, in mathematics, requires a proof of P.
One might consider an assertion as felicitous relying on the hypothesis that its preparatory conditions are satisfied and, if it is later discovered that some condition was, contrary to the hypothesis, not satisfied, one then considers that the original act was actually a misfire. A properly mathematical assertion of a proposition that is later found to be false or whose proof is later discovered to be wrong is like a testament that is for some time taken to be authentic and later discovered to be actually a fraud: the testament is then considered as an infelicitous speech act.
These terms also appear in logic and its applications such as formal semantics.
Axioms and postulates are, of course, not included here. They must be treated separately, and it is far from clear that they are assertions.
At first sight, mathematical assertions require belief with maximum degree of strength, since the mathematical proposition is a necessary truth, and mathematics is the field par excellence in which there can be absolute certainty. However, this is not so, since many times there is a remaining doubt as to whether a proof was correctly carried out. It is far from obvious to understand on what the belief in mathematical propositions relies on. But normally if the belief is less than absolute, the remaining doubt is connected with the preparatory conditions rather than with the content itself. We aim to further explore this topic in future work.
One alternative suggestion is that this vocabulary (‘Theorem’, ‘Lemma’, etc.) is only a way of indicating the basis for the justification of the corresponding proposition instead of different forms of assertion. But this seems incorrect. In mathematics, the only admissible justification is proof, and if the function of these terms were only to indicate that there is a proof of the corresponding proposition, there would be no difference between them. But, intuitively, there is a difference between calling something a ‘Lemma’, a ‘Theorem’, a ‘Corollary’, etc.
Typically there are certain minimal stylistic conventions for IBMAs, e.g., the illocutionary force device is usually written in boldface, while the propositional part is in italics.
Such a view, as we will show in the subsequent discussion, turns out to be problematic.
Again, the case of Axioms is special, and will be left for a future occasion.
Austin recognized a special kind of performative verbs that he calls “expositives” and so described their function:
They make plain how our utterances fit into the course of an argument or conversation, how we are using these words or, in general, are expository (Austin 1962, p. 152).
Expositives are used in acts of exposition involving the expoundings of views, the conducting of arguments, and the clarifying of usages and of references (Austin 1962, p. 161).
Our view is that something resembling the illocutionary force of Austin’s expositives is going on in IBMAs besides the pure assertive act.
After Frege, hardly anyone used the symbol ‘\(\Vdash\)’, but some indicator to the same effect is always present.
We use quotation marks because, as a realist, Frege does not think that concepts are actually created.
There are other also less ceremonial ways for defining new elements of a mathematical discourse, like “let X be such and such”. Here we deal only with Definitions that bear this name. However, we acknowledge that there are also some borderline cases in which it is open for debate whether certain mathematical expression constitute definitions; e.g., see the large debate concerning the definitional status (if any) of Church-Turing thesis (Kleene 1952; Shapiro 2006).
According to Searle’s (1989) account of performatives, any performative utterance in natural language is a declaration. Not so in mathematics; only few and very specific performative formulae are allowed. We are aware that the interpretation of the creative role that is given to mathematical declarations can influence one’s view on mathematical ontology. However, for the time being we will restrict ourselves to a linguistic analysis and leave the ontological question for a future occasion. We shall only anticipate that, in our view, the recognition of the fundamental components of mathematical language will not determine a specific view on the ontology of mathematics, but will be informative in understanding the tension and the reasons of several philosophies of mathematics and their ontological perspectives.
According to Austin’s original classification, ‘define’ belongs to the class of performative verbs that he calls “Expositives”. But, on the other hand, given his characterization of the class that he calls “Exercitive” (“It is a decision that something is to be so, as distinct from a judgment that it is so” (1962, p. 155)), ‘define’ could also belong to this latter class, having thus a double illocutionary force.
This taxonomy could be continued including also aspects of intentionality (like the aim of an author to match a pre-formal notion), but we leave the task of a complete classification of definitions to another time. We thank an anomymous referee for the useful comments and suggestion on this topic.
Earlier in this paper we presented Frege as advocating that contentual definitions are essential to mathematics. We must mention that in some of his late writings (e.g., “Logic in Mathematics”, written in 1914), he seems to hold a different view on definitions, i.e., that the task of contentual definitions is relegated to a conceptual work previous to the construction of the formal system, and, hence, contentual definitions have no place inside the formalized theory. Frege comes to the conclusion that only purely stipulative (i.e., purely arbitrary) definitions deserve to be called definitions. This represents a change with respect to the perspective of his earlier writings (e.g., in “Boole’s Logical Calculus and the Concept-script”, from 1880/81) in which he talks about fruitful definitions as those that yield many interesting results (meaning by that that they are able to derive independently known results) and that “what we may discover in them has a far higher claim on out attention than anything that our everyday trains of thought might offer” (Frege 1981, p. 33); but arbitrary stipulations (i.e., pure abbreviations) cannot be fruitful in this sense.
Some authors propose tests to check the material adequacy of definitions (although the claim that the definition passes these tests is not necessarily what is asserted). E.g., Tarski’s Convention T is proposed as a test to check whether his definition of truth is materially adequate. But material adequacy is not, strictly speaking, a success condition of a contentual definition, since definitions might be successful and, yet, materially inadequate. The purely declarative part is arbitrary and a definition that is not materially adequate is, nevertheless, a successful declaration. The assertive part contains (or so we propose) the claim that the stipulation is correct or materially adequate. In our example, that Tarski’s definition captures the intuitive (classical) notion of truth. But if Tarki’s definition had not passed the Convention-T test (being, therefore, inadequate), it would still nevertheless be a successful assertive declaration-only the asserted content is not true. Success conditions and adequacy conditions are not the same; the assertion of a falsity can be a successful assertion anyway.
Sometimes it is suggested that, as a preparatory condition of a Carnapian explication, one must show that a notion is somewhat defective or unclear. But although this is normally done in the meta-theoretical work as motivating a sharper definition, nobody seems to do that inside the theory (i.e., no proof is required within the theory that a concept is originally fuzzy or ambiguous to proceed to a definition that explicates or sharpens that same concept). Normally only definitions sharpening concepts take place inside mathematical theories, but it is unusual the occurrence of proofs showing that concepts are defective and in need of such definitions.
As we said, directive acts must also part of mathematics but, for reasons of space, we leave them out for now.
Whether they are declarations, assertive declarations or directive speech acts requires a deeper discussion.
Maybe conjectures are, in part, the expression of a commitment to find a proof in the future and, in that sense, commissive acts are after all present in mathematics as well, contrary to what we said at the outset. This, too, requires a deeper discussion.
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The first author was supported by a FAPESP Research Grant (n. 2018/17011-9), the second by the Austrian Science Fund FWF (Project M 2461) and the third by a FAPESP Jovem Pesquisador Grant (n.2016/25891-3).
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Ruffino, M., San Mauro, L. & Venturi, G. Speech acts in mathematics. Synthese 198, 10063–10087 (2021). https://doi.org/10.1007/s11229-020-02702-3
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DOI: https://doi.org/10.1007/s11229-020-02702-3