- Email: [email protected]
At least one black sheep: Pragmatics and mathematical language
Journal of Pragmatics xxx (xxxx) xxx
Contents lists available at ScienceDirect
Journal of Pragmatics journal homepage: www.elsevier.com/locate/pragma
Discussion note
At least one black sheep: Pragmatics and mathematical language* Marco Ruffino a, Luca San Mauro b, *, Giorgio Venturi a a b
Department of Philosophy, University of Campinas, Brazil Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Austria
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2019 Received in revised form 25 December 2019 Accepted 23 January 2020 Available online xxx
In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis. © 2020 Elsevier B.V. All rights reserved.
Keywords Pragmatics Philosophy of mathematics The language of mathematics Existential quantifier Parameters
1. Introduction Let us start with a well-known joke. An engineer, a physicist, and a mathematician are on a train heading north, and had just crossed into Scotland. The engineer looks out of the window and says: “Look! Scottish sheeps are black!” The physicist says: “No, no. Some Scottish sheeps are black”. The mathematician looks up from the newspaper, turns their head, and says calmly: “There is at least one field, containing at least one sheep, of which at least one side is black”. : mathematicians are The joke (as in the case of most mathematical jokes) is based on the following over-used cliche (professionally) incapable of detaching themselves from literal meaning. Or, in other words, mathematical language lacks a feature that, according to many linguists and philosophers, characterizes most linguistic phenomena: a pragmatic dimension. This purported deficit in pragmatic aspects is so intertwined with a certain naïve description of mathematical language that, to our knowledge, it has been scarcely challenged philosophically. In this paper, we raise this challenge. We argue that pragmatic phenomena occur also in mathematics, in a peculiar fashion. We provide two main examples of this fact.
* The first author was supported by CNPq (Brazil) research grant 301721/2013-0. The second author was supported by Austrian Science Fund FWF through project M 2461. The third author was supported by FAPESP Jovem Pesquisador grant n.2016/25891-3. * Corresponding author. E-mail addresses: ruffi[email protected] (M. Ruffino), [email protected] (L. San Mauro), [email protected] (G. Venturi).
https://doi.org/10.1016/j.pragma.2020.01.011 0378-2166/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
2
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
1.1. Pragmatics and mathematics Pragmatics and mathematics scarcely overlap in the philosophical literature. On the one hand, philosophers of language rarely consider mathematical language as a possible case-study for testing their hypotheses. On the other hand, some philosophers of mathematics advocate a so-called “practical turn” in the field (see (Mancosu, 2008)), and there has been a growing interest in certain aspects of mathematics that are hardly reducible to its syntax or semantics (see Lakatos' seminal work (Lakatos, 1976), or Hersch (Hersh, 1991)). To name a couple of other examples: first, Rotman (1988) and Ernest (Paul, 2006) have written extensively about a semiotics of mathematics; secondly, the role of drawings and diagrams received much attention in recent years (see, e.g. (Giaquinto, 2007),). Van Bendegem (Paul Van Bendegem, 1982) explicitly addressed the interplay between pragmatics and mathematical practice. That said, it is still quite uncommon to investigate mathematical language with the sort of techniques developed by mainstream philosophy of language (e.g, speech acts, implicatures, pressupositions, and so forth). This is in part understandable. Some of the founding fathers of the philosophy of contemporary mathematics such as Frege, Hilbert and Russell placed a great emphasis on the need for a language free of those aspects of natural language that made it hard to produce and analyze proofs rigorously (e.g., vagueness, context-sensitivity, meaning-shifts, etc.). By placing such an emphasis they gave the impression that ideally mathematics must be done in a language completely “purified” from anything resembling speaker's intention, implicatures, illocutionary force, etc. Something curious, however, is that despite insisting on this contrast, Frege himself recognizes the indispensability of illocutionary force in logic (and, therefore, in mathematics) and incorporates two illocution;ary force indicators in the formal system of Begriffsschrift itself i.e., the assertion-sign w and the definition-sign y. There is no formula in the formal part of Frege that is left without the assertion-sign (being all of them therefore asserted). So, the relation between pragmatics and mathematical language is mostly unexplored. A rather crude way of justifying this lack of philosophical interest is to claim that there is nothing to be studied. This is the position of a recent book by Ganesalingam (2013), in which the following thesis appears: Mathematics do not exhibit any pragmatic phenomena: […] the meaning of mathematical language is always its literal meaning. We shall call this view the Antipragmatic Thesis (APT). Ganesalingam's book is a remarkable work. It proposes what is arguably the most extensive analysis of the semantics of mathematical language in the literature. It combines approaches coming from philosophy, linguistics, and computer science. Nevertheless, pragmatics is far less considered than semantics. In fact, APT is defended in a few pages, and the author concludes that the role of pragmatics in mathematical texts is negligible. In the rest of this paper, we will argue against this view. Pragmatics is a wide domain, and when talking about pragmatic phenomena in language one might be referring to distinct things such as presuppositions, speech acts, implicatures and discourse articulation. In this paper we shall concentrate on some examples that are probably better seen as cases of implicatures in mathematics, although, as we intend to show in later works, other pragmatic phenomena such as speech acts are also widespread.
1.2. Textual mathematics APT is a bold claim. But its plausibility depends on what we mean by “mathematical language”. If we posit that mathematical language is, or should be, the sort of language that Hilbert and Russell were pointing towards, then APT might be regarded as trivially true: that language was, in a sense, designed to be pragmatics-free. However, human mathematicians never use it. Moreover, Ganesalingam's analysis does not refer to such an idealized language, but rather focuses on the language by which mathematicians communicate their results. Yet, this definition would be so inclusive as to make APT patently false. Indeed, mathematicians communicate their results by several different means, some of which incorporate most (if not all) of the pragmatic devices that abound in natural language. To name a few: when delivering a lecture a teacher can rely, for explanatory purposes, on rhetorical figures, and speak metaphorically or even ironically.1 Similarly, a mathematician explaining the idea behind a certain construction will frequently depart from literal meaning if she considers that to be fruitful. The rough intuition here is that the more we move away from the formal side, the more we inevitably get caught up into pragmatics. A narrower definition of APT is that it applies to textual mathematics, that is, the kind of language in which standard mathematical papers are written. This is indeed Ganesalingam's focus, and it will be the focus of our present work as well. This context is better suited for APT since mathematical prose often aims at adhering to literal meaning. But again, textual mathematics, as a whole, is rife with complex linguistic constructions that make any hasty defence of APT problematic, to say the least. For example, it seems hard to account for the meaning of propositional attitude reports such as “we believe that” (that are widespread in mathematical discourse) if no reference to pragmatics is permitted.
1 Indeed, the role of pragmatics in mathematical education has been relatively explored; see (Rowland, 2003), or the recent volume “Philosophy of Mathematical Education” (Paul et al., 2018) where mathematical education and philosophy of mathematics interact.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
3
Then, a supporter of APT might try to restrict again the perspective, and say that the fragment of mathematics which is pragmatics-free is in fact that consisting of “generic (gnomic) sentences in the present simple tense”,2 as Ganesilagam puts it. Or, in other words, the fragment of mathematics which is pragmatics-free is the one consisting, roughly, of definitions, theorems, proofs, and so forth. The latter is a serious restriction. Yet, we aim to show that pragmatics phenomena arise even at the most fundamental level of mathematical discourse: the meaning of some logical constants such as quantifiers. Then, we move to the analysis of their role within proofs. An upshot of considering such distinguishing components of mathematical language (i.e., logical constants and proofs) is that the kind of pragmatic phenomena we unveil, altough similar to others occurring in natural language, will display peculiar characteristics as well. Nonetheless, let us stress that these pragmatic phenomena belong to mathematical language only if we regard the latter as the mixed language, combining natural and formal language, that is used in ordinary mathematics. So, a reader might want to rephrase the content of the present paper by saying that, rather than showing that pragmatic phenomena occur in mathematics, it actually proves that the utterances of mathematicians never lie entirely on the formal side of mathematical language e and therefore they encompass natural language phenomena, such as some pragmatic moves. We will not argue against this way of understading the following observations.
2. The meaning of the existential quantifier Far-side pragmatics (as opposed to near-side pragmatics) is the study of what contributes to the meaning of a sentence, beyond what the sentence literally means. A paradigmatic example in this direction is provided by what is called, after Grice,3 conversational implicatures. Consider the following dialogue.
(1)
A: Is there any wine left? B: Yes, there is a bottle in the kitchen.
B's answer is compatible, according to the literal meaning of the existential quantifier, with a scenario in which there are, in fact, n remaining bottles of wine, for n > 1. Yet, according to the classical view about implicatures, B conveys to A the information “there is precisely one bottle of wine”, altough not literally saying so. Similar examples are widespread in natural language. This motivates the standard idea of distinguishing between the literal meaning of a given sentence, and the (possibly distinct) speaker's meaning expressed by an utterance of the sentence. Like many important philosophical distinctions, the one between literal and speaker's meaning is not clear cut: for instance, it leaves obviously undefined to which extent someone has to depart from conventional meaning to be considered non-literal. Yet, it is convenient to stress that words and sentences often mean more than their literal meaning. Since we will deal with mathematical language (and in fact with the portion of mathematical language that corresponds to textual mathematics), we shall assume that the literal meaning of a mathematical term corresponds to the meaning assigned to it by its formal definition. In (Ganesalingam, 2013), Ganesalingam claims that pragmatics phenomena as such simply do not occur in mathematics. As an illustration for this claim he presents the following sentence.
(2) The group of symmetries of the cube contains a non trivial subgroup.
In natural language the implicature would be that there is exactly one such subgroup (by reasoning similarly to ð1Þ). Yet, mathematicians read ð2Þ as equivalent to
(3) the group of symmetries of the cube contains at least one non trivial subgroup.
This is true. But this does not imply that the speaker's meaning and the literal meaning of mathematical sentences always coincide. Consider the following theorem from computability theory.
(4) Any infinite computably enumerable set contains an infinite computable subset.
2 3
(Ganesalingam, 2013), p. 21. See (GriceColeMorgan, 1975).
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
4
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
The literal meaning of this sentence is equivalent to saying that there is at least one infinite computable subset. Yet, such a sentence is understood by any practitioner as conveying the following information.
(5) There are infinitely many computable subsets of a given computably enumerable set.
Prima facie, ð5Þ holds because any finite modification of a computable set is still computable. But there is something more profound here. Suppose that a reader R, perhaps unfamiliar with the field, fails to go beyond the literal meaning of ð4Þ. Then R might think that proving that any computably enumerable set A contains two infinite computable subsets is ampliative with respect to the information conveyed by ð4Þ. However computable theorists typically choose to present their results as concerning the construction of one set, instead of the construction of infinitely many satisfying the same class of requirements. A similar choice has of course many benefits in terms of exposition (for one, it makes constructions more readable) e yet it would not really be a viable choice, if literal meaning exhausted the meaning of any mathematical sentence. Indeed, by only considering literal meaning, one cannot make sense of the kind of omissions which abound in computability theory. Consider as another example the following classical theorem.
(6) There is a low simple set.
This sentence and its proof are typically stated with no need of specifying that minor modifications of the construction allow to build, in fact, infinitely many such sets. And it is perfectly fine to rely only on ð6Þ if one wants to consider two different low simple sets. The crucial point here is thus that in order to grasp the speaker's meaning encoded by ð4Þ or by ð6Þ, the reader has to be aware of the following two-pronged principle: (a) most properties P investigated in computability theory are such that if X satisfies P, then P is satisfied by infinitely many other Y's, (b) and because of this, in the context of computability theory, “d” often means “there are infinitely many”, even though that is not its literal meaning. Notice that the principle itself is not a theorem of computability theory e how could it be? Rather, it represents the outcome of the historical process of fixing the scope of computability theory.4 Things could have gone differently, producing hypothetical theories in which the meaning of the existential quantifier was restricted to its literal meaning. After all, it is obvious that not all existential quantifiers in mathematics convey the meaning “there are infinitely many”. Consider, for instance, Bertrand's postulate that states that for any integer n > 3 there is a prime number p with n < p < 2n 2. Here the expression “there is”, although being literally equivalent to the one contained in ð4Þ, cannot have any implicature of infinity. 3. Rigid vs nonrigid parameters The next example shows that the emergence of pragmatic phenomena is far from limited to the meaning of the existential quantifier. Consider the following sentence.
(6) Let x be the midpoint of y and z.
Now consider the following two contexts in which two close versions of ð6Þ can appear. (a) One of the equivalent formulations of the Bolzano-Weierstrass theorem states that every sequence s of real numbers which is bounded in a closed interval ½a; b has a convergent subsequence. The standard proof often starts by considering the midpoint c of a and b (hence formulating a version of ð6Þ; call it ð6aÞ), in order to find a subinterval I0 of ½a; b which contains infinitely many elements of s, and then picks the first element x0 of s that is in I0 . By iterating such
4 The case of computability theory is particularly relevant to our interests, since in theory one could eliminate all pragmatics from this field by requiring, for any constructed function, the full description of, say, a Turing-machine that computes it. Yet, the modern approach to computability theory orginates with Post (1944)'s seminal idea of eliminating any such reference to some background model of computation. The interested reader is referred to (San MauroPiazzaPulcini, 2018) for an extensive historical and philosophical recostruction of this point.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
5
Fig. 1. Tangent to a circle through a given point.
process one builds a nested sequence of closed subintervals ðIk Þk2u , and a subsequence ðxk Þk2u of s that must converge on the single element in ∩k2u Ik . (b) Consider the problem of the compass-and-straightedge construction of a tangent to a given circle of centre C through a given point P not belonging to the circle (see Fig. 1). The idea of the proof is the following. We construct the midpoint M of CP (this is where a version of ð6Þ comes in; call it ð6bÞ). Then we draw the circle with center M passing through C. Let Q be any of the two intersections of the two circles. Then PQ is a tangent to the circle with centre C passing through P. We claim that the information conveyed by the description of ð6aÞ and ð6bÞ differs to a degree that cannot be explained in terms of their (almost equivalent) literal meaning. Indeed, with respect to ðbÞ, we have that the term “midpoint” has to be taken in a rigid way. This is because, the reason for which PQ is a tangent to the given circle is that the angle :PQC is a right angle, which is granted by the fact that M is the midpoint of PC. If M was chosen differently, then the construction would have failed. On the other hand, the Bolzano-Weierstrass scenario is rather different. In ðaÞ the choice of c as the midpoint of a and b is nonrigid in the following sense. Suppose that, instead of splitting each interval Ik ¼ ½ik1 ; ik2 in two subintervals of the same ik ik ik ik length, we choose to split it in ik1 ; 2 3 1 and 2 3 1 ;ik2 . Then the construction would still produce a convergent subsequent. Moreover, the modified proof and the original one would be considered by practitioners as the same proof. So altough the literal meaning of ð6aÞ resembles that of ð6bÞ, the speaker's meaning of ð6aÞ conveys different information that, in order to be grasped, needs a certain amount of pragmatic work on the reader's side. Many proofs (if not all) contain nonrigid parameters and mathematicians often equate proofs differing just in minor details,5 by considering them as “the proof of X (via M)”, where X is a nontrivial theorem and M is a method for proving it. So the meaning of a proof, instead of denoting the single proof expressed by its literal meaning, is thus better understood as the composition of its literal meaning together with a class of “similar enough” proofs, where this notion of similarity is highlycontextual.6 But APT cannot account for this phenomenon. 4. Conclusion The examples provided above show that APT is untenable: even if one restricts one's attention to the fragment of mathematical language that seems to be so close to the formal side as to appear immune to pragmatic phenomena, such phenomena occur. That is to say, literal meaning is simply not sufficiently fine-grained to encode all possible shades of meaning provided by different mathematical contexts. Yet, after refuting APT a lot of philosophical work is still to be done. In fact, we believe that we have only scratched the surface. In future work, we aim to test to which extent pragmatic phenomena unveiled by philosophers of language apply also to the case of mathematics; in particular, in (Ruffino et al., 2019) we investigate the role of speech acts in mathematics.
5
Obviously different contexts might consider different proof-theoretic components as “minor”. Blass, Dershowitz, and Gurevich (Blass et al., 2009) offer a nice discussion on how two different algorithms can be regarded as “the same” relative to a given context e thus showing that the speaker's meaning of algorithm is often (much) more than its crude literal meaning. 6
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
6
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
Acknowledgments We are grateful to the reviewers for their comments and suggestions. References Blass, Andreas, Dershowitz, Nachum, Gurevich, Yuri, 2009. When are two algorithms the same? Bull. Symbolic Logic 15 (2), 145e168. Ganesalingam, Mohan, 2013. The Language of Mathematics. Springer. Giaquinto, Marcus, 2007. Visual Thinking in Mathematics. Oxford University Press. Grice, P., 1975. Logic and conversation. In: Cole, Morgan (Eds.), Syntax and Semantics 3: Speech Acts. Academic Press, pp. 41e58. Hersh, Reuben, 1991. Mathematics has a front and a back. Synthese 88 (2), 127e133. Lakatos, Imre, 1976. Proofs and Refutations: the Logic of Mathematical Discovery. Cambridge University Press. Mancosu, Paolo (Ed.), 2008. The Philosophy of Mathematical Practice. Oxford University Press, Oxford. Paul, Ernest, 2006. A semiotic perspective of mathematical activity: the case of number. Educ. Stud. Math. 61 (1e2), 67e101. Paul Van Bendegem, Jean, 1982. Pragmatics and mathematics or how do mathematicians talk? Philosophica 29. €ller, Regina (Eds.), 2018. The Philosophy of Paul, Ernest, Skovsmose, Ole, Paul van Bendegem, Jean, Bicudo, Maria, Miarka, Roger, Kvasz, Ladislav, Mo Mathematics Education. Springer. Post, Emil L., 1944. Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50 (5), 284e316. Rotman, Brian, 1988. Toward a semiotics of mathematics. Semiotica 72 (1e2), 1e36. Rowland, Tim, 2003. The Pragmatics of Mathematics Education: Vagueness and Mathematical Discourse. Routledge. Ruffino, Marco, San Mauro, Luca, Venturi, Giorgio, 2019. Speech Acts in Mathematics. Submitted. San Mauro, Luca, 2018. Church-Turing thesis, in practice. In: Piazza, Pulcini (Eds.), Truth, Existence and Explanation. Springer, pp. 225e248. Marco Ruffino is professor of philosophy at the University of Campinas (UNICAMP, Brazil) and the editor of Manuscrito, the Brazilian international journal for Analytic Philosophy. He is author of many contributions to important philosophical journals, like Mind, Erkenntnis, and Synthese. His area of specialization is in the areas of philosophy of language and history of philosophy; with particular emphasis in the history of analytic philosophy, and on Frege and Wittgenstein. Giorgio Venturi is professor of philosophy at the University of Campinas (UNICAMP, Brazil). In 2014 he obtained a PhD in mathematics from the Universite Paris 7 and a PhD in philosophy from the Scuola Normale Superiore. He publishes regularly on the main logic journals, like the Journal of Symbolic Logic, the Review of Symbolic Logic, and Studia Logica. His areas of specialization are set theory, modal logic, philosophy of mathematics, and the foundations of mathematics. Luca San Mauro is a Lise Meitner postdoctoral fellow at the Institute of Discrete Mathematics and Geometry of Vienna University of Technology and an adjunct professor of logic at the University of Siena. In 2016 he obtained a PhD in philosophy from the Scuola Normale Superiore. He publishes regularly on the main logic journals, such as the Journal of Symbolic Logic, the Journal of Logic and Computation, and the Archive for Mathematical Logic. His area of specialization are computability theory, computational learning, and philosophy of mathematics.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
Contents lists available at ScienceDirect
Journal of Pragmatics journal homepage: www.elsevier.com/locate/pragma
Discussion note
At least one black sheep: Pragmatics and mathematical language* Marco Ruffino a, Luca San Mauro b, *, Giorgio Venturi a a b
Department of Philosophy, University of Campinas, Brazil Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Austria
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2019 Received in revised form 25 December 2019 Accepted 23 January 2020 Available online xxx
In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis. © 2020 Elsevier B.V. All rights reserved.
Keywords Pragmatics Philosophy of mathematics The language of mathematics Existential quantifier Parameters
1. Introduction Let us start with a well-known joke. An engineer, a physicist, and a mathematician are on a train heading north, and had just crossed into Scotland. The engineer looks out of the window and says: “Look! Scottish sheeps are black!” The physicist says: “No, no. Some Scottish sheeps are black”. The mathematician looks up from the newspaper, turns their head, and says calmly: “There is at least one field, containing at least one sheep, of which at least one side is black”. : mathematicians are The joke (as in the case of most mathematical jokes) is based on the following over-used cliche (professionally) incapable of detaching themselves from literal meaning. Or, in other words, mathematical language lacks a feature that, according to many linguists and philosophers, characterizes most linguistic phenomena: a pragmatic dimension. This purported deficit in pragmatic aspects is so intertwined with a certain naïve description of mathematical language that, to our knowledge, it has been scarcely challenged philosophically. In this paper, we raise this challenge. We argue that pragmatic phenomena occur also in mathematics, in a peculiar fashion. We provide two main examples of this fact.
* The first author was supported by CNPq (Brazil) research grant 301721/2013-0. The second author was supported by Austrian Science Fund FWF through project M 2461. The third author was supported by FAPESP Jovem Pesquisador grant n.2016/25891-3. * Corresponding author. E-mail addresses: ruffi[email protected] (M. Ruffino), [email protected] (L. San Mauro), [email protected] (G. Venturi).
https://doi.org/10.1016/j.pragma.2020.01.011 0378-2166/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
2
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
1.1. Pragmatics and mathematics Pragmatics and mathematics scarcely overlap in the philosophical literature. On the one hand, philosophers of language rarely consider mathematical language as a possible case-study for testing their hypotheses. On the other hand, some philosophers of mathematics advocate a so-called “practical turn” in the field (see (Mancosu, 2008)), and there has been a growing interest in certain aspects of mathematics that are hardly reducible to its syntax or semantics (see Lakatos' seminal work (Lakatos, 1976), or Hersch (Hersh, 1991)). To name a couple of other examples: first, Rotman (1988) and Ernest (Paul, 2006) have written extensively about a semiotics of mathematics; secondly, the role of drawings and diagrams received much attention in recent years (see, e.g. (Giaquinto, 2007),). Van Bendegem (Paul Van Bendegem, 1982) explicitly addressed the interplay between pragmatics and mathematical practice. That said, it is still quite uncommon to investigate mathematical language with the sort of techniques developed by mainstream philosophy of language (e.g, speech acts, implicatures, pressupositions, and so forth). This is in part understandable. Some of the founding fathers of the philosophy of contemporary mathematics such as Frege, Hilbert and Russell placed a great emphasis on the need for a language free of those aspects of natural language that made it hard to produce and analyze proofs rigorously (e.g., vagueness, context-sensitivity, meaning-shifts, etc.). By placing such an emphasis they gave the impression that ideally mathematics must be done in a language completely “purified” from anything resembling speaker's intention, implicatures, illocutionary force, etc. Something curious, however, is that despite insisting on this contrast, Frege himself recognizes the indispensability of illocutionary force in logic (and, therefore, in mathematics) and incorporates two illocution;ary force indicators in the formal system of Begriffsschrift itself i.e., the assertion-sign w and the definition-sign y. There is no formula in the formal part of Frege that is left without the assertion-sign (being all of them therefore asserted). So, the relation between pragmatics and mathematical language is mostly unexplored. A rather crude way of justifying this lack of philosophical interest is to claim that there is nothing to be studied. This is the position of a recent book by Ganesalingam (2013), in which the following thesis appears: Mathematics do not exhibit any pragmatic phenomena: […] the meaning of mathematical language is always its literal meaning. We shall call this view the Antipragmatic Thesis (APT). Ganesalingam's book is a remarkable work. It proposes what is arguably the most extensive analysis of the semantics of mathematical language in the literature. It combines approaches coming from philosophy, linguistics, and computer science. Nevertheless, pragmatics is far less considered than semantics. In fact, APT is defended in a few pages, and the author concludes that the role of pragmatics in mathematical texts is negligible. In the rest of this paper, we will argue against this view. Pragmatics is a wide domain, and when talking about pragmatic phenomena in language one might be referring to distinct things such as presuppositions, speech acts, implicatures and discourse articulation. In this paper we shall concentrate on some examples that are probably better seen as cases of implicatures in mathematics, although, as we intend to show in later works, other pragmatic phenomena such as speech acts are also widespread.
1.2. Textual mathematics APT is a bold claim. But its plausibility depends on what we mean by “mathematical language”. If we posit that mathematical language is, or should be, the sort of language that Hilbert and Russell were pointing towards, then APT might be regarded as trivially true: that language was, in a sense, designed to be pragmatics-free. However, human mathematicians never use it. Moreover, Ganesalingam's analysis does not refer to such an idealized language, but rather focuses on the language by which mathematicians communicate their results. Yet, this definition would be so inclusive as to make APT patently false. Indeed, mathematicians communicate their results by several different means, some of which incorporate most (if not all) of the pragmatic devices that abound in natural language. To name a few: when delivering a lecture a teacher can rely, for explanatory purposes, on rhetorical figures, and speak metaphorically or even ironically.1 Similarly, a mathematician explaining the idea behind a certain construction will frequently depart from literal meaning if she considers that to be fruitful. The rough intuition here is that the more we move away from the formal side, the more we inevitably get caught up into pragmatics. A narrower definition of APT is that it applies to textual mathematics, that is, the kind of language in which standard mathematical papers are written. This is indeed Ganesalingam's focus, and it will be the focus of our present work as well. This context is better suited for APT since mathematical prose often aims at adhering to literal meaning. But again, textual mathematics, as a whole, is rife with complex linguistic constructions that make any hasty defence of APT problematic, to say the least. For example, it seems hard to account for the meaning of propositional attitude reports such as “we believe that” (that are widespread in mathematical discourse) if no reference to pragmatics is permitted.
1 Indeed, the role of pragmatics in mathematical education has been relatively explored; see (Rowland, 2003), or the recent volume “Philosophy of Mathematical Education” (Paul et al., 2018) where mathematical education and philosophy of mathematics interact.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
3
Then, a supporter of APT might try to restrict again the perspective, and say that the fragment of mathematics which is pragmatics-free is in fact that consisting of “generic (gnomic) sentences in the present simple tense”,2 as Ganesilagam puts it. Or, in other words, the fragment of mathematics which is pragmatics-free is the one consisting, roughly, of definitions, theorems, proofs, and so forth. The latter is a serious restriction. Yet, we aim to show that pragmatics phenomena arise even at the most fundamental level of mathematical discourse: the meaning of some logical constants such as quantifiers. Then, we move to the analysis of their role within proofs. An upshot of considering such distinguishing components of mathematical language (i.e., logical constants and proofs) is that the kind of pragmatic phenomena we unveil, altough similar to others occurring in natural language, will display peculiar characteristics as well. Nonetheless, let us stress that these pragmatic phenomena belong to mathematical language only if we regard the latter as the mixed language, combining natural and formal language, that is used in ordinary mathematics. So, a reader might want to rephrase the content of the present paper by saying that, rather than showing that pragmatic phenomena occur in mathematics, it actually proves that the utterances of mathematicians never lie entirely on the formal side of mathematical language e and therefore they encompass natural language phenomena, such as some pragmatic moves. We will not argue against this way of understading the following observations.
2. The meaning of the existential quantifier Far-side pragmatics (as opposed to near-side pragmatics) is the study of what contributes to the meaning of a sentence, beyond what the sentence literally means. A paradigmatic example in this direction is provided by what is called, after Grice,3 conversational implicatures. Consider the following dialogue.
(1)
A: Is there any wine left? B: Yes, there is a bottle in the kitchen.
B's answer is compatible, according to the literal meaning of the existential quantifier, with a scenario in which there are, in fact, n remaining bottles of wine, for n > 1. Yet, according to the classical view about implicatures, B conveys to A the information “there is precisely one bottle of wine”, altough not literally saying so. Similar examples are widespread in natural language. This motivates the standard idea of distinguishing between the literal meaning of a given sentence, and the (possibly distinct) speaker's meaning expressed by an utterance of the sentence. Like many important philosophical distinctions, the one between literal and speaker's meaning is not clear cut: for instance, it leaves obviously undefined to which extent someone has to depart from conventional meaning to be considered non-literal. Yet, it is convenient to stress that words and sentences often mean more than their literal meaning. Since we will deal with mathematical language (and in fact with the portion of mathematical language that corresponds to textual mathematics), we shall assume that the literal meaning of a mathematical term corresponds to the meaning assigned to it by its formal definition. In (Ganesalingam, 2013), Ganesalingam claims that pragmatics phenomena as such simply do not occur in mathematics. As an illustration for this claim he presents the following sentence.
(2) The group of symmetries of the cube contains a non trivial subgroup.
In natural language the implicature would be that there is exactly one such subgroup (by reasoning similarly to ð1Þ). Yet, mathematicians read ð2Þ as equivalent to
(3) the group of symmetries of the cube contains at least one non trivial subgroup.
This is true. But this does not imply that the speaker's meaning and the literal meaning of mathematical sentences always coincide. Consider the following theorem from computability theory.
(4) Any infinite computably enumerable set contains an infinite computable subset.
2 3
(Ganesalingam, 2013), p. 21. See (GriceColeMorgan, 1975).
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
4
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
The literal meaning of this sentence is equivalent to saying that there is at least one infinite computable subset. Yet, such a sentence is understood by any practitioner as conveying the following information.
(5) There are infinitely many computable subsets of a given computably enumerable set.
Prima facie, ð5Þ holds because any finite modification of a computable set is still computable. But there is something more profound here. Suppose that a reader R, perhaps unfamiliar with the field, fails to go beyond the literal meaning of ð4Þ. Then R might think that proving that any computably enumerable set A contains two infinite computable subsets is ampliative with respect to the information conveyed by ð4Þ. However computable theorists typically choose to present their results as concerning the construction of one set, instead of the construction of infinitely many satisfying the same class of requirements. A similar choice has of course many benefits in terms of exposition (for one, it makes constructions more readable) e yet it would not really be a viable choice, if literal meaning exhausted the meaning of any mathematical sentence. Indeed, by only considering literal meaning, one cannot make sense of the kind of omissions which abound in computability theory. Consider as another example the following classical theorem.
(6) There is a low simple set.
This sentence and its proof are typically stated with no need of specifying that minor modifications of the construction allow to build, in fact, infinitely many such sets. And it is perfectly fine to rely only on ð6Þ if one wants to consider two different low simple sets. The crucial point here is thus that in order to grasp the speaker's meaning encoded by ð4Þ or by ð6Þ, the reader has to be aware of the following two-pronged principle: (a) most properties P investigated in computability theory are such that if X satisfies P, then P is satisfied by infinitely many other Y's, (b) and because of this, in the context of computability theory, “d” often means “there are infinitely many”, even though that is not its literal meaning. Notice that the principle itself is not a theorem of computability theory e how could it be? Rather, it represents the outcome of the historical process of fixing the scope of computability theory.4 Things could have gone differently, producing hypothetical theories in which the meaning of the existential quantifier was restricted to its literal meaning. After all, it is obvious that not all existential quantifiers in mathematics convey the meaning “there are infinitely many”. Consider, for instance, Bertrand's postulate that states that for any integer n > 3 there is a prime number p with n < p < 2n 2. Here the expression “there is”, although being literally equivalent to the one contained in ð4Þ, cannot have any implicature of infinity. 3. Rigid vs nonrigid parameters The next example shows that the emergence of pragmatic phenomena is far from limited to the meaning of the existential quantifier. Consider the following sentence.
(6) Let x be the midpoint of y and z.
Now consider the following two contexts in which two close versions of ð6Þ can appear. (a) One of the equivalent formulations of the Bolzano-Weierstrass theorem states that every sequence s of real numbers which is bounded in a closed interval ½a; b has a convergent subsequence. The standard proof often starts by considering the midpoint c of a and b (hence formulating a version of ð6Þ; call it ð6aÞ), in order to find a subinterval I0 of ½a; b which contains infinitely many elements of s, and then picks the first element x0 of s that is in I0 . By iterating such
4 The case of computability theory is particularly relevant to our interests, since in theory one could eliminate all pragmatics from this field by requiring, for any constructed function, the full description of, say, a Turing-machine that computes it. Yet, the modern approach to computability theory orginates with Post (1944)'s seminal idea of eliminating any such reference to some background model of computation. The interested reader is referred to (San MauroPiazzaPulcini, 2018) for an extensive historical and philosophical recostruction of this point.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
5
Fig. 1. Tangent to a circle through a given point.
process one builds a nested sequence of closed subintervals ðIk Þk2u , and a subsequence ðxk Þk2u of s that must converge on the single element in ∩k2u Ik . (b) Consider the problem of the compass-and-straightedge construction of a tangent to a given circle of centre C through a given point P not belonging to the circle (see Fig. 1). The idea of the proof is the following. We construct the midpoint M of CP (this is where a version of ð6Þ comes in; call it ð6bÞ). Then we draw the circle with center M passing through C. Let Q be any of the two intersections of the two circles. Then PQ is a tangent to the circle with centre C passing through P. We claim that the information conveyed by the description of ð6aÞ and ð6bÞ differs to a degree that cannot be explained in terms of their (almost equivalent) literal meaning. Indeed, with respect to ðbÞ, we have that the term “midpoint” has to be taken in a rigid way. This is because, the reason for which PQ is a tangent to the given circle is that the angle :PQC is a right angle, which is granted by the fact that M is the midpoint of PC. If M was chosen differently, then the construction would have failed. On the other hand, the Bolzano-Weierstrass scenario is rather different. In ðaÞ the choice of c as the midpoint of a and b is nonrigid in the following sense. Suppose that, instead of splitting each interval Ik ¼ ½ik1 ; ik2 in two subintervals of the same ik ik ik ik length, we choose to split it in ik1 ; 2 3 1 and 2 3 1 ;ik2 . Then the construction would still produce a convergent subsequent. Moreover, the modified proof and the original one would be considered by practitioners as the same proof. So altough the literal meaning of ð6aÞ resembles that of ð6bÞ, the speaker's meaning of ð6aÞ conveys different information that, in order to be grasped, needs a certain amount of pragmatic work on the reader's side. Many proofs (if not all) contain nonrigid parameters and mathematicians often equate proofs differing just in minor details,5 by considering them as “the proof of X (via M)”, where X is a nontrivial theorem and M is a method for proving it. So the meaning of a proof, instead of denoting the single proof expressed by its literal meaning, is thus better understood as the composition of its literal meaning together with a class of “similar enough” proofs, where this notion of similarity is highlycontextual.6 But APT cannot account for this phenomenon. 4. Conclusion The examples provided above show that APT is untenable: even if one restricts one's attention to the fragment of mathematical language that seems to be so close to the formal side as to appear immune to pragmatic phenomena, such phenomena occur. That is to say, literal meaning is simply not sufficiently fine-grained to encode all possible shades of meaning provided by different mathematical contexts. Yet, after refuting APT a lot of philosophical work is still to be done. In fact, we believe that we have only scratched the surface. In future work, we aim to test to which extent pragmatic phenomena unveiled by philosophers of language apply also to the case of mathematics; in particular, in (Ruffino et al., 2019) we investigate the role of speech acts in mathematics.
5
Obviously different contexts might consider different proof-theoretic components as “minor”. Blass, Dershowitz, and Gurevich (Blass et al., 2009) offer a nice discussion on how two different algorithms can be regarded as “the same” relative to a given context e thus showing that the speaker's meaning of algorithm is often (much) more than its crude literal meaning. 6
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011
6
M. Ruffino et al. / Journal of Pragmatics xxx (xxxx) xxx
Acknowledgments We are grateful to the reviewers for their comments and suggestions. References Blass, Andreas, Dershowitz, Nachum, Gurevich, Yuri, 2009. When are two algorithms the same? Bull. Symbolic Logic 15 (2), 145e168. Ganesalingam, Mohan, 2013. The Language of Mathematics. Springer. Giaquinto, Marcus, 2007. Visual Thinking in Mathematics. Oxford University Press. Grice, P., 1975. Logic and conversation. In: Cole, Morgan (Eds.), Syntax and Semantics 3: Speech Acts. Academic Press, pp. 41e58. Hersh, Reuben, 1991. Mathematics has a front and a back. Synthese 88 (2), 127e133. Lakatos, Imre, 1976. Proofs and Refutations: the Logic of Mathematical Discovery. Cambridge University Press. Mancosu, Paolo (Ed.), 2008. The Philosophy of Mathematical Practice. Oxford University Press, Oxford. Paul, Ernest, 2006. A semiotic perspective of mathematical activity: the case of number. Educ. Stud. Math. 61 (1e2), 67e101. Paul Van Bendegem, Jean, 1982. Pragmatics and mathematics or how do mathematicians talk? Philosophica 29. €ller, Regina (Eds.), 2018. The Philosophy of Paul, Ernest, Skovsmose, Ole, Paul van Bendegem, Jean, Bicudo, Maria, Miarka, Roger, Kvasz, Ladislav, Mo Mathematics Education. Springer. Post, Emil L., 1944. Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50 (5), 284e316. Rotman, Brian, 1988. Toward a semiotics of mathematics. Semiotica 72 (1e2), 1e36. Rowland, Tim, 2003. The Pragmatics of Mathematics Education: Vagueness and Mathematical Discourse. Routledge. Ruffino, Marco, San Mauro, Luca, Venturi, Giorgio, 2019. Speech Acts in Mathematics. Submitted. San Mauro, Luca, 2018. Church-Turing thesis, in practice. In: Piazza, Pulcini (Eds.), Truth, Existence and Explanation. Springer, pp. 225e248. Marco Ruffino is professor of philosophy at the University of Campinas (UNICAMP, Brazil) and the editor of Manuscrito, the Brazilian international journal for Analytic Philosophy. He is author of many contributions to important philosophical journals, like Mind, Erkenntnis, and Synthese. His area of specialization is in the areas of philosophy of language and history of philosophy; with particular emphasis in the history of analytic philosophy, and on Frege and Wittgenstein. Giorgio Venturi is professor of philosophy at the University of Campinas (UNICAMP, Brazil). In 2014 he obtained a PhD in mathematics from the Universite Paris 7 and a PhD in philosophy from the Scuola Normale Superiore. He publishes regularly on the main logic journals, like the Journal of Symbolic Logic, the Review of Symbolic Logic, and Studia Logica. His areas of specialization are set theory, modal logic, philosophy of mathematics, and the foundations of mathematics. Luca San Mauro is a Lise Meitner postdoctoral fellow at the Institute of Discrete Mathematics and Geometry of Vienna University of Technology and an adjunct professor of logic at the University of Siena. In 2016 he obtained a PhD in philosophy from the Scuola Normale Superiore. He publishes regularly on the main logic journals, such as the Journal of Symbolic Logic, the Journal of Logic and Computation, and the Archive for Mathematical Logic. His area of specialization are computability theory, computational learning, and philosophy of mathematics.
Please cite this article as: Ruffino, M et al., At least one black sheep: Pragmatics and mathematical language, Journal of Pragmatics, https://doi.org/10.1016/j.pragma.2020.01.011