David Pimm represents a pretty interesting link between relevance theory and the ideas in Where Mathematics Comes From. In fact, he seems to set some of them up quite nicely, so it’s surprising that they don’t talk about him.
He’s referenced in Martin Schiralli and Nathalie Sinclair – A Constructive Response to ‘Where Mathematics Comes From’ (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.136.3658&rep=rep1&type=pdf) as a proponent of the centrality of metaphor to mathematics. Their paper is an interesting critique of WMCF:
“We will also argue that Lakoff and Nuñez’s methodology of “mathematical idea analysis” is insufficient to adequately describe the nature of mathematical concepts; for this, further empirical research is necessary, especially research that can probe the very idiosyncratic nature of students’ individual conceptions.” p. 80
“WMCF does not differentiate the term ‘mathematics,’ nor indicate whether metaphor might function differently depending on whether one is learning, doing, or using mathematics.” p. 81.
They distinguish Conceptual Mathematics – the public discipline – from the way an individual represents these concepts to themself.
“Lakoff and Nuñez do not distinguish IM from CM. In fact, they imply that ideational and conceptual mathematics will be isomorphic by claiming that the metaphors on which mathematics is based are not at all arbitrary. That is, grounding metaphors are forced on us by our physical nature, and metaphorical mappings, blends and special cases have a stable, precise structure (see p. 375).In contrast, we propose that at least two diversifying factors come into play during the evolution of ideational mathematics which make WMCF’s implication untenable. First, as mentioned above, metaphor is not the only meaning-making strategy used in mathematics. Second, metaphorical mappings can also be made to and from the ongoing experiences (including non-mathematical ones) of the mathematician.” p. 84“We do not deny the power and pervasiveness of metaphor; rather, we seek to understand the processes by which metaphors are constructed by learners. These processes may have much in common with the triggering mechanisms discussed above. In fact, the basic human acts of understanding identified by Sierpinska (1994), based partly on the empirical work of Vygotsky, might provide some insight into these processes. The four acts of understanding – identification, discrimination, generalisation and synthesis – operate from a very early age; however, both the objects of the acts and the rules governing their use, become more sophisticated with age and experience.” p. 88
They criticise WMCF for its exclusive use of ‘mathematical idea analysis’, a “linguistics-based technique – based almost entirely on standard utterances in textbooks and curricula – of revealing the metaphorical sources of mathematical ideas”. That paper’s a good read.
Dierdre Wilson’s Parallels and differences in the treatment of metaphor in relevance theory and cognitive linguistics is a contribution to an effort to reconcile the Cognitive Linguistics view that metaphorical speech reveals underlying cognition with the relevance theory view that:
“there is a continuum of cases between literal talk, loose talk, hyperbole and metaphor, none of which is necessarily a surface reflection of any pre-existing conceptual mapping.” p. 42
“… while cognitive linguists tend to assume that understanding utterances is simply a matter of applying general-purpose cognitive and linguistic abilities to the communicative domain, relevance theorists have argued that understanding utterances involves special-purpose inferential procedures that apply only in the communicative domain.” p. 52
“… hearers understand linguistic metaphors by using linguistic and contextual clues to create new ‘ad hoc’ (occasion-specific) concepts, which are typically not identical to any of the concepts linguistically encoded by the metaphorically-used word or phrase, although they inherit some of their inferential properties from those concepts.”p. 42
“Relevance theory’s treatment of metaphor is part of a more general approach to lexical pragmatics which is based on the following assumptions. First, the lexical meaning of a word is merely a clue to the speaker’s meaning, and the concept communicated by use of a word typically differs from the lexical meaning. Second, metaphor is just one of many ways in which lexical meanings can be modified in use. The concept communicated by use of a word may be narrower (more specific) or broader (more general) than the lexical meaning (or it may be narrower in some respects and broader in others, as is often the case in metaphor). Third, there is a continuum of cases of broadening, ranging from strictly literal use, through various shades of approximation to hyperbole and metaphor, with no sharp cut-off point between them. Fourth, all these cases are interpreted in the same way: there are no special pragmatic principles or mechanisms that apply only to metaphors. And fifth, contrary to what is generally assumed in Gricean pragmatics and philosophy of language, the concept communicated by use of a word contributes to what the speaker is taken to have asserted (i.e. the truth-conditional content of the utterance), and not only to what is implicated (Wilson & Carston 2007; Sperber & Wilson 2008). Since metaphorical uses of language – just like strictly literal uses – contribute to truth-conditional content and fall within the scope of logical connectives, they cannot be dismissed as marginal to the concerns of linguistics proper.” p. 44
” The explanation suggested by relevance theory is that lexical meanings are adjusted in order to satisfy expectations of relevance.” p. 47
This description of ad-hoc concepts echoes Schiralli and Sinclair’s quotation of Wittgenstein:
“Importantly here, Wittgenstein (1953) shows that there need not be a common set of necessary and sufficient conditions respecting all legitimate uses of a word. And as words ‘map’ concepts in the patterns of their uses, there need not be a single ‘abstractable’ entity (image or form) for a word/concept at work at all in much productive abstract thought. To think abstractly is to think with concepts. And thinking with concepts is not the same as thinking about the ideas or images those concepts may occasion in us.” p. 83
Some of Wilson’s ideas might satisfy Schiralli and Sinclair’s desire “… to understand the processes by which metaphors are constructed by learners.” Wilson argues that relevance theory addresses two important problems with Cognitive Linguistics:
“In the first place, cognitive linguists face a major challenge in explaining how hearers not only understand most metaphorical utterances, but typically understand them in the way the speaker intended. … In the second place, although cognitive linguists and relevance theorists have both emphasised the importance of inference in metaphor interpretation, cognitive linguists face a major challenge in explaining how the inferences that hearers draw in the course of utterance comprehension are properly warranted.” p. 54
She argues that the sorts of ‘conceptual mappings’ that they describe might be set up in the following way:
“According to relevance theory, the lexical meaning of virtually every word in an utterance is contextually adjusted in order to satisfy expectations of relevance…
Repeated encounters with linguistic metaphors linking two conceptual domains (e.g. the domains of marriage and voyages, or women and flowers) may lead to the setting up of systematic cross-domain correspondences of the type familiar from cognitive linguistics, so that thoughts of marriage may automatically activate aspects of our encyclopaedic information about journeys, and thoughts of women may automatically activate aspects of our encyclopaedic knowledge of flowers, just as cognitive linguists predict. These crossdomain correspondences would in turn facilitate the production and interpretation of new linguistic metaphors based on the same conceptual activation patterns, resulting in thematically-related clusters of linguistic metaphors, just as cognitive linguists predict.” p. 52
There’s a good response to Wilson’s paper – Coupling of metaphoric cognition and communication: A reply to Deirdre Wilson by Raymond W. Gibbs, JR. and Markus Tendahl – which gives some evidence for metaporical mappings in non-linguistic domains, and argue for a close coupling between thought and language rather than claiming that processes are exclusive to language.
“The question is, “do we think of disagreeable things as DIRTY because there are linguistic metaphors that make this connection, or are the linguistic metaphors simply an expression of a preexisting mental connection?” (Stefanowitsch 2011: 302). Yet this sort of “chicken/egg” issue need not be resolved only one way or the other.” p. 605
“… as is the case with many private emotional experiences (Fridlund 1994), people’s thinking without speaking episodes are implicitly social both for purposes of communicating with ourselves and possibly interacting with others at a later time.” p. 606
Back to Pimm – his approach might be nicely combined with relevance theory, as he talks about how a more experienced mathematician might treat one thing as another to reduce cognitive processing. He talks, for instance, about treating a symbol as an object, and some of the linguistic twists used by teachers that belie this attitude. A pupil might treat the equivalence as literal, and sometimes come a cropper as a result, but a more experienced mathematician uses contextual knowledge to narrow down the scope of the equivalence, qualifying it and discarding it later on. You say it’s an object – but only in these ways and situations! Those which are relevant, because they reduce processing relative to results. This example refers to a person’s self-explanation or understanding, so I suppose I’m with Gibbs and Tendhal right now.
On p. 99 – 106, Pimm considers a number of ‘structural metaphors’ in mathematics, such as that of a complex number as a vector. Pupils noticed the resemblance of their diagramming of a complex number to a vector. They then wanted to map over extra characteristics from their understanding of vectors, like using the term resultant, but this was gently discouraged as this term is not conventionally used for complex numbers. Therefore an incomplete mapping was built up. This teaching-metaphor may well be more heavy-handed than what Wilson had in mind.
A pupil learning how to understand a piece of mathematics at first won’t know which aspects of the expressions will behave in the way expected – in the case of the Leibniz notation for the derivative, dy/dx, they don’t yet know that the superficial resemblance to a fraction won’t allow them to treat the expression like a fraction – “the metaphor a derivative is a fraction is not completely helpful,” though in other ways (eg. taking reciprocal) it is [p. 159]. However an astute learner will quickly realise that writing a fraction in this way would be highly irrelevant – it would immediately need cancelling, so why write it like that in the first place? They will therefore immediately remove some properties from their ad-hoc concept.
I’m not sure about some of this. I want to consider some of these examples further, and see what I can pick apart.