Abstract
The concept of linguistic infinity has had a central role to play in foundational debates within theoretical linguistics since its more formal inception in the mid-twentieth century. The conceptualist tradition, marshalled in by Chomsky and others, holds that infinity is a core explanandum and a link to the formal sciences. Realism/Platonism takes this further to argue that linguistics is in fact a formal science with an abstract ontology. In this paper, I argue that a central misconstrual of formal apparatus of recursive operations such as the set-theoretic operation merge has led to a mathematisation of the object of inquiry, producing a strong analogy with discrete mathematics and especially arithmetic. The main product of this error has been the assumption that natural, like some formal, languages are discretely infinite. I will offer an alternative means of capturing the insights and observations related to this posit in terms of scientific modelling. My chief aim will be to draw from the larger philosophy of science literature in order to offer a position of grammars as models compatible with various foundational interpretations of linguistics while being informed by contemporary ideas on scientific modelling for the natural and social sciences.
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Notes
However, the incompleteness proof did not destroy proof theory itself. Gentzen developed a proof-theoretic approach without the limitations of the Hilbert programme and finitism in full view of Gödel’s incompleteness. His more specific aim was to prove the consistency of logical deduction within arithmetic.
I will return to this analogy in Sect. 8. On the philosophical side, I think that mathematisation is related to the treatment of the rules posited initially for the sake of modelling a phenomenon and then eventually for their own sake. A similar Wittgensteinian diagnosis of the issue within the context of the foundations of linguistics can be found in Wright (1989). Simply put, it is a problem of rule-following. The problem of mathematisation can essentially be viewed as the progression and instantiation of the myth of the autonomy of rules or “the image of a rule as a rail laid to infinity” (Wright 1989: p. 238).
Although as early as Syntactic Structures, Chomsky distinguishes between recursion and computation in constructing a non-recursive grammar of a small fragment of English. I thank an anonymous referee for pointing this out to me.
The history of the term “recursion” in linguistics is extremely messy. Briefly, the idea is that recursive functions introduce a property of self-reference. This involves two steps. One which specifies the condition of termination of the recursion or the base case and the recursive step which reduces all other cases to the base. For some early attempts at formal definition, see Chomsky (1959) and Chomsky and Miller (1963) for an elaboration of the term language and recursion in linguistics. For a development of these ideas within recursive characterisations of particular linguistic constructions see Langendoen (2008).
Notice, even Chomsky’s famous (1956) disavowal of the relevance of stochastic grammar formalisms, in which approximation through continuous mathematics is the goal, can be seen as motivated by mathematisation. The statistical methods of continuous mathematics do not generally make a mathematical object of the target domain but rather treat it as physical process capable of “approximate” characterisation.
As pointed out to me by an anonymous reviewer, Merge itself should not be thought of as yielding infinite output, as a source of recursion or link to arithmetic but rather recursion must be discovered in the atoms and inherited by the objects constructed by the Merge operation. There could be two reasons for this caution. Firstly, in allowing infinitely many Merges you can generate infinite objects, but \(\aleph _{0}\) is not an integer itself. Secondly, if we have a constraint such that Merge(\(\alpha ,\beta \)) can only be applied if there is neither an \(\alpha \) nor a \(\beta \) present in either \(\alpha \) or \(\beta \), Merge will run out of options resulting in a finite number of contructable objects.
Pullum and Scholz (2010) convincingly argue, no such proof can be given sans the induction axiom and successor function in the case of natural language.
This situation then moves into the realm of different cardinalities for different languages, some transfinite others finite. I thank an anonymous reviewer for directing me towards this possibility.
Yablo (2013) offers a related account of the three grades of mathematical involvement for scientific explanation. Yablo’s three grades are roughly and respectively defined as follows: on grade one, mathematics has a descriptive role (something like the first grade on my view but more limited), on grade two it has a structural role and on the third grade it has a substantive role. He attempts to capture the substantive role in terms of a modal notion of extricability. We can think of extricability in terms of logical subtraction. “Logical subtraction sometimes yields a well-defined remainder, surely. Snow is cold and white – Snow is cold = Snow is white, I assume” (Yablo 2013: p. 1014).
See Weisberg (2013) for a discussion on how the large scale Bay Area Model of San Fransciso assisted scientists in rejecting the proposal to build a dam in the Bay Area.
Another way to think of what a model is involves an analogy with fictional worlds, pretenses or ways that the world could have been (Frigg 2010). This view breaks down the connection with model theory in mathematics. In this way models are akin to the fictional worlds of Sherlock Holmes or Luke Skywalker. Counterfactual analyses are also generally connected to the type of representation involved in modelling. For instance, Giere (1988) affirms that model systems are systems which would be concrete if they were in fact real.
As opposed to the then popular “syntactic” accounts in which scientific theories were considered to be consistent sets of sentences in formal languages (“theories” in the logical sense).
See Thomson-Jones (2005) for general discussion. Also see Stokhof and van Lambagen (2011) for discussion with relation to linguistics and Nefdt (2016a) for a response.
Here they borrow from Savitch (1993) who shows why we might assume that languages are (essentially) infinite despite having no evidence for them not being simply largely finite. Savitch’s paper is a formal attempt at capturing parsimony judgements in grammars, i.e. we treat finite sets as essentially infinite if this allows us to get simpler descriptions than we would if we treated them as finite.
In fact, adherence to a strong interpretation of C3 can lead us astray in some cases. There is a school of thought which takes infinity or fixed cardinality not only to be a modelling choice, as in the previous section, but to be a feature of the particular mathematical model used in linguistic theory, namely Post canonical production systems. Thus, infinity is an artefact of the model. It is obvious that not all the artefacts of models should receive interpretation in the target system. Especially if productivity facts can be captured by alternative formalisms which do not posit the putative property. For a specific example, Sampson (2001) criticises Chomsky’s problematic “undue preoccupation with strings” in The Logical Structure of Linguistic Theory (1975). He points out that treating the syntax via derivations of strings and sets of strings is an unnecessary detour when phrase-structure grammars could be characterised with well-formedness conditions on trees directly. Furthermore, the derivational alternative forces certain untoward consequences.
Chomsky’s approach forces him to impose two quite arbitrary restrictions on phrase-structure rules, namely, that no rule may rewrite any symbol A as either the null string, or as a sequence including A. Both of these forbidden types of rule frequently seem appropriate in describing real language, and under the alternative view of phrase-structure grammars there is no objection to them (Sampson 2001: p. 156).
In this case, the model has features that the real world does not. In many other cases, the target has features which outstrip the models. Trying to find a home for every feature of the model as a mechanism or constraint might turn out to be deeply problematic.
Of course, this is not always the case. The consistency of the Continuum hypothesis and its negation do not establish existence.
Incidentially, this rare sequence of word order can be found in some languages along the Amazon basin.
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Acknowledgements
I would like to thank Josh Dever, Ephraim Glick, Patrick Greenough, Geoff Pullum, Kate Stanton, Zoltán Szabó, Bernhard Weiss and two anonymous referees for their insightful comments on various drafts of this paper. I would also like to thank audiences at the Arché Research Centre and the joint Semantics Seminar at the linguistics and philosophy departments at Yale University for their excellent comments and suggestions on this research.
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Nefdt, R.M. Infinity and the foundations of linguistics. Synthese 196, 1671–1711 (2019). https://doi.org/10.1007/s11229-017-1574-x
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DOI: https://doi.org/10.1007/s11229-017-1574-x