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Auteur: Anna-Maria DI SCIULLO

The great leap forward and the emergence of complex numerals

Abstract/Résumé: I argue that the ability for the human mind to compute complex numerals is a consequence of the great leap from finite and continuous systems, such as the gestural system, to systems of discrete infinity, such as language, mathematics and music. I further argue that the sub-systems of the brain sub-serving grammar on the one hand, and mathematics on the other provide complementary dimensions of complex numerals. 1. The set of complex numerals is unbounded (Zwicky1963, Brainerd 1971, Radzinsky 1991, contra Merrifield 1968, Greenberg 1978, a.o.). This set is derived by Merge, the central operator of the Language Faculty, a dyadic, recursive and unbounded operation, deriving ordered pairs, according to Chomsky (2008). I focus on the recursive properties of coordinate and prepositional part-whole structures in complex numerals from different languages, where F is a functional head (conjunction, preposition, cardinal affix, Case) and Num is a lexical or a complex numeral. Simplex and complex numerals merge with a functional head at each step of the derivation, as proposed in Di Sciullo (2012). This is expected in a model where two maximal projections may never merge directly, but only indirectly via a functional projection. 2. The sub-system of the brain sub-serving mathematical reasoning provides a computation of complex numerals. I argue that while the cardinality of a complex numeral is located at the periphery of each sub-constituent, the arithmetic operators (+, x) are legible at the language-mathematics interface as features of the functional heads F asymmetrically relating the parts of complex numerals. 3. The derivation of complex numerals by the Language Faculty is subject to principles of efficient computation, such as phases and ‘pronounce the minimum’ Chomsky (2005, 2011). I argue further that principles of symmetry breaking are also part of these principles, ensuring that only the configurations whose components are asymmetrically related are legible by the external systems. 4. Brain imaging results indicate that processing hierarchically structured mathematical formulae and processing complex syntactic hierarchies in language activate different areas of the brain (Friederici et al 2011, Friedrich & Friederici 2009). The fact that complex numerals are composed of asymmetrical substructures headed by F heads suggests that they are processed by the part of the brain that sub-serves language. The fact that these F heads correspond to arithmetic operators, even if not pronounced, indicates that the algebraic computation of complex numerals is provided by the part of the brain that sub-serves mathematical reasoning.