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Auteur: Koji ARIKAWA

Variable elimination in language system and mathematics

Abstract/Résumé: Alfred Russel Wallace (1823–1913), the co-founder of the theory of evolution, was puzzled by the fact that the "gigantic development of the mathematical capacity is wholly unexplained by the theory of natural selection." Noam Chomsky suggests that the mathematical capacity is derivative from language. Gaussian elimination in the computational procedure of mathematics (CM) is the most elementary algorithm for solving simultaneous linear equations such as (1) ax + by = c and (2) dx + ey = f. I propose that CM has AGREE (MATCH + VALUE + DELETE) as CHL (contra Chomsky). To eliminate the variable x in (2), we first eliminate the coefficient d. We seek a multiplier M (matchmaker) that satisfies d – aM = 0. This 0 becomes the new coefficient for x in (2). Since 0x = 0, x is eliminated. Eliminating x, values y ( "irreflexive" elimination). There is no need to eliminate every x ("partial" elimination). Back substituting y's value, we get x's value. In CHL, there are three types of uF: (i) uPHI-set in V/v* and T/C, (ii) CASE in DP, and (iii) EPP in heads. uPHI has an interpretable matching counterpart PHI in DP; it is valued by PHI ("direct" back substitution?) and deleted because of redundancy ("reflexive" elimination; look-ahead?) When uPHI is deleted, every relevant uPHI is deleted ("total" elimination). CASE does not have an interpretable matching counterpart, but is valued by a head ("direct" back substitution?) and deleted as a reflex (?) of uPHI-deletion. EPP does not have an interpretable matching counterpart and is not valued; it is deleted because of FI (look-ahead?). Suppose that CHL and CM share an elimination process. Consider the v*P phase with a two-place transitive V, leaving aside EPP (coefficient?). CHL solves the simultaneous equations: (1') auPHI + bCASE = c and (2') duPHI + eCASE = f. To eliminate uPHI in (2'), we first eliminate d. We seek M such that d – aM = 0. Suppose that M = PHI in object DP; then d – aPHI = 0. This 0 becomes the new coefficient for uPHI in (2'). Since 0uPHI = 0, uPHI is eliminated. Eliminating uPHI, values CASE, which is a linear algebraic expression of CASE as a "reflex" of uPHI-deletion. Elimination in CM fails when the elimination of d in (2) produces 0y = f', where f' ≠ 0. The two lines (each expressing (1) and (2)) become parallel, i.e., no intersection (solution). This corresponds to (2'), where d = e = 0 and both uPHI and CASE are eliminated. A candidate head is defective V (unaccusative/passive/participle) with incomplete-PHI, which does not enter into the agreement process (enables successive cyclic movement). Elimination in CM may produce infinitely many solutions if (1) = (2). The two lines pile up. This corresponds to (1') = (2'). A candidate is EPP, which is satisfied by anything and in any way (obligatorily or optionally).