Translation, Quotation and Truth :: Mathematical Mathematics Math Essays

Translation, Quotation and Truth ABSTRACT: If logical truth is truth due solely to syntactic form, then mathematics is distinct from logic, even if all mathematical truths are derivable from definitions and logical principles. This is often obscured by the plausibility of the Synonymy Substitution Principle that is implicit in the Fregean conception of analyticity: viz., that synonyms are intersubstitutable without altering sentence sense. Now, unlike logical truth, mathematical truth is not due to syntax, so synonym interchange in mathematical truths preserves sentence syntax, sense, and mathematical necessity. Mathematical necessity, therefore, differs from both logical and lexical necessity. Alonzo Church's Translation Test is supposed to evaluate alleged necessary equivalences. The basic idea is that (correct) translation is meaning preserving, so if some disparity between the alleged equivalents appears under translation, the sentences are inequivalent. The Test translations are essentially synonym substitutions. The translation sentence is to have the syntax of the original, and to differ only morphemically, by synonym subbing. Supposedly, whether the translations are inter-language or intra-language has no logical import. Either way, the underlying and seemingly unassailable principle is that synonym subbing transmits sentence sense. This Synonym Substitution Principle extends to subbing of coreferring proper names by taking their meaning to be their reference. The Test's derivation from this Principle should spark suspicions for two reasons. First, the Principle seems to entail the Test's inutility, for, presumably, any disparity appearing under translation but not appearing in the original must be a discrepancy in and of the translation. If the Principle is platitudinous, the Test should be profitless. This isn't the old Paradox of Analysis. Synonymies aren't analyses. Genuine analyses are translanguistic necessary truths knowable a priori; synonymies are contingent notational equivalences knowable only a posteriori. Unlike analysis, the assumption that translation can be informative for a competent bilingual needs justification. The Test's pet targets are peculiarly Carnapian theses: e.g., that belief attributions like Mother believes Roger is nearby are semantic equivalents of speech attributions like Mother accepts "Roger is nearby". The Test's plausibility and popularity may owe much to its confirming incredulity about independently refutable theses. We knew pre-Church that my dog Mother's believing her master is nearby doesn't entail any wagging at an English sentence. Another curiosity in the Test is that translation seems to reveal disparities not otherwise evident solely with talk about talk and thought, in the relation between direct and indirect discourse. The Test gets touted only with instantiations of alleged logical or analytical principles wherein, allegedly, a quotation embedding sentence mutually entails a quotationless sentence.