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We explore a quantitative interpretation of 2-dimensional intuitionistic typetheory (ITT) in which the identity type is interpreted as a "type ofdifferences". We show that a fragment of ITT, that we call difference typetheory (dTT), yields a general logical framework to talk about quantitativeproperties of programs like approximate equivalence and metric preservation. Todemonstrate this fact, we show that dTT can be used to capture compositionalreasoning in presence of errors, since any program can be associated with a"derivative" relating errors in input with errors in output. Moreover, afterrelating the semantics of dTT to the standard weak factorization systemssemantics of ITT, we describe the interpretation of dTT in some quantitativemodels developed for approximate program transformations, incrementalcomputing, program differentiation and differential privacy.

From Identity to Difference: A Quantitative Interpretation of the Identity Type