In this paper we devise some technical tools for dealing withproblems connected with the philosophical view usually calledmathematical instrumentalism. These tools are interesting intheir own right, independently of their philosophicalconsequences. For example, we show that even though the fragmentof Peano's Arithmetic known as IΣ₁ is a conservativeextension of the equational theory of Primitive RecursiveArithmetic (PRA), IΣ₁ has a super-exponentialspeed-up over PRA. On the other hand, theories studied inthe Program of Reverse Mathematics that formalize powerfulmathematical principles have only polynomial speed-up overIΣ₁.
We propose a logic for reasoning about metric spaces with theinduced topologies. It combines the ‘qualitative’ interior andclosure operators with ‘quantitative’ operators ‘somewhere in thesphere of radius r,’ including or excluding the boundary. Wesupply the logic with both the intended metric space semantics anda natural relational semantics, and show that the latter (i)provides finite partial representations of (in general) infinitemetric models and (ii) reduces the standard‘ε-definitions’ of closure and interior to simpleconstraints on relations. These features of the relationalsemantics suggest a finite axiomatisation of the logic and providemeans to prove its EXPTIME-completeness (even if the rationalnumerical parameters are coded in binary). An extension withmetric variables satisfying linear rational (in)equalities isproved to be decidable as well. Our logic can be regarded as a‘well-behaved’ common denominator of logical systems constructedin temporal, spatial, and similarity-based quantitative andqualitative representation and reasoning. Interpreted on the realline (with its Euclidean metric), it is a natural fragment ofdecidable temporal logics for specification and verification ofreal-time systems. On the real plane, it is closely related toquantitative and qualitative formalisms for spatial representationand reasoning, but this time the logic becomes undecidable.
"A logic for metric and topology." J. Symbolic Logic 70 (3) 795 - 828, September 2005. https://doi.org/10.2178/jsl/1122038915