Quantity Representation and Geometry

Suppeshas developedacharacteristicandconceptuallyunifiedapproach to the theoriesof physicalgeometry andphysicalquantity, basedon ideas andformal techniques which he has also helped to develop within the representationaltheory ofmeasurement (RTM). These interrelatedideas, techniques and results are directly relevant to the philosophyof physics,butremainmostlyunknown withinthatcontext. HereIaimto explaintheseideasand theirrelevance ina clearandsystematic manner, andalso to extendthis general researchprograminseveral ways. IncontrasttoSuppesIstresstheneedforaunified,realistandrevisionistapproachto thetheoryofphysicalquantity,as opposedtohis piecemeal,empiricistandconservative approach.Ialsostress theneedforintrinsicformalizationofwholequantitative theories, as opposed to his hybrid extrinsic set-theoreticapproach. As new contributions to this ongoing program of researchIoutline anRTM-style representation theoryfor differential geometry,leading toaxiomaticRiemanniangeometry. Thisalsoinvolvesan extensionof theRTM theoriesofuniquenessandmeaningfulness tosituationsinvolving nonaturalsymmetry oruniquenessgroup.Ialsodescribesomenew typesofrelationist geometrical theory,developedusing this sameRTM-based technicalapparatus. 1. THEORY OFPHYSICALQUANTITY Suppes' work on geometryand space-time is closely linked tohis systematic work on representational theory of measurement(RTM), in ways explainedbelow. This work is summarized in the three-volume Foundations ofMeasurement (Krantz etal,1971;Suppes et al,1989; Luce et al, 1990) co-authored with Krantz, Luce and Tversky, cited below as FM-1,etc. In contemporary philosophy of physical science surprisingly little attention is devoted to the generaltheory of physicalquantities,physical measurement,and the methodology of quantitative physical theory. Historically these topics were much moreactively discussed: recall for exampletheEudoxian theoryofquantity,orGalileo's turn from the qualitative method of Aristotle to thequantitative method of Archimedes, and theassociated philosophical theory ofprimary and secondaryqual-