1. General Ontology with 3 Categories entitiesthings facts formsparticulars universals atomic facts molecular facts

2. Special Ontology of Determinables (Dimensions) Determinables such as length are monadic universals of the second order (being a length). The maximally specific determinates (a length of 2 cm) are monadic universals of the first order. There are quantitative determinables. Their maximally specific determinates are called quantities.

3. Quantitative Determinables and Serial Order The distinction between quantitative and non-quantitative determinables is not clear-cut. A necessary condition for a determinable to be quantitative is, at any rate, that its maximally specific determinates are serially ordered. A serial order is based ontologically on a conjunctive fact which has relational atomic facts as with the same two-place universals of the second order as constituents, provided that relation is necessarily asymmetric and transitive. A two-place universal U 2 is asymmetric iff the general fact (x)(y) (U 2 xy → ¬ U 2 yx) obtains. The transitivity is grounded correspondingly. It entails that in every given case of two particulars a and b it is necessary that ¬ U 2 ba if U 2 ab. Now the determinable length can be quantitative because (the length) because the relation ‘longer’ which holds between its maximally specific determinates is asymmetric and transitive.

4. Quantitative Determinables and Spatial Wholes Traditionally, quantities were taken to have parts. Now, if quantities are universals, then they cannot have spatial parts. But the particulars which exemplify (have) the quantities can (e.g. spatial parts of a rule do have length as the rod as a whole). It is crucial for the paradigm of measurement (for extensive m. or measurement by counting units) that certain for certain determinables spatial wholes as well as their parts exemplify maximally specific determinates of it and that the determinate exemplified by the whole comes in the series of determinates after the determinates exemplified by its spatial parts. In some cases determinables for which this is the case are used a surrogates to measure causally connected determinables for which this not the case. For example, length is used as surrogate to measure temperature.

5. Measurement by Counting Units A unit is a maximally specific determinate which is introduced by a measuring community by making a prototype particular or adopting a naturally given one which exemplifies that determinate. Then duplicates of the prototype are needed which exemplify the same determinate as it. The duplicates are concatenated into a spatial whole which as such has to have the same determinate as the particular to be measured. Finally the number of duplicates in the whole are counted which furnishes the measurement value.

6. An Example Consider a rigid rod which is 3 cm long. We need 3 duplicates of the cm- prototype. They might be cm-segments on a ruler. Now, the rod to be measured and the three segments together a placed beside the rod to be measured and are seen to be congruent. The fact ascertained is the following: for exactly one monadic universal X 1 of the first order which exemplifies the monadic universal of the second order L 2 of being a length both r (the rod) and some x (which has the cm- duplicates d1, d2 and d3 and no other cm-duplicate as spatial parts) exemplify X 1. (1 X 1 ) (L 2 (X 1 ) & X 1 r & (1 ≤ x) X 1 x & S1(d1 x) & S 1 (d2x) & S 1 (d3, x) & (0 y)(y ≠ d1 & y ≠ d2 & y ≠ d3 → ¬S 1 (y x))). If one symbolises the length quantity of the cm-prototype by “C “ one can ground the length measurement of our example on a simpler molecular fact: (1 X 1 ) L 2 (X 1 ) & X 1 r & (1 ≤ x ) X 1 x &( (3 y) Cy & Syx. 3x belongs to the category of forms and is the quantifier ‘for exactly three”. It shows why the number 3 can be used here as a measurement value and implies that natural numbers are quantifiers.

7. Numerical Representation Measurement is the numerical representation of determinates by numbers. It is particularly called for when a determinable has many maximally specific determinates. Such a determinable is challenge for the human capacity of discrimination and recognition. Numerical representation has the advantage that it allows calculation, i.e., the use of arithmetic to draws conclusions concerning the determinables and the dependence between different determinables. However, not all arithmetical operations are admissible with respect to all quantitative determinables and to all representations (scales). It depends on the structure of the determinable and the informativeness of the measuring operations.

8. Quantities are Mind-Independent Quantities are not man-made as positivists claim who dominate the theory of measurement since Ernst Mach and particularly since D. Scott and P. Suppes paper applying model theory to measurement. Positivist hold that quantitativeness is essentially based on conventions, particularly concerning the choice of a prototype and concerning the choice of a concatenation operation. Objection: the choice of prototype is merely the choice of an object of comparison and does not determine the property indicated by the comparison (as the terms “Prussian blue” or “King’s blue” do no create the shade of colour referred to). Brian Ellis who advocates the positivist analysis argues that in length measurement duplicates of the prototype could be placed in right angles to each other rather then placing them end to end as usual. Objection: The property kind of the spatial wholes (the hypotenuse of the triangle) measured by that operation is not a length but rather the distance between two ends of an angle.

9. Measurement Error The realist view of measurement advocated here allows to distinguish between the quantity the measured particular exemplifies and the result of a measuring operation. Thus it can explicate measurement error. The dominant representationalist theory of measurement cannot because it is clearly operationalist. It defines quantities by measurement operations. That is very strange and mostly goes unnoticed although operationalism in general is thoroughly discredited in the philosophy of science.