Foundations of Measurement Ch 3 &

4

Foundations of Measurement Ch 3 &

4April 4, 2011

presented by Tucker LentzApril 4, 2011

presented by Tucker Lentz

AgendaAgenda11:00 Chapter 3:

Extensive Measurement

12:40 break

12:50 Chapter 4: Difference Measurement

11:00 Chapter 3: Extensive Measurement

12:40 break

12:50 Chapter 4: Difference Measurement

Ch 3: Extensive Measurement

Ch 3: Extensive Measurement

Closed Extensive Structure

closed, i.e., when any two objects can be concatenated

Connectivity: for all a, b A, a ∈ ≿b or b a (p14)≾

7373

No matter what the difference is between c and d, as long as a strictly exceeds b, there is some integer that when multiplied by the difference between a and b will swamp the difference between c and d.

there are no negative nor zero elements

Closed Extensive Structure

7474

Formal Proof of Theorem 1

8080

<A, , ○> is a ≿ simply ordered group iff

<A, > is a simple order≿<A, ○> is a group

If a b, then a○c b○c and c○a c○b. ≿ ≿ ≿<A, , ○> is also ≿ Archimedean if (with the

identity element e) a e, then na b, for ≻ ≻some n.

Theorem 5 (Holder's Theorem) An Archimedean simply ordered group is isomorphic to a subgroup of <R, ≥, +>, and the isomorphism is unique up to scaling by a positive constant.

LEMMA 1: There is no anomalous pair. (p 77)

LEMMA 2: Every element is either positive, null or negative. (p 78)

LEMMA 3: ⟨A, , ◦⟩ is weakly ≿commutative (p 78)

LEMMA 4: The relation ≈ on A x A is an equivalence relation (p 79)

Formal Proof of Theorem 1

Formal Proof of Theorem 1

7979

Formal Proof of Theorem 1

8181

Lemma 6 and Theorem 2.5 prove the existence ofa real-valued function ψ on D such that such that for all [a, b], [c, d] ∈ D

8080

Formal Proof of Theorem 1

Define ϕ on A as follows: for all a ∈ A, ϕ(a) = ψ([2a, a]). We verify that ϕ has the desired properties.

Informal Proof Sketch

Informal Proof SketchSelect any e in A; this will be the unit. For any other a in A, and for any positive integer n, the Archimedean axiom guarantees that there is an integer m for which me na. Let m≻ n be the least integer for which this is true, namely, mne na (m≻ ≿ n - l)e. Thus, mn copies of e, are approximately equal to n copies of a. As we select n larger and larger, the approximation presumably gets closer and closer and, assuming that the limit exists, it is plausible to define

7575

When concatenation is not closed

B is the subset of A x A that contains the pairs that can be concatenated (82)

8484

Extensive structure with no essential maximum

The new associativity: if a, b can be concatenated, and a ◦ b can be concatenated with c, then any concatenation of a , b and c is allowed (82)

8484

Extensive structure with no essential maximum

Commutativity and monotonicity: If a and c can be concatenated, and a strictly exceeds b, then c and b can be concatenated, and the concatenation of a and c must exceed the concatenation of c and b. (83)

8484

Extensive structure with no essential maximum

Solvability postulate: there is no smallest element A that can be concatenated. (83)

8484

Extensive structure with no essential maximum

Positivity

8484

Extensive structure with no essential maximum

Archimedean axiom: In the earlier structure we defined na and then had for any b in A, there is an n such that na b. However, because ≿of the restrictions on B, we may not arbitrarily concatenate elements in A. So axiom 6 defines a strictly bounded standard sequence, and assumes it is finite. (83-84)

8484

Extensive structure with no essential maximum

8585

Some Empirical Interpretations

New concatenation operation: a * b is the hypotenuse of a right triangle formed by rods a and bThis results in a structure ⟨A, , *⟩ that satisfies ≿the axioms of definition 3, and the resulting ψ is proportional to ϕ2

8888

Some Empirical Interpretations

“To most people, the new interpretation seems much more artificial than the original one. In spite of this strong feeling, neither Ellis nor the authors know of any argument for favoring the first interpretation except familiarity, convention, and, perhaps, convenience...”

8888

Some Empirical Interpretations

8989

Some Empirical Interpretations

8989

Some Empirical Interpretations

9090

⟨A, , ◦⟩≿⟨A, ’, ‖⟩≿

Velocity

• In a Newtonian universe, a closed extensive structure ⟨A, , ◦⟩ could ≿represent velocity

• For a relativistic universe we need to introduce extensive structures with essential maxima

Essential Maxima in Extensive Structures

9292

Essential Maxima in Extensive Structures

9292

Essential Maxima in Extensive Structures

9292

Essential Maxima in Extensive Structures

9292

• Theorem 7 gives the representation required for relativistic velocities

Non-Additive Representations

• right-angled concatenation and relativistic velocity are two examples of non-additive representations

• a third: Consider a positive extensive structure with a scale ϕ additive over ◦; then ψ = exp ϕ is an alternative scale, which is multiplicative over ◦.

101000

Non-Additive Representations

101000

Conventionality of Representations

101022

“...despite its great appeal and universal acceptance, the additive representation is just one of the infinitely many, equally adequate representations that are generated by the family of strictly monotonic increasing functions from the reals onto the positive reals...”

Extensive Measurement in the Social Sciences

• In many cases, there is no concatenation operation appropriate to the entities of interest in social science

• However, an empirical concatenation operation is not necessary for fundamental measurement

• There are cases where it works, e.g., subjective probability (ch 5) and risk (3.14.1)

Risk

121255

Risk

121266

Risk

121277

Risk

121277

Limitations of Extensive Measurement

While I am certain we won’t have time to discuss it, I highly recommend reading the 2 page discussion of this topic (pages 130-132).

EndEnd