Why do we use numbers in science to represent
empirical phenomena? Why do we use these numbers in some ways
rather than others? Broadly speaking, these are the kinds of
questions this seminar addresses.
Consider a mundane example. Suppose you're describing two
sculptures you recently saw. You wouldn't describe one of them
simply as having a height of "2'', without specifying whether
this height was in inches, feet, meters, etc. But while
comparing their heights, you might say that one of them was
"twice'' as tall as the other, without specifying your units
of measurement you used.
 Question: What is it about an empirical
phenomenon like "lengths" that makes it impossible for us
to assign individual numbers to them without specifying
our units of measurement?
 This question assumes that such an assignment really
is impossible; this assumption is true, in a very
strong sense.
 Question: Why can lengths be compared without
reference to how they were measured?
 Such comparisons can't be made willynilly: the ratio
between measurements of lengths doesn't depend on
whether we used feet or meters, but the difference (via
subtraction) between them does. Why?
Answers to these questions and many others will also help us
address some much deeper issues, such as:
 Question: Why do the empirical sciences use only
a few different scales of measurement (ratio, interval,
etc.)?
 Short answer: Amazingly, the mathematical
universe simply doesn't contain very many such scales!
(Think in the neighborhood of 3 to 7 possible scales,
depending on how you count.)
Unless noted otherwise, references below are
to: Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A.
(1971).
Foundations of Measurement Volume 1: Additive
and Polynomial Representations. Academic Press, Inc.,
San Diego.
Week

Topic

1

Introduction
 Optional: [paper]
Hand (1996): "Statistics and the theory of
measurement"
 Optional: [paper]
Swoyer (1991): "Structural representation and
surrogative reasoning"
[Background: Chap. 1 [chapter] Introduction
Chap. 2: [chapter] Construction of Numerical Functions
 Theorems 4, 4': Main isomorphism theorem
 Theorem 5: Holder's theorem
 Theorem 2: Orderisomorphism theorem
 [Theorem 6: counterpoint: absolute scale example]

2

Chap. 2 continued
Chap. 3: Extensive Measurement
 Theorem 1: Closed extensive structures
 Theorem 3: Extensive structures with no
essential maximum
 Ellis' alternative construction
 Alternative numerical representations
 [Theorems 89: standard sequences]

3

Chap. 3 continued

4

Chap. 4: Difference Measurement
 Theorem 1: Positivedifference structures
 Theorem 2: Algebraicdifference structures
 [Theorem 3: ratio and difference representations]
 [Theorem 6: Absolutedifference structures]

5

Chap. 5: Probability Representations
 Theorem 2: Qualitative Probability
 Nonsufficiency of Qualitative Probability
 Luce's Axiom
 Theorem 4: Countable additivity
 [Theorem 7: Qualitative Conditional Probability]

6

Chap. 6: Additive Conjoint
Measurement
 Theorem 2: Additive Conjoint Structures
 [Theorem 13: ncomponent additive conjoint
structures]
 [Chap. 7, Theorem 1: decomposable structures]
 [Chap. 9, Section 4: Applications]
 Optional: [paper] Simonov et al (2014): "On an algebraic definition of laws"

7


8

Selected Topics
 Theorem 2 concluded
 [paper]
Narens & Luce (1986): "Measurement: The
theory of numerical assignments"

9

Selected Topics
 [paper]
Narens (1981a): "A general theory of ratio scalability with remarks
about the measurementtheoretic concept of meaningfulness "

10

Selected Topics
 [paper] Narens (1981a) continued (emphasis on sections 12)
 [paper]
Narens (1988): "Meaningfulness and the Erlanger program of Felix Klein"

Finals Week

 [paper]
Luce & Narens (1992): "Intrinsic
Archimedeanness and the continuum"

 Optional: [paper]
Luce (1997): "Several unresolved conceptual
problems of mathematical psychology"
 Optional: [paper]
Luce (1996): "The ongoing debate between
empirical science and measurement theory"
 Fun: [exhibition] Archimedes palimpsest at the Huntington Library
Further References
 Hand, David J. (2004). Measurement theory and practice: The world through quantification. London: Hodder.
 Matthews, Robert (2007). The measure of mind: Propositional attitudes and their attribution. Oxford: OUP.
 Narens, Louis (1985). Abstract measurement theory. Cambridge, MA: MIT.
 Narens, Louis (2002). Theories of meaningfulness. Mahwah, NJ: Laurence Erlbaum.
 Narens, Louis (2007). Introduction to the theories of meaningfulness and the use of symmetry in science. Mahwah, NJ: Laurence Erlbaum.
 Suppes, Patrick (2002). Representation and invariance of scientific structures. Stanford: CSLI.
 Trout, J.D. (1998). Measuring the intentional world: Realism, naturalism, and quantitative methods in the behavioral sciences. Oxford: OUP.