Society for Mathematical Psychology

1
Jonathan Barzilai

• Society for Mathematical Psychology
• Annual Meeting
• 27 July 2007

2
Background

• Helmholtz 1887.
• Campbell 1920.
• Ferguson et al. 1940.
• Von Neumann and Morgenstern 1944.
• The end product of measurement is a set of scales
  ( functions, mappings).

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a1 . . . an
23 . . . 97
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Empirical
Mathematical
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Agreed

• Campbell and vNM
• The purpose of measurement is to enable
  mathematical operations in a mathematical system
  M which is a model ( reflection) of an empirical
  system E.
• Operations in E are reflected into M.

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The Fundamental Problem

• The operations are the key and the applicability
  of mathematical operations in the model M is the
  fundamental problem of measurement theory.
• To re-emphasize The operations are in E and in
  M.

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vNMs by-pass

• Campbells objection is (correctly) rejected
  using a uniqueness argument.
• The uniqueness argument is correct but the
  applicability of operations is ignored and
  remains an open problem.

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Blindly Following vNM

• Uniqueness replaces applicability of operations.
• Nothing in the literature about the conditions
  for applicability of x. Not even errors.

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The Consequences

• No basis for the operations of game theory, e.g.
  V(S)V(T) fundamental errors in utility theory.
• x not applicable to any scales in literature.
• No basis for mathematical operations in
  economics, psychology, etc.

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The Conditions

• The mathematical system M must be a field, a
  one-dimensional vector space, or a
  one-dimensional affine space.
• The empirical objects correspond to scalars,
  vectors, or points.
• Multiple sets and mixed operations.

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The Homomorphisms

• For massm(3) m(1) m(2) and m(2) 1.41 m(1)
• Two equations, not one corresponding to addition
  and multiplication.
• Multiplication is mixed vector space, not group
  m(ka) k m(a) rather than m(ka) m(k)
  m(a).

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Fundamental Errors

• These Structures cannot be modelled by groups.
• They cannot be modelled by the Scott Suppes
  model.

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Two Elementary Errors

• Extensive measurement half a vector space -
  the scalars are lost and with them
  multiplication.
• Difference measurement the vectors and the
  scalars are lost, addition and multiplication are
  lost. The difference of two points is not a
  point.

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Missing

• Necessary conditions for the applicability of
  addition and multiplication are
• Is there one reference?

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Thank you!