[FOM] Resources on the empirical foundations of mathematics

José Félix Costa fgc at math.ist.utl.pt
Fri Jan 13 04:49:36 EST 2006

Ben Crowell: «In cosmology without a cosmological constant, finite volume
also implies finite time before the universe collapses and time ends.»

It is the common view...
It depends on accepting or not the standard model.
Mathematically speaking it is possible to investigate models where Crowell
statement may well be misinterpreted.
E.g., the famous Milne's model has finite volume, infinite mass and has no
big crunch.
Milne's theory is ABSOLUTELY FASHINATING from the mathematical point of
view: it's goal is to DEDUCE from logic out of the cosmological principle
the most we can. Even gravity is not needed. In Milne's theory, GRAVITY is
just a reinterpretation of his dynamic equations.

Milne's universe:

-- it has infinite mass
-- it is finite
-- it is bounded
-- his words: «it has a large number of particles in the field of view of
any observer, merging towards the limit of visibility into a continuous
background» : : it resembles Escher's 'Butterflies'
-- particles are free and manifest behaviour that simulates the effect of

Milne's universe can be transformed mathematically into a universe:

-- infinite and unbounded
-- that obeys to the Robertson-Walker's metric
-- it expands
-- it is homogeneous and isotropic
-- it has negative curvature
-- it corresponds to a null cosmologic constant
-- it expands at zero deceleration

We shouldn't forget Milne's model, although its significance (according with
the astronomer Edward Harrison) is still obscure.

J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm

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