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We live in a space, we get about in it. We also quantify it, we
think of it as having dimensions. Ever since Euclid's ancient
geometry, we have thought of bodies occupying parts of this space
(including our own bodies), the space of our practical orientations
(our 'moving abouts'), as having three dimensions. Bodies have
volume specified by measures of length, breadth and height. But how
do we know that the space we live in has just these three
dimensions? It is theoreti cally possible that some spaces might
exist that are not correctly described by Euclidean geometry. After
all, there are the non Euclidian geometries, descriptions of spaces
not conforming to the axioms and theorems of Euclid's geometry. As
one might expect, there is a history of philosophers' attempts to
'prove' that space is three-dimensional. The present volume surveys
these attempts from Aristotle, through Leibniz and Kant, to more
recent philosophy. As you will learn, the historical theories are
rife with terminology, with language, already tainted by the as
sumed, but by no means obvious, clarity of terms like 'dimension',
'line', 'point' and others. Prior to that language there are
actions, ways of getting around in the world, building things,
being interested in things, in the more specific case of
dimensionality, cutting things. It is to these actions that we must
eventually appeal if we are to understand how science is grounded."
The three spatial characteristics of length, height and depth are
used in the same unreflective way by laymen, technicians and
scientists alike to describe the forms, positions and measure of
bodies and hollow bodies. But how do we know that the space we live
in has just these three dimensions? The question has occupied
philosophers and scientists since antiquity. The answers proposed
have become ever more presumptuous and have increasingly lost sight
of everyday intuitions and have sacrificed explanatory power. In
Euclid's Heritage Janich shows that all explanations of
three-dimensionality hinge on an unreflective geometrical language
which seems to accept the lack of an alternative for the three
sorts of entities -- points, lines and planes -- that bound the
three extended entities -- lines, planes and solids. This is a
Euclidean heritage in a dual sense: Euclid himself adopted a
geometrical language from the art of figure drawing, and left a
tradition of doing geometry as planimetry and of doing stereometry
by rotating plane figures. The systematic approach offered here
starts out from operational definitions of the spatial forms --
plane, straight edge and perpendicularity -- and proofs that only
three planes can intersect pairwise orthogonally. This is the
constructive solution in the frame theory of action, providing an
unequivocal characterisation of spatial relations in the physical
world. The traditional order of geometric concepts turns out to be
the most important obstacle to the methodical ordering of everyday
scientific concepts.
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