> what does "scalar" mean then? that something has no dimensions, or no
> extension?

    It means no dimension.  One has to have 1 or more dimensions to have
extension in that/those dimensions.  You can only have 0 or more extension
if you have a dimension to have the extension in.

    Mathematically a scalar is a point or single number, no matter
how long like pi etc.

    One asks how thick is a line?  One might be tempted to say, zero
thickness, but this is incorrect, as a line doesn't have a thickness
dimension in which to have zero thickness.  One has to HAVE a dimension
before one can have zero extension.

    A 2x2 2 dimensional piece of gold is not zero thick in the third
dimension.  If it were, it would be a 2x2x0 3 dimensional piece of gold,
which would actually be no gold at all as 2x2x0 = 0.

    A 2x2 2 dimensional piece of gold however is definitely some gold,
albeit 2 dimensional :)

    Same with a scalar, it is not a 3 dimensional 0x0x0 entity, nor
a 2 dimensional 0x0 entity nor even a line of 0 length.  It is
a zero dimensional entity.

    In APL, a programming langauge, the AxBxC is called the rho of the
object, denoted {a,b,c}.

    A 3 dimensional object has 3 members in its rho, one for
each dimension.  Thus a {2,3,4} is a 2 by 3 by 4 object.

    A 2 dimensional object has 2 members i.e. {2,3} which is
a 2 by 3 object.

    A 1 dimensional object has 1 member, ie {3} which is a 3 unit

    A 0 dimensional object has 0 members in its rho, its rho
is the empty set.  {zip}

    Thus a {0,0,0} 3 dimensional object, which is in fact
a nothing of no substance, is not the same as a {zip} scalar
object of definite substance.

    People have a hard time picturing a scalar universe, as their
consciousness is capable of only picturing 3D spaces.  So they mock
up a 0x0x0 'point' of 3D space and miss entirely what an infinite
scalar universe might be like.


Sun Jan 29 15:08:12 EST 2017