[FOM] The observational standpoint for numbers

Giovanni Lagnese lagnese at ngi.it
Wed Jan 25 13:50:53 EST 2006

Up to the nineteenth century, euclidean geometry occupied a privileged
position among all possible geometries, because it was considered the
geometry of the physical space, in virtue of the apparent correspondence
between outer and inner experience of space.
Such a standpoint can be called observational standpoint.

The observational standpoint for space was definitively proved untenable
thanks principally to Albert Einstein.

Today an observational standpoint for the natural numbers is still
In fact, most mathematicians believe in a correspondence between the
properties of the natural numbers of our intuition and the properties of the
natural numbers as combinatorial/discrete aspects of reality.
If a property P of the natural numbers is demonstrated from axioms
intuitively "seen" to be true, most people believe that P is true also as a
combinatorial property of the natural world.
Inner computations are considered to be corresponding to
combinatorial/discrete properties of reality, because there is the
conviction that there are "truths of computation" which are independent of
any hypothesis.

I think that the observational standpoint is wrong for natural numbers as it
is wrong for geometry.

Do someone agree with me?


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