Continuity of Spacetime

Spacetime is indeterminately continuous or discrete.

First, we will present the central impetus for pursuing this thesis,

and then we will adjust our understanding of space and ultimately

reconstruct much of calculus in a form which more closely models

physical reality.

We construct two different kinds of random length. One which

is a product, and one which is a sum. These lengths are L_∞ and L‘_∞

respectively.

L_∞ is defined as an ordinary length segment and to each point

we assign a probability that each point exists.

L‘_∞ is defined very differently. It is constructed using infinite

sums of subintervals of an ordinary length intermingled with

infinitely many subintervals of a random portion. As the number

of subintervals goes to infinity, L_∞ and L‘_∞ start to look very similar.

The distinction between L_∞ and L‘_∞ is that one is a PRODUCT and

the other is a SUM. Yet, the expected lengths are necessarily equal :

E (L_∞) = E (L‘_∞ )

The position of this post, in light of the wave-particle duality,

is that the physicist observes expected length and so the

continuity and discreteness of spacetime is therefore

indeterminate. This is not the same as saying "we simply

don't know". In fact we do know, and what we now know is

that it is in fact indeterminate.

These views are practically proven by the fact that if

someone hands you an expected length, there is no way to

know if the underlying length is of the form L_∞ or L‘_∞ . QED

We will show that all of calculus can be conformed to

satisfy this view, and that ultimately this is the best possible

answer to the age old question as to whether space is

continuous or discrete.