Existing math
does allow dividing by zero.
The branch of math starting with calculus is based almost
entirely on the concepts of limits and derivatives.
The limit allows
you to examine things that otherwise would have no
answer, such as dividing by zero. The limit does not
guarantee an answer, but allows you to consider if there
is one, and find it when it exists.
The derivative at its most basic, determines the slope of a curve
at a point, and in so doing
makes use of limits to explicitly divide by zero in a specific,
meaningful, controlled way. The point is considered to be a very short line segment with both end points on the curve. As the length of
the segment approaches the limit of zero,
the end points become coincident. The results of the division by
zero involved in calculating the slope of the segment
is defined on smooth continuous curves, and undefined at sharp corners
and other discontinuities.
The catch here is that the use of the limit only gives meaning to division by zero in the context of a larger expression,
and even then only when a variable is approaching zero, and only if the value of the expression is the same when approaching zero from both sides.
Dividing by the constant zero is not helped, and still has
no meaning.