Finding good replacements for CW complexes in the purely algebraic case is a subject of current research.

One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.

This purely algebraic attack conveys no geometric intuition.

As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings.

The Arab mathematicians were able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically.

The Whitehead problem was the first purely algebraic problem to be proved undecidable.

Applying the above rule, solving any linear differential equation is thus reduced to a purely algebraic problem.

Finally one can take these two properties as basics and give purely algebraic definition of "symmetry"

This allows to reduce the original geometric problem to a purely algebraic problem.

Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field.