  Questions concerning origin of mathematical knowledge and roles of language
and intuition (imagery) in mathematical thoughts are long standing and widely
debated. By introspection, mathematicians usually have some beliefs regarding
these questions. But these beliefs are usually in a big contrast with the
recent cognitive theoretic findings concerning mathematics. Contemporary
cognitive science opens new approaches to reformulate the fundamental questions
concerning mathematics and helps mathematicians break through the Platonic
beliefs about the essence and sources of mathematical knowledge. In this
article, we introduce and discuss mathematical thinking activities and
fundamental processes such as symbolic/formal and visual/spatial ones. Two
different aspects of mathematics should be separated in mathematical cognition.
One aspect considers mathematics as an explicit crystallized knowledge. The
other aspect considers mathematics as an ongoing and transient mental
processing. The cognitive processes and corresponding tasks involved in these
aspects are different. Ongoing mathematical activities both elementary and
advanced, demand working memory resources. Using dual-task techniques, we
design some pilot experiments to differentiate the symbolic/formal and
visual/spatial processes. Using this memory theoretic approach, we explain the
crucial roles of language-based processes such as verbal articulation and
instructive speech and also visuo-spatial intuition such as spatial imagery and
mental movement in various aspects of mathematics.
