  We propose an axiomatic foundation of mathematics based on the finite
sequence as the foundational concept, rather than based on logic and set, as in
set theory, or based on type as in dependent type theories. Finite sequences
lead to a concept of pure data, which is used to represent all mathematical
objects. As an axiomatic system, the foundation has only one axiom which
defines what constitutes a valid definition. Using the axiom, an internal
true/false/undecided valued logic and an internal language are defined, making
logic and language-related axioms unnecessary. Valid proof and valid
computation are defined in terms of equality of pure data. An algebra of pure
data leads to a rich theory of spaces and morphisms which play a role similar
to the role of Category Theory in modern Mathematics. As applications, we
explore Mathematical Machine Learning, the consistency of Mathematics and
address paradoxes due to Godel, Berry, Curry and Yablo.
