Algebra and pattern matching are basically the same thing, aren't they? To find a solution to an algebraic problem, you pattern match on the problem structure and see what falls out. To solve a pattern matching problem, you perform algebraic transformations to isolate the pattern being matched.

>Algebras, like numbers, does not exist outside our heads

Models are called models because they are not the things they model. Anything that is a model is an approximation. This doesn't mean they don't exist outside our heads. If that were true, there wouldn't be any computers.