Mathematics describes objects that have all their implementation and representation details abstracted away. For example, there are numerous numeral systems for representing the natural numbers, all of them equivalent. In category theory all objects are considered up to isomorphism and representations can be described as isomorphisms.
An important part of any computer algebra system is the system for selecting canonical representations for equivalence classes. In a Lisp all the way down system, representations can be described directly in terms of the physical hardware, for example, the exact bit layout of a number could be described.
Lisp all the way down systems allow you to convert abstract to the concrete, ideally using the related areas of decision making, optimisation, planning, and goal seeking along the way.
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