Horizontal mathematics is like a maze. It is flat and expansive. Its concepts have limited complexity and abstraction or universality. Proofs are obtained by joining together clever tricks and ad hoc techniques. Clear insight and intuition can be quite lost. As a body of knowledge it is fragmented and specialized. This is particularly so when the classical negation rule (or indeed any negation rule) is used.
Vertical mathematics is architectural and structured. It soars upwards. Its concepts are built upon each other as a winding staircase of ever higher abstraction and universality. Yet this universality is natural and intuitive and has a powerful unifying aspect. Proofs follow almost immediately from the gradual unpacking of definitions, much like a computation. Their is an inherent necessity in the best proof. Negation rules are rarely used. Clear intuition and insight into the concepts and their mutual relation and unfolding is obtainable. Concepts reach such high levels of abstraction that they can mirror each other and subsume each other.
Hegel's attack on mathematical knowledge in his Preface to the Phenomenology of Spirit is very much weakened if instead of horizontal mathematics we consider vertical mathematics, for instance category theory.
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