Hegel deals here with the question: how can we know that Science is true? The fact that Science offers a new model of the world means that it cannot depend on previous models to validate itself. In other words, Consciousness must assume all phenomena are unverifiable, and doubt them. This assumption implies knowledge of the truth of the assumption, such that Consciousness has produced true knowledge, or content. If we decide to continue discovering truth, and not self-absorbedly admire our superiority, a criterion will be needed for when that goal is achieved, and Hegel explains how it arises. Consciousness knows some things and knows that it knows them. Truth means that an object’s relation to itself is the same relation it has to Consciousness. To determine truth, an objective observer would simply compare an object against the way it appears to Consciousness and be done with it; but Consciousness cannot step outside itself. It can, however, compare the object (as it appears) against how its knowledge of the object (as it appears) appears. If they don’t match, Consciousness should calibrate its knowledge to the object. Adapting this knowledge to what our object is will change the object it refers to; the two are tightly bound. Now, there’s a new set of an object and a knowledge, and Consciousness can practice verification again! The beauty of the verification process, or experience, is twofold. First, it requires no explicit memory of previous experiences because their falsity is inscribed in the objects you’re using; but they still snowball forward kernels of truth starting from the first doubt. Moreover, because experience is self-contained and will produce a more-truthful result without being directed, Consciousness needs no overarching awareness of where it is going; it is merely immersed in the experience of getting there and experiences itself as casually stumbling onto new objects of knowledge. Hegel calls this the Science of the experience of consciousness. The centerpiece of Hegel’s vision is that he doesn’t even have to know the outcome of the dialectic! All he knows is that his process inches towards there, and Consciousness will know when it gets there.
Hegel’s quasi-mathematical manipulation implies some notions of equality that aren’t necessarily justified. This reading reminds me of a video I recently watched about a computer science concept called zero-knowledge proof; it means demonstrating something is true without giving away what it is. It uses advanced statistics to approximate certainty. If Hegel is working off mathematical assumptions already, should a similar approach not also provide the bootstrapping engine he seeks? In a sense, his work is a mathematical proof by induction, where the base case and the method are demonstrated, but not each individual step. This is why his transitive logic and his understanding of equality are the heart of his achievement.