As best I understand it, in Part IV of Process & Reality, Whitehead attempted to account for how geometrical measurement of the physical world is possible without any empirical presuppositions. He was worried that physics had not yet fully accounted for its own experimental practices and so searched for a presuppositionless mathematical starting point for measurement. He worried about the infinite regress of instruments needed to verify a measurement: the ruler used to measure a particular region of space itself needs to be measured by another ruler to confirm its accuracy, which itself needs to be measured by a third ruler, and so on. Whitehead thus sought a purely mathematical derivation of all the geometrical elements, including definitions of points, lines, and planes, definitions that again do not depend upon actually having to measure something. That said, Whitehead is not eliminating experience or perception as such from his derivations. He is doing projective geometry, which forgoes the need for empirical measurement while still relying on subtler mathematical intuitions regarding certain relations, like inclusion and exclusion, etc. Whitehead’s is in this sense still an experientially grounded mathematical scheme, but grounded in a purified mode of experience that is far more generic than the normal way we perceive the spatiotemporal world around us through our species specific sensory organs. As Whitehead tells us earlier in PR, “philosophy is the self-correction by consciousness of its own initial excess of subjectivity” (PR 14). 

Some readers of Whitehead may feel the desire throw his book across the room and challenge the very basis of what he is trying to do, and even to question his motivations for doing it. Is metaphysics even possible?! Who dares try to utter the ultimate? Perhaps there is an irresolvable tension or even conflict between system and freedom. Schelling is another process philosopher who dwells on precisely the paradox between systematic necessity and personal freedom. In his 1801 essay titled “Presentation of My System of Philosophy” he appears to quite grandiosely identify a product of his individual consciousness with the universal system of reality. There is something to such identification, of course (atman is Brahman); but such an inflated view of one’s own philosophy can easily slide into closed narcissistic exclusivism. In his more mature works, Schelling (using the example of his friend Hegel) becomes a good deal more skeptical of the possibility of individually attaining the Absolute once and for all.

Whitehead is no less immodest than Schelling or Hegel in his speculative wagers, but he would insist that we never lose the ability to laugh even at our own most serious ideas about God, the universe, and ultimate reality (in his Dialogues with Lucien Price, Whitehead said: “The total absence of humor from the Bible is one of the most singular things in all literature”). 

Is the so-called “organic realism” or “philosophy of organism” just Whitehead’s personal perspective on things? I don’t think so. We can say for sure that his intention at least is not to offer us his own idiosyncratic artistic rendering or mystical vision of reality. He is rather intending to do metaphysics, that is, he is striving to articulate the most generic, universal, and common features of our shared experience, and to do so with as much logical rigor and scientific adequacy as he can muster. He is trying to reveal the structure and dynamics of reality through the medium of a strange invented language that he admits is almost entirely ill fitted to the task. Flawed and clumsy as it is, he hopes the lexicon of his open system traces the branches of the world-tree we call reality well enough to guide us at least a few steps forward along the philosophic path toward its roots. He was not deluded enough to hope or believe that his precise categories and definitions, if found useful, would remain unchanged as they are carried forward and applied by others. 

I’ll let Whitehead speak for himself on these issues (from Process & Reality, 4-5): 

Philosophers can never hope finally to formulate these metaphysical first principles. Weakness of insight and deficiencies of language stand in the way inexorably. Words and phrases must be stretched towards a gen­erality foreign to their ordinary usage; and however such elements of lan­guage be stabilized as technicalities, they remain metaphors mutely ap­pealing for an imaginative leap.

There is no first principle which is in itself unknowable, not to be cap­tured by a flash of insight. But, putting aside the difficulties of language, deficiency in imaginative penetration forbids progress in any form other than that of an asymptotic approach to a scheme of principles, only de­finable in terms of the ideal which they should satisfy.

The difficulty has its seat in the empirical side of philosophy. Our datum is the actual world, including ourselves; and this actual world spreads itself for observation in the guise of the topic of our immediate experience. The elucidation of immediate experience is the sole justification for any thought; and the starting-point for thought is the analytic observation of components of this experience. But we are not conscious of any clear-cut complete analysis of immediate experience, in terms of the various details which comprise its definiteness. We habitually observe by the method of difference. Sometimes we see an elephant, and sometimes we do not. The result is that an elephant, when present, is noticed.  Facility of observa­tion depends on the fact that the object observed is important when present, and sometimes is absent.

The metaphysical first principles can never fail of exemplification. We can never catch the actual world taking a holiday from their sway. Thus, for the discovery of metaphysics, the method of pinning down thought to the strict systematization of detailed discrimination, already effected by antecedent observation, breaks down. This collapse of the method of rigid empiricism is not confined to metaphysics. It occurs whenever we seek the larger generalities. In natural science this rigid method is the Baconian method of induction, a method which, if consistently pursued, would have left science where it found it. What Bacon omitted was the play of a free imagination, controlled by the requirements of coherence and logic. The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation. The reason for the success of this method of imaginative rationalization is that, when the method of difference fails, factors which are constantly present may yet be observed under the influence of imaginative thought. Such thought supplies the differences which the direct observation lacks. It can even play with in­consistency; and can thus throw light on the consistent, and persistent, elements in experience by comparison with what in imagination is incon­sistent with them. The negative judgment is the peak of mentality. But the conditions for the success of imaginative construction must be rigidly adhered to.”