Here, finally, is the dilemma which the proposed grounding of reality brings us. Self-necessity demands an ungrund so featureless that it is difficult –without ambiguity- to assert that it exists at all. Yet it has been listed among the ‘non-negotiable’ demands that its raison d’etre shall be a resounding eternal life. Our problem, then is figuring out how we bridge the daunting chasm yawning between the two. We need a metamathematics adequate to handle the unfolding of the Ungrund. When this has run its course, what we end up with –at the moment before the initiation of cosmic existence at the ‘Big Bang’ is an articulate domain of potentiality ready to be drawn upon as needed. This same metamathematics must also underwrite the two mysterious eschatological benchmarks that bracket organic evolution. Finally, it must tell us something about what we are to anticipate in eternal life. If nothing else, it remedies an omission that has hamstrung so many of those seeking to integrate the otherwise discordant relationships between finite time and eternity. The assumption –normally asserted sotto voce-is that both can be subsumed within an encompassing linearity; Plotinus is very explicit about the embarrassments and contradictions to which this leads. Metamathematics executes a breakout from this claustrophobias if ‘linearitis’ that undermines both eschatology and speculations over the character and dynamics of eternal life.

        How is this to be thought of? In particular, to what extent may we expect it to be patterned on what we are all very familiar with? As has always been true of the putting together of the parade of classical world-views stretching from Plato to Whitehead, the philosopher finds himself in need of the kind of formal support that only the hybrid disciplines of Logic and mathematics can supply. However, this is no job for a Leibniz –though even he sensed that the ‘calculus ratiocinator’ of conventional formalization was in need of complementation of something higher to bring the broader domains of reasoning and linguistics under control. He sought to fill this gap with his characteristica universalis –though he was to become skeptical as to whether this was going to be able to deliver the goods.

              Figure 2 hints at what is really needed. The left-hand image may be thought of as a diagrammatic implementation of the hint already offered by Köstler et. al., of a discipline whose articulate structure hangs from an apex through a ‘relaxation’ divergence of progressive differentiations. The image on the right is that of the ortho-mathematical paradigm with which we have all been familiar from Euclid on. Such systems are grounded upon a parsimonious set of axioms, postulates and other primitives; what this grounding supports is an ascending and expanding divergence of theorems and other ascending in an expanding series of ever more complex theorems, statements and constructions.

           Figures 2a, 2b and 2c provide 'visualization' analogues of three 'topological features vital to the constitution of Metamatematics.  The first should be viewed in conjunction with Figure 2.  At all times, the subjective and objective aspects of the moment are anamorphically related to one another.  In Figure 2a shows this in concentric form where 'meaning' of the perceptual image is drawn out of it to surround it.  Of course, the meaning of what is perceived is not another version of the perceptual image but something much more abstract;  the 'double objective' character of the figure is simply a dodge to convey the topological character of the relationship.  The Mandelbrot Set, shown in Figure 2b -that disturbs the sleep of some of us- is but a doppolganger, within the Ortho-mathematical domain, of its role within Metamathematics.  Its Divergent, concentric constitution shown in figure 1 has this .Mandelbrot Property, that some analogy or facsimile of the whole  is consistently preserved as differentiations become increasingly delimited.  Figure 2c hints at how I believe finitude is folded back into Infinity -that is, the way in which its 'open' aspect reside within its encompassing 'finished' form.  

          This is an appropriate place to discuss the need for a second mathematical extension that is much closer to home –in its familiar ‘ortho’ as opposed to ‘meta’ character. We are in urgent need of extensions to the lex naturalis specifically in support of life and mind. These are demanded by the twin bankruptcies of the ‘Artificial Intelligence’ initiative and Neo-Darwinism as a sufficient explanation of organic evolution. Attempts to account for human behaviour, in its very mundane, ongoing manifestations and accomplishments, armed with nothing more than the present scope of natural law provides confront us with stupendous shortfalls –and of more kinds than one. This chasm aside, there’s no way we can endow machines with the way in which humans think and address everyday problems –and I will not even mention its inability to account for what Gödel’s theorem demands.

          The problem with neo-Darwinism is its failure to account for the origin of the wonderful stuff that is offered to the process of natural selection. How is it that scattered, largely Random and independent mutational changes can lead to the very cooperative configurations displayed by organisms on every side; equally –the more we ascend the phylogenetic ladder is the ‘top-down’ organization so characteristic of living forms. Minds are just phenomenological entities, but domains, which a unifying agent has, rather than is 

          The lower part of Figure 6 backs away to display the fields needed to permeate each pyramid, if it is to do its job smoothly and correctly, while Figure 7 Shows its ultimate cortex-wide reach. Without out this, Sherrington’s ‘enchanted loom’ of the cortical configuration of some 10¹⁰ neurones would jam up after no more than a few passes of the shuttle.

          Finally, Figure 5 rounds out the discussion over the need for both kinds of extensions to the canon of mathematics and the lex naturalis that they uphold. Metamathematics is needed in support of the subjective interior of agency, ‘meaning’ and intentionality; while exotic orthomathematics is needed for their objective expression –both in ‘real world’ terms of perception/praxial negotiations and of explicitly grasped inward turning odysseys of reason and speculation. As is clear in the figure, both subjective and objective fields stand orthogonal to the crass material cortex –to which, and with which they remain closely integrated, and interfaced. No matter what mind/brain theory one may adopt, there can be no dodging of the dependence of mind and agency upon the substratum that the cortex provides –in the stream of conscious thought of every kind. What the diagram also hints at is the way in which these two fields couple to each other. It is through the advanced topological feature of an anamorphic transduction –in which mind turns itself inside out in order to enter the objective realm of its experience. This can be vaguely sensed through an exercise of an appropriate introspection. On the one hand, in ourselves, we stand above and encompass what is below, yet we also enter into it from within –to which this aspect of our experience we are properly captive. This same anamorphic singularity appears again and again, to provide the kind of infolding needed to be sequestered within the ungrund to maintain its ultimately simple and integral nature.

          As will be examined further below, the basic inorganic realm of cosmic materiality is itself in urgent need of extension to finesse the ‘black holes’ catastrophe. I take it to be the case –as evidently, did also Einstein’ that such singularities have no place in nature but are limited to the realm of Platonic transcendence. No one doubts that when any mass moves within its Schwartzchild radius there will an exotic and drastic contraction. What needs to be defined is an extension to the equations of General Relativity that will halt this contraction at a new state of matter. What is called for is the formulation of a novel phase change similar to that experienced by water vapour when it moves within its dew point value. There is a dramatic contraction which halts at a new phase –a drop of water of greatly reduced but finite dimensions. I suspect that this new state of matter will be increasingly called upon as the cosmos converges to its final terminus..