Metamathematics Summary            

              The basic point is this.  The larger Reality includes `categories that lie totally outside the standard mathematical/physical canon;  but has but to mention consciousness, value, intentionality, qualia and the open-ended notion of 'meaning.  Many of these are vague and in poor focus -are 'intrinsic' rather than articulate.  Regardless, we know from quotidian experience that these are grounded upon a formalism of some sort;  were this not the case it would not be possible to reason about then, or indeed to use natural  language at all.   This automatically leads us to anticipate that the underlying formal canon will be 'soft' or elliptical or incomplete.  Given that the larger Reality resides within the Ungrund -a domain so abstract that it stands above all modes and categories.  An essential service, therefore, that Metamathematics must deliver is that  of unfolding and unraveling this diaphanous domain to deliver the multimodal richness that we find both within and beyond ourselves. 

             It is not easy to speculate just how a metamathematician should conduct himself in the pursuit of this discipline.  Evidently though some kind of process of contemplation; and about the only principle of guidance which would be available would be a sense of convergence and consolidation.  Given that virtually all creative mathematicians are Platonists, perhaps all that is called for is a shift in the mode of Platonic access –from the articulate to the intuitive.  However, where anticipations of how much –and what kind of- progress may be hoped for- we must not lose sight  of the fact that what we are confronted by is not a problem  -as normally understood- but rather, a mystery.  A problem is something that can be confronted and resolved –at least in principle.  But a mystery can never be surrounded, or made captive, or its contents ever plumbed to the depths.  However, -and it is here that a bright ray of hope enters- this mystery contains a problem, inasmuch as interfaces are possible;  windows of a sort can be opened through which something can be seen, though what is viewed can never be brought into sharp focus.  It is enough to hope that what can be glimpsed will be sufficient for our immediate needs

            Perhaps what has done more for me, to bring everything into a single focus has been a prolonged immersion in the writings of Plotinus, Eckhart, Cusanus, Böhme, Bruno –not forgetting some exposure to Taoism and the Bhagavad Gita.  What these authors rub home is that that we are confronted by a problem in mathematics –or metamathematics as we had better call it -for it bears little or no relationship to its Bourbakian counterpart.  The system’s grounding  is not a set of axioms or other primitives upon which it rests,  but rather a unitary invariance of the All from which it hangs.  Such notions are implicit within the contemplations of Cusanus -a philosopher way ahead of his times;  he was to cast a long shadow: 

"......The central fact about Cusanus......is that he is a creative mathematical mind who has in him already the modern idea of mathematics as the 'science of the infinite'.  This idea, in itself, undercuts radically the conventional and rather simple-minded notion, entertained by the scholastics, of mathematics as the science of magnitude, 'that is, of more and less'.......But if....mathematics is the science of the infinite (which is indeed what the great Greek mathematicians had discovered, and Aristotle had tried to cover up), why, then, his metaphysical emotions are apt to respond.  To a mind with medieval training, infinity participates in the divine essence, and should be understood to be somehow at the core of all being. Mathematics ceases to be a science of mere abstractions and becomes a possible avenue to true knowledge of reality."                               Giorgio de Santayana

             The question of categories clear is one which has to be addressed in any comprehensive philosophical overview.  Some examples are given in Figure 2.  Both Eastern and Western sets are included;  one senses a semi-systematic difference between Eastern and Western proposals;  in particular contrast the I Chink with Kant’s listing;  clearly Kant’s has formal overtones less  obvious in the Eastern counterpart.  Can the ‘Eternal Verities’ of Beauty, Truth and Goodness be mapped into McDougal’s triumvirate of Affection, Cognition and Conation?  Perhaps, with a little stretching..  Beauty may be aligned with affection –taken at its best.  Truth, in this context, connotes more than the absence of error or falsehood;  it means rather the appreciation of formal excellence, hence lying somewhere within the cognitive sphere of influence.  Again, goodness, surely, is more than a state, being a dynamic reaching forwards into a higher state of excellence. –hence placing it at the apex of conation. How is this refractory area to be engaged?  It is one thing to discuss problems of modes and categories through the medium of Natural Language –which indeed can be regarded as a ‘soft’ formalism.  If Leibniz’s dream of a Characteristica Universalis were at hand, then its supposedly ‘sound’ foundations –as some analogue of conventional mathematics- might offer us the needed structuring tools to make sense of this recalcitrant area.  However, except perhaps for some of the more vocal advocates of the ‘Artificial Intelligence’ initiative, little hope is held up for such developments to be forthcoming. 

.         But times have changed -due to developments in logic, mathematics and language that were to emerge mostly during the 20th Century.  To mention a few;  Cantor’s Alephs,  the Second Order Predicate Calculus, Geometric Topology, Gödel’s theorem, Recursive Functions and  our insights into the nature of language –started within Port Royal, to be elaborated by Wilhelm von Humboldt a century later, and to receive its coup de grace, in our own times, by Noam Chomsky.  And finally, the Mandelbrot set –something to keep us awake at nights!!   The concept of ‘symmetry breaking’ has been very much in evidence –many centuries before it was to be taken up by physicists.  Many of its transactions are non destructive in the sense that its products do not replace or obliterate their sources –which remain intact;  evolutions are largely epigenetic  rather than transient  Had they been available to Cusanus, who can say where Metamathematics might have been today     

           Now let me attempt to outline the form which metamathematics has come to take within my philosophical overview.  Take a look at figure 3 –that has been put together, both to display its intrinsic characteristics, and also the ways in which it contrasts with the standard mathematical canon.   Its most striking characteristic is the way in which it takes its origin from the higher Unity –the ‘All’ of logicians in stark contrast to the integer of number theorists.  As with conventional mathematics, its essential nature is that of a master invariant that supports a vast envelope of covariant possibilities.  But the form of the invariant/covariant relationship is very different in the two cases;  in the metamathematical canon it hangs from its unitary apex from which the covariant envelope emerges through a progressive differentiation of subdivisions –an expanding divergence that contained within  the Invariant Source. Metamathematics is ‘top-down’ while that of its ortho counterpart is bottom up.  A final way of putting this is to say that metamathematics must be viewed by looking downwards into its interior  it rather than bottom-up and at.  I have been at pains to convey this distinction in Figure 3.  It is most important to distinguish between two ways in which ‘top-down’ may be taken.  That given above isn’t accepted by the intellectual;  for him ‘top down’ is interpreted to mean standard mathematical configurations ‘viewed’ from above.  It has nothing to do which foundations remain secure as classically understood.  It is simply a matter of posture –of where one has chosen to stand.   

          Metamathermatics takes its emergence more or less directly from the Nameless All of the Ungrund while conventional mathematics is built upon a small set of axioms, postulates and other primitives. Both Meta- and conventional mathematics stem from a common source;  there is no question of their being two disjunct entities sealed off in private formal domains. Further, it would be much nearer the truth to say that conventional mathematics emerged from its Meta-counterpart than to attempt to depict Metamaths as an extension of ordinary mathematical analysis.

          Figure 4 is of great related interest for the way in which it shows how both conventional and metamathematics are needed by minds in action.  This figure depicts the mind caught at a typical conscious existential moment.  The upper, metamathematical stratum is supportive of the Agency of the Psyche, the gamut of qualia, values, and the vague intuitions of 'meaning'. 

       It must not be forgotten that there is a lex moralis every bit as much as a lex naturalis.  There are absolutes within ethics, even though the unfocussed form of any ‘top down’ regimen is not of ‘normative’ constitution, i.e. is something capable of a definitive list of dos and don’ts –however useful some of these may be as guide lines.  I shall have a great deal more to say about this in the concluding sections of this document. The lower, stratum houses the objective, more or less articulate aspects of mind and the world beyond which it knows.  Though grounded upon conventional mathematics as understood as the 'bottom-up' domain of formalism, mind could not function in the absence of extensions of the conventional bottom-up domain.  Contending with this 'Exotic' domain -as we might call- raises no daunting problems of a philosophical or metaphysical nature.  It is constrained to the same basic parameters and categories of its classical counterpart.  It simply underwrites legitimate science -mostly that supportive of the organic province of life, mind and evolution.  It may be regarded as something that arises from conventional matter as a kind of phase change -when the underlying  molecular configuration is appropriately constituted.   

        Let us look for a moment at that incredible miracle of Natural Language.  Clearly it is formal in nature -it follows vaguely understood rules which are soft and imprecise.  The syntactical component of any language seeks to preserve this;  it is 'bottom-up' in nature in contrast to the 'top-down' nature of the semantic component.  Yet it has proved impossible to define an exhaustive syntax, or to cleanly separate it from semantics.  Roger Penrose insists that the discipline is essentially open and elliptical and cannot be brought into effective operation save by an appeal to a connection to an extended Platonic  domain.  This makes of him a Platonist in spades;  I fully support Penrose over this matter.  One might say that in operation, the flow of speech -to be properly understood- must forever run within a Gödelian influx -however slender this may be much of the time.