The need for a ‘transfigural’ mathematical approach to understanding the evolutionary origins and diversification of natural flow-form

By Lere. O. Shakunle and Alan D.M. Rayner

(lereshak@yahoo.de; a.d.m.rayner@bath.ac.uk)

Summary

In this short communication, we address the following fundamental issues:

1. Why the foundations of classical and modern mathematics are paradoxical and inadequate as a basis for understanding the origins and dynamics of natural evolutionary processes.

2. Why new, more naturally representative foundations are needed.

3. What fundamental geometric principles these new foundations need to be based upon.

4. How new understandings of the natural origin and interplay of spherical, spiral, linear and annular flow-form can emerge from these principles.

We go on to provide a brief introduction to how the foundations of the transfigural mathematical approach resolve and clarify these issues.

Introduction

The foundations of both classical and modern mathematics are paradoxical because they arise from an attempt to freeze-frame continuous natural fluidity within fixed three- or four-dimensional structure, which is represented primarily either as a cube extended to infinity or the surface of a sphere. Closely associated with this rigid framing is the representation of tangible form as an assembly of particulate (atomic) point-masses devoid of space. The latter, inescapably discontinuous entities constitute the discrete numerical and geometric ‘figures’ that are conventionally added to, subtracted from, divided and multiplied by one another to represent or reconstruct reality as if from a set of primary building blocks that can only be moved around by ultimately inscrutable physical force.

As a calculating tool, and so long as conditions remain constant, such representation has a huge variety of applications. Even curved (non-linear) trajectories or surfaces can seemingly be differentiated into and integrated from discrete infinitesimal constituent units derived by linear regression and approximation. But deep problems and paradoxes emerge as soon as this methodology is used to try to extrapolate to the past origins or future outcomes of truly natural evolutionary processes of cumulative transformation and diversification. These difficulties arise from what is ignored in the initial abstraction of explicit, tangible ‘figures’ out of the context of their implicit, intangible, ‘ground’ of non-resistive spatial presence that they both include and are included by (which is thereby treated paradoxically as both physical ‘absence’ and ‘distance’).

Without including the intangible ground, particulate figures cannot in themselves be used to account for their own natural evolutionary origins and future behaviour because they can neither be counted ‘down’ to absolute zero nor counted ‘up’ to absolute infinity, but necessarily begin and end at a fixed, dimensionless ‘point’ or boundary limit. What truly is continuous, and hence a limitless, invisible (i.e. ‘appearing’ transparent or dark, depending on situation) indivisible presence that does not stop start at physical ‘boundaries’ is mentally either omitted entirely or cut up into discrete packages coextensive with each figure. In other words, space is either excluded or occluded by an excommunicative mental act of axiomatic definition that imposes complete closure or enclosure around a self-contained singularity of ‘One’ as an individual or group that has no access from or to others in its neighbourhood. Correspondingly all conventional mathematical proofs are self-fulfilling prophecies of the initial discontinuous framing of the figures they refer to – i.e. they are ‘self-referential’ in a way that is exposed by the paradox of the Cretan liar in which an inhabitant of Crete claims that all Cretans are liars (Dawson, 2009).

New Foundations – the transfigural mathematical approach

To overcome these problems at a fundamental level, nothing less than a new kind of mathematics is needed, based on fluid foundations that dynamically include space in figure and figure in space and so incorporate both zero and infinity into natural flow-form. This is the basis of transfigural mathematics, first developed in 1985 by Lere Shakunle (see Shakunle, 1993). Conventionally founded ‘reaction-diffusion’ models of autocatalytic growth may, however, to some extent both anticipate these new foundations and provide a conceptual platform from which the need for them can be appreciated (cf Davidson et al, 1996).

At the philosophical core of transfigural mathematics is the recognition of two distinct but mutually inclusive and co-creative kinds of dynamic influence at work in the cosmos: figural and transfigural. Figural influence is energetic, resistive and brings local distinctiveness – but never complete discreteness – to natural, variable viscous flow-form. Transfigural influence is non-resistive, spatial and present everywhere, without limit. Transfigural influence can be thought of implicitly as spreading radially, from ‘zero identity centres’ (called ‘zeroids’ in transfigural mathematics, see below) distributed everywhere, although this radial influence cannot be manifested explicitly without figural influence. Figural influence can be thought of as distributed tangentially with respect to transfigural influence, in fluid boundaries or energetic interfacings that distinguish but do not isolate inner world from outer world. Whereas transfigural influence is receptive, figural presence which includes space that includes it through the transfigural is reflective and responsive, so providing scope for a vast variety of flow-form to emerge as energetic/fluid configurations of space in figure and figure in space.

The transfigural emergence* geometry of natural flow-forms

The underlying dynamic geometry of natural flow form and transfigural mathematics is hence primarily non-linear and space-including, i.e. the inverse of abstract Euclidean and non-Euclidean geometries. Through the interplay of inseparable transfigural and figural, radial and tangential dynamic influence, spiral, linear and annular form emerges fluidly from primarily spherical form as a local figural inclusion of non-local transfigural influence. Hence, it is possible to account for anything from the Earthly growth forms of fungi – as they germinate from spores, radiate, network, form fairy rings and mushroom (e.g. Davidson et al, 1996, 1997) – to the orbital systems of atomic, planetary and galactic flow-form as expressions of transfigural (gravitational) in figural (energetic) influence (Shakunle & Rayner, 2009).

Transfigural space is not only gravitational but it is also a dynamic presence of love and creative potential.

With this understanding, the opposition of ‘scientific’ doctrines of evolution by natural selection and ‘big bang’ cosmology to the ‘religious’ doctrine of ‘spontaneous creation’ by a forceful father-figure is shown to be no more than the product of a collision between two intransigent objective orthodoxies built from discontinuous logical foundations. Correspondingly, the open-ended understanding of evolution through natural inclusion (as ‘the co-creative, fluid dynamic transformation of all through all in receptive spatial context’, Rayner, 2006, 2007, 2008; cf Rayner 2004), which arose from an appreciation of transfigurality, is radically different from the closed-ended materialistic notion of evolution ‘by means of natural selection, or the preservation of favoured races in the struggle for life’ (Darwin, 1859). The natural inclusional perception is fully compatible with a spiritual and soulful perception of life on Earth in which ‘figural’ couples fluidly and inextricably with ‘transfigural’ in natural flow-form. There is no justification, given this inclusional and transfigural view, either for the supposition that everything is predetermined by the initial conditions of its origin, or for the belief that waywardness must be eliminated as the ‘means’ to achieve the ‘end’ of a desirable future. Instead, the ‘present’ endlessly includes the presence of the ‘past’ in the coming of the ‘future’.

The Transfigural Middle – From the Number that Excludes To the Number that Includes Its Neighbourhood In the Core of Its Self-Identity

The discontinuous foundations of classical and modern mathematics, respectively underpinning Peano’s arithmetic and Cantor’s sets, cannot accommodate the recognition that where everything flows it is possible to be dynamically distinct but not permanently discrete. Even Brouwer’s intuitionistic mathematics for infinite sets based on ‘actual infinity’, cannot do this.

Although intuitionists explicitly reject the applicability of the discontinuous ‘law of the excluded middle’ (LEM) or ‘tertium non datur’ (‘the third is not allowed’, there can only be two alternative possibilities of present or absent, true or false) to ‘actual infinity’, they accept the discreteness of finite numbers as constructions of the human mind that collectively make up a ‘process’ or ‘potential infinity’. The problem here is that infinity, whether potential or actual, is an enigma when perceived conventionally in terms of a discrete ‘whole’ and its ‘parts’. By its very name, it cannot all be comprehended, because no human life is long enough to reach to its endings, which are always openings. In other words, infinity is open throughout its length, breadth and depth and this is irrespective of whether it is given once as ‘actual’ or constructed from finite intervals as a ‘potential infinity’.

The solution to this enigma, provided by Transfigural Mathematics (TfM) entails showing, through the concept of ‘relative infinity’, how the potential is a flow in the actual such that the dichotomy of actual and potential is removed. Space is a vital inclusion of the flow. This is possible through local figural in the nonlocal transfigural in transSpace. Although fundamentally different from intuitionism in its space-including foundations, relative infinity provides a way of understanding how the potential infinity that can be constructed lives inside the once-given actual infinity.

Relative infinity originated from the condition of the flowcurves of zerospirals (see below). It is an infinity that flows into and out from the pure infinities of the inner in the outer infinity. Being the flow of the figural of here-there in the transfigural of there-beyond, it transcends all construction. A relative infinity is what can be envisaged clearly but cannot be reached. An illustration of relative infinity can be found in the pursuit of Woland by Ivan in Master and Margarita (p. 49 – 51) whom Ivan could see before him but could not reach. It is infinity here and there in the infinity there and beyond. Relative infinity is a given that can be lived. It is life here and there; it is the dream here and realization of it there; it is the goal in hand here whose attainment is there; it is what urges on that cannot be reached. It is the future impinging on the present.

To return to mathematics and its logic, it was not only intuitionism that tried but failed to break through or somehow get around the law of excluded middle in order to account for infinity. So too did the sets of Georg Cantor and their ‘transfinities’. A set could be thought of as a Many that is made into One by connecting them up in contiguous series. But each of the many remains defined axiomatically according to the LEM as a part that is not another part of a whole that is not another whole. Perhaps Cantor himself might have foreseen that set theory is susceptible to the liar paradox, which shows the inherent conflict between part and whole to be found in any logical system built on the dichotomy between One and Many, even when the latter are forced to connect into the former as figures and figure alone.

In transfigural mathematics, the paradox that originates from a set being One that is Many is solved with a fluid logic number as One-in-Other fluid logic numbers. Here, the Other involves, if, for example, the One is a person, other persons of society and the world, nature – fauna and flora - and space, one as home and the other as habitat. If One is a tree, the Other are the other trees of forest and ecosystem, nature which includes ecosystem and space and so includes the tree and the forest and human beings.

One-in-Other as a flow translates therefore to One knowing what it is through others, alongside One, and through these others, to and through All with All also reaching inside One. This also translates to the origins of human identity, the place of the person in society and the world and how human beings are in Nature and Nature in them and how all are included in Space, a common habitat that includes them. The middle, in this transfigural outlook, is not only included but every fluid logic number which is distinct, flows in what comes before and after it that also flow in it. What results is every fluid logic number is a flowform. Every fluid logic number is the included middle that gathers in from, gives out to and derives its identity from its natural neighbourhood.

The Logic of the Transfigural Middle

In the intersection of conventional sets through the union operation

A ∩ B (with ∩ as AND )(*)

both A and B are two different things being brought together, neither of which is known for what it is. Even where we have,

A ∩ A(#)

and substitute A with a number such as 2 and ∩ with logic ‘+’ in which (#) is 4 split in two, the problem of what 4 is remains unsolved.

Formalists, for whom mathematics is a game of symbols without meaning, do not concern themselves with what a number really is. To them, numbers are just like the point of Euclid – lifeless. One can understand their fear and dilemma as they confront living beings, for which their idea of identity in simple terms of ‘presence’ or ‘absence’ (0 or 1) is meaningless, however much it may help in solving problems based on axioms that are packaged in water-tight rules. Yet to a breathing, living world as we have it in transfigural mathematics of breathing-points, identity matters. Indeed it is at the heart of human being and becoming.

Things cannot be formalized in transfigural mathematics in the same way as in classical, modern and intuitionistic mathematics. This is because the nonlocal transfigural T which is trans-linear (nonlinear in linear and beyond) is always changing the local-in-nonlocal figural f a flow into, out of and deep within itself. Hence, the figural is always becoming whilst the transfigural is inexhaustible in its transfigural creative potential. Nonetheless, for the sake of getting the feel of one of the fundamental differences between transfigural mathematics and the classical, modern and intuitionistic mathematics – which also differ in some fundamental respects from one another – the transfigural versions of (*) and (#) above will be presented in the following account.

In a transfigural setting, both (*) and (#) are very interesting. For (*), A and B conventionally represent discrete and different units or connected groups of units, but in transfigural mathematics,

A ∩ B

mean one and the same identity once “∩” is substituted with “T” for “T(ransfigural)” such that we have the set-theoretic (*) as transfigural:

ATB(§)

which is valid for every fluid logic number.

In (§) we have the figural (AT, BT) and the transfigural (TT) = (T) which is the creative potential zero (0).

By substituting “┬” for T in the figural (AT, BT), we have figural (A┬, B┬) . This is written as figural (┬,┴) respectively. From this follows that we now have the generalized form of (§) as:

fln(T) = ┬T┴

with fln from (f luid logic number) which can be written simply as:

fln(T) = xTy

=(xT, (TT), yT)(§i)

= (┬(T)┴)(§j)

in which (§i) are the numbers of fln(T) and (§j) is the logic of fln(T).

With T=1,2,.. we have (§i) as the numbers of fluid logic numbers fln(1,2,..). If T=1, then (§i) are the numbers of fln(1). These numbers are unique to fln(1). With (§j) we are dealing with transfigural logic. This logic is valid for every fluid logic number.

From fln(T) we have it that for every classical 1, the transfigural is 012 and for every classical 2, the transfigural is 123 and so on with the logic, their basic fluid constitution being the same.

Correspondingly, the distinct but not discrete figural (┬, ┴) flows embody the transfigural (T) space within that is continuous with the space without that includes the figural as One-in-All. Hence, what we have in (§) is the flow of a fluid logic number inside itself and through the figural (┬, ┴) in transfigural (T), in the others before and after it and through them in all that includes space everywhere.

Suppose two numbers are different, what do we have? This simply gives:

A(T)B

in which A is a flow in B through and in (T) which includes the space that includes them.

An example of A(T)B is given in (§k) later. There we have, for example:

fln(k) = (┬(T)i┴┬(T)j┴)

= (┬(T)i┴(T)k┬(T)j┴)

in which we have:

fln(1) = (┬(T)i┴)

fln(2) = (┬(T)j┴)

therefore (refer (§k)):

fln(k) = fln(1,2)

= (┬(T)i┴(T)k┬(T)j┴)

In the case of (#)

A ∩ A

if for example A is 1 or 2 or any number, there are different types of 1s and 2s and so on in fluid logic numbers. For example, fln(1), fln(37), fln(100) are all 1. This means in for (#) we are dealing transfigurally with different As such that we shall have (#) as:

Ai(T)Aj

in which a type of Ai which may be the watertype Ai is a flow inside watertype Aj which Ai and Aj are different kinds water with Aj having more folds (heavier) than Ai. By their internal constitution (transfigural logic) Ai and Aj are equal but different (in their folds).

From this follows that we have:

Ai(T)Aj = watertypei(Transfigural for all all Watertypes)watertypej

This (T) is the transfigural of all types of water. The (T)k in fln(k) above (to be met again later) is the transfigural across and through different things. For example all types of water as Ai,j,.. and all types of tree as Bi,j,.. have (T)k whilst all types of trees, as it for water above, have in them (T). (T) and (T)k include what is before and after them which include space that includes them.

Now, since there is no oppositional OR in transfigural logic, what happens where we represent (*) as,

A U B (with U replacing OR)

In other words, what kind of relationship exists transfigurally between A and B in (*)? Here it is vitally important not to confuse transfigural logic with the purely connective, relational logic of a structural field of pre-defined space. In transfigural logic, what is between one identity and another is not a connective field but a pool of space in which each flows in the other’s influence. But for the sake of providing a gateway from conventional set theory, we will stay with this question and give the transfigural answer which is samedifference.

The symbol for samedifference is which is “same and different” which is shortened to read samedifferent such that we have (*) in transfigural mathematics as

which reads, “A samedifferent B”.

What does “A samedifference B” mean? It means that if A and B are fluid logic numbers, they are the same through their basic constitution which is transfigural logic (§j) but are different through (§i) which are their figural numbers that produce the same logic. In other words they are the same in their basic form of dynamic inclusion of transfigural space, but unique in the figural (alpha-omega) neighbourhood from which they derive their local identity.

In terms of language, the concept of samedifference enables every fluid logic number to be recognized as a story within context: a unique dynamically distinguishable identity that cannot be extricated from the common space in which it flows in the other’s influence. It is naturally distinct but not absolutely definable as an independent entity. This corresponds with William Wordsworths recognition (Wordsworth, 1815) that:

“In nature everything is distinct, yet nothing defined into absolute, independent singleness”

This samedifference of ‘dynamically distinct but not rigidly discrete’, in which one is a flow in the other, therefore provides grounds for mutual respect for difference as the source of diversity and its beatification, without contradicting what is common to all.

Samedifference – Its Sustaining Spirit

In the oppositional logic of OR, it is lose, out, win, remain. There are no such grounds for conflict in samedifference. Underlying samedifference is the spirit of sharing: giving, receiving and passing on in a natural flow of energy. There is no mutually exclusive game of winners and losers. In samedifference, the greatest pleasure comes from bringing joy to the heart of the other through sharing in the gift and promise of life as it waxes and wanes, like a child belonging in the community of a village that belongs in the community of the world and ultimately limitless cosmos that is its home.

Questions of Difference in Sameness

Aristotle raised a question about the supposedly uniform and spatially separated atoms of Democritus, which shows the discrete limit of ancient thinking and the Greek foundations of objective science that persist to this day:

“if all of them [the indivisibles (atoms)] are uniform in nature, what is it that separates them from one another? Or why, when they come in contact, do they not coalesce into one, as drops of water run together when drop touches drop…”

[Needham, p. 5]

The central concerns of this question are (a) what is it that separated them [atoms which are uniform] from one another, and (b) why…do they [the atoms] not coalesce into one? With respect to (a), it is the condition and nature of the fluid logic numbers of transfigural mathematics that one is a flow in the other can only be possible by virtue of their dynamic inclusion of space in space (i.e. relative infinity) and so there is no absolute separation between one fluid logic number and any other. With respect to (b) a fluid logic number is the same, through transfigural logic, in its local-in-nonlocal form as any other. There can be no hierarchy of greater and lesser when every figural form transfigurally includes the space of zero and infinity: in this sense, an elephant is equal to a human being and a bird to an elephant. The transfigural logic which accounts for their fundamental equality also ensures that fluid logic numbers do not coalesce into one. To show how this is the case, we need (§j) again.

We have, for every fluid logic number the transfigural logic:

fln(T) = (┬(T)┴)(§j)

In (§j) (┬) is the alpha, (┴) is the omega both of which are the figural. The transfigural is the included middle (T).

Now let T be T=1,2,…this would give

fln(1,2) = (┬(T)┴, (┬(T)┴)

which, with fln(1,2) as fln(1,2=k) we have:

fln(k) = (┬(T)i┴┬(T)j┴)

= (┬(T)i┴(T)k┬(T)j┴)(§k)

in which we have,

(T)i,k,j = (T)

How do we get Tk in (§k)? The answer is in the bidirectional and reciprocal Alpha-Omega nature and condition of the figural, whereby for every figural (┬, ┴) there is an included middle transfigural (T). This is what we have in (§j). And so, in (§k), fln(1) and fln(2) as flowforms do not coalesce in one but flow as one in the other even if they are drops of water!

Mathematical Identity and Natural Identity

There is a disparity between what conventional mathematics calls identity and the natural identity of things. Unfortunately, those sciences whose prime concern is to observe and understand truly natural identity, don’t take this into account. Whereas conventional psychology founds its concept of identity individualistically, in the discrete person, conventional sociology treats the individual as a member of a discrete collective set or group. Similarly conventional ecology divides itself between the autecology of individual species and synecology of ecosystems. This is because in the mathematics of social, psychological and ecological research, the discrete 1 of classical mathematics still reigns supreme as the basic unit of quantitative and qualitative methodologies and bounded theory.

A glaring example of the disparity between mathematical identity and identity of things can be found in the difference between ordinary and scientific perceptions of water (Brakel, 1997):

“The problem between the manifest and scientific is illustrated well by Hare’s (1984) defence of ordinary, liquid water as not supervening on (scientific) H2O – in solid, liquid, or vapour form. Hare says there is no such supervenience or dependence, first, because, ‘water’ and H2O are words with different meanings (senses)..”p.257”

Brakel goes on to conclude:

“I must conclude therefore that ordinary (somewhat vague, but no less objective) concepts like ‘water’, and even ‘(pure) subatance’ are somewhat much more better entrenched than ‘atoms’ and ‘molecules’. If the question of priority must be raised – I don’t say it should – it is the manifest [water] that is prior to the scientific [water] and not the other way round.” (p.259)

At the core of this disparity is the difference between what is considered to be mathematical continuity and natural continuity. Talk about flow in conventional mathematics is in terms of infinitesimal distances between contiguous points along a line that when run together like the frames of a cine film make the line look like a flow that in reality it is not. The conventional mathematical ‘continuity’ of points set alongside one another is not natural continuity and so conventional mathematical flow is not natural flow.

Transfigural mathematics is about naturally continuous form, identity, distinction, and space. This is made possible by taking account of how Nature achieves continuity and flow through the inclusion of transfigural space in fluid logic numbers. In this case, natural (transfigural) points and lines are neither Euclidean nor non-Euclidean. A point is a breathing-point (opening out, out-folding, folding in the deep and in depth) and a line a pointline (as channel), that is, a line as a point in relative infinity. At the centre of the point is transfigural space through and in which the figural alpha and omega domains form and reform into what can be experienced through the senses and reached through the human imagination. This (T) is the seat of identity of things, the middle that includes and is included by fluid numbers. How might we envisage it?

Figural-Transfigural Interplay – The Breathing-Point!

A breathing-point is a flow-channel that is both a line in a point and a point in a line. It arises from the spiral flow and counterflow of alpha and omega figural domains of the fluid logic numbers of transfigural mathematics into and out from the core ‘zero identity’ at its heart, and so is known as a ‘zero spiral’. A simple example of such an energetic configuration of space in figure and figure in space is shown below.

Used only with express written permission

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Gödel, Truth and the Foundations Crises of Mathematics

Before Gödel’s famous theorem was born unto the world, the foundations of mathematics had been rocked by antinomies and paradoxes that made the sea become suddenly turbulent just as the weather became inclement in the world of mathematics. The battle of the intellect to make exactitute return to mathematics was still raging when exactitude and sure-footedness were shattered by Kurt Gödel. Morris Kline devoted his book, Mathematics, the Loss of Certainty, to the foundation crises of mathematics.

At the heart of most if not all of the paradoxes and antinomies in mathematics are the numbers, mostly what are ironically called the ‘natural numbers’. They were central to Gödel’s Incompleteness Theorem and Cantor’s Continuum Hypothesis which Gödel and Cohen also gave a try. Central to the crises of foundations of mathematics is the conflict between One and Many, Part and Whole which the logics – binary, fuzzy, four-valued logic, many-valued logic and quantum logic – could not be bridge. These problems have their source in human beings.

One and Many is about One and Other, Day and Night, Mind and Brain, Man and Woman, Heaven and Earth, indeed it is about contraries like cold and hot, wet and dry, which are seen as closed boundaries that do not permit of a flow from one to the other. And of course, were there to be no flow from one to the other, there could be no ‘human being’ at the biological level. It takes woman and man to get the child such that right from the beginning of life, identity has been building from one on the right, the other on the left of the One. And so, the three as one is a biological reality. It is also a social reality. A psychological reality too because human survival depends on the community. No human being can survive loneliness in thought. Space pools all together inexorably.

As a universal principle, it is easy to see that every number, like everything, derives its awareness of self through one in the others which involves the pool of space. This is central to the transfigural outlook.

It is not possible to get to the train of thought that led Gödel to reach the conclusion that the mathematical methods in place since the time of Euclid were inadequate for discovering all that is true of natural numbers. With this his discovery undercut the foundations on which mathematics had been built up to the last century [Dawson, Jr. 2009]

What is certain is that to Gödel, numbers are not just mere symbols on the paper as formalists would like to have them and other entities of mathematics nor can they be what the mind can construct as intuitionists demand when the mind cannot reach its own depth. As a Platonist, it was easy to Gödel to see the shortcomings of various approaches to the ‘natural’ numbers especially where one tries to bring them to the plane of human existence. By seeing the numbers from the high heavens of Platonism, it was possible to feel their sufferings of identity below.

But, contends, the philosophy of transfigural mathematics, human beings belong here and there whilst, as given, the entities of mathematics in Platonism belong to the beyond. To know what numbers are requires that we put them in our place which is the best way to get the feel of and feel for others. We are between heaven and earth just as the biological person is between mother and father, and human being between man and woman, day between night time and daytime and so every number is between what is alongside it together and through which it is what it is. So numbers belong here and there as figurals which is why it is possible to have access to them at all and there and beyond which is why they have, as every thing in and of this world, their included middle transfigural (T) which is the seat of the creative pool in which all of creativity finds its expression and discovers or invents challenges that shall remain forever here, there – and beyond.

Conclusion

The transfigural approach is rooted in life which is not possible without the others of other lives, the natural environment that Nature provides and the spatial room that is given as habitat by Space – all out of love. And so, for mathematics to reflect life – and love – numbers, point and the line, the soul of mathematics, must have the qualities of life and love.

In that case, and this is the case with Transfigural Mathematics, every number is a flow of love in life and flow of life in love. This is the flow of being of love in the becoming of life in nature and the flow of becoming of life in nature in the being of love in us and in everything. The becoming of love that is life involves empathy, kindness, yes it is also that smile that comes naturally as greetings from the heart to the world and nature. Becoming is change-in-permanence and Being is permanence-in-change, that is life in love and love in life, indeed of alpha in omega and omega in alpha. In the middle that includes them and which they include is the Transfigural, the seat of the creative potential.

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Wordsworth, W. (1815) Essay Supplementary to Preface

* transfigural emergence is not about the open ends of infinity from which the point in line and line in point emerge as contained in Basic Concepts of Transfigural Geometry (below) transfigural emergence is primordial emergence from the womb of infinity to the relative infinity of becoming in being to the submergence in the same womb of infinity.

Acknowledgement

Many Thanks to Rev. Roy Reynolds for the love and care that he made to bear on his going through and making suggestions for fine-tuning here and there in the paper. In spite of the other priorities waiting for his immediate attention, he put the paper on do-in-now list and sent his edited version to us as urgently as he could make it.