Transfigural Mathematics -
Number, Space and Infinity
By Lere O. Shakunle
1.0. What is it to be natural?
Life flows. Nature Flows. River flows. Air flows. Thoughts flow. What all these flows have together is that they fold. That's what it means to be natural.
Characters flow in a novel. Life flows in awareness. Nature flows in expression. What these things have in common is space. That's what we have been saying about what it means to be natural.
Being flows to exist. Life flows to be aware. Nature flows to express itself. What all these things have in common is the creative potential, the zeropoint of a number that flows and folds thus balancing the odds of One and Many. That's what we have been saying and shall continue to say in what follows about what it means to be natural.
1.1. How natural are ‚natural numbers'?
A natural number, based on what we have come to know about what it means to be natural, should flow. Central to this, we demand that it flows and folds at its odd-balancing centre which is the zeropoint - we shall come to this very soon - and flows at the local centres, the alpha and omega subset subdomains, which could be drawn apart but which as flows can never be severed. The unnatural numbers - Penrose1 [31] wrote them as ‚natural numbers' if only to point the attention to the fact that there is something wrong with the term - or the succession 0,1,2,3,.. do not flow since to do so would mean to be one of those things about what it means to be natural. They are none of them. So what are they then?
They are counting numbers which result from the culture of the eyes that need to separate for their own convenience without the need to ask about what things really are. It is the Mind that should address that deeper question. It is Life that should express its answer. This is what the numbers discussed in this section of our tripartite essay sets out to do.
1.2. Nature, Life, and Numbers
In his book for which he was interviewed2, Alan Rayner showed how even trees which are discreet points in space are connected, how the riverbank is a meeting-melting points of the world of the river and that of the land, indeed how a boundary is dynamic which is the source of his context dynamic theory.
This paradigm, which questions the selfish-gene outlook has implications not only for how we look at life but demonstrates the implications of a gene isolated from the environment of its neighbours and Nature for interpersonal, social, organizational and international relations. The natural numbers which we shall henceforth refer to as ‚natural number' have no place, even when reduced to fractions which should bridge the neighbours, for context dynamics as paradigm nor for inclusionality as philosophy. Alan's theory shows the weaknesses and failures of the classical number theory in general as this relates to grappling with the deep message of a non-discretist outlook in biology. This is how he came to the fluid logic numbers.
Let us try to understand the problem. Life, Nature and Being flow. Indeed, as Heraclitus says, everything flows. Yes Heraclitus. We can as well begin to use him as our inroad to the world of fluid logic numbers. From Heraclitus, „Logos and the Unity of Opposites" we have, among others, his most famous passage where we find, (i) that the world is in constant change (different and different water flows) and (ii) the world is unified whole (the river) which is constant yet contains this perpetual change made possible by flow. And as what served as the forerunner of Plato's „The Divided Line" Heraclitus talks about anything which differs as paired opposites, such as hot-cold, but that these paired opposites can be viewed as one, unified whole3.
In life and nature, there are these opposites everywhere. They are however not in the binary sense as was understood by binary thinking but in the sense of two sides of a single coin. At this point however, something happens. And this has to do with Truth and Falsehood which is what binary thinking made of opposites. In the world outlook of Heraclitus, it is not possible to apply True-False dichotomy because it is not possible to unify Truth and Falsehood. In that case, it is also not possible to unify Good and Bad. What are we then left with to avoid monism - Leibnz's view of a world of discrete, undivided entities, the ‚monads'? This answer is provided by fluid logic numbers.
Fluid logic numbers, as we shall see, are threesome oneness. They are based on Truth with different domains. These are called alpha and omega domains which represent the negative and the positive sides of the same thing, in this case, Truth. The centrepoint of these domains is called the Zeropoint. This is the seat of Creative Potential which is composed of zeroids. What this means transfigurally is that it is only true statements that are acceptable in the logic of fluid logic numbers. This leads to the Existence Criteria. These say that a thing exists if it meets the following conditions:
(i) It is true
(ii) It flows (by virtue of which)
(iii) It is dynamic
Both (i) and (ii) consitute the source of ‚dynamic existence' which we agreed to be the leitmotif of our group. There was a lot of discussion in the group which consists of myself, Alan who wrote the first part of this paper and Ted whose paper follows this one before coming to see that dynamic existence which says that what exists is dynamic on the condition that it flows unifies our metaphors. In that context, life and nature exist and are dynamic becaue they flow. This is what the fluid logic numbers set out to demonstrate. Before then we shall take a short excursion in the origins of ‚natural numbers'
2.0. Peano Arithmetic and Numbers
More than any other works [Frege4, Dedekind5], Peano's Axioms6 constituted the foundation of what are called ‚natural numbers' but which, as we shall see, do not fulfill the conditions of being natural. These axioms are at the basis of his arithmetic. This arithmetic and the implications of its axioms are to be found in different forms in what is today known as Number Theory and in all of classical and modern mathematics. And since unnatural numbers are primary, indeed since they are the foundations of any other numbers and beyond that since together with point and line they constitute the soul of mathematics therefore are to mathematics what Nature is to physics, it is being true to history, ancient and contemporary, that they constitute the foundations of and the source of inspiration in mathematics. Even set theory was inspired by them. It sets out to capture their mystery and demystify the riddles of their fundamental properties such as membership and infinity.
The informal presentation of the five axioms of Peano for ‚natural numbers' is of the form:
1: 0 is a natural number.
2: If a is a natural number then so is a+1.
3:If you can prove something about a and
that implies that you can prove
it for a+1,
and if you can prove the very same thing for 0 ,
then will this
hold for all natural numbers.
4: If a+1=b+1 then a=b.
5: You can not add 1 to a natural number
to get 0.
What comes to mind straightaway from these axioms is „What is a natural number?" „What is?" is concerned with identity of things. Now let us put this question to a gene. We would say that a gene is itself and its environment. This is a gene that flows. It is not and cannot be selfish. Let us put this question to human beings. We would say that human beings are what they are and Nature. This is a being that flows from self to Nature and since Nature flows too, we would have Nature as itself and the World which includes the being and all things. This is what is meant by being natural. What we got from these axioms is a discrete number of the form 0 for axiom 1 and 1 for axiom 2 such that in the end we have what we learned by rote in the school, 0,1,2,3.... in which 0 normally doesn't feature. The questions we put to the gene, the being and Nature and the answers we get prepare us for the fluid logic numbers and their spiral geometry which is composed of the geometries of zero spirals. Alan calls them non-Euclidian geometry. I call them Post-Euclidean geometry to say goodbye to Euclid.
3.0. Sets and Infinity
One of the problems which unnatural numbers present us with is how to cope with the infinity of them. The other one, which is equally fundamental, is identity of numbers or what numbers are by which we mean natural numbers. One easy way which classical mathematics found in this respect was to call 0 and 1 identities and give conditions for being so. And then one can ask, what about the other numbers? Do they lack identity or what? Now let us dwell a bit on infinity. It is not possible, based on the ‚natural numbers' of Peano to count these numbers to the end. They are like pebbles by the river side. They can keep the head busy with counting for a lifetime.
There must be a way to grapple with the infinity of unnatural numbers, argued the founder of Set Theory, Georg Cantor. In this case he has the infinities of unatural numbers in mind that is -( and +(. That's why his transfinities begin with the unnatural numbers and continue to omega and from there to aleph ending in the Absolute [Infinity]. What emerges in the end are hierachies of infinities in which one is greater than the other(s). This legitimizes „greater than" or „less than" which Cantor wanted to eliminate with his sets. In other words, there must be a way to manage if to tame this infinity of unnatural numbers is not possible. If we go the way of Peano, this is not possible. To overcome the infinity that cannot be counted to the end, Georg Cantor invented his transfinities even as he allows a set to contain as many as the mind might think of.
The battle that was let loose by Peano arithmetic which are contained in his axioms was found to be more than enumeration. It is a battle between One and Many. In this context, I mean in the form we got it from Peano, 2 is greater than 1 and 3 is greater than 1 or 2. This means One is less than Many. And then what you get is that One is a world apart from Many since One is not equal to, cannot be equal to Many in the outlook of Peano that produced his axioms. If One is not the same as Many, we can then say that One is the Other of Many and Many is the Other of One. What results is the division of consciousness into a battle field of One and Other which is the origin of binary thinking of two worlds, two ideologies with no sands of compromise between them. Set was born to put an end to this conflict. It achieved quite a lot but couldn't.
3.1. What is a Set?
Set theory sets out to bridge the gulf between One and Many. In Cantor's words:
A set is a Many that allows itself to be thought of as a One7 (p.42)
How is this possible? If we deal with Peano's numbers this is not possible since it is stepwise growth and more fundamentally many of such numbers cannot be thought of as a one. We have from Aristotle two types of infinities. These are (i) Actual, and (ii) Potential Infinities. Actual infinity is presented to the mind at once while potential infinity is forever becoming. Cantor based his set theory on actual infinity. This infinity is presented to the mind at once. This means the mind can think of actual infinity which is composed of many as a one.
What does this mean mathematically? Sets begin with null set which is {} such that we have:
0 = {} or Φ
1 = {0} or {Φ}
2 = {0,1}or {Φ,1} (*)
John Barrow gives a pictorial example,
Define the number zero, 0, to be empty set,..because it has no numbers. Now define 1 to be the set containing 0; that is, simply the set {?} which contains only member....that is {?}. It is important to see that this is by no means the same thing as empty set. ...Carrying on this way we have 2 to be the set {0,1}.... (p.166)8
such that when we continue in this way we get the number N which is defined to be:
the set containing 0 and all the numbers smaller than N, so N = {0, 1, 2,....,N-1)9 (167)
which we see to be the case even with 2 for which the last number is 2-1 in (*) above. What we get is a far cry to Peano! In set theory we would say we have a number defined by all its predecessors without itself and its successors. In set theory we would say we know where we are coming from but may not know where we are going. In Peano's arithmetic we don't know where we are if we are anywhere at all. Even then sets as numbers are richer than all that. They confront us with the mystery of Nothing(ness). Now let us look at the empty set again. Is it really empty? Something is enclosed in something even though we may not see it! All this takes us to introduction of Infinity Theory which is the physics of zero spirals10. The (0) of fluid logic numbers, as we shall see, with their zeroids tell us that there is Something out of and inside Nothing.
3.2. Infinity and Paradox
Cantor's creation is neat. If this is so, how come the paradox into a world built on solid foundations? The Liar Paradox was the single challenge which Cantor could not solve. It is this paradox that inspired works by Frege, Russell and Whitehead. It originated from two concepts in set theory. These are (i) counting, and (ii) membership. The issue of Many as possible of being thought as a One could be seen to be so in the nature of actual infinity and how the mind sees it. Not so with membership which, once related to counting, brings the conflict between One and Many, therefore One and Infinity, through the back door. There was a town called Crete.Truth is unknown to the people of Crete. Simply put, they are all liars, the Cretans. And then comes a man who tells us that all Cretans are liars. True and False at the same time we would tell him. If it is true what he said it must be false since whatever any Cretan says is a lie. The problem arises from a kind of All or None logic of membership. The One is not Many which is what this Cretan was telling us when he says he is speaking the truth but once we say this One is Many based on a logic that is binary as set theory - a thing is a member or not - then the soil is created for a battle royal between One and Many. The problem is compounded by the fact that we are dealing with counting numbers in set theory. This is also a bit of the apron string of Peano. Be all as it may, set theory is still goodbye to Peano.
Intuitionistic mathematics which was founded by Brouwer11 came to say, okay, I have to tell you Cantor that the problem lies in the law of excluded middle, that is, in either-or that is being applied to infinite sets and before you go, I hasten to add that this actual infinity has to be shown the exit too. Potential infinity which involves the step-wise contruction of numbers should be able to tame the paradoxes. Intuitionistic mathematics with its abrogation of the law of excluded middle for infinite sets and set-theoretic approach that is based on potential infinity could not tame nor manage the paradoxes either.
4.0. Fluid Logic Numbers of Transfigural Mathematics
Thesis : A fluid logic number(12,13,14) is One that is Many and Many that are One
4.1. Definition of Fluid Logic Numbers
A fluid logic number is itself and the number before and after it.
In other words, a number that derives its identity from its neighbours is a
fluid logic number.
In this case, we have the generalisation of unnatural numbers of classical mathematics as,
cln(x)i=1,2,....,n = 1, 2,......,n (#i)
The generalisation of fluid logic numbers which we have shown fulfil the conditions of numbers being natural gives,
fln(x)i =1,2,....,Ψ (= 012, 123,...,∞-1(Ψ)∞+1 (#ii)
with ( as inner infinity. This is the seat of zeroids.
From (#ii) we have, for inner infinity,
n -1 = ∞-1 : outer infinity on the minus side of zero
For example, for fln(1)=012, n-1=01,12
such that we have 01=12 for alphas.
(refer zero spiral below)
n+1 = ∞+1 : outer infinity on the plus side of zero
Equally for fln(1)=012, n+1 = 21, 10
for which we have, for example, 10=21 for omegas.
(refer zero spiral below)
Ψ : Inner Infinity
This is (0) logically which is 11 in 012, 22 in 123...
with 11 = 22 =.....
In what we have above, the mathematical symbols of „greater than" and „less than" do not apply. This is because alpha 01 is the same (in Form (logic)) as even though different (in Content (mathematics)) from 12. By the same token omega 10 is the same (in Form (logic)) as even though different (in Content (mathematics)) from 21. What this means is that as Form, alpha 01 and 12 are the same but as Content, they differ. The same holds for omega 10 and 21. When we bring the form (logic) and the content (mathematics) together what we get is equality. That is (01 + 21 =22) and (12 + 10 = 22) respectively. This makes the transformation of one into the other possible.
Putting (#i) and (#ii) together we have,
fln(xqy = n)
with
x = n-1
q = (n)
y = n+1
for which we have the classical ‚cln':
cln(n=1,2,3....)
such that
fln(1) = 012, fln(2) = 123,....., fln(n)= n-1nn+1.
For now, there are three types of zero spirals. Zero spirals derive from [a] fluid logic number(s). This results from,
fln(1)j = 012
= (01 (11) 21) (i)
and
fln(1)k = 012
= (12 (11) 21)
(ii)
from which we have the following numbers:
01, 12 : alpha numbers or simply alphas which is α
11 : identity I which contains the zeroids Z
21, 10 : omega numbers or simply omegas which is ω
From these numbers which differ for every fluid logic number follow the logic:
fln(1)j = 012
= (01 (11) 21)
= ( 0 - 1 (1 - 1) 2 - 1 )
= ( -1 (0) +1)
which is the same for every fluid logic number. The (0), which is the world within the worlds of the general domain (-1 (0) +1) is called the Zeropoint or the Creative Potential. It is the Identity I of the general domain whose local domains are the alpha domain -1 and omega domain +1 respectively. The Zeropoint is the seat of zeroids Z. The table below takes us a step further.
4.2. Naturalness of Numbers
At the beginning we gave the conditions for naturalness. A number that fulfills these conditions is therefore a natural number. That this is the case with fluid logic numbers is what we set out to show in what follows.
5.0. Space, Fluid Logic Numbers and Zero Spirals
Fluid logic numbers through their algebras and the geometries of the zero spirals show how space are the stuff of what things are made of even as they show the alone cannot understand not to talk of grappling with the mysteries of the inner infinities called the zeroids and what happens to the outer infinities of alpha and omega numbers through or as a result of them. The matter is deep. Only creativity can deal with this mystery to the level of the infinity that is human.
Let me try to explain the magnitude of the challenge. Zeroids change the configurations of alpha and omega folds that flow into them. What could be the configurations of the zeroids at any monent in time? Once they enter, how do the alpha and omega folds come out? And when they do, is the zero spiral still symmetric as we have in this introduction to them? What are the possible configurations of the zeroids? Can they all be accessible to creative imagination - as human beings? My answer to the last question is no.
Be all this as it may fluid logic numbers, their algebras and geometries show that without a flow of knowledge that involves a flow of reasoning into feelings and feelings into reasoning with one and both flowing into language and from their flowing into how we relate to one another, natue and the world, and so doing dispensing with packaging of knowledge that is called specialisation, it is not possible to have a glimpse not to talk of immersion in the what things really are and for that long, for as long as appearance parades as the master of ceremony, shall the gate of truth remains closed while knowledge remains a ghost of its real self.
5.1. The Zeroplines
I call the inner infinities the zeroids. Zeroids show the limit of our knowledge of things. Zeroids constitute the inner life of the zeropoint. They are the seat of the Creative Potential. Create! The challenge. Show us what you think we are. Show us the nature of our numbers, our folds, yes and how it comes to pass that the negative integers enter and come as positive integers in cycles of everlasting renewal.
Zero spirals result from the spreading out of the alpha, omega and identity numbers of a fluid logic number. Thus, from (i) and (ii) above, we have,
fln(1)q = ((01 (11) 21) (10 (11) 12))
which, by removing the brackets and speading out, gives,
fln(1)q = 011121 101112
which is called the Ideal number of fln(1) = 012. The first zero spiral results from the ideal of every logic number. From fln(1) we get two of the folds of a zero spiral called Zeroplines (from ZERO PointLINE Spirals):
From these folds we have 11=(0) in the middle. This is the inner infinity (. One of the postulates of zero spirals says that a line is a curve at infinity. In the folds that we have above, the line curves for the fold on the left at 0 and at 1 on the right. For the fold on the right, the line curves at 1 on the left and at 2 on the right. These are the positions of infinities. These infinities are outside the zeropoint (0) in the middle. This is what leads to alpha subset domain as outer infinity minus of the inner infinity zero and omega subset domain as outer infinity plus of the inner infinity zero. In other worlds, the identity domain (0) has on the left the local domain alpha and on the right, the local domain omega. These are -1 and +1 respectively. As infinities, they are (-1 and (+1. By bringing the two outer infinities together, we have inner infinity, that is:
Ψ = ∞-1∞+1
(#i)
which is the same thing as
0 = -1+1
(#ii)
It is crucial to remember that (Ψ) is the region of the Zeroids Z which take is into the inner dimensions. This means ∞-1∞+1 does not exhaust (Ψ). What this means is that (Ψ)=(0)which is the inner infinity, manifests through the outer infinities (-1(+1 = -1+1 of every fluid logic number. On the other hand, since none of them is separate transfigurally, -1 and +1 also manifests as a result of (0), that is,
∞-1∞+1 =Ψ (#iii)
and
-1+1 = (0) (#iv)
respectively.
Arithmetically for fluid logic number 012, we have for (#i - #iv)
01 + 10 = 11 = A which is still inside 012
01 + 21 = 22 = B which takes us into 123
12 + 10 = 22 = B which takes us into 123
12 + 21 = 33 = C which takes us into 234
Since we every identity is equal to any other identity and alpha to any other alpha and omega equals any other omega we then have,
11 = 22 = 33 = 1-1=2-2=3-3 = (0)
In all this we should not forget that the numbers above flow into one another and that everything is happening inside one fluid logic number which takes us into the other fluid logic numbers. On its own, 012 flows into 123, 123 into 234 and so on. All of them have the same logic but their numbers are not the same. That means, they are equal but different. Since these infinities are inner and outer space respectively, that all this means is that every logic number breathes in and out of itself into other numbers while the others also breathe in and out into it. This means for every point which is a line - which is the source of pointline, that is a line that is point by folding on itself and inside itself -- there are outer and inner infinities. These are inner space-worlds and outer space-worlds. The ‚inner infinity space-worlds' and ‚outer infinity space-worlds' breath into one another and are nested in an ‚infinite series' of ‚relative infinities', each with their own unique dynamic fulcral points, lines and planes. What this means is that we have inner infinity breathing in and out into the outer infinities and outer infinities breathing out and into the inner infinity.
Outer infinity minus is the infinity outside zero on the left side. It is the alpha domain. Outer infinity plus is the infinity outside zero on the right side. It is the omega domain. Together they are called outer infinities which means outer space of the inner infinities called inner space.
From the two folds above we have,
The superimposition of the two folds give the first zero spiral called Zeroplines.
Zeroplines
The zeropoint (0), that is the identity I, is the very midpoint which is the seat of zeroids, Z. The alpha domains are (01, 12) while the omega domains are (21, 10) respectively. In all this we do not forget that we are talking about a fluid logic number. Thus in every fluid logic number, the alpha and omega flow into the identity zeropoint which contains the zeroids - these are the numbers and geometric configurations of the zero (0) really - from which they flow out and spead out to infinity. The place where they fold and return to the zeropoint is called the outer infinities. Inside the zeropoint itself which is the abode of the zeroids we have the inner infinities. Thus the inner flows into outer while the outer flows back into inner such that at any point on the line there are zeroids that change inner into outer and outer into inner which gives the inner-outer, outer-inner unity.
5.2. Space, Creativity and Identity
The centre of the zeroplines is the zeropoint. Zeropoint, which is called the Creative Potential is composed of Zeroids. Zeroids are what we get inside the zeropoint (0) which is the world within the world of a zeroplines. Zeroids are transfigurative in the sense that they change radically what get into them by changing its configuration depending on their configuration when either the alpha and omega curves and the curls of their local folds fall into them. Thus they can be symmetric and asymmetric. In fact inside the zeroids is not symmetric at all even though the transfiguration of what gathers into and spread out from them may be symmetric.
5.3. Space, Origin and Fluid Logic Numbers
What we get from the zero spirals are as follows:
(i) space is the stuff of things
(ii) everything is a flow
: which follows from (i)
(iii) everything is dynamic
: which follows from (ii)
(iv) what exists flows
: which follows from (i-iii)
(v) things are equal but different
This follows from the fact that every fluid number is equal to any other fluid logic number therefore applies to (i - iv) (refer below)
5.3.1. Fluid Logic Numbers - All in One - One is All.
Every logic number is contained in every other logic number. This means that every logic number carries the stamp of the Many even as the Many also carry the stamp of One. Thus a fluid logic number is One that is Many and Many that are One.
6.0. Flows, Folds and Configurations
In fluid logic numbers and the zero spirals, everything flows. Flows give rise to folds at the local and identity domains respectively. But do the folds give rise to configurations inside the zeroids? This is not easy to answer. What we can say is that the zeropoint manifests through the local domains alpha and omega. What we can also say is that without the zeropoint there are nor local domain alpha and local domain omega to talk about. What we have is the indivisibility of chicken and egg. The zeroids give the impression of auto-creation or auto-generation but even then the gathering of the alpha and omega folds also change their configurations too.
6.1.ShPLP - Origin and Implications
The idea of Shakulean Point, Line and Plane - ShPLP - which led to the idea of dynamic fulcrum were all from Alan. He brought them into our group while pondering the condition of a point on the lifeline of a zero spiral. In this point we have motion at rest since we are dealing with a breathing space. In fact every zero spiral is a line that rolls back on itself at infinity. The centre of this line is everywhere. Each point has its own centre inside itself which is the seat of zeroids while the zeropoint, is also the centre of the line that spreads out and gathers into the zeroidal infinities. These ideas also lead to a Shakunlean Point, ShP, as a dynamic fulcrum such that a line is composed of dynmaic fulcra.
6.3. ShPLP and Relative Infinity
The fact that a line becomes a curve at infinity in the sense that it folds back on itself gives rise to the idea of relative infinity in zero spirals. In this context, a human being is a kind of infinity which is equal but different from the infinity of being a flower. And this is so because we cannot produce the blossom of the flower while the flower may not understand a bit of our hit parade. So everything lives within the possibilities, strengths and limitations of its own infinity which is what Creative Potential is all about.
7.0. Conclusion - Space and What Things Are
Fluid logic numbers and spiral geometry which is composed of the geometries of the zero spirals, apart from their algebras, geometries and physics are rooted in and provide solutions to the fundamental questions of philosophy of being, theory of knowledge through their philosophy of mathematics whose thesis is:
Every thing its own mathematics
The fluid logic numbers of Transfigural Mathematics and their spiral geometry take us into Heraclitus, The Divided Line of Plato15, Tsuan and far beyond. Indeed these numbers, mumbers that can claim to be natural, constitute the return to the lost paradise of One that is Other and Other that is One. What we have is One Otherself. This is existence. This is dynamics. This is flow. It is Being, World and Nature as Many that are One, and One that is Many.
References
1. Penrose, Roger (1989) : The Emperor's New Mind. Concerning Computers, Minds, and The Laws of Physics (Oxford University Press, New Yor 1989, p.31)
2. Rayner, Alan D.M.(1997): Breaking Boundaries - Building Niches Interview based on Alan Rayner's Book, Degrees of Freedom. Living in Dynamic Boundaries by Lere O. Shakunle for Transfigural Mathematics, Vol.4.No.1.1998, pp 69 - 77
3. Hooker, Richard. (1995). Translation of Heraclitus' Logos and the Unity of Opposites
4. Frege, G. Grundlagen der Arithemetik
5. Dedekind.....Was sollen oder sind die Zahlen?
6. Peano's Axioms and their Arithmetic
7. Rucker, Rudy (1983) : Infinity and The Mind.
The Science and Philosophy of The Infinite
(Bantam Books, New York)
8. Barrow, John (2000): The Book of Nothing
(Jonathan Cape, London)
9. Barrow, John (2000): The Book of Nothing - ibid-
10. Shakunle, Lere O.: Introduction to Infinity Theory
(The Matran School, Berlin 1995)
11. Hersh, Reuben (1997): What is Mathematics Really?
(Oxford University Press, New York p.138 - 140)
12. Shakunle, Lere O. Foundations of Logic Numbers, Vols. I., II., III.
(The Matran School, Gottigen, Ulm, Berlin 1987 - 1996)
13. Shakunle, Lere O. Introduction to Logic Numbers, Vol.1., II (in preparation)
(The Matran School, Berlin 1995 - 1997)
14. Shakunle Lere O.: Water Logic - Zero Spirals and Fluid Numbers Invited Talk at The Language of Water. A 4-Day International Science-Art Conference and 4-week Art Exhibition Exploring and Celebrating Water as the Dynamic Context of Life organized by Alan Rayner and Caroline Way, Bath April 2001
15. Shakunle, Lere O., Bulckens, Anne: The Divided Line of Plato and Logic Numbers. International Congress of Mathematicians, Berlin August 18-27,1998
Glossary of Terms
α : alpha number(s) or alpa
ω : omega number(s) or omega
Z : zeroids (numbers of zero); inner infinities
(This is Z with dot on top)
I : identity (number)
(This is I with dot on top)
-∞ : all negative integers represented by -1 in the logic of
fluid logic numbers - outer infinities minus (outer dimensions) -
dependence and interdependent
+∞ : all positive integers represented by +1 in the logic of
fluid logic numbers - outer infinities plus (outer dimensions) -
dependence and interdependent
(Ψ) : totality of integers with the zeroids Z as inner
dimensions (with their infinities)
- independent, dependent and seat of interdependence
Epilogue: I must Thank you Alan for your helping to find the right names for some of my ideas. For example fluid logic numbers used to be called logic numbers before the Language of Water Conference which you organized with Caroline Way in Bath at which I gave an Invited Talk. The Shakulean Point, Line and Plane which in your musings gave us the dynamic fulcra and which helped to pack my ideas in what I call ShPLP also came from you. And you too, Ted, it has been quite an exploration with you on geometry, existence, dynamics and questions again, and answers again until we eventually got to what you called our leitmotif, dynamic existence.
In fact you put the nature of the tension that sometimes befogged out path succintly in your letter which I got while working on this section of the homepage. In that message with the title, 'Binary and
Ternary Evolution of Meaning' you wrote, among other equally
important issues which I could quote but choose to use what follows:
...This 'relativistic' view of ternary dynamics seems fit with what we have been doing, ... two of us push on each other relative to the enveloping environmental ideas, ... this puts stress on the enveloping environmental ideas which the environmental ideas absorb but which they will rebound back from, ... but by changing the shape of the subject, ... as the rebound comes (we could stay locked in head-to-head conflict), when the squeeze comes, we slip onward and upward out of it. So the evolution of 'meaning for us' a la vygotsky's concept development research, could be split into binary dynamic and ternary dynamic modes, with the western civilization having gotten stuck on the binary dynamic, ... and ourselves, ... with considerable effort, ... trying to get (back) into the ternary dynamical mode which is the innate way of nature and evolution while the binary mode is a simplified abstraction...
You also showed Ted, in the epilogue to your essay how the wrong-right whose generalisation is true-false split consciousness, the offspring of binary thinking, has disorganized the hub of human relations beyond repair with the absolutism of "I know you are wrong and I am right" that pollutes the atmosphere to feel and corrupts the heart to love.With this you brought us back to the rough and tough path we have used together to reach thus far.
We did learn a lot somewhere along the line from one another. Transfigurally and inclusionally what we learned boils down to feeling into what the other person is saying, a relativism of infinity that makes us co-travellers in the voyage of discovery and revelation of truth that give us the wings to fly back home to our common indivisible humanity so we can now build knowledge on solid foundations. What a prize to win!
It has been quite a journey with you, Alan and Ted. To all of us, I would say, with our kind of spirit and chemistry, we are simply lucky to come together.
Salut!
© Shakunle, Lere O. 2002