Apeiron Centre (AC): The Center of Infinity orthe Infinite Center launched in Paris in 2002,  is the first philosophical organization in the world that studies the idea of infinity in intimate contact with the finite as presented in our physical universe, society, and mind. Far from being an unsolvable absurdity, the principle of infinity, which is the most radical of ideas, can resolve the problems of life and answer the questions of our finite, inadequate world.

Cosmology/Philosophy

 

Infinity as a Complex Idea 

Anaximander, the Greek Ionian philosopher (sixth century BCE), asserted that the principle and substance of all existing physical bodies is the Infinite One, which he called the Infinite (to apeiron). He defined the Infinite as that which is intermediate between two elements and hence is internally unbounded without internal distinctions. He equally defined the Infinite as that which is a mixture (synthesis) of all elements. As an intermediate thing, the Infinite is neither finite nor infinite and therefore is indeterminate, or impartial. As a synthesis of elements, the Infinite is both finite and infinite and therefore complex and balanced.  Following Anaximander’s powerful insight, which was shared by Pythagoras, Anaxagoras, Democritus, and Plato, and which is in opposition to the finite-analytic ontology of Aristotle and Kant, the Apeiron Centre assumes the following about the physical body:

  1. The physical body, which we define as the sum total of an infinite number of parts and we number by the real infinite whole 1, has the power of continuous life and motion. We call real physical body, the physical universe and anything in the physical universe.
  2. The Infinite is not the impossibility or collapse of the real physical body; on the contrary, the Infinite is the very immanent (intrinsic, essential or proper) nature of the physical body endowed with continuous life and motion.
  3. The founding principle of the Infinite is the synthetic principle of equivalence, which stipulates the equality of unequal things — for example, of the limited and the unlimited, of the whole and its proper part, of the one and the infinitely many. Because on the circle the opposites are united, we represent one-dimensionally the Infinite by a complex circle, which is simultaneously closed and open, limited and unlimited without contradiction or paradox. We call it the infinite circle. 
  4. The synthetic equivalence principle is, in its complete form, the principle of highest unity-reality-truth, of supreme justice-love-good, as well as the ideal principle of continuous life and motion. Continuous motion is independent of generation and destruction, of external causality and temporal order, and continuous motion has maximum speed.
  5. The discovery of our real and true nature, namely that of being a physical body endowed with continuous life and motion, becomes the driving force, constitutive principle, and accessible telos of human growth (gnostic thesis). In this sense, we may comprehend our growth as the inductive process of connecting our finite body with the infinite physical universe. This connection will in turn reveal:
    • the hidden infinity of our finite body and
    • the hidden finitude of the infinite physical universe.
  6. If continuous life is independent of generation and destruction, then our finite body in its complete development needs neither generation nor destruction, neither external causality nor temporal order for its continuous life and motion. More specifically, if our complete finite body has access to the immanent infinity of its finite sensibility, then there is no need of an external cause, a transcendent God, an artefact, or a violent chronological big bang for originating and sustaining its life and motion. An accomplished finite body is therefore a naturally self-ordering, self-originating, and self-sustaining body in a self-originating, self-ordering, and self-sustaining physical universe; this constitutes the absolute autonomy of the complete finite body, which, as a real physical body, is free of constraining time and external force.

The standard position of mainstream empirical science is to exclude the infinite and its inverse, nothing (zero), which is the infinite according to the division. Nature or the physical body is simple and uniquely finite and nothing finite is infinite! There is no infinite body, no Infinite One! 

This Aristotelian finite-analytic worldview is incapable of understanding the physical world’s indeterminate and complex nature and generates in science infinitely regressing absurdities. Given the crisis in the finite-analytic foundation of empirical science, the new trend is to return to the first thinkers of Greece (the school of Ionia), India (the Upanishads) and China (the school of Tao), who laid down simultaneously and independently the first spiritual foundations of humanity. These three regions formed at the dawn of philosophy (sixth to fifth century BCE) the axis of the apeiron, or Infinite One. According totheir first intuitions about the physical world, which were the most powerful and true, the Infinite One is the principle and substance, the source and destination of the physical universe. As a complex principle, the Infinite One ensures the unity of the physical whole’s opposite poles, the finite and the infinite, as well as the permanence of its life and motion. As a complex substance of all things, the Infinite One allows us to affirm without absurdity that (a) everything finite is infinite, (b) that all finite quantities have infinite magnitudes, (c) that a finite spherical body of radius 1 is simultaneously infinite (according to the extension and the division).

Finally, we call infinite-synthetic paradigm,the emerging non-Aristotelian model of the physical universe according to which everything is complex, verifies synthetic principles of existence, and is both finite and infinite. Because the infinite-synthetic paradigm abolishes one-way time and its analytic hierarchy generating unsolvable conflicts, it is regarded as the natural response to society’s contradictions and evils. 

Cosmology/Society

 

Infinity in Globalization

How is it possible to bring meaning, unity and hope to a world that is drifting and whose individuals are mutually diverging, similar to the galaxies of our observable universe, repelling each otherin a war of all against all?

That which is internally unbounded without internal distinctions and boundaries is the infinite according to the quality of Anaximander. It is also the infinite of the globalization of our human society, which is a process of boundary-breaking and community making. Breaking political, economic, social, cultural, racial, genetic, organic, psychical, and spatiotemporal boundaries permits us to discover the hidden unity of apparently different things, that which is common and universal among individuals and makes of them a permanent unified whole — that is, a boundless community that exists and continues to exist in space and time because it obeys unifying universal principles.

The organization of our society according to common interests and universal laws reflecting the unifying principles of the infinite universe is the common and universal program of globalization and of scientific progress thought of as an inductive process of universalization-globalization.

A society without internal boundaries and yet bounded by unifying principles is a spherical body governed by the equivalence principle, according to which the distribution of wealth — of matter, energy, and information on the spherical surface of the body, is continuous, universal, and uniform, the same everywhere in all directions at all times.

If time and history generate opaque and hermetic boundaries among the individuals of the world, then breaking the boundaries through scientific progress and successive globalization will bring in the ultimate future:

i) the end of time taken as the generator of hermetic boundaries 

 ii) the complete development of the universal  as an idea and as experience, and 

 iii) the emergence of a universal community composed of simultaneous members in continuous communication

Since the time of the ancient Greek philosophers, we have known that life, society,and the physical world exist and can continue to exist if, and only if, they obey unifying principles that (a) hold the world together, (b) allow communication between its members, and (c) stabilize its disruptive variation and imbalance. 

 

Destructive Globalization

If we fail in our globalization process, human society will decompose itself into discontinuous, isolated individual parts increasingly diverging from each other and in Hobbesian war of all against all. It will then be the global collapse of all civilization. We have failed in our globalization (a case of destructive globalization) (a) when we lose our individual identity without expanding it; (b) when we open our particular limits that separate us without fixing our universal limits that unifies us; (c) when we increase society’s disruptive mobility without enhancing her cohesive stability; (d) when we develop artificial power at the expense of our natural power; (e) when we concentrate wealth without distributing it; (f) when we compress the entire scattered world within our mind without composing it; (g) when we unify things without discerning them, and discern things without comprehending them.

 

Ultimate Globalization

The last boundaries that scientific progress is breaking are the genetic boundaries between individual organisms, the organic boundaries between individual organisms and non-organic matter, the material boundaries between matter, space, light and mind, and the psychical boundaries between conscious and unconscious self. 

In the first case, science breaks the genetic boundaries between individual species by communicating genes among microbes, plants and animals. This in turn creates a community of organisms that are complex wholes having their individual genomes increasingly extended and enriched through the continuous exchange of their genes. 

In the second case, science breaks the boundaries between organic and no-organic matter by translating the genetic code of the individual organism into the universal language of and b. The universal matrix a = b of the cell’s genome, which is simultaneously the matrix of the mind and of the physical universe, will in turn, provide the unifying framework for obtaining a universal cognition and communication across categories: organic and non-organic matter, space, light and mind. The universal matrix or equation a = b, will celebrate the unity and continuity of space, matter, light and mind. It will describe how the variation of a unit of space generates material, electromagnetic and mental effects as well as how the variation of mind generates electromagnetic, material and spatial effects.

In the third case, empirical science breaks the psychical boundaries between the finite conscious and the infinite unconscious in order to establish a continuous communication between them such that the infinite forces of the unconscious infinitely extend the finite mental faculties of the conscious independently of external cause¾of an artificial machine or a transcendent God.

The meaning of globalization, which is an inductive process of empirical science moving from the finite individual to the infinite universal, is to recompose the wholeness of the living body, which is decomposed by the corrupting force of time. In fact, time decomposes the wholeness of the living body into an aggregate of isolated parts (the individuals) deprived of unity, continuity and communication and which we call contradictories occurring successively in time. For instance, at the beginning of our observable world, the corrupting force of time decomposed the spatiotemporal living whole into the conflicting opposite parts of space and time, where space is the horizontal axis of simultaneity and coexistence, and time is the vertical axis of succession and impossible coexistence. 

Time also decomposed the electro-gravitational living whole into the conflicting parts of light and matter (the cosmologists call this the decoupling of light and matter). Matter was decomposed into the conflicting parts of gravitational and anti-gravitational matter. Then through Darwinian evolution, time decomposed the living whole (taken as a community of communicating wholes, bodies or cells), into an aggregate of isolated individual species in competition.

If we think of time as being a deviation from the physical world’s natural state, which is a community of living wholes in continuous communication coexisting independently and prior of time, then inductive globalization can be regarded as a restoring force aiming to mechanically restore the physical world’s natural state of continuity. Thus, the ultimate end of inductive globalization undertaken by science is to arrive at zero time, that is to say, at zero corruption or zero decomposition, where life, society and the physical world emerge as a permanent community of communicating wholes connected by the equivalence principle.

 

Apeiron Centre as a Cosmopolitan Humanist platform  

Cosmic flower by Kahori Maki

Mind is infinite and self-ruled

Anaxagoras

In this confusing transition period where our old civilization is declining and the new civilization, not yet properly understood and named, is arising, the Apeiron Centre is also a cosmopolitan humanist platform that promotes in a coherent way the complex and unifying idea of humanity recently attacked by technological fundamentalism as well as what is best within the human, which is the infinite mind ⎯ the Greek infinite Nous.

We state unequivocally that there is no future for humanity if it follows the militant path of technological fundamentalism which, as we speak, is systematically tearing asunder all traces of cosmopolitan humanism and its traditions. The Center seeks to make that which is best within the human, the infinite mind, ⎯ Greek Nous ⎯ the centerpiece for the formulation and evaluation of alternatives that have bearing upon humanity’s fate. In accordance with this principle, the Center does not seek to find ways to make Nous useful but to find ways to bring all that is useful to the service of Nous as end.

Throughout history, humanity recognized its finitude and incompleteness and creatively used linear time and artificial tools to transcend them step by step with the distant aim to approach and eventually attain the completeness of the physical universe free of the artificiality of linear time. This transcendental movement never reduced humanity to the manifestations of its incompleteness, but instead ascended to nous where it discovered that power within us that can integrate, as a solution, our state of incomplete partiality with its potential complete wholeness.  

The transcendental progression of humanity did not and does not now require its external transformation into an alien being involving its annihilation, something that is militantly advocated today by post-modern relativists. We oppose attempts that would define humanity as a fragmented aggregate whose unity is to be found in shared biases and common gratifications free of culture. Instead what is called for is humanity’s internal self-transformation.  Proceeding from humanity’s low state of incompleteness, a characteristic of our finite body, the progression seeks that high state of completeness wherein the human infinite mind reflects the infinite physical universe. We find this view ensconced in thinkers from Heraclitus, Pythagoras, Plato, Epicurus and can perhaps best be summarized by Aristotle’s idea that nous in its activity becomes in thought what the universe is in its being. Throughout history humanists have sought to take these ideas as guiding principles for shaping politics, moral training and education in a manner consistent with human nature. 

We follow in the footsteps of the tradition that was set by Plato and adopted by his philosophical heirs:  that the human being, who in Greek is called anthrôpos (ἂνθρωπος → ἀναθρών  α όπωπε ), has been so named because of all animals it is the only one that 

contemplates upwards (anathrei) what it has seen (opôpe)” (Cratylus 399c3-6). 

In contradistinction to the word “human” whose Latin root is “humus” meaning earth, the Greek word anthrôpos emphasizes the vertical and therefore transcendental and spiritual dimension of the human. It is this infinite spiritual power within us that can lead humans beyond their terrestrial and biological limitations to discover the unlimited reality of external and internal worlds. And in so doing, the human being has the power to become a complete whole that can unfold akin to a cosmic flower in its infinite potential.

 

Infinity in Political Economy

Will economic growth grounded in the law of diminishing costs bring about the end of work, capital, matter and time, regarded as costs in the production of wealth?

Time is not only a principle and force of corruption that determines the particular individual; it is also a cost in the production of wealth. In the postmodern phase of political economy, the law of diminishing costs, dramatically poses the problem of how to produce a finite amount of wealth with minimum cost, that is, with minimum time, and hence with minimum work, capital and resources (matter), which in our time-conditioned society are compressed forms of time. The limiting point of this indefinite diminution of costs is the power to produce a good without engaging the above costs, in other words, free of cost.

Is such an ideal state possible? In so far as the law of diminishing costs is a logical consequence of infinite Nature’s principle of least action, according to which every physical being behaves in a way that minimizes its action, then the above ideal state is a natural end that will be necessarily realized in our particular case because it already governs the entire timeless physical world.

According to Aristotle, an accomplished being possesses its good without exercising action, whereas an imperfect being possesses its good by means of multiple actions exactly  the way the human body does.  A body can remain healthy without any exercise, another body will obtain its health only after vigorous effort, whereas another body will have no health  no matter how vigorous the effort may be. (see On Heavens 11,12 292a, 25).

The above reflections lead us to conclude that the intrinsic meaning of economic growth is to obtain wealth and health at zero cost, namely with zero work, in zero time. In fact, the possession of wealth and health by every human being must be spontaneous, effortless and natural, like respiration. The moment that we add work and effort to respiration, we are by definition, ill!    

Only within the context of a global federal government that unifies the scattered members of the Euclidean world into a universally communicating spherical whole, can effortless possession of wealth and health by every human being, be a universal human end and right.

 

On Infinite Public Deficit and Its Self-payment

Is there any end to society’s endless growth of public debt in an indefinitely expanding world? And if there is an end, is this end the collapse of society or is it society’s ultimate liberation from public debt generating external constraint?

There is a tremendous amount of emotion involved in people’s attitude toward public debt. Like sex and religion, public debt is a popular subject. But what is the philosophical meaning of public debt? When do we have public debt? Is public debt vicious or virtuous? 

We have public debt when there is a difference and an inequality between public expenditure (consumption) and public income.  If public consumption C is greater or increases faster than public income I such as C > I, we have a public deficit entailing a public debt. In a finite Euclidean society ruled by the analytic principle of inequality and linear time whose economic manifestation is the law of increasing deficit either because of increasing consumption or decreasing income, public deficit and debt necessarily grow in proportion and have a depressive effect on society. 

The growth of the public debt proceeds by constantly deferring its payment, which constitutes a debt added to the initial debt. We have thus the increasingly depressive case in which income, wealth, and energy decrease with time, thereby putting an increasing burden and constraint on society and on later generations. We call this, the law of diminishing returns, an economic expression of the analytic principle of inequality and temporal order that rules our finite mortal society situated in an indefinitely expanding hyperbolic or Euclidean world.  

Now, is there any limit to society’s seemingly endless growth of public deficit and debt in a world expanding without limit? And if there is an end, is it the collapse of society because of the exhaustion of its income and available energy? Or is it the definite payment of the public debt and society’s ultimate liberation from obligation and constraint? 

In a spherical world free of linear time, there is no problem of accumulating a public deficit. At the beginning the public debt increases in proportion to the public deficit; at a certain point, because of the curvature of the sphere, the public debt varies in inverse proportion to the public deficit. We have then the case, where the public deficit continues to increase without limit, while the public debt diminishes to zero at the antipode. The infinite public deficit entails a zero public debt freeing society from all burden. The same thing happens with spacetime. At the beginning time increases in proportion to space; at a certain point, because of the curvature   of the sphere, time varies in inverse proportion to space. We have then the case, where space continues to increase without limit, while time shrinks to zero at the sphere’s diametrically opposite point. Because on the timeless sphere every point is simultaneously its antipodal point, and every sense is inverted into its opposite sense without absurdity, we can also assert that at every moment the increasing public debt and time are decreasing and hence self-paid and that the spherical world has no hole, no time no constrain, no burden.

What does this mean to the economy of our time-conditioned Euclidean society which is inherently programmed to increase without limit its public deficit and debt and to generate an ever-larger hole within its public wealth? It means that the solution to society’s infinite public debt occurs beyond our Euclidean time-conditioned world. Indeed, from a higher dimension we observe the whole of our Euclidean world, that is, our Euclidean time-conditioned world as a timeless spherical whole that spontaneously and continuously pays off the endless increase of the public debt of our incessantly expanding and consuming Euclidean world. This payment is realized by using infinite Nature’s renewable resources, which are infinite and free from impurities (waste) because they are continuously recycled. 

The state in which we have an unlimited public consumption with a limited public income or, to put it another way, a limited public consumption with a zero public income, we call infinite public deficit entailing zero public debt and maximum wealth.  It is a maximum generalization of our actual telecommunication system that enables each individual to have an unlimited communication with a finite cost (for example, with a finite payment), which is equivalent to the complementary case of a limited communication with a zero cost (for example, with a zero payment). The state of maximum wealth constitutes the first step toward the civilization of infinity occurring on the surface of a permanently rotating spherical world and whose balanced equalitarian structures reflect the eternity of the spherical whole. Indeed, the equalitarian structures of such an ideal civilization operate analogically to the spherical whole, that is, continuously in a costless, debtless and effortless way free of the second law of thermodynamics.

 

Conclusion

Instead of trying to understand why there is an unlimited public deficit in a society expanding in the infinite universe without limit, and determine how to convert it into a public debt that shrinks to zero without affecting the unlimited expansion, the experts curtail the expansion in the hope that the deficit too will be curtailed. What they create, however, is a contracting recession that instead of eliminating the deficit eliminates life itself!

We take the infinitist point of view in which deficit in an expanding economy is virtuousand constructive under the condition that it is paid off by infinite Nature’s renewable resources. So what we propose here, then, is to get out of the Euclidean-finitist-analytic paradigm, governed by the second law of thermodynamics and observe human society as an infinite spherical whole endowed with the endless recycling capacity of infinite Nature. In such an ideal society expansion and consumption are unlimited because a finite and constant amount of resources is continuously recycled. 

What should economic authorities actually do? They should eliminate all the public debt of human society by creating a global monetary surplus whose value would be guaranteed by Nature’s renewable resources. Because infinite Nature taken as a spherical whole or universe is the foundation of all things, it is necessarily our only real and inexhaustible guarantor.

 

Ancient Thinkers

Anaxagoras (fifth century BCE). Greek Ionian thinker who asserted that since opposites come out of one another (for instance, something comes out of nothing and vice versa), they must have been present in one another always. It follows that everything is present in everything and that external or determinate causality upon which empirical science is grounded is an illusion.

Anaximander (sixth century BCE). Greek Ionian philosopher who defined the infinite (to apeiron) as “that which is internally unbounded without internal distinctions”, that is, that which is indeterminate, infinite in quality,beyond categories and hence all-inclusive. The infinite is neither finite nor infinite and therefore is indeterminate and impartial. The infinite is both finite and infinite and therefore complex. Because the infinite is the common substance and principle from which all things originate and to which all things return, its form is spherical.

Aristotle(fourth century BCE). Greek philosopher and farther of analytic logic. He employed  analytic principles of thought to show the absurdities of infinity and motion.  If infinity and motion are real properties of the physical world, how can we save infinity, motion and the physical world from destructive analytic criticism? 

Chinese thinkers of the school of Tao (sixth to fifth century BCE). Tao, the way and principle of heavens, is the unity and interpenetration of opposites called contraries. Every being in the infinite universe is a complex indeterminate whole, which is both female (yin) and male (yang) and neither female (yin) nor male (yang).

Democritus (fourth century BCE). Greek Ionian philosopher who asserted that atoms are finite bodies that move with infinite speed in the infinite void. Moreover, he asserted thatthere is only one kind of natural or original motion, namely that of vibration. However the observed one-way backward (or forward) and outward acceleration of our observable universe contradicts the natural and symmetric motion of vibration. How can we solve this conflict?

Plato (fifth-fourth century BCE). Following the Pythagoreans, Plato asserted in his Philibus that nature has united in every being the finite and the infinite (not-finite). However, in virtue of what principle is this primitive unity of opposites true? How can every being be both finite and infinite while avoiding absurdity? Is this unity of opposites a principle of the mind (faculty of synthetic reason) or a principle of the physical world?  Is it an essential or an accidental, a real or an apparent principle of nature?

Pythagoras (sixth century BCE). Greek philosopher from the island of Samos who asserted that two principles, the limited and the unlimited, govern all bodies, that everything numbered by the whole 1 is a complex unity of opposites, of the limited and the unlimited. In the Pythagorean system, the unlimited is an even, female number and the limited is an odd, male number.

Indian thinkers of the Upanishads. Indic sacred scriptures were principally written between ninth and fifth century BCE. Their exposed doctrine asserts the unity of the inner self or soul (atman) with the outer ultimate reality (Brahman) through contemplation and meditation. 

If the outer ultimate reality (defined as the sum total of an infinite number of outer realities) has an infinite extension and the inner self or soul (defined as the sum total of an infinite number of inner realities) has a zero extension, then from their unity emerges the real magnitude 1 of the physical whole 1, which is the ratio between its greatest outer part and its smallest inner part. 

 

Glossary  

Analytic logic. The science of the principles that govern the object experienced as a finite individual part, having at one time a unique determination or contradictory determinations and at different times consecutive determinations. The father of analytic logic is Aristotle. 

Analytic principles are the principles of reflexive identity, contradiction, excluded third, and temporal (irreflexive) order. The cognitive faculty of analytic principles we call understanding.

Anaxagoras (5th century BCE). Greek thinker who affirmed that since opposites come out one another (for example, since something comes out of nothing and vice versa) they must have been present in one another always. It follows that everything is present in everything and that determinate causality upon which empirical science is grounded, is an illusion. 

Anaximander(6th century BCE). Greek Ionian philosopher who defined the apeiron as “that which is internally unbounded without internal distinctions”, i.e. that which is indistinct, infinite in kind, beyond categories and hence all-inclusive.

Aristotle (4th century BCE). Greek philosopher and father of analytic logic used analytic principles of thought to show the absurdities of infinity and motion.  If infinity and motion are real properties of the physical world, how can we save infinity and motion from destructive analytic criticism? 

Artificial.That which is unnatural or non-spontaneous, and depends upon the external cause for its existence and motion. The artificial is ephemeral and its fate is the collapse. The end of the artificial is the natural. For instance, the end of artificial intelligence is natural intelligence, namely the power of self-ordering and self-origination independent of external cause such as a transcendent God or an artificial machine.

Self-causality.According to the ancient Greek philosopher, Anaxagoras self-causality is a principle of the infinite universe and the infinite mind. Because everything is both cause and effect of itself, both before and after itself, that is, self-caused and self-ordered or self-ruled, we do not need an external cause, an artificial machine, a violent big bang, a transcendent God for originating, ordering and sustaining the universe and the mind.

Continuous motion.  Immanent and essential property of the real physical body thought of as an infinite whole and numbered by the real 1. Continuous motion is spontaneous, free of external cause, circular (reversible), permanent, uniform and has a maximum speed equal to the speed of light, which we designate by the real 1. What is the founding principle of continuous motion? What form must the real physical body have to receive continuous motion? 

Contradiction. The  impossible unity and coexistence of opposite things. Contradiction is a quality of our incomplete observable universe, which is the observable part of our physical universe. What are the organizing principles of our incomplete observable world? 

Contrariety.  The necessary unity and coexistence of opposite things. Contrariety is a quality of the complete physical universe. What are the organizing principles of our complete physical universe?

Cosmic acceleration. Our observable universe is in a phase of accelerating expansion. The fundamental quest in rational cosmology is to identify the origin, meaning and limit of this cosmic acceleration. We coincide mankind’s accelerating growth with the observable universe’s accelerating expansion.

Democritus (4th century BCE).According to the Greek Ionian philosopher Democritus, there is only one kind of natural or original motion, namely that of symmetric vibration. However the backward (or forward) and outward acceleration of our observable universe contradicts the natural motion of vibration. How can we solve this conflict?

Equivalence principle. An intellectual principle of our faculty of synthetic reason that stipulates the equality of unequal things; for instance the equality of matter and energy, of whole and part, of one and many. But if no unequal things are equal, how then is it possible to equalize unequal things without absurdity, illusion or schizophrenia? Can this intellectual principle be at the same time an objective and constitutive principle of our real, physical world? 

Euclidean geometry. The geometry of the time-conditioned, observable part of the real, physical universe. This observable part called the observable or sensible universe is a finite variable increasing (or decreasing) indefinitely in time. In Euclidean space any pair of points  are determinate and unequal; Euclidean space is therefore discontinuous.

The founding principle of the physical universe. The universal principle or common point that unites our observable world’s infinite multiplicity of scattered parts enables immediate communication across its distant regions, stabilizes its disruptive variation and assigns meaning and limit to its indefinite acceleration. What is the logical formulation and geometric expression of this founding principle?

Globalization. The internally unbounded without internal distinctions and rigid boundaries is the infinite of Anaximander. It is also the infinite of globalization, which is a process of boundary-breaking and community making, that is, a process aiming at discovering that which is common and universal among individual things and makes of them a permanent whole — a community whose members continuously communicate. Globalization can be equally thought of as an inductive process of moving from the individual to the universal, from the local to the global, of opening local boundaries and fixing universal boundaries. We distinguish different kinds of globalization, such as socio- economic-political, cultural, genetic, psychical and spatiotemporal. 

We have destructive globalization (a) when we lose our individual identity without expanding it; (b) when we open our particular limits without fixing our universal limits; (c) when we increase society’s disruptive mobility without enhancing her cohesive stability; (d) when we develop artificial power at the expense of our natural power; (e) when we concentrate wealth without distributing it; (f) when we compress the entire world within our mind without comprehending it and unify things without discerning them. In this case, human society will decompose itself into discontinuous, isolated individual parts increasingly diverging from each other and in Hobbesian war of all against all. 

God. The idea of a constant and maximum limit to the infinite series of varying  (increasing- decreasing) parts. What is the nature of this maximum limit? Is it accessible or inaccessible by the varying parts? 

Hubble constant. The Hubble constant Hdenotes the rate of cosmic acceleration; it links v, the speed of a galaxy, with d, its distance from us on the surface of the earth: v = Hd.

In the Hubble observable universe of radius r = 1.5 × 1010 LY (Light Years), the Hubble constant H is roughly: k = v /d = 30km/s/1.5 × 106 LY (Heidman Jean. Extragalactic Adventure. Cambridge: Cambridge University Press, 1982. p62). What is the nature of the Hubble constant? Is the Hubble constant or varying? Does the Hubble constant change with time or with space?

Infinite (apeiron). Aristotle divided the infinite into the infinite as complete whole (infinite whole, actual infinite, absolute infinite) and the infinite as an incomplete part (potential infinite, relative infinite, indefinite). 

The infinite whole or the infinite as a maximum is the infinite, which has nothing external to itself and beyond itself. It is a proper infinite, free of time (of succession, comparison, subordination, computation) and the laws of arithmetic. We also have an infinite whole when we think of the whole’s infinite multiplicity of parts at once, as one thing and with maximum speed. Does the infinite whole exist? Can we experience the infinite whole? What is the founding principle of the infinite whole?

Potential infinite is the infinite, which always has something external to itself and beyond itself. It is an improper infinite, which in reality is a finite part indefinitely varying in time and obeying the laws of arithmetic. What are the organizing principles of the indefinitely varying finite part?

Infinite body. Rejected by Aristotle as an absurdity, the infinite body was regarded by Greek Ionian philosophy (cf. Anaximander, Democritus) as the body of the physical universe and of its composed atoms. 

Infinite nature.By infinite nature, we comprehend the infinite totality of things connected by the equivalence principle. Infinite nature is also the existence of things according to the synthetic principle of equivalence. 

Infinite One. That which is neither finite nor infinite and therefore is indeterminate or impartial. The Infinite One is equally that which is both finite and infinite and therefore is complex. The Greek Ionian philosopher Anaximander(6th century BCE) regarded the impartial and synthetic Infinite One as the primitive substance, number and principle of the real physical body. By physical body, we mean the physical universe and any member of the physical universe. 

Infinite sensibility.The faculty of perceiving the infinite whole, which is the world’s infinite multiplicity of parts at once with unity and limit. If no finite sensibility is infinite, how then is it possible to have an infinite sensibility without absurdity? Is it possible to have a synthetic sensibility, which is both finite and infinite?

The principle of least action.The perfection of the physical universe requires that the laws of motion and rest must be such as to obtain an effect, with minimum action and in minimum time. The principle of least action is derived from Aristotle’s original idea of immobile action where the ideal being (for instance god), has the power to impart movement to something other than itself without being moved. How is such as case logically possible?

Metaphysics.The science of the real, physical body (or physical being-thing-object) thought of as an infinite whole and defined as the sum total of an infinite number of parts. The physical body, which we number by the real whole 1, designates the physical universe and any member of the physical universe. 

Plato (5th-4th century BCE). Following the Pythagoreans, Plato asserted in Philebus that nature has united in every being the finite and the infinite (not-finite). However, in virtue of what principle this primitive unity of opposites is true? How can every being be both finite and infinite (not finite) while avoiding absurdity? Is this unity of opposites an essential or an accidental, a real or an apparent, an objective or a subjective state of nature?

Physical body.The real physical body thought of as an infinite whole is the sum total of an infinite number of parts. The physical body, which we number by the real whole 1, designates the physical universe and any member of the physical universe. 

Singularity (cosmological).A  region where the curvature of space-time is infinite and the size of space-time is zero. What are the fundamental properties of cosmological singularity? Is singularity destructive chaos or the source of continuous life and motion? 

Spherical geometry.The geometry of the real physical universe numbered by the real whole 1 and defined as the sum total of an infinite number of parts. How is it possible for a unit sphere of finite size 1 to contain actually an infinite number of parts of positive size? 

Supreme good.The principle of equivalence, which is simultaneously a principle of justice, love and truth as well as the principle of continuous life and motion. Can we think of our cosmic acceleration as the realization of the supreme good? Where is the place of the supreme good relative to us here and now taken as the center of the universe? 

Synthetic logic.The science of the principles that govern the being, thing, or object thought of as an infinite whole, having at one time contrary determinations. The cognitive faculty of synthetic logic is the synthetic reason (defined as the faculty of proportion). Synthetic logic is the negation of the principles of analytic logic. This latter logic regards the real, physical object as a simple, finite, individual part varying in time. Is the physical object a simple finite individual or a complex infinite whole? Is the finite individual conditioned by linear time a real or illusory property of the physical object?

Universal community. A multiplicity of isolated parts unified into a whole by a universal principle and a common point. The unified whole is endowed with permanent life and is composed of parts in continuous communication.  A universal community has a universal reason, a universal sense, a universal will. 

Universe. Analogical to Plato’s intelligible/sensible division of the world, the universe is divided intoreal, physical universe defined as the sum total of its infinite number of parts, and sensible or observable universedefined as the finite, observable and variable part of the constant physical universe. This variable part indefinitely accelerates in time. What are the fundamental properties and organizing principles of these two universes? 

Universal sensibility (see also infinite sensibility).The synthetic faculty of sensing the universe’s infinite multiplicity of scattered parts as one thing, immediately with maximum speed and hence as an infinite whole unified by a universal principle or a common point. Is it possible to have electromagnetic (photonic) signals that travel the universe and our body, both at finite and infinite speeds and avoid absurdity.

© Apeiron Centre, 2002

THE FOUNDER

Ion Soteropoulos

Ion Soteropoulos

Ion Soteropoulos is an independent research philosopher and scientific metaphysician who is passionately devoted to creative and transdisciplinary work. In 2002 she founded Apeiron Centre, the first philosophical organization in the world that studies the idea of infinity in dynamic connection with the finite as presented in our physical universe, society and physical body.

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Question

What law will get us to the farthest stars and to our deepest brain and mind?