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Coincidentia Oppositorum

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Coincidentia oppositorum is a Latin phrase meaning coincidence of opposites. It is a neoplatonic term attributed to 15th century German polymath Nicholas of Cusa in his essay, De Docta Ignorantia (1440). Mircea Eliade, a 20th-century historian of religion, used the term extensively in his essays about myth and ritual, describing the coincidentia oppositorum as "the mythical pattern". Psychiatrist Carl Jung, philosopher and Islamic Studies professor Henry Corbin as well as Jewish philosophers Gershom Scholem and Abraham Joshua Heschel also used the term. In alchemy, coincidentia oppositorum is a synonym for coniunctio. For example, Michael Maier stresses that the union of opposites is the aim of the alchemical work. Or, according to Paracelsus' pupil, Gerhard Dorn, the highest grade of the alchemical coniunctio consisted in the union of the total man with the unus mundus ("one world").

The term is also used in describing a revelation of the oneness of things previously believed to be different. Such insight into the unity of things is a kind of transcendence, and is found in various mystical traditions. The idea occurs in the traditions of Tantric Hinduism and Buddhism, in German mysticism, Taoism, Zen and Sufism, among others.

 

One such traditionally transmitted truth devalued by empiricism and historicism is, of course, the coincidentia
oppositorum itself, and one of the traditions which has transmitted it is the Indian, which
has distinguished two aspects of Brahman: apara and para, "inferior" and "superior," visible and invisible,
manifest and nonmanifest. In other words, it is always the mystery of a polarity, all at once a biunity and a
rhythmic alteration, that can be deciphered in the different mythological, religious, and philosophical
''illustrations": Mitra and Varuna, the visible and invisible aspects of Brahman, Brahman and Maya, purusa
and prakrti, and later on Siva and Sakti, or samsara and Nirvana.
But some of these polarities tend to annul themselves in a coincidentia oppositorum, in a paradoxical unity-
totality. That it is not only a question of metaphysical speculations but also of formulas with the help of
which India tried to circumscribe a particular mode of existence, is proved by the fact that coincidentia
oppositorum is implied in jivanmukta, the "liberated in life," who continues to exist in the world even
though he has attained final deliverance; or the "awakened one" for whom Nirvana and samsara appear to
be one and the same thing. Now, however one may conceive the Absolute, it cannot be conceived except as
beyond contraries and polarities. The summum bonum is situated beyond polarises. (Quest, 169)
 

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Georg  Cantor  (1845-1918)  is remembered  chiefly  for his creation of

transfinite set  theory,  which revolutionized mathematics  by making

possible  a  new, powerful approach  to  understanding  the nature of the

infinite. But Cantor's concerns extended well  beyond  the  purely  techni-

cal content of his  research,  for he  responded seriously  to criticism from

philosophers  and  theologians  as he  sought  to advance and to refine his

transfinite set  theory.  He was also  keenly  aware of the  ways  in which his

work  might  in turn aid and  improve  both  philosophy  and  theology.

Prompted by  a  strong  belief in the role set  theory  could  play  in  helping

the Roman Catholic Church to avoid  misinterpreting  the nature of in-

finity,  he undertook an extensive  correspondence  with Catholic theo-

logians,  and even addressed one letter and a number of his  pamphlets

directly  to  Pope  Leo XIII.

 

 Cantor's Radical Innovation.-Cantor's introduction of the actual

infinite was a radical  departure  from traditional  practice,  even  dogma.

It was an idea which  mathematicians, philosophers,  and  theologians  in

general  had  repudiated  since the time of Aristotle.  Philosophers  and

mathematicians  rejected completed  infinities  largely  because of the  ap-

parent logical paradoxes they seemed to  generate. Theologians

represented  another tradition of  opposition  to the actual  infinite, regard-

ing  it as a direct  challenge  to the  unique  and absolute infinite nature of

God. St.  Thomas,  in  particular,  had  argued against  the  possibility  of  any

absolute  infinity,  and  yet  Cantor's new  theory produced precisely  what

Thomas had denied.

 

Cantor claimed  reality  for both the  physical  and ideal halves of his

approach  to the number  concept.  The dual  realities,  in  fact,  were  always

found in a  joined sense,  insofar as a  concept possessing  an immanent

reality always possessed  a transient  reality  as well.25 It was one of the

most difficult  problems  of  metaphysics  to find the  determining  features

of the connection between the two kinds of  reality.  Cantor ascribed the

coincidence of the dual  aspects  of the  reality  of numbers to the  unity  of

the universe itself. This left mathematicians with the  important  result

that it was  possible  to  study only  the immanent  realities,  without  having

to confirm or conform to any subjective content. This placed

mathematics  apart  from all other sciences for which natural  phenomena

provided objective, physical objects  and events for scientific  scrutiny.  It

gave  mathematics an  independence  that was to  imply great  freedom for

mathematicians in the creation of mathematical  concepts.

 

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Absolute (n. das Absolute; or adj. absolut) Hegel’s use of the

term ‘absolute’ is the source of a great deal of confusion. Nevertheless, it is

the term most commonly associated with his philosophy. Hegel frequently

uses it as an adjective, for example in ‘Absolute Idea’, ‘Absolute Knowing’,

‘Absolute Religion’ and ‘Absolute Spirit’. He utilizes the substantive ‘the

Absolute’ less frequently.

The term ‘absolute’ has a long history in German philosophy. Nicholas of

Cusa in his Of Learned Ignorance (De Docta Ignorantia, 1440) used the

term absolutum to mean God, understood as a being that transcends all

finite determinations: the coincidentia oppositorum (coincidence of

opposites). Schelling’s use of ‘Absolute’ is remarkably similar to Cusa’s. For

Schelling, the Absolute is the ‘indifference point’ beyond the distinction of

subject and object, or any other distinction. In the famous Preface to The

Phenomenology of Spirit Hegel rejects this conception of the Absolute,

referring to it derisively as ‘the night in which all cows are black’ (Miller, 9;

PG, 13). Hegel means that when the Absolute is conceived simply as the

transcendent unity of all things (or as the cancellation of all difference) it

really amounts to an idea devoid of all content. It is terribly easy to say ‘in

this world definite distinctions abide – but in the Absolute all is one.’ But

what does this really mean?

One might think this would lead Hegel to reject the idea of an Absolute

altogether, but he does not. The reason is that Hegel saw the aim of

philosophy itself as knowledge of the Absolute, where this is understood, in

very broad terms, as the ultimate ground or source of all being. It is this

knowledge that was sought right from the beginning of the Western

tradition in the Pre-Socratic philosopher Thales who declared that ‘water’ is

the source of all that is. According to Hegel, the trouble with Schelling is

not that he has conceived of an Absolute, but that he has misconceived it.

(In fact, one might say that from the standpoint of a Hegelian, Hegel’s great

achievement in the history of philosophy is to have arrived at a proper

understanding of the Absolute.)

Hegel retains the idea of the Absolute, and even agrees with Schelling’s

description of it as somehow overcoming the subject-object distinction. For

Hegel, however, the Absolute is the whole. The Absolute is not something

that transcends existence; it is the whole of existence itself understood as a

system in which each part is organically and inseparably related to every

other. However, one might ask, how does this conception of the Absolute

as ‘the whole’ show how it is ‘the ultimate ground or source of all being’?

The answer is simple: Hegel’s philosophy attempts to show how the being

of each finite thing in existence just is its place in the whole, as part of the

system of reality itself. The Absolute, however, is not the whole of reality

conceived as a static, block universe. Instead, Hegel argues that the

Absolute is active and dynamic, continually replenishing or reconstituting

itself through the finite beings that make up the infinite whole.

In short, things exist in order that the whole may be complete – for all

things are what they are in virtue of their place within the systematic

totality of existence. Thus, Hegel believes that he has answered the age-old

question ‘What is being?’ Fundamentally, to be is to be the whole or

Absolute, but the finite things of our experience can be said to derive a kind

of being from their place within the whole. However, another of the age-

old metaphysical questions is ‘Why is there anything at all, rather than

nothing?’, or ‘Why does existence exist at all?’ Hegel may say that to be is

to be a moment or aspect of the whole, but we might ask him why this

whole exists in the first place. Hegel does have an answer to this, and it

constitutes the most important idea in his philosophy, as well as the true

understanding of the specific sense in which the whole is dynamic and

active. Very simply, Hegel believes that existence exists in order to achieve

consciousness of itself.

Hegel sees all of existence as a kind of ‘great chain of being’, culminating

in the achievement of self-awareness in human beings. Human beings are

creatures of nature, but we stand at the apex of nature because we

subsume within ourselves the non-living, mineral and chemical world, as

well as the vegetative and animal functions (growth, repair, nutrition,

sensation, self-motion, etc.). In addition to this, we actualize a function not

to be found in lower nature: we are self-aware. Our quest for self-

awareness displays itself pre-eminently in our striving to understand

ourselves as a species and our place in nature through science and

philosophy. Because we are ourselves creatures of nature, we can say that

our self-awareness is, in fact, the self-awareness of nature – of all of

existence. Thus, in human beings, existence itself reaches a kind of closure

or completion: existence rebounds upon itself and knows itself. We can also

understand this in terms of the ‘overcoming’ of the subject-object distinc-

tion. In our self-awareness, subject has become object: we subjects become

objects to ourselves. At the same time, in our self-awareness object has

become subject: our object is in fact the subject – and nature, the objective

world itself, has achieved subjectivity through us.

The Absolute, for Hegel, is thus the whole of the objective world

understood as a system perpetually giving rise to the conditions necessary

for it to confront itself as object, and thus achieve closure. In the Phenome-

nology Hegel tells us that the Absolute is, ‘the process of its own becoming,

the circle that presupposes its end as its goal, having its end also as its

beginning; and only by being worked out to its end, is it actual’ (Miller, 10;

PG, 14). Further, the Absolute must be conceived in ‘the whole wealth of

the developed form. Only then is it conceived and expressed as an actuality’

(Miller, 11; PG, 15).

Hegel’s description of the Absolute begins with his Logic, which offers a

kind of skeletal account of the whole itself. It does not discuss the specific

members of the systematic whole; instead it is an account of the funda-

mental ideas or categories that make the whole organic and systematic,

and which are being ‘realized’ or expressed in concrete existence all around

us. The system of ideas in the Logic culminates in ‘Absolute Idea’ – which is

absolute because it in a sense ‘contains’ all the preceding, fundamental

ideas. Because there is no further idea that can encompass it, Absolute Idea

therefore exhibits one of the classical characteristics of the absolutum: it is

‘unconditioned’ (or uncomprehended) by any finite determinations external

to it. Further, the Absolute Idea is conceived by Hegel as a self-related idea.

Indeed, it is the idea of idea itself. Nevertheless, Hegel points out that it is

only an idea: it lacks concrete being or expression.

Hegel’s Philosophy of Nature shows how all of nature may be understood

as an expression of the categories of the Logic – and as a hierarchy of forms

in which the self-relation described abstractly in Absolute Idea is being pro-

gressively realized in the flesh. However, true self-relatedness is not

achieved in what we think of as the natural world, but only in human

thought. Hegel refers to human nature as Spirit and shows how the highest

expression of our humanity is what he calls Absolute Spirit: the achieve-

ment of self-relatedness as self-consciousness, through art, religion and

philosophy. Thus, we may say that the Absolute is achieved or consum-

mated through the realization of Absolute Idea in Absolute Spirit, with all

non-human, lower nature understood as a dim approximation to Absolute

Spirit, and as a set of necessary conditions for its achievement. (The other

conditions for the achievement of Absolute Spirit being historical, and

cultural.

 

Magee, Glenn Alexander,

The Hegel dictionary

 

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Duhovit video o Kurtu Gedelu, najvećem matematičaru 20 veka. Posle relaksirajućeg uvoda autor daje odličan prikaz Gedelovog života i rada, posebno njegov doprinos logici i nauci. U drugom delu je najzanimljivije, prikaz Gedelovih filozofskih stajališta i potrage za Apsolutom.

 

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The Limits of Understanding

 

This statement is false. Think about it, and it makes your head hurt. If it’s true, it’s false. If it’s false, it’s true. In 1931, Austrian logician Kurt Godel shocked the worlds of mathematics and philosophy by establishing that such statements are far more than a quirky turn of language: he showed that there are mathematical truths which simply can't be proven. In the decades since, thinkers have taken the brilliant Godel's result in a variety of directions—linking it to limits of human comprehension and the quest to recreate human thinking on a computer. In this full program from the 2010 Festival, leading thinkers untangle Godel's discovery and examine the wider implications of his revolutionary finding.

Participants: Gregory Chaitin, Mario Livio, Marvin Minsky, Rebecca Newberger Goldstein

 

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Gerald Sacks, logičar, profesor na Harvardu i MIT-u, dugogodišnji prijatelj i saradnik Kurta Gedela priča o svom kolegi i prijatelju:

 

 

 

Ono što je jako zanimljivo su filozofske teme koje je Gedel pokretao u raspravama sa svojim kolegama (posebno o Apsolutu) ali koje su imale slab odjek jer jednostavno nije imao adekvatnog sagovornika u tom mnoštvu pozitivista. Gedel je bio do srži neoplatoničar, veliki Lajbnicov poštovalac...

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Dva najbolja prijatelja, Ajnštajn i Gedel, najveći fizičar i najveći logičar 20. veka. Svi su želeli da znaju šta su ova dva genija pričala tokom svojih šetnji u najprestižnijem Institutu za napredna istraživanja u Prinstonu. 

 

kurt-goedel-albert-einstein.jpg

 

TIME BANDITS

 

What were Einstein and Gödel talking about?

 

BY JIM HOLT

 

In 1933, with his great scientific discoveries behind him, Albert Einstein came to America. He spent the last twenty-two years of his life in Princeton, New Jersey, where he had been recruited as the star member of the Institute for Advanced Study. Einstein was reasonably content with his new milieu, taking its pretensions in stride. “Princeton is a wonderful piece of earth, and at the same time an exceedingly amusing ceremonial backwater of tiny spindle-shanked demigods,” he observed. His daily routine began with a leisurely walk from his house, at 115 Mercer Street, to his office at the institute. He was by then one of the most famous and, with his distinctive appearance—the whirl of pillow-combed hair, the baggy pants held up by suspenders—most recognizable people in the world.

A decade after arriving in Princeton, Einstein acquired a walking companion, a much younger man who, next to the rumpled Einstein, cut a dapper figure in a white linen suit and matching fedora. The two would talk animatedly in German on their morning amble to the institute and again, later in the day, on their way homeward. The man in the suit may not have been recognized by many townspeople, but Einstein addressed him as a peer, someone who, like him, had single-handedly launched a conceptual revolution. If Einstein had upended our everyday notions about the physical world with his theory of relativity, the younger man, Kurt Gödel, had had a similarly subversive effect on our understanding of the abstract world of mathematics.

 
 

Gödel, who has often been called the greatest logician since Aristotle, was a strange and ultimately tragic man. Whereas Einstein was gregarious and full of laughter, Gödel was solemn, solitary, and pessimistic. Einstein, a passionate amateur violinist, loved Beethoven and Mozart. Gödel’s taste ran in another direction: his favorite movie was Walt Disney’s “Snow White and the Seven Dwarfs,” and when his wife put a pink flamingo in their front yard he pronounced it furchtbar herzig—“awfully charming.” Einstein freely indulged his appetite for heavy German cooking; Gödel subsisted on a valetudinarian’s diet of butter, baby food, and laxatives. Although Einstein’s private life was not without its complications, outwardly he was jolly and at home in the world. Gödel, by contrast, had a tendency toward paranoia. He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.

Although other members of the institute found the gloomy logician baffling and unapproachable, Einstein told people that he went to his office “just to have the privilege of walking home with Kurt Gödel.” Part of the reason, it seems, was that Gödel was undaunted by Einstein’s reputation and did not hesitate to challenge his ideas. As another member of the institute, the physicist Freeman Dyson, observed, “Gödel was . . . the only one of our colleagues who walked and talked on equal terms with Einstein.” But if Einstein and Gödel seemed to exist on a higher plane than the rest of humanity, it was also true that they had become, in Einstein’s words, “museum pieces.” Einstein never accepted the quantum theory of Niels Bohr and Werner Heisenberg. Gödel believed that mathematical abstractions were every bit as real as tables and chairs, a view that philosophers had come to regard as laughably naïve. Both Gödel and Einstein insisted that the world is independent of our minds, yet rationally organized and open to human understanding. United by a shared sense of intellectual isolation, they found solace in their companionship. “They didn’t want to speak to anybody else,” another member of the institute said. “They only wanted to speak to each other.”

People wondered what they spoke about. Politics was presumably one theme. (Einstein, who supported Adlai Stevenson, was exasperated when Gödel chose to vote for Dwight Eisenhower in 1952.) Physics was no doubt another. Gödel was well versed in the subject; he shared Einstein’s mistrust of the quantum theory, but he was also skeptical of the older physicist’s ambition to supersede it with a “unified field theory” that would encompass all known forces in a deterministic framework. Both were attracted to problems that were, in Einstein’s words, of “genuine importance,” problems pertaining to the most basic elements of reality. Gödel was especially preoccupied by the nature of time, which, he told a friend, was the philosophical question. How could such a “mysterious and seemingly self-contradictory” thing, he wondered, “form the basis of the world’s and our own existence”? That was a matter in which Einstein had shown some expertise.

 
 
 

A century ago, in 1905, Einstein proved that time, as it had been understood by scientist and layman alike, was a fiction. And this was scarcely his only achievement that year, which John S. Rigden skillfully chronicles, month by month, in “Einstein 1905: The Standard of Greatness” (Harvard; $21.95). As it began, Einstein, twenty-five years old, was employed as an inspector in a patent office in Bern, Switzerland. Having earlier failed to get his doctorate in physics, he had temporarily given up on the idea of an academic career, telling a friend that “the whole comedy has become boring.” He had recently read a book by Henri Poincaré, a French mathematician of enormous reputation, which identified three fundamental unsolved problems in science. The first concerned the “photoelectric effect”: how did ultraviolet light knock electrons off the surface of a piece of metal? The second concerned “Brownian motion”: why did pollen particles suspended in water move about in a random zigzag pattern? The third concerned the “luminiferous ether” that was supposed to fill all of space and serve as the medium through which light waves moved, the way sound waves move through air, or ocean waves through water: why had experiments failed to detect the earth’s motion through this ether?

Each of these problems had the potential to reveal what Einstein held to be the underlying simplicity of nature. Working alone, apart from the scientific community, the unknown junior clerk rapidly managed to dispatch all three. His solutions were presented in four papers, written in the months of March, April, May, and June of 1905. In his March paper, on the photoelectric effect, he deduced that light came in discrete particles, which were later dubbed “photons.” In his April and May papers, he established once and for all the reality of atoms, giving a theoretical estimate of their size and showing how their bumping around caused Brownian motion. In his June paper, on the ether problem, he unveiled his theory of relativity. Then, as a sort of encore, he published a three-page note in September containing the most famous equation of all time: E = mc2.

All of these papers had a touch of magic about them, and upset deeply held convictions in the physics community. Yet, for scope and audacity, Einstein’s June paper stood out. In thirty succinct pages, he completely rewrote the laws of physics, beginning with two stark principles. First, the laws of physics are absolute: the same laws must be valid for all observers. Second, the speed of light is absolute; it, too, is the same for all observers. The second principle, though less obvious, had the same sort of logic to recommend it. Since light is an electromagnetic wave (this had been known since the nineteenth century), its speed is fixed by the laws of electromagnetism; those laws ought to be the same for all observers; and therefore everyone should see light moving at the same speed, regardless of the frame of reference. Still, it was bold of Einstein to embrace the light principle, for its consequences seemed downright absurd.

Suppose—to make things vivid—that the speed of light is a hundred miles an hour. Now suppose I am standing by the side of the road and I see a light beam pass by at this speed. Then I see you chasing after it in a car at sixty miles an hour. To me, it appears that the light beam is outpacing you by forty miles an hour. But you, from inside your car, must see the beam escaping you at a hundred miles an hour, just as you would if you were standing still: that is what the light principle demands. What if you gun your engine and speed up to ninety-nine miles an hour? Now I see the beam of light outpacing you by just one mile an hour. Yet to you, inside the car, the beam is still racing ahead at a hundred miles an hour, despite your increased speed. How can this be? Speed, of course, equals distance divided by time. Evidently, the faster you go in your car, the shorter your ruler must become and the slower your clock must tick relative to mine; that is the only way we can continue to agree on the speed of light. (If I were to pull out a pair of binoculars and look at your speeding car, I would actually see its length contracted and you moving in slow motion inside.) So Einstein set about recasting the laws of physics accordingly. To make these laws absolute, he made distance and time relative.

It was the sacrifice of absolute time that was most stunning. Isaac Newton believed that time was regulated by a sort of cosmic grandfather clock. “Absolute, true, mathematical time, of itself, and from its own nature, flows equably without relation to anything external,” he declared at the beginning of his “Principia.” Einstein, however, realized that our idea of time is something we abstract from our experience with rhythmic phenomena: heartbeats, planetary rotations and revolutions, the ticking of clocks. Time judgments always come down to judgments of simultaneity. “If, for instance, I say, ‘That train arrives here at 7 o’clock,’ I mean something like this: ‘The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events,’ ” Einstein wrote in the June paper. If the events in question are at some distance from one another, judgments of simultaneity can be made only by sending light signals back and forth. Working from his two basic principles, Einstein proved that whether an observer deems two events to be happening “at the same time” depends on his state of motion. In other words, there is no universal now. With different observers slicing up the timescape into “past,” “present,” and “future” in different ways, it seems to follow that all moments coexist with equal reality.

Einstein’s conclusions were the product of pure thought, proceeding from the most austere assumptions about nature. In the century since he derived them, they have been precisely confirmed by experiment after experiment. Yet his June, 1905, paper on relativity was rejected when he submitted it as a dissertation. (He then submitted his April paper, on the size of atoms, which he thought would be less likely to startle the examiners; they accepted it only after he added one sentence to meet the length threshold.) When Einstein was awarded the 1921 Nobel Prize in Physics, it was for his work on the photoelectric effect. The Swedish Academy forbade him to make any mention of relativity in his acceptance speech. As it happened, Einstein was unable to attend the ceremony in Stockholm. He gave his Nobel lecture in Gothenburg, with King Gustav V seated in the front row. The King wanted to learn about relativity, and Einstein obliged him.

In 1906, the year after Einstein’s annus mirabilis, Kurt Gödel was born in the city of Brno (now in the Czech Republic). As Rebecca Goldstein recounts in her enthralling intellectual biography “Incompleteness: The Proof and Paradox of Kurt Gödel” (Atlas/Norton; $22.95), Kurt was both an inquisitive child—his parents and brother gave him the nickname der Herr Warum, “Mr. Why?”—and a nervous one. At the age of five, he seems to have suffered a mild anxiety neurosis. At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged.

Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of nineteen-twenties Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.

Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid. But how could one demonstrate that mathematics could not be reduced to the artifices of logic? Gödel’s strategy—one of “heart-stopping beauty,” as Goldstein justly observes—was to use logic against itself. Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of double speak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another. In fact, as Gödel showed, a numerical formula could even be made to say something about itself. (Goldstein compares this to a play in which the characters are also actors in a play within the play; if the playwright is sufficiently clever, the lines the actors speak in the play within the play can be interpreted as having a “real life” meaning in the play proper.) Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. The conclusion—that no logical system can capture all the truths of mathematics—is known as the first incompleteness theorem. Gödel also proved that no logical system for mathematics could, by its own devices, be shown to be free from inconsistency, a result known as the second incompleteness theorem.

Wittgenstein once averred that “there can never be surprises in logic.” But Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. “Are we to think that 2 + 2 is not 4, but 4.001?” Russell asked decades later in dismay, adding that he was “glad [he] was no longer working at mathematical logic.” As the significance of Gödel’s theorems began to sink in, words like “debacle,” “catastrophe,” and “nightmare” were bandied about. It had been an article of faith that, armed with logic, mathematicians could in principle resolve any conundrum at all—that in mathematics, as it had been famously declared, there was no ignorabimus. Gödel’s theorems seemed to have shattered this ideal of complete knowledge.

That was not the way Gödel saw it. He believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.” It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives. Some thinkers (like the physicist Roger Penrose) have taken this theme further, maintaining that Gödel’s incompleteness theorems have profound implications for the nature of the human mind. Our mental powers, it is argued, must outstrip those of any computer, since a computer is just a logical system running on hardware, and our minds can arrive at truths that are beyond the reach of a logical system.

Gödel was twenty-four when he proved his incompleteness theorems (a bit younger than Einstein was when he created relativity theory). At the time, much to the disapproval of his strict Lutheran parents, he was courting an older Catholic divorcée by the name of Adele, who, to top things off, was employed as a dancer in a Viennese night club called Der Nachtfalter (the Moth). The political situation in Austria was becoming ever more chaotic with Hitler’s rise to power in Germany, although Gödel seems scarcely to have noticed. In 1936, the Vienna Circle dissolved, after its founder was assassinated by a deranged student. Two years later came the Anschluss. The perilousness of the times was finally borne in upon Gödel when a band of Nazi youths roughed him up and knocked off his glasses, before retreating under the umbrella blows of Adele. He resolved to leave for Princeton, where he had been offered a position by the Institute for Advanced Study. But, the war having broken out, he judged it too risky to cross the Atlantic. So the now married couple took the long way around, traversing Russia, the Pacific, and the United States, and finally arriving in Princeton in early 1940. At the institute, Gödel was given an office almost directly above Einstein’s. For the rest of his life he rarely left Princeton, which he came to find “ten times more congenial” than his once beloved Vienna.

“There it was, inconceivably, K. Goedel, listed just like any other name in the bright orange Princeton community phonebook,” writes Goldstein, who came to Princeton University as a graduate student of philosophy in the early nineteen-seventies. (It’s the setting of her novel “The Mind-Body Problem.”) “It was like opening up the local phonebook and finding B. Spinoza or I. Newton.” Although Gödel was still little known in the world at large, he had a godlike status among the cognoscenti. “I once found the philosopher Richard Rorty standing in a bit of a daze in Davidson’s food market,” Goldstein writes. “He told me in hushed tones that he’d just seen Gödel in the frozen food aisle.”

So naïve and otherworldly was the great logician that Einstein felt obliged to help look after the practical aspects of his life. One much retailed story concerns Gödel’s decision after the war to become an American citizen. The character witnesses at his hearing were to be Einstein and Oskar Morgenstern, one of the founders of game theory. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the United States Constitution. On the eve of the hearing, he called Morgenstern in an agitated state, saying he had found an “inconsistency” in the Constitution, one that could allow a dictatorship to arise. Morgenstern was amused, but he realized that Gödel was serious and urged him not to mention it to the judge, fearing that it would jeopardize Gödel’s citizenship bid. On the short drive to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried to distract Gödel with jokes. When they arrived at the courthouse, the judge was impressed by Gödel’s eminent witnesses, and he invited the trio into his chambers. After some small talk, he said to Gödel, “Up to now you have held German citizenship.”

No, Gödel corrected, Austrian.

“In any case, it was under an evil dictatorship,” the judge continued. “Fortunately that’s not possible in America.”

“On the contrary, I can prove it is possible!” Gödel exclaimed, and he began describing the constitutional loophole he had descried. But the judge told the examinee that “he needn’t go into that,” and Einstein and Morgenstern succeeded in quieting him down. A few months later, Gödel took his oath of citizenship.

Around the same time that Gödel was studying the Constitution, he was also taking a close look at Einstein’s relativity theory. The key principle of relativity is that the laws of physics should be the same for all observers. When Einstein first formulated the principle in his revolutionary 1905 paper, he restricted “all observers” to those who were moving uniformly relative to one another—that is, in a straight line and at a constant speed. But he soon realized that this restriction was arbitrary. If the laws of physics were to provide a truly objective description of nature, they ought to be valid for observers moving in any way relative to one another—spinning, accelerating, spiralling, whatever. It was thus that Einstein made the transition from his “special” theory of relativity of 1905 to his “general” theory, whose equations he worked out over the next decade and published in 1916. What made those equations so powerful was that they explained gravity, the force that governs the over-all shape of the cosmos.

Decades later, Gödel, walking with Einstein, had the privilege of picking up the subtleties of relativity theory from the master himself. Einstein had shown that the flow of time depended on motion and gravity, and that the division of events into “past” and “future” was relative. Gödel took a more radical view: he believed that time, as it was intuitively understood, did not exist at all. As usual, he was not content with a mere verbal argument. Philosophers ranging from Parmenides, in ancient times, to Immanuel Kant, in the eighteenth century, and on to J. M. E. McTaggart, at the beginning of the twentieth century, had produced such arguments, inconclusively. Gödel wanted a proof that had the rigor and certainty of mathematics. And he saw just what he wanted lurking within relativity theory. He presented his argument to Einstein for his seventieth birthday, in 1949, along with an etching. (Gödel’s wife had knitted Einstein a sweater, but she decided not to send it.)

What Gödel found was the possibility of a hitherto unimaginable kind of universe. The equations of general relativity can be solved in a variety of ways. Each solution is, in effect, a model of how the universe might be. Einstein, who believed on philosophical grounds that the universe was eternal and unchanging, had tinkered with his equations so that they would yield such a model—a move he later called “my greatest blunder.” Another physicist (a Jesuit priest, as it happens) found a solution corresponding to an expanding universe born at some moment in the finite past. Since this solution, which has come to be known as the Big Bang model, was consistent with what astronomers observed, it seemed to be the one that described the actual cosmos. But Gödel came up with a third kind of solution to Einstein’s equations, one in which the universe was not expanding but rotating. (The centrifugal force arising from the rotation was what kept everything from collapsing under the force of gravity.) An observer in this universe would see all the galaxies slowly spinning around him; he would know it was the universe doing the spinning, and not himself, because he would feel no dizziness. What makes this rotating universe truly weird, Gödel showed, is the way its geometry mixes up space and time. By completing a sufficiently long round trip in a rocket ship, a resident of Gödel’s universe could travel back to any point in his own past.

Einstein was not entirely pleased with the news that his equations permitted something as Alice in Wonderland-like as spatial paths that looped backward in time; in fact, he confessed to being “disturbed” by Gödel’s universe. Other physicists marvelled that time travel, previously the stuff of science fiction, was apparently consistent with the laws of physics. (Then they started worrying about what would happen if you went back to a time before you were born and killed your own grandfather.) Gödel himself drew a different moral. If time travel is possible, he submitted, then time itself is impossible. A past that can be revisited has not really passed. And the fact that the actual universe is expanding, rather than rotating, is irrelevant. Time, like God, is either necessary or nothing; if it disappears in one possible universe, it is undermined in every possible universe, including our own.

Gödel’s conclusion went almost entirely unnoticed at the time, but it has since found a passionate champion in Palle Yourgrau, a professor of philosophy at Brandeis. In “A World Without Time: The Forgotten Legacy of Gödel and Einstein” (Perseus; $24), Yourgrau does his best to redress his fellow-philosophers’ neglect of the case that Gödel made against time. The “deafening silence,” he submits, can be blamed on the philosophical prejudices of the era. Behind all the esoteric mathematics, Gödel’s reasoning looked suspiciously metaphysical. To this day, Yourgrau complains, Gödel is treated with condescension by philosophers, who regard him, in the words of one, as “a logician par excellence but a philosophical fool.” After ably tracing Gödel’s life, his logical achievements, and his friendship with Einstein, Yourgrau elaborately defends his importance as a philosopher of time. “In a deep sense,” he concludes, “we all do live in Gödel’s universe.”

Gödel’s strange cosmological gift was received by Einstein at a bleak time in his life. His quest for a unified theory of physics was proving fruitless, and his opposition to quantum theory alienated him from the mainstream of physics. Family life provided little consolation. His two marriages had been failures; a daughter born out of wedlock seems to have disappeared from history; of his two sons one was schizophrenic, the other estranged. Einstein’s circle of friends had shrunk to Gödel and a few others. One of them was Queen Elisabeth of Belgium, to whom he confided, in March, 1955, that “the exaggerated esteem in which my lifework is held makes me very ill at ease. I feel compelled to think of myself as an involuntary swindler.” He died a month later, at the age of seventy-six. When Gödel and another colleague went to his office at the institute to deal with his papers, they found the blackboard covered with dead-end equations.

After Einstein’s death, Gödel became ever more withdrawn. He preferred to conduct all conversations by telephone, even if his interlocutor was a few feet distant. When he especially wanted to avoid someone, he would schedule a rendezvous at a precise time and place, and then make sure he was somewhere far away. The honors the world wished to bestow upon him made him chary. He did show up to collect an honorary doctorate in 1953 from Harvard, where his incompleteness theorems were hailed as the most important mathematical discovery of the previous hundred years; but he later complained of being “thrust quite undeservedly into the most highly bellicose company” of John Foster Dulles, a co-honoree. When he was awarded the National Medal of Science, in 1975, he refused to go to Washington to meet Gerald Ford at the White House, despite the offer of a chauffeur for him and his wife. He had hallucinatory episodes and talked darkly of certain forces at work in the world “directly submerging the good.” Fearing that there was a plot to poison him, he persistently refused to eat. Finally, looking like (in the words of a friend) “a living corpse,” he was taken to the Princeton Hospital. There, two weeks later, on January 14, 1978, he succumbed to self-starvation. According to his death certificate, the cause of death was “malnutrition and inanition” brought on by “personality disturbance.”

A certain futility marked the last years of both Gödel and Einstein. What may have been most futile, however, was their willed belief in the unreality of time. The temptation was understandable. If time is merely in our minds, perhaps we can hope to escape it into a timeless eternity. Then we could say, like William Blake, “I see the Past, Present and Future, existing all at once / Before me.” In Gödel’s case, Rebecca Goldstein speculates, it may have been his childhood terror of a fatally damaged heart that attracted him to the idea of a timeless universe. Toward the end of his life, he told one confidant that he had long awaited an epiphany that would enable him to see the world in a new light, but that it never came. Einstein, too, was unable to make a clean break with time. “To those of us who believe in physics,” he wrote to the widow of a friend who had recently died, “this separation between past, present, and future is only an illusion, if a stubborn one.” When his own turn came, a couple of weeks later, he said, “It is time to go.” 

 

 

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Po Gedelu vreme je iluzija, realno ne postoji. Ajnštajn je bio šokiran ovim zaključkom ali je prihvatio Gedelovu teoriju.

 

A World Without Time: The Forgotten Legacy of Godel and Einstein

 

If Einstein had succeeded in transforming time into space, Godel would perform a trick yet

more magical: He would make time disappear. Having already rocked the mathematical

world to its foundations with his incompleteness theorem, Godel now took aim at Einstein

and relativity. Wasting no time, he announced in short order his discovery of new and

unsuspected cosmological solutions to the field equations of general relativity, solutions in

which time would undergo a shocking transformation. The mathematics, the physics and

the philosophy of Godel's results were all new. In the possible worlds governed by these

new cosmological solutions, the so-called rotating or Godel universes, it turned out that

the space-time structure is so greatly warped or curved by the distribution of matter that

there exist timelike future-directed paths by which a spaceship, if it travels fast

enoughóand Godel worked out the precise speed and fuel requirements, omitting only the

lunch menuócan penetrate into any region of the past, present or future.

Godel, the union of Einstein and Kafka, had for the first time in human history proved,

from the equations of relativity, that time travel was not a philosopher's fantasy but a

scientific possibility. Yet again he had somehow contrived, from within the very heart of

mathematics, to drop a bomb into the laps of the philosophers. The fallout, however, from

this mathematical bomb was even more perilous than that from

the incompleteness theorem. Godel was quick to point out that if we can revisit the past,

then it never really "passed." But a time that fails to pass is no time at all. Einstein saw at

once that if Godel was right, he had not merely domesticated time: he had killed it. Time,

"that mysterious and seemingly self-contradictory being," as Godel put it, "which, on the

other hand, seems to form the basis of the world's and our own existence," turned out in

the end to be the world's greatest illusion. In a word, if Einstein's relativity was real, time

itself was merely ideal. The father of relativity was shocked. Though he praised Godel for

his great contribution to the theory of relativity, he was fully aware that time, that elusive

prey, had once again slipped his net.

But now something truly amazing took place: nothing. Although in the immediate

aftermath of Godel's discoveries a few physicists bestirred themselves to refute him and,

when this failed, tried to generalize and explore his results, this brief flurry of interest

soon died down. Within a few years the deep footprints in intellectual history traced by

Godel and Einstein in their long walks home had disappeared, dispersed by the harsh winds

of fashion and philosophical prejudice. A conspiracy of silence descended on the Einstein-

Godel friendship and its scientific consequences.

 

51Emb4kO-oL._SX322_BO1,204,203,200_.jpg

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Теорема непотпуности

 

„Достигнуће Курта Гедела у модерној логици је сингуларно и монументално — заиста, оно је више од споменика, то је међаш који ће остати видљив далеко у простору и времену... Природа и могућности логике су сигурно потпуно промењене Геделовим достигнућем.“

Џон фон Нојман

 

 

1931. док је још боравио у Бечу, Гедел је објавио своје теореме о непотпуности у раду О формалној неодређености поставки у „Принципима математике“ и односним системима (нем. Uber formal unentscheidbare Sätze der 'Principia Mathematica' und verwandter Systeme). У том раду је доказао да за сваки израчунљив аксиоматски систем који је довољно снажан да опише аритметику природних бројева (на пример Пеанове аксиоме или Зермело-Френкел теорија скупова са аксиомом избора), важи:

  1. ако је систем конзистентан, он не може бити потпун.
  2. конзистентност аксиома не може бити доказана унутар система.

Ове теореме су окончале пола века дуге покушаје да се пронађе скуп аксиома довољних за заснивање целокупне математике, који су почели радом Фрегеа а кулминирали у делу Principia Mathematica Расела и Вајтхеда и Хилбертовим формализмом.

Основна идеја која лежи у срцу теореме о непотпуности је прилично једноставна. Гедел је у суштини конструисао формулу која тврди да је недоказива у датом формалном систему. Ако би била доказива, онда би била нетачна, што представља контрадикцију идеји да су у конзистентном систему доказиви искази увек тачни. Стога ће увек постојати бар један истинит али недоказив исказ.

wiki

 

Kako je Gedel srušio pozitivističke snove svojim teoremama o neodlučivosti:

 

 

 

Bear in mind also what Godel proved and what he did not. He did not discover some deep and mysterious mathematical proposition that no formal system was powerful enough to count among its theorems. That would have demonstrated the existence of an absolutely unprovable mathematical proposition, something that Godel, like Hilbert, was deeply skeptical of. Rather, what he showed is that in any particular formal system of sufficient strength, given the limitations imposed on such a system insofar as it is truly formal, there would always be some formula which, while intuitively true, could not be proved in or relative to that system. And the same holds for its negation. But the formula would be a perfectly ordinary, though complex, mathematical proposition, which nevertheless, because of its form, slipped through the net of the given formal system. That very formula, however, could always be proved in a more inclusive formal system; only that new formal system, in turn, would be unable to prove some new formula, which was nevertheless intuitively true. And so on. There was, then, no "supervirus" that affected all formal systems. Instead, for each particular formal system, there would be some perfectly ordinary bug or virus that rendered that system incomplete.

Palle Yourgrau

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Gödel left in his papers a  fourteen-point outline of his philosophical beliefs, that are dated around 1960. They show his deep belief in the rational structure of the world. Here are his 14 points:

  1. The world is rational.
  2. Human reason can, in principle, be developed more highly (through certain techniques).
  3. There are systematic methods for the solution of all problems (also art, etc.).
  4. There are other worlds and rational beings of a different and higher kind.
  5. The world in which we live is not the only one in which we shall live or have lived.
  6. There is incomparably more knowable a priori than is currently known.
  7. The development of human thought since the Renaissance is thoroughly intelligible (durchaus einsichtige).
  8. Reason in mankind will be developed in every direction.
  9. Formal rights comprise a real science.
  10. Materialism is false.
  11. The higher beings are connected to the others by analogy, not by composition.
  12. Concepts have an objective existence.
  13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
  14. Religions are, for the most part, bad– but religion is not.

Based on this far-reaching rational belief, he thought he can proof the existence of God, in a modified (and logically consistent) version of Anselm’s ontological proof.  I admit that his argument needs a lot of explanation. Gödel was a theist, not a pantheist, and he also rejected Einsteins idea of an impersonal God. He saw himself in the tradition of Leibniz, not Spinoza. It seems that Gödel worked on the “ontological proof” for a long time, and finally he handed it over to a friend in 1970. It was published in 1987.

 

Notes:

In: Wang, Hao. A Logical Journey: From Gödel to Philosophy. A Bradford Book, 1997. Print. p.316.

 

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At the heart of Godel and Einstein's opposition to positivism was their unfashionable

realism, their reluctance to make ontology, the theory of what is, subservient to

epistemology, the theory of what can be known. At bottom, the positivist mentality

consists in deriving ontology from epistemology. This was the source of Ernst Mach's posi-

tivistic objection to atomic theory, since individual atoms will never be directly

encountered by humans. But the springs of Mach's philosophy ran deeper. His rejection of a

reality "beyond" what appears to human sensibility was a simplified version of Kant's

philosophy. The "Copernican Revolution" in epistemology inaugurated by Kant consisted in

the radical doctrine that the known must conform to the knower. The hard-nosed,

ultraempiricist Mach had derived his positivism from Kant, who was not a realist but an

idealist, albeit of the deep, German, transcendental variety, not (as Kant saw it) of the

shallow British strain of George Berkeley.

Still, idealism is idealism, whether British or German. Although Kant, unlike Mach,

recognized the existence of a reality beyond what appears to us, he made it clear that the

objects of science are not the "things in themselves" that lie behind "the appearances," but

rather the appearances themselves. This was a doctrine rejected by both Einstein and

Godel. Godel made his objections explicit: Whereas it was a "fruitful viewpoint [to make] a

distinction between subjective and objective elements in our knowledge (which is so

impressively suggested by Kant's comparison with the Copernican system), [when such a

doctrine] appears in the history of science, there is at once a tendency to exaggerate it

into a boundless subjectivism. . . . Kant's doctrine of the unknowability of the things in

themselves is one example. ..."

Godel, however, was not through with Kant. In an essay written in 1961 but never

published, he noted that it was "a general feature of Kant's assertions that literally

understood they are false, but in a broader sense contain deeper truths." He had in mind

Kant's doctrine

that in proving geometrical theorems we always need new geometrical intuitions. This,

Godel pointed out, is provably false. But if we substitute "mathematical" for "geometrical,"

the result is a truth that follows directly from Godel's incompleteness theorem. What was

needed, then, for the continual development of mathematics (and, one might add,

philosophy), was "a procedure that should produce in us a new state of consciousness in

which we describe in detail the basic concepts we use in our thought, or grasp other basic

concepts hitherto unknown to us." This he claimed to have found in the later

"transcendental phenomenology" of Edmund Husserl. "Transcendental phenomenology," he

wrote in a draft of a letter to the mathematician-philosopher Gian-Carlo Rota, "carried

through, would be nothing more nor less than Kant's critique of pure reason transformed

into an exact science," which "far from destroying traditional metaphysics . . . would

rather prove a solid foundation for it." In Husserl, Godel thought he had found a form of

idealism that, though derived from Kant's, was not incompatible with realism. That Husserl

shared Godel's disdain for unreconstructed Kantianism is apparent from a remark he made

in 1915: "German idealism has always made me want to throw up."

 

Einstein's objections, in turn, to the new quantum mechanics in particular his formulation

of the EPR paradox reflected a rejection of the Kantian turn in epistemology in its

simplified reconstruction by pos-itivists like Mach. The uncertainty principle, after all, is

an example of the same tendency to draw ontological conclusions from epistemologi-cal

premises, in this instance, from our inability in principle to know simultaneously the

position and velocity of a subatomic particle, to the nonexistence of such a combined

state. Not only did Einstein reject this reasoning, he resisted what he took to be

Heisenberg's more fundamental belief that we should abandon the very idea of "quantum

reality." For Einstein, as for Godel, philosophy without ontology was an illusion, and

physics without philosophy reduced to engineering. (And for Einstein, engineering was a

poor substitute for physics. When his eldest son, Hans, decided on an engineering career,

his father wrote that he was pleased that Hans had found a subject to concentrate on but

also that

"what he is interested in isn't really important, even if it is, alas, engineering. One cannot

expect one's children to inherit a mind.") :lolol:

 

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Volterovski oklevetani Sokrat zapadne filozofije: Gottfried Wilhelm von Leibniz

 

Gottfried_Wilhelm_von_Leibniz.jpg

 

Skrajnuti genije, po rečima Gedela ''zato da ga ljudi ne bi proučavali i naučili da misle''

 

Instant monadologija Lajbniza za neupućene:

 

 

Monadology

Gottfried Leibniz
1714
Squashed version edited by Glyn Hughes © 2011

1. The Monad, of which we shall here speak, is nothing but the simple substance, that which makes up all compounds. By 'simple' is meant 'without parts.'

3. These Monads are the real atoms of nature, which make up things.

4. Monads cannot fail. No simple substance can be destroyed by natural means.

5. Neither can any truly simple substance come into being by being formed from the combination of parts.

7. Monads have no window, through which anything could come in or go out. Neither substance nor accident can come into a Monad from outside.

8. Yet Monads must have some qualities, otherwise they would not exist.

9. Each Monad must be different from every other. For in nature there are never two beings perfectly alike.

10. Every Monad is continuously changing.

11. Thus changes in Monads must come from their internal principle, since nothing external can influence their inner being.

12. Besides the principle of the change, there must be a particular series of changes, which constitutes the specific nature of the simple substances.

13. Every change takes place over some period of time.

14. The brief condition in which many things are represented within the simple substance may be called Perception, which is dimmer than Apperception or Consciousness. Descartes is defective, for he treats as non-existent those perceptions of which we are not consciously aware. This has led many to believe that that there are no souls in animals. Like the uneducated crowd, they have confused a coma and death, and fallen into the old prejudices of souls entirely separate from bodies and of souls being mortal.

15. The internal principle which produces the change from one perception to another may be called Appetition.

17. Supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as a mill. But, on examining its interior, we should find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for.

18. All simple substances or created Monads might be called Entelechies, for they have in them a certain perfection (echousi to enteles); and a certain self-sufficiency (autarkeia) which makes them the sources of their internal activities and, so to speak, incorporeal automata.

19. If we are to give the name of Soul to everything which has perceptions and desires, then all simple substances or created Monads might be called souls; but as feeling is more than a bare perception, I think that the name of Monads or Entelechies should be given to simple substances which have perception only, and that the name of Souls should be given to those in which perception is accompanied by memory.

20. When we swoon or fall into dreamless sleep, our soul does not perceptibly differ from a bare Monad.

22. And as every present state of a simple substance is naturally a consequence of its preceding state, in such a way that its present is pregnant with its future;

23. And as, on waking from stupor, we are conscious of our perceptions, we must have had perceptions immediately before we awoke; for one perception can come only from another perception, as a motion can come only from motion.

25. We see also that nature has given heightened perceptions to animals, from the care she has taken to provide them with organs, which collect rays of light, or undulations of the air.

26. Memory provides the soul with a kind of consecutiveness. For instance, when a stick is shown to dogs, they remember the pain it has caused them, and howl and run away.

29. But it is the knowledge of necessary and eternal truths that distinguishes us from the mere animals and gives us Reason and the sciences, raising us to the knowledge of ourselves and of God. And it is this in us that is called the rational soul or mind.

30. It is also through the knowledge of necessary truths, and through their abstract expression, that we rise to acts of reflexion, which make us think of what is called I. And these acts of reflexion furnish the chief objects of our reasonings.

31. Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed to the false;

32. And that of sufficient reason, in virtue of which we hold that there can be no fact real or existing, no statement true, unless there be a sufficient reason

33. There are also two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible: truths of fact are contingent and their opposite is possible. When a truth is necessary, its reason can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary.

34. Thus in Mathematics speculative Theorems are reduced by analysis to Definitions, Axioms and Postulates.

35. In short, there are simple ideas or primary principles of which no definition can be given and which cannot be proved, and indeed have no need of proof; whose opposite involves an express contradiction.

38. Thus the final reason of things must be in a necessary substance, which we call God.

42. It follows also that created beings derive their perfections from the influence of God, but that their imperfections come from their own nature.

48. In God there is Power, Knowledge, whose content is the variety of the ideas, and Will, which makes changes or products according to the principle of the best.

49. A created thing is said to act on other things in so far as it has perfection, and to suffer to be itself acted upon in so far as it is imperfect. Thus activity is attributed to a Monad, in so far as it has distinct perceptions, and passivity in so far as its perceptions are confused.

50. And one created thing is more perfect than another, in this, that there is found in the more perfect that which serves to explain a priori what takes place in the less perfect, and it is on this account that the former is said to act upon the latter.

51. But in simple substances the influence of one Monad upon another can have its effect only through the mediation of God.

53. Now, as in the Ideas of God there is an infinite number of possible universes, and as only one of them can be actual, there must be a sufficient reason for God to decide upon one thing rather than another.

54. And this reason can be found only in the fitness or in the degrees of perfection.

56. Now the connexion of all created things to each and of each to all, means that each simple substance has relations which express all the others, and, consequently, that it is a perpetual living mirror of the universe.

57. And as the same town, looked at from various sides, appears quite different; so from the point of view of each Monad it is as if there were so many different universes.

58. And by this means there is obtained as great variety as possible, and as much perfection as possible.

61. All is a plenum (and thus all matter is connected together) and in the plenum every motion has an effect upon distant bodies in proportion to their distance, so that each body not only is affected by those which are in contact with it and in some way feels the effect of everything that happens to them, but also is affected by bodies adjoining itself. This inter-communication of things extends to any distance, however great. And consequently every body feels the effect of all that takes place in the universe, so that he who sees all might read in each what is happening everywhere, and even what has happened or shall happen, observing in the present that which is far off as well in time as in place. But a soul can read in itself only that which is there represented distinctly; it cannot all at once unroll everything.

62. Thus, although each created Monad represents the whole universe, it represents more distinctly the body which specially pertains to it, and of which it is the entelechy.

63. The Monad which is the entelechy or soul of a living body is, like every Monad, a mirror of the universe, and as the universe is ruled according to a perfect order, there must also be an extent to which that perfection is represented in the soul.

64. Thus the organic body of a living being is a kind of divine machine, which infinitely surpasses all artificial automata. For a machine made by men is not a machine in each of its parts. But the machines of nature, living bodies, are still machines in their smallest parts ad infinitum. It is this that constitutes the difference between the skill of nature and craft skill, that is to say, between the divine art and ours.

65. Each portion of matter is not only divisible to infinity, as the ancients realised, but is actually sub-divided without end, of which each has some motion of its own.

66. Whence it appears that in the smallest particle of matter there is a world of creatures, living beings, animals, entelechies, souls.

67. Each portion of matter may be conceived as like a garden full of plants and like a pond full of fishes. But each branch of every plant, each member of every animal, each drop of its liquid parts is also some such garden or pond.

68. And though the earth and the air which are between the plants of the garden, or the water which is between the fish of the pond, be neither plant nor fish; yet they also contain plants and fishes, but mostly so minute as to be imperceptible to us.

69. Thus there is nothing fallow, nothing sterile, nothing dead in the universe, no chaos, no confusion save in appearance, somewhat as it might appear to be in a pond at a distance.

70. Hence it appears that each living body has a dominant entelechy, which in an animal is the soul; but the members of this living body are full of other living beings, plants, animals, each of which has also its dominant entelechy or soul.

73. It follows from this that there never is absolute birth [generation] nor complete death. What we call births [generations] are developments and growths, while what we call deaths are envelopments and diminutions.

74. Philosophers have been much perplexed about the origin of forms. But nowadays it is become known, through careful studies of plants and animals, that the organic bodies of nature are never products of chaos or putrefaction, but always come from seeds, in which there was undoubtedly some being already formed; and it is held that not only the organic body was already there before conception, but also a soul in this body, and, in short, the animal itself. Something like this is seen apart from birth, as when worms become flies and caterpillars become butterflies.

77. Thus it may be said that not only the soul, being a mirror of an indestructible universe, is indestructible, but also the animal itself.

78. These principles have given me a way of explaining the union or rather the mutual agreement [conformite] of the soul and the organic body. The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other in virtue of the pre-established harmony between all substances, since they are all representations of one and the same universe.

81. According to this system bodies act as if (to suppose the impossible) there were no souls, and souls act as if there were no bodies, and both act as if each influenced the other.

82. Thus, animals and souls come into being when the world begins and no more come to an end that the world does. The spermatic animalcules have merely ordinary souls; but when those which are chosen through conception, their sensuous souls are raised to the rank of reason.

83. Minds are also images of the Deity or Author of nature Himself, capable of knowing the system of the universe, and to some extent of imitating it, each being like a small divinity in its own sphere.

84. It is this that enables spirits [or minds- esprits] to enter into a kind of fellowship with God.

85. Whence the totality of all spirits must compose the City of God, that is to say, the most perfect State that is possible.

86. This City of God is a moral world in the natural world.

88. A result of this harmony is that things lead to grace by the very ways of nature, and that this globe, for instance, must be destroyed and renewed by natural means at the very time when the government of spirits requires it, for the punishment of some and the reward of others.

90. Under this perfect government no good action would be unrewarded and no bad one unpunished. This it is which leads wise and virtuous people to devote their energies to everything which appears in harmony with the presumptive or antecedent will of God, and yet makes them content with what God actually brings to pass by His secret will. If we could sufficiently understand the order of the universe, we should find that it exceeds all the desires of the wisest men, and that it is impossible to make it better than it is, not only as a whole and in general but also for ourselves in particular. It is this attachment to the Author of all which can alone make our happiness.

 

i još kraće objašnjenje:

 

 

The first and biggest problem about understanding Leibniz is to get a proper context. Some of your remarks suggest to me that you missed the point altogether, although I have to add at once that you’re not alone.

Par. 1 of the Monadology gives you an unambiguous clue. Monads are simple substances. Therefore they have no parts, are not divisible. There is no bargaining with this definition, accordingly you must resist thinking of them as things in any way whatever (cf. Par. 2-3).

You should now consider where in the world you find such monads? Exactly nowhere. Because individually they are nothings and don’t actually exist. Leibniz tells you, after all, that they ‘strive for existence’. Consequently they represents a potential for existence. In order to actualise this potential, they must aggregate. Monads can only exist as a plurality, more exactly: a collective.

They are not souls either. Your terms you have used, ‘basic souls’ and ‘inanimate monads’ are self-contradictions, as you should now see. Leibniz says quite plainly that you can think of them ‘like souls’ if you wish (having his Christian readers in mind, specifically Remond and the Prince of Savoy); but when he speaks of them philosophically, he calls them ‘entelechies’.

So what is this potential and how can it be actualised? Fundamentally a monad (aka entelechy) is a point of FORCE either passive or active. Its properties are appetition and perception which represent agency. These properties confer on it the perception of other monads as inert or striving entities, but necessarily ‘other’. The actualisation proceeds by monads with varying degrees of agency (from zero to full) combining and as a collective ‘mirroring’ the perceptions of all other monads. This mirroring can be conceived as the beginning of actual perception, depending on the quality of the collective as a whole. All collectives initially form secondary matter (mass); and now it depends on the preponderance of active or passive force whether such an aggregate forms material substances with minimal agency (e.g. rocks) or maximal agency (e.g. animal bodies). The former give off phenomena and are characterised by inertia (their agency is resistance), the latter turn into organic machines and require a highly developed ‘dominant monad’ to organise them as living existents.

This is probably very confusing, but if you read slowly and try to grasp each point in turn, you may find the answer to your question in there. Unfortunately you don’t get such a description from the Monadology. Readers of Leibniz have been misled for centuries into believing that this work ‘is’ the philosophy of Leibniz. That’s like saying you need only to read ‘The Tempest’ to understand everything about Shakespeare. Untrue in both cases.

So pars. 67-68 have to be understood in the context of the infinite continuity of the world in the large and the small. Monads being zero-dimensional points of force, they construct a cosmos of infinite dimensionality. When you look through a microscope (Leibniz is telling you) you see a whole world of life and matter which is like your world. If those creatures looked into a microscope they would see yet another world throbbing with life and matter. And so on to infinity. Yes: it is a kind of fractal world. Leibniz understood this long before we discovered real fractals and it is not far-fetched to see the Mandelbrot/Julia Set unfolding in that imagery.

 

Sam Lajbiz bi bio oduševljen ovom skraćenom verzijom, on je voleo priručnike for dummies, i sam ih je pravio za aristokratiju koju je podučavao raznoraznim stvarima.

Edited by slow

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