Simulacra and Simulation (French: Simulacres et Simulation) is a 1981 philosophical treatise by Jean Baudrillard seeking to interrogate the relationship among reality, symbols, and society.

Simulacra are copies that depict things that either had no reality to begin with, or that no longer have an original.^[1] Simulation is the imitation of the operation of a real-world process or system over time.^[2]

...The simulacrum is never that which conceals the truth—it is the truth which conceals that there is none. The simulacrum is true.^[3]

Simulacra and Simulation is most known for its discussion of symbols, signs, and how they relate to contemporaneity (simultaneous existences). Baudrillard claims that our current society has replaced all reality and meaning with symbols and signs, and that human experience is of a simulation of reality. Moreover, these simulacra are not merely mediations of reality, nor even deceptive mediations of reality; they are not based in a reality nor do they hide a reality, they simply hide that anything like reality is relevant to our current understanding of our lives. The simulacra that Baudrillard refers to are the significations and symbolism of culture and media that construct perceived reality, the acquired understanding by which our lives and shared existence is and are rendered legible; Baudrillard believed that society has become so saturated with these simulacra and our lives so saturated with the constructs of society that all meaning was being rendered meaningless by being infinitely mutable. Baudrillard called this phenomenon the "precession of simulacra".

"Simulacra and Simulation" breaks the sign-order into 4 stages:

1. The first stage is a faithful image/copy, where we believe, and it may even be correct, that a sign is a "reflection of a profound reality" (pg 6), this is a good appearance, in what Baudrillard called "the sacramental order".
2. The second stage is perversion of reality, this is where we come to believe the sign to be an unfaithful copy, which "masks and denatures" reality as an "evil appearance—it is of the order of maleficence". Here, signs and images do not faithfully reveal reality to us, but can hint at the existence of an obscure reality which the sign itself is incapable of encapsulating.
3. The third stage masks the absence of a profound reality, where the simulacrum pretends to be a faithful copy, but it is a copy with no original. Signs and images claim to represent something real, but no representation is taking place and arbitrary images are merely suggested as things which they have no relationship to. Baudrillard calls this the "order of sorcery", a regime of semantic algebra where all human meaning is conjured artificially to appear as a reference to the (increasingly) hermetic truth.
4. The fourth stage is pure simulation, in which the simulacrum has no relationship to any reality whatsoever. Here, signs merely reflect other signs and any claim to reality on the part of images or signs is only of the order of other such claims. This is a regime of total equivalency, where cultural products need no longer even pretend to be real in a naïve sense, because the experiences of consumers' lives are so predominantly artificial that even claims to reality are expected to be phrased in artificial, "hyperreal" terms. Any naïve pretension to reality as such is perceived as bereft of critical self-awareness, and thus as oversentimental.

Muhammad, in full Abū al-Qāsim Muḥammad ibn ʿAbd Allāh ibn ʿAbd al-Muṭṭalib ibn Hāshim   (born 570, Mecca, Arabia [now in Saudi Arabia]—died June 8, 632, Medina), founder of the religion of Islam, accepted by Muslims throughout the world as the last of the prophets of God.
Although his name is now invoked in reverence several billion times every day, Muhammad was the most-reviled figure in the history of the West from the 7th century until quite recent times. Because Muhammad is one of the most influential figures in history, his life, deeds, and thoughts have been debated by followers and opponents over the centuries, which makes a biography of him difficult to write. At every turn, both the Islamic understanding of Muhammad and the rationalist interpretation of him by Western scholars, which grew out of 18th- and 19th-century philosophies such as positivism, must be considered. Moreover, on the basis of both historical evidence and the Muslim understanding of Muhammad as the Prophet, a response must be fashioned to Christian polemical writings characterizing Muhammad as an apostate if not the Antichrist. These date back to the early Middle Ages and still influence to some degree the general Western conception of him. It is essential, therefore, both to examine the historical record—though not necessarily on the basis of secularist assumptions—and to make clear the Islamic understanding of Muhammad.

Methodology and terminology

Sources for the study of the Prophet

The sources for the study of Muhammad are multifarious and include, first and foremost, the Qurʾān (or Koran), the sacred scriptures of Islam. Although the Qurʾān is considered by Muslims to be the word of God and not of Muhammad, it nevertheless reveals the most essential aspects associated with Muhammad. There are also the sayings of Muhammad himself (Hadith) and accounts of his actions (Sunnah). Furthermore, there are biographies (sīrah) of him going back to the works of Ibn Isḥāq (c. 704–767) in the 9th-century recensions of Ibn Hishām and Yūnus ibn Bukayr. Works of sacred history by later writers such as al-Ṭabarī and al-Thaʿālibī also contain extensive biographies of Muhammad. Then there are the accounts of the maghāzī (“battles”) that determined the fate of the early Islamic community. The most important of these works is the Kitāb al-maghāzī of al-Wāqidī (747–823). The Kitāb al-ṭabaqāt al-kabīr of Ibn Saʿd (died 844/845) is another important source on the life of Muhammad, his companions, and later figures in Islamic history. Finally, there are oral traditions. Although usually discounted by positivist historians, oral tradition plays a major role in the Islamic understanding of Muhammad, just as it does in the Christian understanding of Jesus Christ or the Jewish understanding of Moses and the other ancient prophets of Israel.
Beyond these there are later Western works, many of which, from the 18th century onward, distanced themselves from the polemical histories of earlier Christian authors. These more historically oriented treatments, which generally reject the prophethood of Muhammad, are coloured by the Western philosophical and theological framework of their authors. Many of these studies reflect much historical research, and most pay more attention to human, social, economic, and political factors than to religious, theological, and spiritual matters. It was not until the latter part of the 20th century that Western authors combined rigorous scholarship as understood in the modern West with empathy toward the subject at hand and, especially, awareness of the religious and spiritual realities involved in the study of the life of the founder of a major world religion.

WO MAJOR ATTEMPTS have been made by Muslim theologians and thinkers to logically justify the cessation of even non-law-bearing prophets. The first relates to the issue of the need for a new teacher. The advent of a perfect teacher and a perfect book, it is argued, obviates the need of any other teacher to follow. Of course, if it can be proved that the presence of a perfect book and the appearance of a perfect teacher are sufficient guarantees against any future moral or spiritual decline, then there is no reason why, after this, another prophet should ever be raised again. Regrettably however, this proposition can neither be proved correct theoretically nor historically.

This contention is insupportable because the bringing of a book of law is not the only function performed by prophets. Prophethood is a thing of many splendours. After the death of a law-bearing prophet, the mere preservation of his book and his traditions cannot offer a sufficient substitute for prophethood itself. The case in point becomes amply clear when we examine the conduct of Muslims after the demise of the Holy Prophetsa. The progressive deterioration of Muslim society should be sufficient to prove this point. The difference between their moral status during the lifetime of the Holy Prophetsa and that of Muslims today defies comparison. The Book however remains the same perfect, unaltered, un-interpolated Book that it was fourteen hundred years ago.

HE SECOND justification in support of the Doctrine of Absolute Finality relates to the idea of the intellectual maturity of man. The chief proponent of this view is no less a person than 'Allamah Iqbal claimed by some to be the greatest Muslim thinker of modern times. This doctrine of maturity is based on the assumption that the Holy Quran was revealed at a time when man had finally reached the ultimate stage of his mental and intellectual maturity. As such, he stood in no further need of day-to-day guidance by any Divine personage as did his ancestors of earlier ages. A beautiful philosophy but how hollow and empty of substance it turns out to be under closer scrutiny. The very premise that man has matured enough to be able to draw his own conclusions and chart his own course of conduct from the principle teachings of a perfect religion is challengeable on many counts.

The view on the Prophets of God (Arabic: نبي) in Ahmadiyya Islam differs with that of Christianity, Orthodox Islam, Zoroastrianism, Judaism, and the Baha'i Faith. Unlike Orthodox Islam, the Ahmadiyya Muslim Community considers the term Messenger (rasul) and Prophet (nabi) as being different aspects of the same office of a Khalifatullah (Representative of God on Earth). According to Ahmadiyya belief, the terms encountered in the Qur'an to signify divinely appointed individuals, namely, Warner (Nazir), Prophet (Nabi), Messenger (Rasul), are generally synonymous. Ahmadis however categorise prophets as law-bearing ones and non-lawbearing ones.

Buber's main proposition is that we may address existence in two ways:

1. The attitude of the "I" towards an "It", towards an object that is separate in itself, which we either use or experience.
2. The attitude of the "I" towards "Thou", in a relationship in which the other is not separated by discrete bounds.

One of the major themes of the book is that human life finds its meaningfulness in relationships. In Buber's view, all of our relationships bring us ultimately into relationship with God, who is the Eternal Thou.
Buber explains that humans are defined by two word pairs: I-It and I-Thou.
The "It" of I-It refers to the world of experience and sensation. I-It describes entities as discrete objects drawn from a defined set (e.g., he, she or any other objective entity defined by what makes it measurably different from other entities). It can be said that "I" have as many distinct and different relationships with each "It" as there are "It"s in one's life. Fundamentally, "It" refers to the world as we experience it.
By contrast, the word pair I-Thou describes the world of relations. This is the "I" that does not objectify any "It" but rather acknowledges a living relationship. I-Thou relationships are sustained in the spirit and mind of an "I" for however long the feeling or idea of relationship is the dominant mode of perception. A person sitting next to a complete stranger on a park bench may enter into an "I-Thou" relationship with the stranger merely by beginning to think positively about people in general. The stranger is a person as well, and gets instantaneously drawn into a mental or spiritual relationship with the person whose positive thoughts necessarily include the stranger as a member of the set of persons about whom positive thoughts are directed. It is not necessary for the stranger to have any idea that he is being drawn into an "I-Thou" relationship for such a relationship to arise. But what is crucial to understand is the word pair "I-Thou" can refer to a relationship with a tree, the sky, or the park bench itself as much as it can refer to the relationship between two individuals. The essential character of "I-Thou" is the abandonment of the world of sensation, the melting of the between, so that the relationship with another "I" is foremost.
Buber's two notions of "I" require attachment of the word "I" to a word partner. The splitting into the individual terms "I" and "it" and "thou" is only for the purposes of analysis. Despite the separation of "I" from the "It" and "Thou" in this very sentence describing the relationship, there is to Buber's mind either an I-Thou or an I-It relationship. Every sentence that a person uses with "I" refers to the two pairs: "I-Thou" and "I-It", and likewise "I" is implicit in every sentence with "Thou" or "It". Each It is bounded by others and It can only exist through this attachment because for every object there is another object. Thou on the other hand, has no limitations. When "Thou" is spoken, the speaker has no thing (has nothing), hence, Thou is abstract; yet the speaker “takes his stand in relation”.
What does it mean to experience the world? One goes around the world extracting knowledge from the world in experiences betokened by "He", "She", and "It". One also has I-Thou relationships. Experience is all physical, but these relationships involve a great deal of spirituality. The twofold nature of the world means that our being in the world has two aspects: the aspect of experience, which is perceived as I-It, and the aspect of relation, which is perceived as I-Thou.

Examples

Buber uses an example of a tree and presents five separate relations:

1. Looking at the tree as a picture with the color and details through the aesthetic perception.
2. Identifying the tree as movement. The movement includes the flow of the juices through the veins of the tree, the breathing of the leaves, the roots sucking the water, the never-ending activities between the tree and earth and air, and the growth of the tree.
3. Categorizing the tree by its type; in other words, studying it.
4. Exercising the ability to look at something from a different perspective. “I can subdue its actual presence and form so sternly that I recognize it only as an expression of law”.
5. Interpreting the experience of the tree in mathematical terms.

Through all of these relations, the tree is still an object that occupies time and space and still has the characteristics that make it what it is.^[1]
If "Thou" is used in the context of an encounter with a human being, the human being is not He, She, or bound by anything. You do not experience the human being; rather you can only relate to him or her in the sacredness of the I-Thou relation. The I-Thou relationship cannot be explained; it simply is. Nothing can intervene in the I-Thou relationship. I-Thou is not a means to some object or goal, but a definitive relationship involving the whole being of each subject.
Like the I-Thou relation, love is a subject-to-subject relationship. Love is not a relation of subject to object, but rather a relation in which both members in the relationship are subjects and share the unity of being.
The ultimate Thou is God. In the I-Thou relation there are no barriers. This enables us to relate directly to God. God is ever-present in human consciousness, manifesting in music, literature, and other forms of culture. Inevitably, Thou is addressed as It, and the I-Thou relation becomes the being of the I-Thou relation. God is now spoken to directly not spoken about.
There is no world that disconnects one from God, a world of It alone, when I-Thou guides one’s actions. "One who truly meets the world goes out also to God." God is the worldwide relation to all relations.

I AND THOU

Martin Buber

General Info

Context

Summary

Important Terms

Philosophical Themes, Arguments, Ideas

Summary and Analysis

Book I, aphorisms 1–8: Basic Words and the Mode of Experience

Part I, aphorisms 9–19: Relation

Part I, aphorisms 19–22: Love and the Dialogical

Part I, aphorisms 23–29: Arguments for the Primacy of Relation

Part II, aphorisms 1–6: The It-World

Part II, aphorisms 6–8

Part II, aphorisms 9–13

Part III, aphorisms 1–4: Encountering the Eternal You

Part III, aphorisms 5–14: What Religion is Not

Part III, aphorisms 15–17: Revelation through Action

Study Tools

Important Quotations

Key Facts

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How to Cite This SparkNote

"The hour of departure has arrived, and we go our ways - I to die, and you to live. Which is better God only knows."

"Thou shouldst eat to live; not live to eat."

"Think not those faithful who praise all thy words and actions; but those who kindly reprove thy faults."


"Having the fewest wants, I am nearest to the gods."


"Regard your good name as the richest jewel you can possibly be possessed of - for credit is like fire; when once you have kindled it you may easily preserve it, but if you once extinguish it, you will find it an arduous task to rekindle it again ..."


"The shortest and surest way to live with honor in the world, is to be in reality what we would appear to be; and if we observe, we shall find, that all human virtues increase and strengthen themselves by the practice of them."


"Remember what is unbecoming to do is also unbecoming to speak of."


"Remember that there is nothing stable in human affairs; therefore avoid undue elation in prosperity, or undue depression in adversity."


"Get not your friends by bare compliments, but by giving them sensible tokens of your love."


"The only good is knowledge and the only evil is ignorance."


"Envy is the ulcer of the soul."


"The unexamined life is not worth living."


"One should never do wrong in return, nor mistreat any man, no matter how one has been mistreated by him."


"Let him that would move the world first move himself."


"It is not living that matters, but living rightly."


"If all misfortunes were laid in one common heap whence everyone must take an equal portion, most people would be contented to take their own and depart."


"If a man is proud of his wealth, he should not be praised until it is known how he employs it."


"I was really too honest a man to be a politician and live."


"If you cannot persuade your country you must do whatever it orders, and patiently submit to any punishment that it imposes."


"One must not even do wrong when one is wronged, which most people regard as the natural course."


"I cannot abandon the principles which I used to hold in the past simply because this accident has happened to me; they seem to me to be much as they were, and I respect and regard the same principles now as before. So unless we can find better principles on this occasion, ..."


"I only wish that ordinary people had an unlimited capacity for doing harm; then they might have an unlimited power for doing good, which would be a splendid thing, if it were so. Actually they have neither. They cannot make a man wise or stupid; they simply act at random."

"I decided that it was not wisdom that enabled poets to write their poetry, but a kind of instinct or inspiration, such as you find in seers and prophets who deliver all their sublime messages without knowing in the least what they mean."


"He is richest who is content with the least, for content is the wealth of nature."


"He is a man of courage who does not run away, but remains at his post and fights against the enemy."


"From the deepest desires often come the deadliest hate."


"False words are not only evil in themselves, but they infect the soul with evil."


"Employ your time in improving yourself by other men's writings, so that you shall gain easily what others have labored hard for."


"Death may be the greatest of all human blessings."


"By all means, marry. If you get a good wife, you'll become happy; if you get a bad one, you'll become a philosopher."


"Beware the barrenness of a busy life."


"Beauty is the bait which with delight allures man to enlarge his kind."


"Beauty is a short-lived tyranny."


"Be slow to fall into friendship; but when thou art in, continue firm and constant."


"Be as you wish to seem."

C
"As to marriage or celibacy, let a man take which course he will, he will be sure to repent."


"As for me, all I know is that I know nothing."


"An honest man is always a child."


"All men's souls are immortal, but the souls of the righteous are immortal and divine."

The predisposition to religious belief is an ineradicable part of human behavior. Mankind has produced 100,000 religions. It is an illusion to think that scientific humanism and learning will dispel religious belief. Men would rather believe than know... A kind of Darwinistic survival of the fittest has occurred with religions... The ecological principle called Gause's law holds that competition is maximal between species with identical needs... Even submission to secular religions such as Communism and guru cults involve willing subordination of the individual to the group. Religious practices confer biological advantage. The mechanisms of religion include (1) objectification (the reduction of reality to images and definitions that are easily understood and cannot be refuted), (2) commitment through faith (a kind of tribalism enacted through self-surrender), (3) and myth (the narratives that explain the tribe's favored position on the earth, often incorporating supernatural forces struggling for control, apocalypse, and millennium). The three great religion categories of today are Marxism, traditional religion, andscientific materialism... Though theology is not likely to survive as an independent intellectual discipline, religion will endure for a long time to come and will not be replaced by scientific materialism.

God and religion[edit]

On the question of God, Wilson has described his position as provisional deism.^[30] He has explained his faith as a trajectory away from traditional beliefs: "I drifted away from the church, not definitively agnostic or atheistic, just Baptist & Christian no more."^[22] Wilson argues that the belief in God and rituals of religion are products of evolution.^[31] He argues that they should not be rejected or dismissed, but further investigated by science to better understand their significance to human nature. In his book The Creation, Wilson suggests that scientists ought to "offer the hand of friendship" to religious leaders and build an alliance with them, stating that "Science and religion are two of the most potent forces on Earth and they should come together to save the creation."^[32]

By chronology[edit]

Earliest prehistory and undated[edit]

The controversy surrounding prehistoric or "primal" matriarchy began in reaction to the book by Bachofen, Mother Right: An Investigation of the Religious and Juridical Character of Matriarchy in the Ancient World, in 1861. Several generations of ethnologists were inspired by his pseudo-evolutionary theory of archaic matriarchy. Following him and Jane Ellen Harrison, several generations of scholars, usually arguing from known myths or oral traditions and examination of Neolithic female cult-figures, suggested that many ancient societies might have been matriarchal, or even that there existed a wide-ranging matriarchal society prior to the ancient cultures of which we are aware. According to Uwe Wesel, Bachofen's myth interpretations have proved to be untenable.^[90] The concept was further investigated by Lewis Morgan.^[91] Many researchers studied the phenomenon of matriarchy afterward, but the basis was laid by the classics of sociology. The notion of a "woman-centered" society was developed by Bachofen, whose three-volume Myth, Religion, and Mother Right (1861) impacted the way classicists such as Harrison, Arthur Evans, Walter Burkert, and James Mellaart^[92] looked at the evidence of matriarchal religion in pre-Hellenic societies.^[93]According to historian Susan Mann, as of 2000, "few scholars these days find ... [a "notion of a stage of primal matriarchy"] persuasive."^[94]

The following excerpts from Lewis Morgan's Ancient Society will explain the use of the terms: "In a work of vast research, Bachofen has collected and discussed the evidence of female authority, mother-right, and of female rule, gynecocracy."^{[page needed]} "Common lands and joint tillage would lead to joint-tenant houses and communism in living; so that gyneocracy seems to require for its creation, descent in the female line. Women thus entrenched in large households, supplied from common stores, in which their own gens so largely predominated in numbers, would produce the phenomena of mother right and gyneocracy, which Bachofen has detected and traced with the aid of fragments of history and of tradition."^{[page needed]}

Kurt Derungs is a non-academic author advocating an "anthropology of landscape" based on allegedly matriarchal traces in toponymy and folklore.^{[citation needed]}

A superorganism is an organism consisting of many organisms. This is usually meant to be a social unit of eusocial animals, where division of labour is highly specialised and where individuals are not able to survive by themselves for extended periods of time. Ants are the best-known example of such a superorganism, while the naked mole rat is a famous example of the eusocial mammal. The technical definition of a superorganism is "a collection of agents which can act in concert to produce phenomena governed by the collective,"^[1] phenomena being any activity "the hive wants" such as ants collecting food or bees choosing a new nest site.

The Gaia hypothesis of James Lovelock^[2] and the work of James Hutton, Vladimir Vernadsky and Guy Murchie, have suggested that the biosphere can be considered a superorganism. This view relates to Systems Theory and the dynamics of a complex system.

Superorganisms are important in cybernetics, particularly biocybernetics. They exhibit a form of "distributed intelligence," a system in which many individual agents with limited intelligence and information are able to pool resources to accomplish a goal beyond the capabilities of the individuals. Existence of such behavior in organisms has many implications for military and management applications, and is being actively researched.^[3]

"The Superorganism promises to be one of the most important scientific works published in this decade. Coming eighteen years after the publication of The Ants, this new volume expands our knowledge of the social insects (among them, ants, bees, wasps, and termites) and is based on remarkable research conducted mostly within the last two decades. These superorganisms—a tightly knit colony of individuals, formed by altruistic cooperation, complex communication, and division of labor—represent one of the basic stages of biological organization, midway between the organism and the entire species. The study of the superorganism, as the authors demonstrate, has led to important advances in our understanding of how the transitions between such levels have occurred in evolution and how life as a whole has progressed from simple to complex forms. Ultimately, this book provides a deep look into a part of the living world hitherto glimpsed by only a very few."

A Confession (Russian: Исповедь [Ispoved']) is a short work (confession) on the subject of melancholia, philosophy and religion by the acclaimed Russian novelist Leo Tolstoy. It was written in 1879 to 1880, when Tolstoy was of late-middle age.[1]

The book is a brief autobiographical story of the author's struggle with a mid-life existential crisis of melancholia. It describes his search for answers to the profound questions: "What will come of my life?" and "What is the meaning of life?", without answers to which life, for him, had become "impossible".

Tolstoy reflects on the arc of his philosophical life till then: His childhood abandonment of his Russian orthodox faith; His mastery of strength, will power, and reason; And how, after he had achieved tremendous financial success and social status, his life became meaningless.

After despairing of his attempts to find answers in science, philosophy, eastern wisdom, and his fellow men of letters, he "confesses" that he found the answer to his trouble in the deep orthodox religious convictions of ordinary citizens. He describes how he subdued his rational scepticism and abhorrence of "superstition within the Christian truths" in order to gain the peace of mind that he needed to furnish his "survival".

The term "a confession" was used because Tolstoy was acutely aware that he was going against the anti-religious / atheistic biases which predominated much of the secular Russian and European elite (by default of the shortcomings of orthodoxy).

In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism.

Background[edit]

The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s. In his biography of Kurt Gödel, John W. Dawson, Jr summarizes the result as follows:

"At issue in the sometimes bitter disputes was the relation of mathematics to logic, as well as fundamental questions of methodology, such as how quantifiers were to be construed, to what extent, if at all, nonconstructive methods were justified, and whether there were important connections to be made between syntactic and semantic notions." (Dawson 1997:48)

Dawson observes that "partisans of three principal philosophical positions took part in the debate" (ibid.) – the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his "school" of collaborators), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L.E.J. Brouwer.

The following sections will expand these disputes noted by Dawson.

Brief history of Brouwer and Intuitionism[edit]

Brouwer in effect founded the mathematical philosophy of intuitionism as a challenge to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others (cf. Kleene (1952), pp. 46–59). As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning.

In 1908:

"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).

"After completing his dissertation (1907: see Van Dalen), Brouwer made a conscious decision temporarily to keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert – the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict – admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid.).

In the later 1920s, Brouwer became involved in a public and demeaning controversy with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer)

Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.^[8]

A major force behind intuitionism was L.E.J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis within this framework.

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.

--------------------------

L. E. J. Brouwer

Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. The most celebrated result was his proof of the topological invariance of dimension. Among his further results, the Brouwer fixed point theorem is also well known. Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.

Brouwer in effect founded the mathematical philosophy of intuitionism as an opponent to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann,John von Neumann and others (cf. Kleene (1952), p. 46–59). As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics.^[6] It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning.

Brouwer was member of the Significs group. It formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group.

In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions.^[7][8][9] Brouwer then "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:

"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).

"After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert—the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict—admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism " (ibid).

He was combative for a young man. He was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

About his last years, Davis (2002) remarks:

"...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financial worries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house." (Davis, p. 100 quoting van Stigt. p. 110.)

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Do you agree... Ten Questions about Intuitionism

This website has as its goal to investigate the opinions of the mathematical community on intuitionism. To this end it presents ten slightly provocative questions and collects answers to these questions from as many people as possible.

If you want to participate in this investigation, send your answers to the ten questions to answers@intuitionism.org. The web site also contains an annotated version of these questions which for each question presents a short explanation of how you might look at it.

Each question should be answered by yes or no or mu (= "the question is meaningless", or "it depends", or maybe, as Bas put it: "explaining how you should look at this question takes so much text that there is no point in doing so"; also use this if you do not know your answer to a question, or do not want to think about it), together with a possibly empty motivation. Try to have as little mus as possible. For each person the answers will be summarized on the main page as a string of +s, -s or #s.

Questions

Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?
Do you agree that the intermediate value theorem does not hold the way that it is normally stated?
Do you agree that there are only three infinite cardinalities?
Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?
Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?
Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?
Do you agree that for any two statements the first implies the second or the second implies the first?
Do you agree that a constructive proof of a theorem gives more insight than a classical proof?
Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?
Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?
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