Nikiya Anton Bettey 2021: 09/17/21

Friday, September 17, 2021

09-17-2021-0328 - Paradox Unsolved Problems Language Linguistics Philosophy Epistemology Observer Observable Universe

The Einstein–Podolsky–Rosen paradox (EPR paradox) is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided by quantum mechanics was incomplete.^[1] In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

The thought experiment involves a pair of particles prepared in an entangled state (note that this terminology was invented only later). Einstein, Podolsky, and Rosen pointed out that, in this state, if the position of the first particle were measured, the result of measuring the position of the second particle could be predicted. If instead the momentum of the first particle were measured, then the result of measuring the momentum of the second particle could be predicted. They argued that no action taken on the first particle could instantaneously affect the other, since this would involve information being transmitted faster than light, which is forbidden by the theory of relativity. They invoked a principle, later known as the "EPR criterion of reality", positing that, "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity". From this, they inferred that the second particle must have a definite value of position and of momentum prior to either being measured. This contradicted the view associated with Niels Bohr and Werner Heisenberg, according to which a quantum particle does not have a definite value of a property like momentum until the measurement takes place.

Locality has several different meanings in physics. EPR describe the principle of locality as asserting that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that energy can never be transmitted faster than the speed of light without violating causality;^[20]^{: 427–428}^[30] however, it turns out that the usual rules for combining quantum mechanical and classical descriptions violate EPR's principle of locality without violating special relativity or causality.^[20]^{: 427–428}^[30] Causality is preserved because there is no way for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, the no-cloning theorem, which makes it impossible for him to make an arbitrary number of copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "−", regardless of whether or not his axis is aligned with Alice's.

As a summary, the results of the EPR thought experiment do not contradict the predictions of special relativity. Neither the EPR paradox nor any quantum experiment demonstrates that superluminal signaling is possible; however, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance".^[b] The conclusion they drew was that quantum mechanics is not a complete theory.^[32]

Quine's classification[edit]

W. V. O. Quine (1962) distinguished between three classes of paradoxes:^[18]^[19]

According to Quine's classification of paradoxes:

A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless. The paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays had he been born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. Monty Hall paradox (or equivalently Three Prisoners problem) demonstrates that a decision that has an intuitive fifty–fifty chance is in fact heavily biased towards making a decision that, given the intuitive conclusion, the player would be unlikely to make. In 20th-century science, Hilbert's paradox of the Grand Hotel, Schrödinger's cat, Wigner's friend or Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end.
A falsidical paradox establishes a result that not only appears false but actually is false, due to a fallacy in the demonstration. The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples of this, often relying on a hidden division by zero. Another example is the inductive form of the horse paradox, which falsely generalises from true specific statements. Zeno's paradoxes are 'falsidical', concluding, for example, that a flying arrow never reaches its target or that a speedy runner cannot catch up to a tortoise with a small head-start. Therefore, falsidical paradoxes can be classified as fallacious arguments.
A paradox that is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling–Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.

A fourth kind, which may be alternatively interpreted as a special case of the third kind, has sometimes been described since Quine's work:

A paradox that is both true and false at the same time and in the same sense is called a dialetheia. In Western logics, it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditions (e.g. in the Mohists,^[20] the Gongsun Longzi,^[21] and in Zen^[22]) and in paraconsistent logics. It would be mere equivocation or a matter of degree, for example, to both affirm and deny that "John is here" when John is halfway through the door, but it is self-contradictory simultaneously to affirm and deny the event.

Ramsey's classification[edit]

Frank Ramsey drew a distinction between logical paradoxes and semantical paradoxes, with Russell’s paradox belonging to the former category, and Liar's paradox and Grelling’s paradoxes to the latter.^[23] Ramsey introduced the by-now standard distinction between logical and semantical contradictions. While logical contradictions involve mathematical or logical terms, like class, number, and hence show that our logic or mathematics is problematic, semantical contradictions involve, besides purely logical terms, notions like “thought”, “language”, “symbolism”, which, according to Ramsey, are empirical (not formal) terms. Hence these contradictions are due to faulty ideas about thought or language and they properly belong to “epistemology”(semantics). ^[24]

In philosophy[edit]

A taste for paradox is central to the philosophies of Laozi, Zeno of Elea, Zhuangzi, Heraclitus, Bhartrhari, Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes in the Philosophical Fragments that:

But one must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to will the collision, although in one way or another the collision must become its downfall. This, then, is the ultimate paradox of thought: to want to discover something that thought itself cannot think.^[25]

In medicine[edit]

A paradoxical reaction to a drug is the opposite of what one would expect, such as becoming agitated by a sedative or sedated by a stimulant. Some are common and are used regularly in medicine, such as the use of stimulants such as Adderall and Ritalin in the treatment of attention deficit hyperactivity disorder (also known as ADHD), while others are rare and can be dangerous as they are not expected, such as severe agitation from a benzodiazepine.^[26]

In the smoker's paradox, cigarette smoking, despite its proven harms, has a surprising inverse correlation with the epidemiological incidence of certain diseases.

Kantian[edit]

Immanuel Kant employs the term "amphiboly" in a sense of his own, as he has done in the case of other philosophical words. He denotes by it a confusion of the notions of the pure understanding with the perceptions of experience, and a consequent ascription to the latter of what belongs only to the former.^[18]

Formal semantics (natural language)

Central concepts

Compositionality Denotation Entailment Extension Generalized quantifier Intension Logical form Presupposition Proposition Reference Scope Speech act Syntax–semantics interface Truth conditions

Topics

Areas

Anaphora Ambiguity Binding Conditionals Definiteness Disjunction Evidentiality Focus Indexicality Lexical semantics Modality Negation Propositional attitudes Tense–aspect–mood Quantification Vagueness

Phenomena

Antecedent-contained deletion Cataphora Coercion Conservativity Counterfactuals Cumulativity De dicto and de re De se Deontic modality Discourse relations Donkey anaphora Epistemic modality Faultless disagreement Free choice inferences Givenness Crossover effects Hurford disjunction Inalienable possession Intersective modification Logophoricity Mirativity Modal subordination Negative polarity items Opaque contexts Performatives Privative adjectives Quantificational variability effect Responsive predicate Rising declaratives Scalar implicature Sloppy identity Subsective modification Telicity Temperature paradox Veridicality

Formalism

Formal systems

Alternative semantics Categorial grammar Combinatory categorial grammar Discourse representation theory Dynamic semantics Generative grammar Glue semantics Inquisitive semantics Intensional logic Lambda calculus Mereology Montague grammar Segmented discourse representation theory Situation semantics Supervaluationism Type theory TTR

Concepts

Autonomy of syntax Context set Continuation Conversational scoreboard Existential closure Function application Meaning postulate Monads Possible world Quantifier raising Quantization Question under discussion Squiggle operator Type shifter Universal grinder

Etymology[edit]

The term existence comes from Old French existence, from Medieval Latin existentia/exsistentia.^[3]

Context in philosophy[edit]

Materialism holds that the only things that exist are matter and energy, that all things are composed of material, that all actions require energy, and that all phenomena (including consciousness) are the result of the interaction of matter. Dialectical materialismdoes not make a distinction between being and existence, and defines it as the objective reality of various forms of matter.^[2]

Idealism holds that the only things that exist are thoughts and ideas, while the material world is secondary.^[4]^[5] In idealism, existence is sometimes contrasted with transcendence, the ability to go beyond the limits of existence.^[2] As a form of epistemological idealism, rationalism interprets existence as cognizable and rational, that all things are composed of strings of reasoning, requiring an associated idea of the thing, and all phenomena (including consciousness) are the result of an understanding of the imprint from the noumenal world in which lies beyond the thing-in-itself.

In scholasticism, existence of a thing is not derived from its essence but is determined by the creative volition of God, the dichotomy of existence and essence demonstrates that the dualism of the created universe is only resolvable through God.^[2] Empiricismrecognizes existence of singular facts, which are not derivable and which are observable through empirical experience.

The exact definition of existence is one of the most important and fundamental topics of ontology, the philosophical study of the nature of being, existence, or reality in general, as well as of the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology deals with questions concerning what things or entities exist or can be said to exist, and how such things or entities can be grouped, related within a hierarchy, and subdivided according to similarities and differences.

Anti-realism is the view of idealists who are skeptics about the physical world, maintaining either: (1) that nothing exists outside the mind, or (2) that we would have no access to a mind-independent reality even if it may exist. Realists, in contrast, hold that perceptions or sense data are caused by mind-independent objects. An "anti-realist" who denies that other minds exist (i. e., a solipsist) is different from an "anti-realist" who claims that there is no fact of the matter as to whether or not there are unobservable other minds (i. e., a logical behaviorist).

Subcategories

This category has the following 3 subcategories, out of 3 total.

Pages in category "Physical paradoxes"

The following 44 pages are in this category, out of 44 total. This list may not reflect recent changes (learn more).

Physical paradox

A

B

C

Carroll's paradox

D

Denny's paradox

E

F

G

H

K

L

M

N

Norton's dome

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Olbers' paradox

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R

Rietdijk–Putnam argument

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U

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Wigner's friend

Z

Categories: Paradoxes Philosophy of physics Thought experiments in physics

https://en.wikipedia.org/wiki/Category:Physical_paradoxes

https://en.wikipedia.org/wiki/EPR_paradox#Paradox

Formulated in 1862 by Lord Kelvin, Hermann von Helmholtz and William John Macquorn Rankine,^[1] the heat death paradox, also known as Clausius's paradox and thermodynamic paradox,^[2] is a reductio ad absurdum argument that uses thermodynamics to show the impossibility of an infinitely old universe.

Assuming that the universe is eternal, a question arises: How is it that thermodynamic equilibrium has not already been achieved?^[3]

This paradox is directed at the then-mainstream strand of belief in a classical view of a sempiternal universe whereby its matter is postulated as everlasting and having always been recognisably the universe. Clausius's paradox is one of paradigm. It was necessary to amend the fundamental cosmic ideas meaning change of the paradigm. The paradox was solved when the paradigm was changed.

The paradox was based upon the rigid mechanical point of view of the second law of thermodynamics postulated by Rudolf Clausiusaccording to which heat can only be transferred from a warmer to a colder object. It notes: if the universe were eternal, as claimed classically, it should already be cold and isotropic (its objects the same temperature).^[3]

Any hot object transfers heat to its cooler surroundings, until everything is at the same temperature. For two objects at the same temperature as much heat flows from one body as flows from the other, and the net effect is no change. If the universe were infinitely old, there must have been enough time for the stars to cool and warm their surroundings. Everywhere should therefore be at the same temperature and there should either be no stars, or everything should be as hot as stars.

Since there are stars and colder objects the universe is not in thermal equilibrium so it cannot be infinitely old.

The paradox does not arise in Big Bang nor modern steady state cosmology. In the former the universe is too young to have reached equilibrium; in the latter including the more nuanced quasi-steady state theory sufficient hydrogen is posited to have been replenished or regenerated continuously to allow constant average density. Star population depletion and reduction in temperature is slowed by the formation or coalescing of great stars which between certain masses and certain temperatures form supernovae remnant nebulae—such reincarnation postpones the heat death as does expansion of the universe.

Hydrostatic paradox[edit]

Wikimedia Commons has media related to Hydrostatic paradox.

Diagram illustrating the hydrostatic paradox

The barometric formula depends only on the height of the fluid chamber, and not on its width or length. Given a large enough height, any pressure may be attained. This feature of hydrostatics has been called the hydrostatic paradox. As expressed by W. H. Besant,^[2]

Any quantity of liquid, however small, may be made to support any weight, however large.

The Dutch scientist Simon Stevin was the first to explain the paradox mathematically.^[3] In 1916 Richard Glazebrook mentioned the hydrostatic paradox as he described an arrangement he attributed to Pascal: a heavy weight $W$ rests on a board with area $A$ resting on a fluid bladder connected to a vertical tube with cross-sectional area α. Pouring water of weight $w$ down the tube will eventually raise the heavy weight. Balance of forces leads to the equation

W={\frac {wA}{\alpha }}.

Glazebrook says, "By making the area of the board considerable and that of the tube small, a large weight $W$ can be supported by a small weight $w$ of water. This fact is sometimes described as the hydrostatic paradox."^[4]

Demonstrations of the hydrostatic paradox are used in teaching the phenomenon.^[5]^[6]

https://en.wikipedia.org/wiki/Vertical_pressure_variation#Hydrostatic_paradox

A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the density and gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite at the singularity point, the laws of normal spacetime break down.^[1]^[2]

https://en.wikipedia.org/wiki/Gravitational_singularity

Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle.^[1] As well as atoms and molecules, the empty space of the vacuum has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions (i.e., leptons and quarks), and force fields, whose quanta are bosons (e.g., photons and gluons). All these fields have zero-point energy.^[2] These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics^[1]^[3] since some systems can detect the existence of this energy. However, this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity.^[1]

https://en.wikipedia.org/wiki/Zero-point_energy

The quantum vacuum state or simply quantum vacuum refers to the quantum state with the lowest possible energy.

Quantum vacuum may also refer to:

Quantum chromodynamic vacuum (QCD vacuum), a non-perturbative vacuum
Quantum electrodynamic vacuum (QED vacuum), a field-theoretic vacuum
Quantum vacuum (ground state), the state of lowest energy of a quantum system
- Quantum vacuum collapse, a hypothetical vacuum metastability event
- Quantum vacuum expectation value, an operator's average, expected value in a quantum vacuum
Quantum vacuum energy, an underlying background energy that exists in space throughout the entire Universe
The Quantum Vacuum, a physics textbook by Peter W. Milonni

ee also[edit]

Zero-point energy, the minimum energy a quantum mechanical system may have
Zero-point field, a synonym for the vacuum state in quantum field theory
Hofstadter zero-point, a special point associated with every plane triangle
Point of origin (disambiguation)
Triple zero (disambiguation)
Point Zero (disambiguation)

https://en.wikipedia.org/wiki/Zero_point

MissingNo. (Japanese: けつばん^[1], Hepburn: Ketsuban), short for "Missing Number" and sometimes spelled without the period, is an unofficial Pokémon species found in the video games Pokémon Red and Blue. Due to the programming of certain in-game events, players can encounter MissingNo. via a glitch.

https://en.wikipedia.org/wiki/MissingNo.

In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting.^[1] They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in Clark Kimberling's Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992.^[1]

https://en.wikipedia.org/wiki/Hofstadter_points

In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. the word Zero-point field is sometimes used as a synonym for the vacuum state of an quantized field which is completely individual.

According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space".^[1]^[2] According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field.^[3]^[4]^[5]

The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, who jointly received the Nobel prize for this work in 1965.^[6] Today the electromagnetic interactions and the weak interactions are unified (at very high energies only) in the theory of the electroweak interaction.

The Standard Model is a generalization of the QED work to include all the known elementary particles and their interactions (except gravity). Quantum chromodynamics(or QCD) is the portion of the Standard Model that deals with strong interactions, and QCD vacuum is the vacuum of quantum chromodynamics. It is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions.^[7]

https://en.wikipedia.org/wiki/Quantum_vacuum_state

In philosophical epistemology, there are two types of coherentism: the coherence theory of truth;^[1] and the coherence theory of justification^[2] (also known as epistemic coherentism).^[3]

Coherent truth is divided between an anthropological approach, which applies only to localized networks ('true within a given sample of a population, given our understanding of the population'), and an approach that is judged on the basis of universals, such as categorical sets. The anthropological approach belongs more properly to the correspondence theory of truth, while the universal theories are a small development within analytic philosophy.

The coherentist theory of justification, which may be interpreted as relating to either theory of coherent truth, characterizes epistemic justification as a property of a belief only if that belief is a member of a coherent set. What distinguishes coherentism from other theories of justification is that the set is the primary bearer of justification.^[4]

As an epistemological theory, coherentism opposes dogmatic foundationalism and also infinitism through its insistence on definitions. It also attempts to offer a solution to the regress argument that plagues correspondence theory. In an epistemological sense, it is a theory about how belief can be proof-theoretically justified.

Coherentism is a view about the structure and system of knowledge, or else justified belief. The coherentist's thesis is normally formulated in terms of a denial of its contrary, such as dogmatic foundationalism, which lacks a proof-theoretical framework, or correspondence theory, which lacks universalism. Counterfactualism, through a vocabulary developed by David K. Lewis and his many worlds theory^[5]although popular with philosophers, has had the effect of creating wide disbelief of universals amongst academics. Many difficulties lie in between hypothetical coherence and its effective actualization. Coherentism claims, at a minimum, that not all knowledge and justified belief rest ultimately on a foundation of noninferential knowledge or justified belief. To defend this view, they may argue that conjunctions (and) are more specific, and thus in some way more defensible, than disjunctions (or).

After responding to foundationalism, coherentists normally characterize their view positively by replacing the foundationalism metaphor of a building as a model for the structure of knowledge with different metaphors, such as the metaphor that models our knowledge on a ship at sea whose seaworthiness must be ensured by repairs to any part in need of it. This metaphor fulfills the purpose of explaining the problem of incoherence, which was first raised in mathematics. Coherentists typically hold that justification is solely a function of some relationship between beliefs, none of which are privileged beliefs in the way maintained by dogmatic foundationalists. In this way universal truths are in closer reach. Different varieties of coherentism are individuated by the specific relationship between a system of knowledge and justified belief, which can be interpreted in terms of predicate logic, or ideally, proof theory.^[6]

https://en.wikipedia.org/wiki/Coherentism

In epistemology, the coherence theory of truth regards truth as coherence within some specified set of sentences, propositions or beliefs. The model is contrasted with the correspondence theory of truth.

A positive tenet is the idea that truth is a property of whole systems of propositions and can be ascribed to individual propositions only derivatively according to their coherence with the whole. While modern coherence theorists hold that there are many possible systems to which the determination of truth may be based upon coherence, others, particularly those with strong religious beliefs, hold that the truth only applies to a single absolute system. In general, truth requires a proper fit of elements within the whole system. Very often, though, coherence is taken to imply something more than simple formal coherence. For example, the coherence of the underlying set of concepts is considered to be a critical factor in judging validity. In other words, the set of base concepts in a universe of discourse must form an intelligible paradigm before many theorists consider that the coherence theory of truth is applicable.^{[citation needed]}

^{https://en.wikipedia.org/wiki/Coherence_theory_of_truth}

Justification (also called epistemic justification) is a concept in epistemology used to describe beliefs that one has good reason for holding.^[1] Epistemologists are concerned with various epistemic features of belief, which include the ideas of warrant(a proper justification for holding a belief), knowledge, rationality, and probability, among others. Loosely speaking, justification is the reason that someone holds a rationally admissible belief (although the term is also sometimes applied to other propositional attitudes such as doubt).

Debates surrounding epistemic justification often involve the structure of justification, including whether there are foundational justified beliefs or whether mere coherence is sufficient for a system of beliefs to qualify as justified. Another major subject of debate is the sources of justification, which might include perceptual experience (the evidence of the senses), reason, and authoritative testimony, among others.

https://en.wikipedia.org/wiki/Justification_(epistemology)

Proof theory is a major branch^[1] of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.

https://en.wikipedia.org/wiki/Proof_theory

Foundationalism concerns philosophical theories of knowledge resting upon justified belief, or some secure foundation of certainty such as a conclusion inferred from a basis of sound premises.^[1] The main rival of the foundationalist theory of justification is the coherence theory of justification, whereby a body of knowledge, not requiring a secure foundation, can be established by the interlocking strength of its components, like a puzzle solved without prior certainty that each small region was solved correctly.^[1]

https://en.wikipedia.org/wiki/Foundationalism

Infinitism is the view that knowledge may be justified by an infinite chain of reasons. It belongs to epistemology, the branch of philosophy that considers the possibility, nature, and means of knowledge.

https://en.wikipedia.org/wiki/Infinitism

In epistemology, the regress argument is the argument that any proposition requires a justification. However, any justification itself requires support. This means that any proposition whatsoever can be endlessly (infinitely) questioned, resulting in infinite regress. It is a problem in epistemology and in any general situation where a statement has to be justified.^[1]^[2]^[3]

The argument is also known as diallelus^[4] (Latin) or diallelon, from Greek di allelon"through or by means of one another" and as the epistemic regress problem. It is an element of the Münchhausen trilemma.^[5]

https://en.wikipedia.org/wiki/Regress_argument

In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world.^[1]

Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on one hand, and things or facts on the other.

https://en.wikipedia.org/wiki/Correspondence_theory_of_truth

Analytic philosophy is a branch and tradition of philosophy using analysis which is popular in the Western World and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United States, Canada, Australia, New Zealand and Scandinavia and continues today. Writing in 2003, John Searle claimed that "the best philosophy departments in the United States are dominated by analytic philosophy."^[1]

Central figures in this historical development of analytic philosophy are Gottlob Frege, Bertrand Russell, G. E. Moore, and Ludwig Wittgenstein. Other important figures in its history include the logical positivists (particularly Rudolf Carnap), W. V. O. Quine, Saul Kripke, and Karl Popper.

Analytic philosophy is characterized by an emphasis on language, known as the linguistic turn, and for its clarity and rigor in arguments, making use of formal logic and mathematics, and, to a lesser degree, the natural sciences.^[2]^[3]^[4] It also takes things piecemeal, in "an attempt to focus philosophical reflection on smaller problems that lead to answers to bigger questions."^[5]^[6]

Analytic philosophy is often understood in contrast to other philosophical traditions, most notably continental philosophies such as existentialism, phenomenology, and Hegelianism.^[7]

https://en.wikipedia.org/wiki/Analytic_philosophy

A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be scientific, belong to a non-scientific discipline, or no discipline at all. Depending on the context, a theory's assertions might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings.

In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific knowledge,^[1] in contrast to more common uses of the word "theory" that imply that something is unproven or speculative (which in formal terms is better characterized by the word hypothesis).^[2]Scientific theories are distinguished from hypotheses, which are individual empirically testable conjectures, and from scientific laws, which are descriptive accounts of the way nature behaves under certain conditions.

Theories guide the enterprise of finding facts rather than of reaching goals, and are neutral concerning alternatives among values.^[3]^: 131 A theory can be a body of knowledge, which may or may not be associated with particular explanatory models. To theorize is to develop this body of knowledge.^[4]^: 46

The word theory or "in theory" is sometimes used erroneously by people to explain something which they individually did not experience or test before.^[5] In those instances, semantically, it is being substituted for another concept, a hypothesis. Instead of using the word "hypothetically", it is replaced by a phrase: "in theory". In some instances the theory's credibility could be contested by calling it "just a theory" (implying that the idea has not even been tested).^[6] Hence, that word "theory" is very often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for doing, which is opposed to theory.^[6] A "classical example" of the distinction between "theoretical" and "practical" uses the discipline of medicine: medical theory involves trying to understand the causes and nature of health and sickness, while the practical side of medicine is trying to make people healthy. These two things are related but can be independent, because it is possible to research health and sickness without curing specific patients, and it is possible to cure a patient without knowing how the cure worked.^[a]

https://en.wikipedia.org/wiki/Theory

A scientific theory is an explanation of an aspect of the natural world and universe that has been repeatedly tested and verified in accordance with the scientific method, using accepted protocols of observation, measurement, and evaluation of results. Where possible, theories are tested under controlled conditions in an experiment.^[1]^[2] In circumstances not amenable to experimental testing, theories are evaluated through principles of abductive reasoning. Established scientific theories have withstood rigorous scrutiny and embody scientific knowledge.^[3]

A scientific theory differs from a scientific fact or scientific law in that a theory explains "why" or "how": a fact is a simple, basic observation, whereas a law is a statement (often a mathematical equation) about a relationship between facts. For example, Newton’s Law of Gravity is a mathematical equation that can be used to predict the attraction between bodies, but it is not a theory to explain how gravity works.^[4] Stephen Jay Gould wrote that "...facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts."^[5]

The meaning of the term scientific theory (often contracted to theory for brevity) as used in the disciplines of science is significantly different from the common vernacular usage of theory.^[6]^{[note 1]} In everyday speech, theory can imply an explanation that represents an unsubstantiated and speculative guess,^[6] whereas in science it describes an explanation that has been tested and is widely accepted as valid.^{[citation needed]}

The strength of a scientific theory is related to the diversity of phenomena it can explain and its simplicity. As additional scientific evidence is gathered, a scientific theory may be modified and ultimately rejected if it cannot be made to fit the new findings; in such circumstances, a more accurate theory is then required. Some theories are so well-established that they are unlikely ever to be fundamentally changed (for example, scientific theories such as evolution, heliocentric theory, cell theory, theory of plate tectonics, germ theory of disease, etc.). In certain cases, a scientific theory or scientific law that fails to fit all data can still be useful (due to its simplicity) as an approximation under specific conditions. An example is Newton's laws of motion, which are a highly accurate approximation to special relativity at velocities that are small relative to the speed of light.^{[citation needed]}

Scientific theories are testable and make falsifiable predictions.^[7] They describe the causes of a particular natural phenomenon and are used to explain and predict aspects of the physical universe or specific areas of inquiry (for example, electricity, chemistry, and astronomy). As with other forms of scientific knowledge, scientific theories are both deductive and inductive,^[8] aiming for predictiveand explanatory power. Scientists use theories to further scientific knowledge, as well as to facilitate advances in technology or medicine.

https://en.wikipedia.org/wiki/Scientific_theory

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Kripke.

https://en.wikipedia.org/wiki/Epistemic_modal_logic

In mathematics, a homogeneous binary relation R on a set X is reflexive if it relates every element of X to itself.^[1]^[2]

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

https://en.wikipedia.org/wiki/Reflexive_relation

Logical atomism is a philosophical view that originated in the early 20th century with the development of analytic philosophy. Its principal exponent was the British philosopher Bertrand Russell. It is also widely held that the early works^[a] of his Austrian-born pupil and colleague, Ludwig Wittgenstein, defend a version of logical atomism. Some philosophers in the Vienna Circle were also influenced by logical atomism (particularly Rudolf Carnap, who was deeply sympathetic to some of its philosophical aims, especially in his earlier works).^[1] Gustav Bergmann also developed a form of logical atomism that focused on an ideal phenomenalisticlanguage, particularly in his discussions of J.O. Urmson's work on analysis.^[2]

https://en.wikipedia.org/wiki/Logical_atomism

In Philosophy, logical holism is the belief that the world operates in such a way that no part can be known without the whole being known first.

Theoretical holism is a theory in philosophy of science, that a theory of science can only be understood in its entirety, introduced by Pierre Duhem. Different total theories of science are understood by making them commensurable allowing statements in one theory to be converted to sentences in another.^[1]^: II Richard Rorty argued that when two theories are incompatible a process of hermeneutics is necessary.^[1]^: II

Practical holism is a concept in the work of Martin Heidegger than posits it not possible to produce a complete understanding of one's own experience of reality because your mode of existence is embedded in cultural practices, the constraints of the task that you are doing.^[1]^: III

Bertrand Russell concluded that "Hegel's dialectical logical holism should be dismissed in favour of the new logic of propositional analysis."^[2] and introduced a form of logical atomism.^[3]

https://en.wikipedia.org/wiki/Logical_holism

The doctrine of internal relations is the philosophical doctrine that all relations are internal to their bearers, in the sense that they are essential to them and the bearers would not be what they are without them. It was a term used in British philosophy around in the early 1900s.^[1]^[2]

https://en.wikipedia.org/wiki/Doctrine_of_internal_relations

An a priori probability is a probability that is derived purely by deductive reasoning.^[1] One way of deriving a priori probabilities is the principle of indifference, which has the character of saying that, if there are N mutually exclusive and collectively exhaustiveevents and if they are equally likely, then the probability of a given event occurring is 1/N. Similarly the probability of one of a given collection of K events is K / N.

One disadvantage of defining probabilities in the above way is that it applies only to finite collections of events.

In Bayesian inference, "uninformative priors" or "objective priors" are particular choices of a priori probabilities.^[2] Note that "prior probability" is a broader concept.

Similar to the distinction in philosophy between a priori and a posteriori, in Bayesian inference a priori denotes general knowledge about the data distribution before making an inference, while a posteriori denotes knowledge that incorporates the results of making an inference.^[3]

https://en.wikipedia.org/wiki/A_priori_probability

An uninformative prior or diffuse prior expresses vague or general information about a variable. The term "uninformative prior" is somewhat of a misnomer. Such a prior might also be called a not very informative prior, or an objective prior, i.e. one that's not subjectively elicited.

Uninformative priors can express "objective" information such as "the variable is positive" or "the variable is less than some limit". The simplest and oldest rule for determining a non-informative prior is the principle of indifference, which assigns equal probabilities to all possibilities. In parameter estimation problems, the use of an uninformative prior typically yields results which are not too different from conventional statistical analysis, as the likelihood function often yields more information than the uninformative prior.

Some attempts have been made at finding a priori probabilities, i.e. probability distributions in some sense logically required by the nature of one's state of uncertainty; these are a subject of philosophical controversy, with Bayesians being roughly divided into two schools: "objective Bayesians", who believe such priors exist in many useful situations, and "subjective Bayesians" who believe that in practice priors usually represent subjective judgements of opinion that cannot be rigorously justified (Williamson 2010). Perhaps the strongest arguments for objective Bayesianism were given by Edwin T. Jaynes, based mainly on the consequences of symmetries and on the principle of maximum entropy.

https://en.wikipedia.org/wiki/Prior_probability#Uninformative_priors

In Roman mythology, Veritas (Classical Latin: [ˈweː.rɪ.t̪aːs]), meaning truth, is the goddess of truth, a daughter of Saturn (called Cronus by the Greeks, the Titan of Time, perhaps first by Plutarch), and the mother of Virtus. She is also sometimes considered the daughter of Jupiter (called Zeus by the Greeks),^[1] or a creation of Prometheus.^[2]^[3] The elusive goddess is said to have hidden in the bottom of a holy well.^[4] She is depicted both as a virgin dressed in white and as the "naked truth" (nuda veritas) holding a hand mirror.^[5]^[6]^[7]

Veritas is also the name given to the Roman virtue of truthfulness, which was considered one of the main virtues any good Roman should possess. The Greek goddess of truth is Aletheia (Ancient Greek: Ἀλήθεια). The German philosopher Martin Heidegger argues that the truth represented by aletheia (which essentially means "unconcealment") is different from that represented by veritas, which is linked to a Roman understanding of rightness and finally to a Nietzschean sense of justice and a will to power.^[8]

In Western culture, the word may also serve as a motto.

Goddess of truth
Veritas depicted on the monument to Pope Alexander VII

Statue of Veritas outside the Supreme Court of Canada

https://en.wikipedia.org/wiki/Veritas

Saturn (Latin: Sāturnus [saːˈturnus]) was a god in ancient Roman religion, and a character in Roman mythology. He was described as a god of generation, dissolution, plenty, wealth, agriculture, periodic renewal and liberation. Saturn's mythological reign was depicted as a Golden Age of plenty and peace. After the Roman conquest of Greece, he was conflated with the Greek Titan Cronus. Saturn's consort was his sister Ops, with whom he fathered Jupiter, Neptune, Pluto, Juno, Ceres and Vesta.

Saturn was especially celebrated during the festival of Saturnalia each December, perhaps the most famous of the Roman festivals, a time of feasting, role reversals, free speech, gift-giving and revelry. The Temple of Saturn in the Roman Forum housed the state treasury and archives (aerarium) of the Roman Republic and the early Roman Empire. The planet Saturn and the day of the week Saturday are both named after and were associated with him.

https://en.wikipedia.org/wiki/Saturn_(mythology)

In ancient Roman religion, Ops or Opis (Latin: "Plenty") was a fertility deity and earth goddess of Sabine origin.

https://en.wikipedia.org/wiki/Ops

Via et veritas et vita (Classical Latin: [ˈwɪ.a ɛt ˈweːrɪtaːs ɛt ˈwiːta], Ecclesiastical Latin: [ˈvi.a et ˈveritas et ˈvita]) is a Latin phrase meaning "the way and the truth and the life". The words are taken from Vulgate version of John 14:6, and were spoken by Jesus Christ in reference to himself.

These words, and sometimes the asyndetic variant via veritas vita, have been used as the motto of various educational institutions and governments.

https://en.wikipedia.org/wiki/Via_et_veritas_et_vita

Asyndeton (UK: /æˈsɪndɪtən, ə-/, US: /əˈsɪndətɒn, ˌeɪ-/;^[1]^[2] from the Greek: ἀσύνδετον, "unconnected", sometimes called asyndetism) is a literary scheme in which one or several conjunctions are deliberately omitted from a series of related clauses.^[3]^[4]Examples include veni, vidi, vici and its English translation "I came, I saw, I conquered". Its use can have the effect of speeding up the rhythm of a passage and making a single idea more memorable. Asyndeton may be contrasted with syndeton (syndetic coordination) and polysyndeton, which describe the use of one or multiple coordinating conjunctions, respectively.

More generally, in grammar, an asyndetic coordination is a type of coordination in which no coordinating conjunction is present between the conjuncts.^[5]

Quickly, resolutely, he strode into the bank.

No coordinator is present here, but the conjoins are still coordinated.

https://en.wikipedia.org/wiki/Asyndeton

In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.

Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the posterior probability distribution, which is the conditional distribution of the uncertain quantity given the data.

Similarly, the prior probability of a random event or an uncertain proposition is the unconditional probability that is assigned before any relevant evidence is taken into account.

https://en.wikipedia.org/wiki/Prior_probability

Argument–deduction–proof distinctions originated with logic itself.^[1] Naturally, the terminology evolved.

An argument, more fully a premise–conclusion argument, is a two-part system composed of premises and conclusion. An argument is valid if and only if its conclusion is a consequence of its premises. Every premise set has infinitely many consequences each giving rise to a valid argument. Some consequences are obviously so, but most are not: most are hidden consequences. Most valid arguments are not yet known to be valid. To determine validity in non-obvious cases deductive reasoning is required. There is no deductive reasoning in an argument per se; such must come from the outside.

Every argument's conclusion is a premise of other arguments. The word constituent may be used for either a premise or conclusion. In the context of this article and in most classical contexts, all candidates for consideration as argument constituents fall under the category of truth-bearer: propositions, statements, sentences, judgments, etc.

https://en.wikipedia.org/wiki/Argument–deduction–proof_distinctions

Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.^[1]

Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.

Deductive reasoning ("top-down logic") contrasts with inductive reasoning ("bottom-up logic"): in deductive reasoning, a conclusion is reached reductively by applying general rules which hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) remains. In deductive reasoning there is no uncertainty.^[2] In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules resulting in a conclusion that has epistemic uncertainty.^[2]

The inductive reasoning is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.

Deductive reasoning differs from abductive reasoning by the direction of the reasoning relative to the conditionals. The idea of "deduction" popularized in Sherlock Holmes stories is technically abduction, rather than deductive reasoning. Deductive reasoninggoes in the same direction as that of the conditionals, whereas abductive reasoning goes in the direction contrary to that of the conditionals.

https://en.wikipedia.org/wiki/Deductive_reasoning

Abductive reasoning (also called abduction,^[1] abductive inference,^[1] or retroduction^[2]) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peircebeginning in the last third of the 19th century. It starts with an observation or set of observations and then seeks the simplest and most likely conclusion from the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as inference to the best explanation,^[3] although not all usages of the terms abduction and inference to the best explanation are exactly equivalent.^[4]^[5]

In the 1990s, as computing power grew, the fields of law,^[6] computer science, and artificial intelligenceresearch^[7] spurred renewed interest in the subject of abduction.^[8] Diagnostic expert systemsfrequently employ abduction.^[9]

https://en.wikipedia.org/wiki/Abductive_reasoning

The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter.^[1] The opposite of the closed-world assumption is the open-world assumption (OWA), stating that lack of knowledge does not imply falsity. Decisions on CWA vs. OWA determine the understanding of the actual semantics of a conceptual expression with the same notations of concepts. A successful formalization of natural language semantics usually cannot avoid an explicit revelation of whether the implicit logical backgrounds are based on CWA or OWA.

Negation as failure is related to the closed-world assumption, as it amounts to believing false every predicate that cannot be proved to be true.

https://en.wikipedia.org/wiki/Closed-world_assumption

Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive ${\mathrm {not}}~p$ (i.e. that $~p$ is assumed not to hold) from failure to derive $~p$ . Note that ${\mathrm {not}}~p$ can be different from the statement $\neg p$ of the logical negation of $~p$ , depending on the completeness of the inference algorithm and thus also on the formal logic system.

Negation as failure has been an important feature of logic programming since the earliest days of both Planner and Prolog. In Prolog, it is usually implemented using Prolog's extralogical constructs.

More generally, this kind of negation is known as weak negation,^[1]^[2] in contrast with the strong (i.e. explicit, provable) negation.

https://en.wikipedia.org/wiki/Negation_as_failure

A non-monotonic logic is a formal logic whose conclusion relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusion(s) based on further evidence.^[1] Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to a theory never produces a pruning of its set of conclusions. Intuitively, monotonicity indicates that learning a new piece of knowledge cannot reduce the set of what is known. A monotonic logic cannot handle various reasoning tasks such as reasoning by default (conclusions may be derived only because of lack of evidence of the contrary), abductive reasoning (conclusions are only deduced as most likely explanations), some important approaches to reasoning about knowledge (the ignorance of a conclusion must be retracted when the conclusion becomes known), and similarly, belief revision (new knowledge may contradict old beliefs).

https://en.wikipedia.org/wiki/Non-monotonic_logic

Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called monotonic logics in order to differentiate them from non-monotonic logics.

https://en.wikipedia.org/wiki/Monotonicity_of_entailment

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rulesclosely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.

Motivation[edit]

Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935.^[1] His proposals led to different notations such as Fitch-style calculus (or Fitch's diagrams) or Suppes' method for which Lemmon gave a variant called system L.

Natural deduction in its modern form was independently proposed by the German mathematician Gerhard Gentzen in 1934, in a dissertation delivered to the faculty of mathematical sciences of the University of Göttingen.^[2] The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper:

Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens".^[3]
(First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)

Gentzen was motivated by a desire to establish the consistency of number theory. He was unable to prove the main result required for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system, the sequent calculus, for which he proved the Hauptsatz both for classical and intuitionistic logic. In a series of seminars in 1961 and 1962 Prawitz gave a comprehensive summary of natural deduction calculi, and transported much of Gentzen's work with sequent calculi into the natural deduction framework. His 1965 monograph Natural deduction: a proof-theoretical study^[4] was to become a reference work on natural deduction, and included applications for modal and second-order logic.

In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly. The system presented in this article is a minor variation of Gentzen's or Prawitz's formulation, but with a closer adherence to Martin-Löf's description of logical judgments and connectives.^[5]

Judgments and propositions[edit]

A judgment is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it.^[6] Thus "it is raining" is a judgment, which is evident for the one who knows that it is actually raining; in this case one may readily find evidence for the judgment by looking outside the window or stepping out of the house. In mathematical logic however, evidence is often not as directly observable, but rather deduced from more basic evident judgments. The process of deduction is what constitutes a proof; in other words, a judgment is evident if one has a proof for it.

The most important judgments in logic are of the form "A is true". The letter A stands for any expression representing a proposition; the truth judgments thus require a more primitive judgment: "A is a proposition". Many other judgments have been studied; for example, "A is false" (see classical logic), "A is true at time t" (see temporal logic), "A is necessarily true" or "A is possibly true" (see modal logic), "the program M has type τ" (see programming languages and type theory), "A is achievable from the available resources" (see linear logic), and many others. To start with, we shall concern ourselves with the simplest two judgments "A is a proposition" and "A is true", abbreviated as "A prop" and "A true" respectively.

The judgment "A prop" defines the structure of valid proofs of A, which in turn defines the structure of propositions. For this reason, the inference rules for this judgment are sometimes known as formation rules. To illustrate, if we have two propositions A and B (that is, the judgments "A prop" and "B prop" are evident), then we form the compound proposition A and B, written symbolically as " $A\wedge B$ ". We can write this in the form of an inference rule:

{\frac {A{\hbox{ prop}}\qquad B{\hbox{ prop}}}{(A\wedge B){\hbox{ prop}}}}\ \wedge _{F}

where the parentheses are omitted to make the inference rule more succinct:

{\frac {A{\hbox{ prop}}\qquad B{\hbox{ prop}}}{A\wedge B{\hbox{ prop}}}}\ \wedge _{F}

This inference rule is schematic: A and B can be instantiated with any expression. The general form of an inference rule is:

{\frac {J_{1}\qquad J_{2}\qquad \cdots \qquad J_{n}}{J}}\ {\hbox{name}}

where each $J_{i}$ is a judgment and the inference rule is named "name". The judgments above the line are known as premises, and those below the line are conclusions. Other common logical propositions are disjunction ( $A\vee B$ ), negation ( $\neg A$ ), implication ( $A\supset B$ ), and the logical constants truth ( $\top$ ) and falsehood ( $\bot$ ). Their formation rules are below.

{\frac {A{\hbox{ prop}}\qquad B{\hbox{ prop}}}{A\vee B{\hbox{ prop}}}}\ \vee _{F}\qquad {\frac {A{\hbox{ prop}}\qquad B{\hbox{ prop}}}{A\supset B{\hbox{ prop}}}}\ \supset _{F}

{\frac {\hbox{ }}{\top {\hbox{ prop}}}}\ \top _{F}\qquad {\frac {\hbox{ }}{\bot {\hbox{ prop}}}}\ \bot _{F}\qquad {\frac {A{\hbox{ prop}}}{\neg A{\hbox{ prop}}}}\ \neg _{F}

Introduction and elimination[edit]

Now we discuss the "A true" judgment. Inference rules that introduce a logical connective in the conclusion are known as introduction rules. To introduce conjunctions, i.e., to conclude "A and B true" for propositions A and B, one requires evidence for "Atrue" and "B true". As an inference rule:

${\frac {A{\hbox{ true}}\qquad B{\hbox{ true}}}{(A\wedge B){\hbox{ true}}}}\ \wedge _{I}$

It must be understood that in such rules the objects are propositions. That is, the above rule is really an abbreviation for:

${\frac {A{\hbox{ prop}}\qquad B{\hbox{ prop}}\qquad A{\hbox{ true}}\qquad B{\hbox{ true}}}{(A\wedge B){\hbox{ true}}}}\ \wedge _{I}$

This can also be written:

${\frac {A\wedge B{\hbox{ prop}}\qquad A{\hbox{ true}}\qquad B{\hbox{ true}}}{(A\wedge B){\hbox{ true}}}}\ \wedge _{I}$

In this form, the first premise can be satisfied by the $\wedge _{F}$ formation rule, giving the first two premises of the previous form. In this article we shall elide the "prop" judgments where they are understood. In the nullary case, one can derive truth from no premises.

${\frac {\ }{\top {\hbox{ true}}}}\ \top _{I}$

If the truth of a proposition can be established in more than one way, the corresponding connective has multiple introduction rules.

${\frac {A{\hbox{ true}}}{A\vee B{\hbox{ true}}}}\ \vee _{I1}\qquad {\frac {B{\hbox{ true}}}{A\vee B{\hbox{ true}}}}\ \vee _{I2}$

Note that in the nullary case, i.e., for falsehood, there are no introduction rules. Thus one can never infer falsehood from simpler judgments.

Dual to introduction rules are elimination rules to describe how to deconstruct information about a compound proposition into information about its constituents. Thus, from "A ∧ B true", we can conclude "A true" and "B true":

${\frac {A\wedge B{\hbox{ true}}}{A{\hbox{ true}}}}\ \wedge _{E1}\qquad {\frac {A\wedge B{\hbox{ true}}}{B{\hbox{ true}}}}\ \wedge _{E2}$

As an example of the use of inference rules, consider commutativity of conjunction. If A ∧ B is true, then B ∧ A is true; this derivation can be drawn by composing inference rules in such a fashion that premises of a lower inference match the conclusion of the next higher inference.

${\cfrac {{\cfrac {A\wedge B{\hbox{ true}}}{B{\hbox{ true}}}}\ \wedge _{E2}\qquad {\cfrac {A\wedge B{\hbox{ true}}}{A{\hbox{ true}}}}\ \wedge _{E1}}{B\wedge A{\hbox{ true}}}}\ \wedge _{I}$

The inference figures we have seen so far are not sufficient to state the rules of implication introduction or disjunction elimination; for these, we need a more general notion of hypothetical derivation.

Hypothetical derivations[edit]

A pervasive operation in mathematical logic is reasoning from assumptions. For example, consider the following derivation:

${\cfrac {A\wedge \left(B\wedge C\right){\hbox{ true}}}{{\cfrac {B\wedge C{\hbox{ true}}}{B{\hbox{ true}}}}\ \wedge _{E1}}}\ \wedge _{E2}$

This derivation does not establish the truth of B as such; rather, it establishes the following fact:

If A ∧ (B ∧ C) is true then B is true.

In logic, one says "assuming A ∧ (B ∧ C) is true, we show that B is true"; in other words, the judgment "B true" depends on the assumed judgment "A ∧ (B ∧ C) true". This is a hypothetical derivation, which we write as follows:

${\begin{matrix}A\wedge \left(B\wedge C\right){\hbox{ true}}\\\vdots \\B{\hbox{ true}}\end{matrix}}$

The interpretation is: "B true is derivable from A ∧ (B ∧ C) true". Of course, in this specific example we actually know the derivation of "B true" from "A ∧ (B ∧ C) true", but in general we may not a priori know the derivation. The general form of a hypothetical derivation is:

${\begin{matrix}D_{1}\quad D_{2}\ \cdots \ D_{n}\\\vdots \\J\end{matrix}}$

Each hypothetical derivation has a collection of antecedent derivations (the D_i) written on the top line, and a succedent judgment (J) written on the bottom line. Each of the premises may itself be a hypothetical derivation. (For simplicity, we treat a judgment as a premise-less derivation.)

The notion of hypothetical judgment is internalised as the connective of implication. The introduction and elimination rules are as follows.

${\cfrac {\begin{matrix}{\cfrac {}{A{\hbox{ true}}}}\ u\\\vdots \\B{\hbox{ true}}\end{matrix}}{A\supset B{\hbox{ true}}}}\ \supset _{I^{u}}\qquad {\cfrac {A\supset B{\hbox{ true}}\quad A{\hbox{ true}}}{B{\hbox{ true}}}}\ \supset _{E}$

In the introduction rule, the antecedent named u is discharged in the conclusion. This is a mechanism for delimiting the scope of the hypothesis: its sole reason for existence is to establish "B true"; it cannot be used for any other purpose, and in particular, it cannot be used below the introduction. As an example, consider the derivation of "A ⊃ (B ⊃ (A ∧ B)) true":

${\cfrac {{\cfrac {{\cfrac {}{A{\hbox{ true}}}}\ u\quad {\cfrac {}{B{\hbox{ true}}}}\ w}{A\wedge B{\hbox{ true}}}}\ \wedge _{I}}{{\cfrac {B\supset \left(A\wedge B\right){\hbox{ true}}}{A\supset \left(B\supset \left(A\wedge B\right)\right){\hbox{ true}}}}\ \supset _{I^{u}}}}\ \supset _{I^{w}}$

This full derivation has no unsatisfied premises; however, sub-derivations are hypothetical. For instance, the derivation of "B ⊃ (A ∧ B) true" is hypothetical with antecedent "A true" (named u).

With hypothetical derivations, we can now write the elimination rule for disjunction:

${\cfrac {A\vee B{\hbox{ true}}\quad {\begin{matrix}{\cfrac {}{A{\hbox{ true}}}}\ u\\\vdots \\C{\hbox{ true}}\end{matrix}}\quad {\begin{matrix}{\cfrac {}{B{\hbox{ true}}}}\ w\\\vdots \\C{\hbox{ true}}\end{matrix}}}{C{\hbox{ true}}}}\ \vee _{E^{u,w}}$

In words, if A ∨ B is true, and we can derive "C true" both from "A true" and from "B true", then C is indeed true. Note that this rule does not commit to either "A true" or "B true". In the zero-ary case, i.e. for falsehood, we obtain the following elimination rule:

${\frac {\perp {\hbox{ true}}}{C{\hbox{ true}}}}\ \perp _{E}$

This is read as: if falsehood is true, then any proposition C is true.

Negation is similar to implication.

${\cfrac {\begin{matrix}{\cfrac {}{A{\hbox{ true}}}}\ u\\\vdots \\p{\hbox{ true}}\end{matrix}}{\lnot A{\hbox{ true}}}}\ \lnot _{I^{u,p}}\qquad {\cfrac {\lnot A{\hbox{ true}}\quad A{\hbox{ true}}}{C{\hbox{ true}}}}\ \lnot _{E}$

The introduction rule discharges both the name of the hypothesis u, and the succedent p, i.e., the proposition p must not occur in the conclusion A. Since these rules are schematic, the interpretation of the introduction rule is: if from "A true" we can derive for every proposition p that "p true", then A must be false, i.e., "not A true". For the elimination, if both A and not A are shown to be true, then there is a contradiction, in which case every proposition C is true. Because the rules for implication and negation are so similar, it should be fairly easy to see that not A and A ⊃ ⊥ are equivalent, i.e., each is derivable from the other.

Consistency, completeness, and normal forms[edit]

A theory is said to be consistent if falsehood is not provable (from no assumptions) and is complete if every theorem or its negation is provable using the inference rules of the logic. These are statements about the entire logic, and are usually tied to some notion of a model. However, there are local notions of consistency and completeness that are purely syntactic checks on the inference rules, and require no appeals to models. The first of these is local consistency, also known as local reducibility, which says that any derivation containing an introduction of a connective followed immediately by its elimination can be turned into an equivalent derivation without this detour. It is a check on the strength of elimination rules: they must not be so strong that they include knowledge not already contained in their premises. As an example, consider conjunctions.

────── u   ────── w
A true     B true
────────────────── ∧I
    A ∧ B true
    ────────── ∧E₁
      A true

⇒

────── u
A true

Dually, local completeness says that the elimination rules are strong enough to decompose a connective into the forms suitable for its introduction rule. Again for conjunctions:

────────── u
A ∧ B true

⇒

────────── u    ────────── u
A ∧ B true      A ∧ B true
────────── ∧E₁  ────────── ∧E₂
  A true          B true
  ─────────────────────── ∧I
       A ∧ B true

These notions correspond exactly to β-reduction (beta reduction) and η-conversion (eta conversion) in the lambda calculus, using the Curry–Howard isomorphism. By local completeness, we see that every derivation can be converted to an equivalent derivation where the principal connective is introduced. In fact, if the entire derivation obeys this ordering of eliminations followed by introductions, then it is said to be normal. In a normal derivation all eliminations happen above introductions. In most logics, every derivation has an equivalent normal derivation, called a normal form. The existence of normal forms is generally hard to prove using natural deduction alone, though such accounts do exist in the literature, most notably by Dag Prawitz in 1961.^[7] It is much easier to show this indirectly by means of a cut-free sequent calculus presentation.

First and higher-order extensions[edit]

Summary of first-order system

The logic of the earlier section is an example of a single-sorted logic, i.e., a logic with a single kind of object: propositions. Many extensions of this simple framework have been proposed; in this section we will extend it with a second sort of individuals or terms. More precisely, we will add a new kind of judgment, "t is a term" (or "t term") where t is schematic. We shall fix a countable set V of variables, another countable set F of function symbols, and construct terms with the following formation rules:

{\frac {v\in V}{v{\hbox{ term}}}}{\hbox{ var}}_{F}

and

{\frac {f\in F\qquad t_{1}{\hbox{ term}}\qquad t_{2}{\hbox{ term}}\qquad \cdots \qquad t_{n}{\hbox{ term}}}{f(t_{1},t_{2},\cdots ,t_{n}){\hbox{ term}}}}{\hbox{ app}}_{F}

For propositions, we consider a third countable set P of predicates, and define atomic predicates over terms with the following formation rule:

{\frac {\phi \in P\qquad t_{1}{\hbox{ term}}\qquad t_{2}{\hbox{ term}}\qquad \cdots \qquad t_{n}{\hbox{ term}}}{\phi (t_{1},t_{2},\cdots ,t_{n}){\hbox{ prop}}}}{\hbox{ pred}}_{F}

The first two rules of formation provide a definition of a term that is effectively the same as that defined in term algebra and model theory, although the focus of those fields of study is quite different from natural deduction. The third rule of formation effectively defines an atomic formula, as in first-order logic, and again in model theory.

To these are added a pair of formation rules, defining the notation for quantified propositions; one for universal (∀) and existential (∃) quantification:

{\frac {x\in V\qquad A{\hbox{ prop}}}{\forall x.A{\hbox{ prop}}}}\;\forall _{F}\qquad \qquad {\frac {x\in V\qquad A{\hbox{ prop}}}{\exists x.A{\hbox{ prop}}}}\;\exists _{F}

The universal quantifier has the introduction and elimination rules:

{\begin{matrix}{\cfrac {}{a{\hbox{ term}}}}{\hbox{ u}}\\\vdots \\{\cfrac {[a/x]A{\hbox{ true}}}{\forall x.A{\hbox{ true}}}}\;\forall _{I^{u,a}}\end{matrix}}\qquad \qquad {\frac {\forall x.A{\hbox{ true}}\qquad t{\hbox{ term}}}{[t/x]A{\hbox{ true}}}}\;\forall _{E}

The existential quantifier has the introduction and elimination rules:

{\frac {[t/x]A{\hbox{ true}}}{\exists x.A{\hbox{ true}}}}\;\exists _{I}\qquad \qquad {\cfrac {{\begin{matrix}\\\\\\\\\exists x.A{\hbox{ true}}\\\end{matrix}}\qquad {\begin{matrix}{\cfrac {}{a{\hbox{ term}}}}{\hbox{ u}}\qquad {\cfrac {}{[a/x]A{\hbox{ true}}}}{\hbox{ v}}\\\vdots \\C{\hbox{ true}}\\\end{matrix}}}{C{\hbox{ true}}}}\exists _{E^{a,u,v}}

In these rules, the notation [t/x] A stands for the substitution of t for every (visible) instance of x in A, avoiding capture.^[8] As before the superscripts on the name stand for the components that are discharged: the term a cannot occur in the conclusion of ∀I (such terms are known as eigenvariables or parameters), and the hypotheses named u and v in ∃E are localised to the second premise in a hypothetical derivation. Although the propositional logic of earlier sections was decidable, adding the quantifiers makes the logic undecidable.

So far, the quantified extensions are first-order: they distinguish propositions from the kinds of objects quantified over. Higher-order logic takes a different approach and has only a single sort of propositions. The quantifiers have as the domain of quantification the very same sort of propositions, as reflected in the formation rules:

{\cfrac {\begin{matrix}{\cfrac {}{p{\hbox{ prop}}}}{\hbox{ u}}\\\vdots \\A{\hbox{ prop}}\\\end{matrix}}{\forall p.A{\hbox{ prop}}}}\;\forall _{F^{u}}\qquad \qquad {\cfrac {\begin{matrix}{\cfrac {}{p{\hbox{ prop}}}}{\hbox{ u}}\\\vdots \\A{\hbox{ prop}}\\\end{matrix}}{\exists p.A{\hbox{ prop}}}}\;\exists _{F^{u}}

A discussion of the introduction and elimination forms for higher-order logic is beyond the scope of this article. It is possible to be in-between first-order and higher-order logics. For example, second-order logic has two kinds of propositions, one kind quantifying over terms, and the second kind quantifying over propositions of the first kind.

Different presentations of natural deduction[edit]

Tree-like presentations[edit]

Gentzen's discharging annotations used to internalise hypothetical judgments can be avoided by representing proofs as a tree of sequents Γ ⊢A instead of a tree of A true judgments.

Sequential presentations[edit]

Jaśkowski's representations of natural deduction led to different notations such as Fitch-style calculus (or Fitch's diagrams) or Suppes' method, of which Lemmon gave a variant called system L. Such presentation systems, which are more accurately described as tabular, include the following.

1940: In a textbook, Quine^[9] indicated antecedent dependencies by line numbers in square brackets, anticipating Suppes' 1957 line-number notation.
1950: In a textbook, Quine (1982, pp. 241–255) demonstrated a method of using one or more asterisks to the left of each line of proof to indicate dependencies. This is equivalent to Kleene's vertical bars. (It is not totally clear if Quine's asterisk notation appeared in the original 1950 edition or was added in a later edition.)
1957: An introduction to practical logic theorem proving in a textbook by Suppes (1999, pp. 25–150). This indicated dependencies (i.e. antecedent propositions) by line numbers at the left of each line.
1963: Stoll (1979, pp. 183–190, 215–219) uses sets of line numbers to indicate antecedent dependencies of the lines of sequential logical arguments based on natural deduction inference rules.
1965: The entire textbook by Lemmon (1965) is an introduction to logic proofs using a method based on that of Suppes.
1967: In a textbook, Kleene (2002, pp. 50–58, 128–130) briefly demonstrated two kinds of practical logic proofs, one system using explicit quotations of antecedent propositions on the left of each line, the other system using vertical bar-lines on the left to indicate dependencies.^[10]

Proofs and type theory[edit]

The presentation of natural deduction so far has concentrated on the nature of propositions without giving a formal definition of a proof. To formalise the notion of proof, we alter the presentation of hypothetical derivations slightly. We label the antecedents with proof variables (from some countable set V of variables), and decorate the succedent with the actual proof. The antecedents or hypotheses are separated from the succedent by means of a turnstile (⊢). This modification sometimes goes under the name of localised hypotheses. The following diagram summarises the change.

──── u₁ ──── u₂ ... ──── u_n
 J₁      J₂          J_n
              ⋮
              J

⇒

u₁:J₁, u₂:J₂, ..., u_n:J_n ⊢ J

The collection of hypotheses will be written as Γ when their exact composition is not relevant. To make proofs explicit, we move from the proof-less judgment "A true" to a judgment: "π is a proof of (A true)", which is written symbolically as "π : A true". Following the standard approach, proofs are specified with their own formation rules for the judgment "π proof". The simplest possible proof is the use of a labelled hypothesis; in this case the evidence is the label itself.

u ∈ V
─────── proof-F
u proof

───────────────────── hyp
u:A true ⊢ u : A true

For brevity, we shall leave off the judgmental label true in the rest of this article, i.e., write "Γ ⊢ π : A". Let us re-examine some of the connectives with explicit proofs. For conjunction, we look at the introduction rule ∧I to discover the form of proofs of conjunction: they must be a pair of proofs of the two conjuncts. Thus:

π₁ proof    π₂ proof
──────────────────── pair-F
(π₁, π₂) proof

Γ ⊢ π₁ : A    Γ ⊢ π₂ : B
───────────────────────── ∧I
Γ ⊢ (π₁, π₂) : A ∧ B

The elimination rules ∧E₁ and ∧E₂ select either the left or the right conjunct; thus the proofs are a pair of projections—first (fst) and second (snd).

π proof
─────────── fst-F
fst π proof

Γ ⊢ π : A ∧ B
───────────── ∧E₁
Γ ⊢ fst π : A

π proof
─────────── snd-F
snd π proof

Γ ⊢ π : A ∧ B
───────────── ∧E₂
Γ ⊢ snd π : B

For implication, the introduction form localises or binds the hypothesis, written using a λ; this corresponds to the discharged label. In the rule, "Γ, u:A" stands for the collection of hypotheses Γ, together with the additional hypothesis u.

π proof
──────────── λ-F
λu. π proof

Γ, u:A ⊢ π : B
───────────────── ⊃I
Γ ⊢ λu. π : A ⊃ B

π₁ proof   π₂ proof
─────────────────── app-F
π₁ π₂ proof

Γ ⊢ π₁ : A ⊃ B    Γ ⊢ π₂ : A
──────────────────────────── ⊃E
Γ ⊢ π₁ π₂ : B

With proofs available explicitly, one can manipulate and reason about proofs. The key operation on proofs is the substitution of one proof for an assumption used in another proof. This is commonly known as a substitution theorem, and can be proved by inductionon the depth (or structure) of the second judgment.

Substitution theorem: If Γ ⊢ π₁ : A and Γ, u:A ⊢ π₂ : B, then Γ ⊢ [π₁/u] π₂ : B.

So far the judgment "Γ ⊢ π : A" has had a purely logical interpretation. In type theory, the logical view is exchanged for a more computational view of objects. Propositions in the logical interpretation are now viewed as types, and proofs as programs in the lambda calculus. Thus the interpretation of "π : A" is "the program π has type A". The logical connectives are also given a different reading: conjunction is viewed as product (×), implication as the function arrow (→), etc. The differences are only cosmetic, however. Type theory has a natural deduction presentation in terms of formation, introduction and elimination rules; in fact, the reader can easily reconstruct what is known as simple type theory from the previous sections.

The difference between logic and type theory is primarily a shift of focus from the types (propositions) to the programs (proofs). Type theory is chiefly interested in the convertibility or reducibility of programs. For every type, there are canonical programs of that type which are irreducible; these are known as canonical forms or values. If every program can be reduced to a canonical form, then the type theory is said to be normalising (or weakly normalising). If the canonical form is unique, then the theory is said to be strongly normalising. Normalisability is a rare feature of most non-trivial type theories, which is a big departure from the logical world. (Recall that almost every logical derivation has an equivalent normal derivation.) To sketch the reason: in type theories that admit recursive definitions, it is possible to write programs that never reduce to a value; such looping programs can generally be given any type. In particular, the looping program has type ⊥, although there is no logical proof of "⊥ true". For this reason, the propositions as types; proofs as programs paradigm only works in one direction, if at all: interpreting a type theory as a logic generally gives an inconsistent logic.

Example: Dependent Type Theory[edit]

Like logic, type theory has many extensions and variants, including first-order and higher-order versions. One branch, known as dependent type theory, is used in a number of computer-assisted proof systems. Dependent type theory allows quantifiers to range over programs themselves. These quantified types are written as Π and Σ instead of ∀ and ∃, and have the following formation rules:

Γ ⊢ A type    Γ, x:A ⊢ B type
───────────────────────────── Π-F
Γ ⊢ Πx:A. B type

Γ ⊢ A type  Γ, x:A ⊢ B type
──────────────────────────── Σ-F
Γ ⊢ Σx:A. B type

These types are generalisations of the arrow and product types, respectively, as witnessed by their introduction and elimination rules.

Γ, x:A ⊢ π : B
──────────────────── ΠI
Γ ⊢ λx. π : Πx:A. B

Γ ⊢ π₁ : Πx:A. B   Γ ⊢ π₂ : A
───────────────────────────── ΠE
Γ ⊢ π₁ π₂ : [π₂/x] B

Γ ⊢ π₁ : A    Γ, x:A ⊢ π₂ : B
───────────────────────────── ΣI
Γ ⊢ (π₁, π₂) : Σx:A. B

Γ ⊢ π : Σx:A. B
──────────────── ΣE₁
Γ ⊢ fst π : A

Γ ⊢ π : Σx:A. B
──────────────────────── ΣE₂
Γ ⊢ snd π : [fst π/x] B

Dependent type theory in full generality is very powerful: it is able to express almost any conceivable property of programs directly in the types of the program. This generality comes at a steep price — either typechecking is undecidable (extensional type theory), or extensional reasoning is more difficult (intensional type theory). For this reason, some dependent type theories do not allow quantification over arbitrary programs, but rather restrict to programs of a given decidable index domain, for example integers, strings, or linear programs.

Since dependent type theories allow types to depend on programs, a natural question to ask is whether it is possible for programs to depend on types, or any other combination. There are many kinds of answers to such questions. A popular approach in type theory is to allow programs to be quantified over types, also known as parametric polymorphism; of this there are two main kinds: if types and programs are kept separate, then one obtains a somewhat more well-behaved system called predicative polymorphism; if the distinction between program and type is blurred, one obtains the type-theoretic analogue of higher-order logic, also known as impredicative polymorphism. Various combinations of dependency and polymorphism have been considered in the literature, the most famous being the lambda cube of Henk Barendregt.

The intersection of logic and type theory is a vast and active research area. New logics are usually formalised in a general type theoretic setting, known as a logical framework. Popular modern logical frameworks such as the calculus of constructions and LFare based on higher-order dependent type theory, with various trade-offs in terms of decidability and expressive power. These logical frameworks are themselves always specified as natural deduction systems, which is a testament to the versatility of the natural deduction approach.

Classical and modal logics[edit]

For simplicity, the logics presented so far have been intuitionistic. Classical logic extends intuitionistic logic with an additional axiomor principle of excluded middle:

For any proposition p, the proposition p ∨ ¬p is true.

This statement is not obviously either an introduction or an elimination; indeed, it involves two distinct connectives. Gentzen's original treatment of excluded middle prescribed one of the following three (equivalent) formulations, which were already present in analogous forms in the systems of Hilbert and Heyting:

────────────── XM₁
A ∨ ¬A true

¬¬A true
────────── XM₂
A true

──────── u
¬A true
⋮
p true
────── XM₃^{u, p}
A true

(XM₃ is merely XM₂ expressed in terms of E.) This treatment of excluded middle, in addition to being objectionable from a purist's standpoint, introduces additional complications in the definition of normal forms.

A comparatively more satisfactory treatment of classical natural deduction in terms of introduction and elimination rules alone was first proposed by Parigot in 1992 in the form of a classical lambda calculus called λμ. The key insight of his approach was to replace a truth-centric judgment A true with a more classical notion, reminiscent of the sequent calculus: in localised form, instead of Γ ⊢ A, he used Γ ⊢ Δ, with Δ a collection of propositions similar to Γ. Γ was treated as a conjunction, and Δ as a disjunction. This structure is essentially lifted directly from classical sequent calculi, but the innovation in λμ was to give a computational meaning to classical natural deduction proofs in terms of a callcc or a throw/catch mechanism seen in LISP and its descendants. (See also: first class control.)

Another important extension was for modal and other logics that need more than just the basic judgment of truth. These were first described, for the alethic modal logics S4 and S5, in a natural deduction style by Prawitz in 1965,^[4] and have since accumulated a large body of related work. To give a simple example, the modal logic S4 requires one new judgment, "A valid", that is categorical with respect to truth:

If "A true" under no assumptions of the form "B true", then "A valid".

This categorical judgment is internalised as a unary connective ◻A (read "necessarily A") with the following introduction and elimination rules:

A valid
──────── ◻I
◻ A true

◻ A true
──────── ◻E
A true

Note that the premise "A valid" has no defining rules; instead, the categorical definition of validity is used in its place. This mode becomes clearer in the localised form when the hypotheses are explicit. We write "Ω;Γ ⊢ A true" where Γ contains the true hypotheses as before, and Ω contains valid hypotheses. On the right there is just a single judgment "A true"; validity is not needed here since "Ω ⊢ A valid" is by definition the same as "Ω;⋅ ⊢ A true". The introduction and elimination forms are then:

Ω;⋅ ⊢ π : A true
──────────────────── ◻I
Ω;⋅ ⊢ box π : ◻ A true

Ω;Γ ⊢ π : ◻ A true
────────────────────── ◻E
Ω;Γ ⊢ unbox π : A true

The modal hypotheses have their own version of the hypothesis rule and substitution theorem.

─────────────────────────────── valid-hyp
Ω, u: (A valid) ; Γ ⊢ u : A true

Modal substitution theorem: If Ω;⋅ ⊢ π₁ : A true and Ω, u: (A valid) ; Γ ⊢ π₂ : C true, then Ω;Γ ⊢ [π₁/u] π₂ : C true.

This framework of separating judgments into distinct collections of hypotheses, also known as multi-zoned or polyadic contexts, is very powerful and extensible; it has been applied for many different modal logics, and also for linear and other substructural logics, to give a few examples. However, relatively few systems of modal logic can be formalised directly in natural deduction. To give proof-theoretic characterisations of these systems, extensions such as labelling or systems of deep inference.

The addition of labels to formulae permits much finer control of the conditions under which rules apply, allowing the more flexible techniques of analytic tableaux to be applied, as has been done in the case of labelled deduction. Labels also allow the naming of worlds in Kripke semantics; Simpson (1993) presents an influential technique for converting frame conditions of modal logics in Kripke semantics into inference rules in a natural deduction formalisation of hybrid logic. Stouppa (2004) surveys the application of many proof theories, such as Avron and Pottinger's hypersequents and Belnap's display logic to such modal logics as S5 and B.

Comparison with other foundational approaches[edit]

Sequent calculus[edit]

The sequent calculus is the chief alternative to natural deduction as a foundation of mathematical logic. In natural deduction the flow of information is bi-directional: elimination rules flow information downwards by deconstruction, and introduction rules flow information upwards by assembly. Thus, a natural deduction proof does not have a purely bottom-up or top-down reading, making it unsuitable for automation in proof search. To address this fact, Gentzen in 1935 proposed his sequent calculus, though he initially intended it as a technical device for clarifying the consistency of predicate logic. Kleene, in his seminal 1952 book Introduction to Metamathematics, gave the first formulation of the sequent calculus in the modern style.^[11]

In the sequent calculus all inference rules have a purely bottom-up reading. Inference rules can apply to elements on both sides of the turnstile. (To differentiate from natural deduction, this article uses a double arrow ⇒ instead of the right tack ⊢ for sequents.) The introduction rules of natural deduction are viewed as right rules in the sequent calculus, and are structurally very similar. The elimination rules on the other hand turn into left rules in the sequent calculus. To give an example, consider disjunction; the right rules are familiar:

Γ ⇒ A
───────── ∨R₁
Γ ⇒ A ∨ B

Γ ⇒ B
───────── ∨R₂
Γ ⇒ A ∨ B

On the left:

Γ, u:A ⇒ C       Γ, v:B ⇒ C
─────────────────────────── ∨L
Γ, w: (A ∨ B) ⇒ C

Recall the ∨E rule of natural deduction in localised form:

Γ ⊢ A ∨ B    Γ, u:A ⊢ C    Γ, v:B ⊢ C
─────────────────────────────────────── ∨E
Γ ⊢ C

The proposition A ∨ B, which is the succedent of a premise in ∨E, turns into a hypothesis of the conclusion in the left rule ∨L. Thus, left rules can be seen as a sort of inverted elimination rule. This observation can be illustrated as follows:

natural deduction		sequent calculus
────── hyp \| \| elim. rules \| ↓ ────────────────────── ↑↓ meet ↑ \| \| intro. rules \| conclusion	⇒	─────────────────────────── init ↑ ↑ \| \| \| left rules \| right rules \| \| conclusion

natural deduction

sequent calculus

 ────── hyp
 |
 | elim. rules
 |
 ↓
 ────────────────────── ↑↓ meet
 ↑
 |
 | intro. rules
 |
 conclusion

⇒

 ─────────────────────────── init
 ↑            ↑
 |            |
 | left rules | right rules
 |            |
 conclusion

In the sequent calculus, the left and right rules are performed in lock-step until one reaches the initial sequent, which corresponds to the meeting point of elimination and introduction rules in natural deduction. These initial rules are superficially similar to the hypothesis rule of natural deduction, but in the sequent calculus they describe a transposition or a handshake of a left and a right proposition:

────────── init
Γ, u:A ⇒ A

The correspondence between the sequent calculus and natural deduction is a pair of soundness and completeness theorems, which are both provable by means of an inductive argument.

Soundness of ⇒ wrt. ⊢: If Γ ⇒ A, then Γ ⊢ A.
Completeness of ⇒ wrt. ⊢: If Γ ⊢ A, then Γ ⇒ A.

It is clear by these theorems that the sequent calculus does not change the notion of truth, because the same collection of propositions remain true. Thus, one can use the same proof objects as before in sequent calculus derivations. As an example, consider the conjunctions. The right rule is virtually identical to the introduction rule

sequent calculus		natural deduction
Γ ⇒ π₁ : A Γ ⇒ π₂ : B ─────────────────────────── ∧R Γ ⇒ (π₁, π₂) : A ∧ B		Γ ⊢ π₁ : A Γ ⊢ π₂ : B ───────────────────────── ∧I Γ ⊢ (π₁, π₂) : A ∧ B

sequent calculus

natural deduction

Γ ⇒ π₁ : A     Γ ⇒ π₂ : B
─────────────────────────── ∧R
Γ ⇒ (π₁, π₂) : A ∧ B

Γ ⊢ π₁ : A      Γ ⊢ π₂ : B
───────────────────────── ∧I
Γ ⊢ (π₁, π₂) : A ∧ B

The left rule, however, performs some additional substitutions that are not performed in the corresponding elimination rules.

sequent calculus		natural deduction
Γ, u:A ⇒ π : C ──────────────────────────────── ∧L₁ Γ, v: (A ∧ B) ⇒ [fst v/u] π : C		Γ ⊢ π : A ∧ B ───────────── ∧E₁ Γ ⊢ fst π : A
Γ, u:B ⇒ π : C ──────────────────────────────── ∧L₂ Γ, v: (A ∧ B) ⇒ [snd v/u] π : C		Γ ⊢ π : A ∧ B ───────────── ∧E₂ Γ ⊢ snd π : B

sequent calculus

natural deduction

Γ, u:A ⇒ π : C
──────────────────────────────── ∧L₁
Γ, v: (A ∧ B) ⇒ [fst v/u] π : C

Γ ⊢ π : A ∧ B
───────────── ∧E₁
Γ ⊢ fst π : A

Γ, u:B ⇒ π : C
──────────────────────────────── ∧L₂
Γ, v: (A ∧ B) ⇒ [snd v/u] π : C

Γ ⊢ π : A ∧ B
───────────── ∧E₂
Γ ⊢ snd π : B

The kinds of proofs generated in the sequent calculus are therefore rather different from those of natural deduction. The sequent calculus produces proofs in what is known as the β-normal η-long form, which corresponds to a canonical representation of the normal form of the natural deduction proof. If one attempts to describe these proofs using natural deduction itself, one obtains what is called the intercalation calculus (first described by John Byrnes), which can be used to formally define the notion of a normal formfor natural deduction.

The substitution theorem of natural deduction takes the form of a structural rule or structural theorem known as cut in the sequent calculus.

Cut (substitution): If Γ ⇒ π₁ : A and Γ, u:A ⇒ π₂ : C, then Γ ⇒ [π₁/u] π₂ : C.

In most well behaved logics, cut is unnecessary as an inference rule, though it remains provable as a meta-theorem; the superfluousness of the cut rule is usually presented as a computational process, known as cut elimination. This has an interesting application for natural deduction; usually it is extremely tedious to prove certain properties directly in natural deduction because of an unbounded number of cases. For example, consider showing that a given proposition is not provable in natural deduction. A simple inductive argument fails because of rules like ∨E or E which can introduce arbitrary propositions. However, we know that the sequent calculus is complete with respect to natural deduction, so it is enough to show this unprovability in the sequent calculus. Now, if cut is not available as an inference rule, then all sequent rules either introduce a connective on the right or the left, so the depth of a sequent derivation is fully bounded by the connectives in the final conclusion. Thus, showing unprovability is much easier, because there are only a finite number of cases to consider, and each case is composed entirely of sub-propositions of the conclusion. A simple instance of this is the global consistency theorem: "⋅ ⊢ ⊥ true" is not provable. In the sequent calculus version, this is manifestly true because there is no rule that can have "⋅ ⇒ ⊥" as a conclusion! Proof theorists often prefer to work on cut-free sequent calculus formulations because of such properties.

Notes[edit]

^ Jaśkowski 1934.
^ Gentzen 1934, Gentzen 1935.
^ Gentzen 1934, p. 176.
^ Jump up to: ^a ^b Prawitz 1965, Prawitz 2006.
^ Martin-Löf 1996.
^ This is due to Bolzano, as cited by Martin-Löf 1996, p. 15.
^ See also his book Prawitz 1965, Prawitz 2006.
^ See the article on lambda calculus for more detail about the concept of substitution.
^ Quine (1981). See particularly pages 91–93 for Quine's line-number notation for antecedent dependencies.
^ A particular advantage of Kleene's tabular natural deduction systems is that he proves the validity of the inference rules for both propositional calculus and predicate calculus. See Kleene 2002, pp. 44–45, 118–119.
^ Kleene 2009, pp. 440–516. See also Kleene 1980.

References[edit]

Barker-Plummer, Dave; Barwise, Jon; Etchemendy, John (2011). Language Proof and Logic (2nd ed.). CSLI Publications. ISBN 978-1575866321.
Gallier, Jean (2005). "Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi". Retrieved 12 June 2014.
Gentzen, Gerhard Karl Erich (1934). "Untersuchungen über das logische Schließen. I". Mathematische Zeitschrift. 39 (2): 176–210. doi:10.1007/BF01201353. S2CID 121546341. (English translation Investigations into Logical Deduction in M. E. Szabo. The Collected Works of Gerhard Gentzen. North-Holland Publishing Company, 1969.)
Gentzen, Gerhard Karl Erich (1935). "Untersuchungen über das logische Schließen. II". Mathematische Zeitschrift. 39 (3): 405–431. doi:10.1007/bf01201363. S2CID 186239837.
Girard, Jean-Yves (1990). Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, England. Archived from the original on 2016-07-04. Retrieved 2006-04-20. Translated and with appendices by Paul Taylor and Yves Lafont.
Jaśkowski, Stanisław (1934). On the rules of suppositions in formal logic. Reprinted in Polish logic 1920–39, ed. Storrs McCall.
Kleene, Stephen Cole (1980) [1952]. Introduction to metamathematics (Eleventh ed.). North-Holland. ISBN 978-0-7204-2103-3.
Kleene, Stephen Cole (2009) [1952]. Introduction to metamathematics. Ishi Press International. ISBN 978-0-923891-57-2.
Kleene, Stephen Cole (2002) [1967]. Mathematical logic. Mineola, New York: Dover Publications. ISBN 978-0-486-42533-7.
Lemmon, Edward John (1965). Beginning logic. Thomas Nelson. ISBN 978-0-17-712040-4.
Martin-Löf, Per (1996). "On the meanings of the logical constants and the justifications of the logical laws" (PDF). Nordic Journal of Philosophical Logic. 1 (1): 11–60. Lecture notes to a short course at Università degli Studi di Siena, April 1983.
Pfenning, Frank; Davies, Rowan (2001). "A judgmental reconstruction of modal logic" (PDF). Mathematical Structures in Computer Science. 11 (4): 511–540. CiteSeerX 10.1.1.43.1611. doi:10.1017/S0960129501003322. S2CID 16467268.
Prawitz, Dag (1965). Natural deduction: A proof-theoretical study. Acta Universitatis Stockholmiensis, Stockholm studies in philosophy 3. Stockholm, Göteborg, Uppsala: Almqvist & Wicksell.
Prawitz, Dag (2006) [1965]. Natural deduction: A proof-theoretical study. Mineola, New York: Dover Publications. ISBN 978-0-486-44655-4.
Quine, Willard Van Orman (1981) [1940]. Mathematical logic (Revised ed.). Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-674-55451-1.
Quine, Willard Van Orman (1982) [1950]. Methods of logic (Fourth ed.). Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-674-57176-1.
Simpson, Alex (1993). The proof theory and semantics of intuitionistic modal logic (PDF). University of Edinburgh. PhD thesis.
Stoll, Robert Roth (1979) [1963]. Set Theory and Logic. Mineola, New York: Dover Publications. ISBN 978-0-486-63829-4.
Stouppa, Phiniki (2004). The Design of Modal Proof Theories: The Case of S5. University of Dresden. CiteSeerX 10.1.1.140.1858. MSc thesis.
Suppes, Patrick Colonel (1999) [1957]. Introduction to logic. Mineola, New York: Dover Publications. ISBN 978-0-486-40687-9.

External links[edit]

Laboreo, Daniel Clemente, "Introduction to natural deduction."
Domino On Acid. Natural deduction visualized as a game of dominoes.
Pelletier, Jeff, "A History of Natural Deduction and Elementary Logic Textbooks."
Levy, Michel, A Propositional Prover.

https://en.wikipedia.org/wiki/Natural_deduction

METHODS OF PROOF

Subcategories

This category has only the following subcategory.

Mathematical induction‎ (1 C, 8 P)

Pages in category "Methods of proof"

The following 11 pages are in this category, out of 11 total. This list may not reflect recent changes (learn more).

Analytic proof

Axiomatic system

Conditional proof

Counterexample

Method of analytic tableaux

Natural deduction

Proof by contradiction

Proof by contrapositive

Proof by exhaustion

Proof of impossibility

RecycleUnits

Categories:

https://en.wikipedia.org/wiki/Category:Methods_of_proof

In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem which was free from intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817).

Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter (Sebastik 2007). In proof theory, an analytic proof has come to mean a proof whose structure is simple in a special way, due to conditions on the kind of inferences that ensure none of them go beyond what is contained in the assumptions and what is demonstrated.

Structural proof theory[edit]

In proof theory, the notion of analytic proof provides the fundamental concept that brings out the similarities between a number of essentially distinct proof calculi, so defining the subfield of structural proof theory. There is no uncontroversial general definition of analytic proof, but for several proof calculi there is an accepted notion. For example:

In Gerhard Gentzen's natural deduction calculus the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule;
In Gentzen's sequent calculus the analytic proofs are those that do not use the cut rule.

However, it is possible to extend the inference rules of both calculi so that there are proofs that satisfy the condition but are not analytic. For example, a particularly tricky example of this is the analytic cut rule, used widely in the tableau method, which is a special case of the cut rule where the cut formula is a subformula of side formulae of the cut rule: a proof that contains an analytic cut is by virtue of that rule not analytic.

Furthermore, structural proof theories that are not analogous to Gentzen's theories have other notions of analytic proof. For example, the calculus of structures organises its inference rules into pairs, called the up fragment and the down fragment, and an analytic proof is one that only contains the down fragment.

References[edit]

Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen Gesellschaft der WissenschaftenVol. V, pp.225-48.
Pfenning (1984). Analytic and Non-analytic Proofs. In Proc. 7th International Conference on Automated Deduction.
Sebastik (2007). Bolzano's Logic. Entry in the Stanford Encyclopedia of Philosophy.

Categories:

https://en.wikipedia.org/wiki/Analytic_proof

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds.^[1] This is a method of direct proof. A proof by exhaustion typically contains two stages:

A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases.
A proof of each of the cases.

The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance.^[2] Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.^[3]

In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching.^{[citation needed]}

^{https://en.wikipedia.org/wiki/Proof_by_exhaustion}

^{Inductive reasoning is a method of reasoning in which a body of observations is synthesized to come up with a general principle.^[1]Inductive reasoning is distinct from deductive reasoning. If the premises are correct, the conclusion of a deductive argument is certain; in contrast, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.^[2]}

^{https://en.wikipedia.org/wiki/Inductive_reasoning#Enumerative_induction}

^{A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.}

^{https://en.wikipedia.org/wiki/Conditional_proof}

^{In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems.^[1] An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication.^[2] A formal proof is a complete rendition of a mathematical proof within a formal system.}

^{https://en.wikipedia.org/wiki/Axiomatic_system}

^{In proof theory, the semantic tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula. The tableau method can also determine the satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure for modal logics (Girle 2000).}

^{https://en.wikipedia.org/wiki/Method_of_analytic_tableaux}

In mathematics, a proof of impossibility, also known as negative proof, proof of an impossibility theorem, or negative result is a proof demonstrating that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general.^[1] Proofs of impossibility often put decades or centuries of work attempting to find a solution to rest. To prove that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory.^[2]Impossibility theorems are usually expressible as negative existential propositions, or universal propositions in logic (see universal quantification for more).

Perhaps one of the oldest proofs of impossibility is that of the irrationality of square root of 2, which shows that it is impossible to express the square root of 2 as a ratio of integers. Another famous proof of impossibility was the 1882 proof of Ferdinand von Lindemann, showing that the ancient problem of squaring the circle cannot be solved,^[3] because the number $π$ is transcendental(i.e., non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century.

A problem arising in the 16th century was that of creating a general formula using radicals expressing the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible,^[4] using concepts such as solvable groups from Galois theory—a new subfield of abstract algebra.

Among the most important proofs of impossibility of the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm at all, with the most famous one being the halting problem. Other similarly-related findings are those of the Gödel's incompleteness theorems, which uncovers some fundamental limitations in the provability of formal systems.^[5]

In computational complexity theory, techniques like relativization (see oracle machine) provide "weak" proofs of impossibility excluding certain proof techniques. Other techniques, such as proofs of completeness for a complexity class, provide evidence for the difficulty of problems, by showing them to be just as hard to solve as other known problems that have proved intractable.

https://en.wikipedia.org/wiki/Proof_of_impossibility

In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradictionused to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction.^[1]^[2] It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.^[3]^[4]

Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists— is incorrect: its correctness produces a contradiction.

An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction.

The earliest uses of the method of infinite descent appear in Euclid's Elements.^[3] A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.^[2]

The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations.^[4]^[5] Two typical examples are showing the non-solvability of the Diophantine equation r² + s⁴ = t⁴ and proving Fermat's theorem on sums of two squares, which states that an odd prime p can be expressed as a sum of two squares when p ≡ 1 (mod 4) (see proof). In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression).

In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doubling function for rational points on an elliptic curve E. The context is of a hypothetical non-trivial rational point on E. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits), so that a "halving" a point gives a rational with smaller terms. Since the terms are positive, they cannot decrease forever.

https://en.wikipedia.org/wiki/Proof_by_infinite_descent

In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction.^[1]^[2]^[3] More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true.^[4]

If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent.^[1] In which case, there may be multiple and more complex ways to structure the argument of the proof.

The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind.

https://en.wikipedia.org/wiki/Minimal_counterexample

Number theory[edit]

In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory and the study of L-functions. The structural result of Mordell, that the rational points on an elliptic curve E form a finitely-generated abelian group, used an infinite descent argument based on E/2E in Fermat's style.

To extend this to the case of an abelian variety A, André Weil had to make more explicit the way of quantifying the size of a solution, by means of a height function – a concept that became foundational. To show that A(Q)/2A(Q) is finite, which is certainly a necessary condition for the finite generation of the group A(Q) of rational points of A, one must do calculations in what later was recognised as Galois cohomology. In this way, abstractly-defined cohomology groups in the theory become identified with descentsin the tradition of Fermat. The Mordell–Weil theorem was at the start of what later became a very extensive theory.

Irrationality of √2[edit]

The proof that the square root of 2 (√2) is irrational (i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the ancient Greeks, and is perhaps the earliest known example of a proof by infinite descent. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.^[6]^[7]^[8] The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example Conway & Guy (1996).^[9]

The ancient Greeks, not having algebra, worked out a geometric proof by infinite descent (John Horton Conway presented another geometric proof by infinite descent that may be more accessible^[10]). The following is an algebraic proof along similar lines:

Suppose that √2 were rational. Then it could be written as

{\sqrt {2}}={\frac {p}{q}}

for two natural numbers, $p$ and $q$ . Then squaring would give

2={\frac {p^{2}}{q^{2}}},

2q^{2}=p^{2},

so 2 must divide p². Because 2 is a prime number, it must also divide p, by Euclid's lemma. So p = 2r, for some integer r.

But then,

2q^{2}=(2r)^{2}=4r^{2},

q^{2}=2r^{2},

which shows that 2 must divide q as well. So q = 2s for some integer s.

This gives

{\frac {p}{q}}={\frac {2r}{2s}}={\frac {r}{s}}

Therefore, if √2 could be written as a rational number, then it could always be written as a rational number with smaller parts, which itself could be written with yet-smaller parts, ad infinitum. But this is impossible in the set of natural numbers. Since √2 is a real number, which can be either rational or irrational, the only option left is for √2 to be irrational.^[11]

(Alternatively, this proves that if √2 were rational, no "smallest" representation as a fraction could exist, as any attempt to find a "smallest" representation p/q would imply that a smaller one existed, which is a similar contradiction.)

https://en.wikipedia.org/wiki/Proof_by_infinite_descent

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."^[1]^{[note 1]} Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".^{[note 2]} (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence.^{[note 3]} In particular, arithmetical is commonly preferred as an adjective to number-theoretic.

https://en.wikipedia.org/wiki/Number_theory

In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and one important conjecture involving L-functions is the Riemann hypothesis and its generalization.

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between L-functions and the theory of prime numbers.

The Riemann zeta function can be thought of as the archetype for all L-functions.^[1]

https://en.wikipedia.org/wiki/L-function

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).

Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions, not just the special case of Dirichlet L-functions.)

https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function $\chi :\mathbb {Z} \rightarrow \mathbb {C}$ is a Dirichlet character of modulus $m$ (where $m$ is a positive integer) if for all integers $a$ and $b$ :^[1]

\chi (ab)=\chi (a)\chi (b);

i.e.

\chi

is completely multiplicative.

\chi (a){\begin{cases}=0&{\text{if }}\;\gcd(a,m)>1\\\neq 0&{\text{if }}\;\gcd(a,m)=1.\end{cases}}

\chi (a+m)=\chi (a)

; i.e.

\chi

is periodic with period

m

The simplest possible character, called the principal character, usually denoted $\chi _{0}$ , (see Notation below) exists for all moduli:^[2]

\chi _{0}(a)={\begin{cases}0&{\text{if }}\;\gcd(a,m)>1\\1&{\text{if }}\;\gcd(a,m)=1.\end{cases}}

Dirichlet introduced these functions in his 1837 paper on primes in arithmetic progressions.^[3]^[4]

https://en.wikipedia.org/wiki/Dirichlet_character

Generalized Riemann hypothesis (GRH)[edit]

The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884.^[1]Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.

The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by

L(\chi ,s)=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}

for every complex number s such that Re s > 1. By analytic continuation, this function can be extended to a meromorphic function(only when $\chi$ is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet character χ and every complex number s with L(χ, s) = 0, if s is not a negative real number, then the real part of s is 1/2.

The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.

Extended Riemann hypothesis (ERH)[edit]

Suppose K is a number field (a finite-dimensional field extension of the rationals Q) with ring of integers O_K (this ring is the integral closure of the integers Z in K). If a is an ideal of O_K, other than the zero ideal, we denote its norm by Na. The Dedekind zeta-function of K is then defined by

\zeta _{K}(s)=\sum _{a}{\frac {1}{(Na)^{s}}}

for every complex number s with real part > 1. The sum extends over all non-zero ideals a of O_K.

The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation to the whole complex plane. The resulting function encodes important information about the number field K. The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζ_K(s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2.

The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.

The ERH implies an effective version^[6] of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in Cis

{\frac {|C|}{|G|}}{\Bigl (}\operatorname {li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},

where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant.

The Artin conjecture[edit]

The Artin conjecture on Artin L-functions states that the Artin L-function $L(\rho ,s)$ of a non-trivial irreducible representation ρ is analytic in the whole complex plane.^[1]

This is known for one-dimensional representations, the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions.^[1] More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is supersolvable or more generally monomial, then all representations are of this form so the Artin conjecture holds.

André Weil proved the Artin conjecture in the case of function fields.

Two-dimensional representations are classified by the nature of the image subgroup: it may be cyclic, dihedral, tetrahedral, octahedral, or icosahedral. The Artin conjecture for the cyclic or dihedral case follows easily from Erich Hecke's work. Langlands used the base change lifting to prove the tetrahedral case, and Jerrold Tunnell extended his work to cover the octahedral case; Andrew Wiles used these cases in his proof of the Taniyama–Shimura conjecture. Richard Taylor and others have made some progress on the (non-solvable) icosahedral case; this is an active area of research. The Artin conjecture for odd, irreducible, two-dimensional representations follows from the proof of Serre's modularity conjecture, regardless of projective image subgroup.

Brauer's theorem on induced characters implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore meromorphic in the whole complex plane.

Langlands (1970) pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL(n) for all $n \geq 1$ . More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL_n(A_Q) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation. The Artin conjecture then follows immediately from the known fact that the L-functions of cuspidal automorphic representations are holomorphic. This was one of the major motivations for Langlands' work.

https://en.wikipedia.org/wiki/Artin_L-function#The_Artin_conjecture

In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.

https://en.wikipedia.org/wiki/Supersolvable_group

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element.^[1] That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.^[1]

Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.

Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.

https://en.wikipedia.org/wiki/Cyclic_group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

https://en.wikipedia.org/wiki/Solvable_group

Tuesday, September 14, 2021

09-14-2021-0302 - Ferredoxins redox iron
 Ferredoxins (from Latin ferrum: iron + redox, often abbreviated "fd") are iron–sulfur proteins that mediate electron transfer in a range of metabolic reactions. The term "ferredoxin" was coined by D.C. Wharton of the DuPont Co. and applied to the "iron protein" first purified in 1962 by Mortenson, Valentine, and Carnahan from the anaerobic bacterium Clostridium pasteurianum.[1][2]
Another redox protein, isolated from spinach chloroplasts, was termed "chloroplast ferredoxin".[3] The chloroplast ferredoxin is involved in both cyclic and non-cyclic photophosphorylation reactions of photosynthesis. In non-cyclic photophosphorylation, ferredoxin is the last electron acceptor thus reducing the enzyme NADP+ reductase. It accepts electrons produced from sunlight-excited chlorophyll and transfers them to the enzyme ferredoxin: NADP+ oxidoreductase EC 1.18.1.2.
Ferredoxins are small proteins containing iron and sulfur atoms organized as iron–sulfur clusters. These biological "capacitors" can accept or discharge electrons, with the effect of a change in the oxidation state of the iron atoms between +2 and +3. In this way, ferredoxin acts as an electron transfer agent in biological redox reactions.
Other bioinorganic electron transport systems include rubredoxins, cytochromes, blue copper proteins, and the structurally related Rieske proteins.
Ferredoxins can be classified according to the nature of their iron–sulfur clusters and by sequence similarity.
https://en.wikipedia.org/wiki/Ferredoxin
Friday, September 17, 202109-17-2021-0056 - Phantom Energy 
 Phantom energy is a hypothetical form of dark energy satisfying the equation of state with . It possesses negative kinetic energy, and predicts expansion of the universe in excess of that predicted by a cosmological constant, which leads to a Big Rip. The idea of phantom energy is often dismissed, as it would suggest that the vacuum is unstable with negative mass particles bursting into existence.[1] The concept is hence tied to emerging theories of a continuously-created negative mass dark fluid, in which the cosmological constant can vary as a function of time.[2][3]
Big Rip mechanism[edit]Main article: Big Rip
The existence of phantom energy could cause the expansion of the universe to accelerate so quickly that a scenario known as the Big Rip, a possible end to the universe, occurs. The expansion of the universe reaches an infinite degree in finite time, causing expansion to accelerate without bounds. This acceleration necessarily passes the speed of light (since it involves expansion of the universe itself, not particles moving within it), causing more and more objects to leave our observable universe faster than its expansion, as light and information emitted from distant stars and other cosmic sources cannot "catch up" with the expansion. As the observable universe expands, objects will be unable to interact with each other via fundamental forces, and eventually, the expansion will prevent any action of forces between any particles, even within atoms, "ripping apart" the universe, making distances between individual particles infinite.
One application of phantom energy in 2007 was to a cyclic model of the universe, which reverses its expansion extremely shortly before the would-be Big Rip.[4]
See also[edit]Quintessence (physics)
References[edit]^ Carroll, Sean (September 14, 2004). "Vacuum stability". Preposterous Universe Blog. Retrieved February 1, 2019.; however, note that the preceding source also contains a link to another blog post on the same blog (a post dated one day earlier!) that also discusses (a lot) the topic of "", etc.; therefore, [feel free to] "see also": Carroll, Sean (September 13, 2004). "Phantom Energy". "Preposterous Universe" [blog]. Archived from the original on October 28, 2014. Retrieved November 2, 2020.
^ Farnes, J. S. (2018). "A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified ΛCDM Framework". Astronomy and Astrophysics. 620: A92. arXiv:1712.07962. Bibcode:2018A&A...620A..92F. doi:10.1051/0004-6361/201832898. S2CID 53600834.
^ Farnes, Jamie (December 17, 2018). "Bizarre 'Dark Fluid' with Negative Mass Could Dominate the Universe".
^ Lauris Baum and Paul Frampton (2007). "Turnaround In Cyclic Cosmology". Phys. Rev. Lett. 98 (7): 071301. arXiv:hep-th/0610213. Bibcode:2007PhRvL..98g1301B. doi:10.1103/PhysRevLett.98.071301. PMID 17359014. S2CID 17698158.
Further reading[edit]Robert R. Caldwell et al.: Phantom Energy and Cosmic Doomsday
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https://en.wikipedia.org/wiki/Phantom_energy

https://en.wikipedia.org/wiki/Sfermion#Sleptons

https://en.wikipedia.org/wiki/Introduction_to_eigenstates

https://en.wikipedia.org/wiki/Spacetime_symmetries

https://en.wikipedia.org/wiki/Soliton

https://en.wikipedia.org/wiki/Replica_trick

https://en.wikipedia.org/wiki/Universe

https://www.livescience.com/strange-theories-about-the-universe.html

https://www.vox.com/2015/6/29/8847863/holographic-principle-universe-theory-physics

https://en.wikipedia.org/wiki/Holographic_principle

The holographic principle is a tenet of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region—such as a light-likeboundary like a gravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation by Leonard Susskind,^[1] who combined his ideas with previous ones of 't Hooft and Charles Thorn.^[1]^[2] As pointed out by Raphael Bousso,^[3] Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. The prime example of holography is the AdS/CFT correspondence.

The holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory.^[4]However, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called "Wheeler's bags of gold". The existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not fully understood yet.^[5]

https://en.wikipedia.org/wiki/Holographic_principle

In physics, black hole thermodynamics^[1] is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black-hole event horizons. As the study of the statistical mechanics of black-body radiation led to the development of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.^[2]

https://en.wikipedia.org/wiki/Black_hole_thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopicconstituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering and mechanical engineering, but also in other complex fields such as meteorology.

https://en.wikipedia.org/wiki/Thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measurements of various thermodynamic properties, but also the application of mathematical methods to the study of chemical questions and the spontaneity of processes.

The structure of chemical thermodynamics is based on the first two laws of thermodynamics. Starting from the first and second laws of thermodynamics, four equations called the "fundamental equations of Gibbs" can be derived. From these four, a multitude of equations, relating the thermodynamic properties of the thermodynamic system can be derived using relatively simple mathematics. This outlines the mathematical framework of chemical thermodynamics.^[1]

https://en.wikipedia.org/wiki/Chemical_thermodynamics

An object with relatively high entropy is microscopically random, like a hot gas. A known configuration of classical fields has zero entropy: there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.

But Jacob Bekenstein noted that this leads to a violation of the second law of thermodynamics. If one throws a hot gas with entropy into a black hole, once it crosses the event horizon, the entropy would disappear. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. One way of salvaging the second law is if black holes are in fact random objects with an entropy that increases by an amount greater than the entropy of the consumed gas.

Bekenstein assumed that black holes are maximum entropy objects—that they have more entropy than anything else in the same volume. In a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The only known limit is gravitational; when there is too much energy the gas collapses into a black hole. Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is directly proportional to the area of the event horizon.^[9] Gravitational time dilation causes time, from the perspective of a remote observer, to stop at the event horizon. Due to the natural limit on maximum speed of motion, this prevents falling objects from crossing the event horizon no matter how close they get to it. Since any change in quantum state requires time to flow, all objects and their quantum information state stay imprinted on the event horizon. Bekenstein concluded that from the perspective of any remote observer, the black hole entropy is directly proportional to the area of the event horizon.

Stephen Hawking had shown earlier that the total horizon area of a collection of black holes always increases with time. The horizon is a boundary defined by light-like geodesics; it is those light rays that are just barely unable to escape. If neighboring geodesics start moving toward each other they eventually collide, at which point their extension is inside the black hole. So the geodesics are always moving apart, and the number of geodesics which generate the boundary, the area of the horizon, always increases. Hawking's result was called the second law of black hole thermodynamics, by analogy with the law of entropy increase, but at first, he did not take the analogy too seriously.

Hawking knew that if the horizon area were an actual entropy, black holes would have to radiate. When heat is added to a thermal system, the change in entropy is the increase in mass-energy divided by temperature:

{\rm {d}}S={\frac {{\rm {\delta }}M\ c^{2}}{T}}.

(Here the term δM c² is substituted for the thermal energy added to the system, generally by non-integrable random processes, in contrast to dS, which is a function of a few "state variables" only, i.e. in conventional thermodynamics only of the Kelvin temperature T and a few additional state variables as, e.g., the pressure.)

If black holes have a finite entropy, they should also have a finite temperature. In particular, they would come to equilibrium with a thermal gas of photons. This means that black holes would not only absorb photons, but they would also have to emit them in the right amount to maintain detailed balance.

Time-independent solutions to field equations do not emit radiation, because a time-independent background conserves energy. Based on this principle, Hawking set out to show that black holes do not radiate. But, to his surprise, a careful analysis convinced him that they do, and in just the right way to come to equilibrium with a gas at a finite temperature. Hawking's calculation fixed the constant of proportionality at 1/4; the entropy of a black hole is one quarter its horizon area in Planck units.^[10]

The entropy is proportional to the logarithm of the number of microstates, the ways a system can be configured microscopically while leaving the macroscopic description unchanged. Black hole entropy is deeply puzzling – it says that the logarithm of the number of states of a black hole is proportional to the area of the horizon, not the volume in the interior.^[11]

Later, Raphael Bousso came up with a covariant version of the bound based upon null sheets.^[12]

Black hole information paradox[edit]

Hawking's calculation suggested that the radiation which black holes emit is not related in any way to the matter that they absorb. The outgoing light rays start exactly at the edge of the black hole and spend a long time near the horizon, while the infalling matter only reaches the horizon much later. The infalling and outgoing mass/energy interact only when they cross. It is implausible that the outgoing state would be completely determined by some tiny residual scattering.^{[citation needed]}

Hawking interpreted this to mean that when black holes absorb some photons in a pure state described by a wave function, they re-emit new photons in a thermal mixed state described by a density matrix. This would mean that quantum mechanics would have to be modified because, in quantum mechanics, states which are superpositions with probability amplitudes never become states which are probabilistic mixtures of different possibilities.^{[note 1]}

Troubled by this paradox, Gerard 't Hooft analyzed the emission of Hawking radiation in more detail.^[13]^{[self-published source?]} He noted that when Hawking radiation escapes, there is a way in which incoming particles can modify the outgoing particles. Their gravitational field would deform the horizon of the black hole, and the deformed horizon could produce different outgoing particles than the undeformed horizon. When a particle falls into a black hole, it is boosted relative to an outside observer, and its gravitational field assumes a universal form. 't Hooft showed that this field makes a logarithmic tent-pole shaped bump on the horizon of a black hole, and like a shadow, the bump is an alternative description of the particle's location and mass. For a four-dimensional spherical uncharged black hole, the deformation of the horizon is similar to the type of deformation which describes the emission and absorption of particles on a string-theory world sheet. Since the deformations on the surface are the only imprint of the incoming particle, and since these deformations would have to completely determine the outgoing particles, 't Hooft believed that the correct description of the black hole would be by some form of string theory.

This idea was made more precise by Leonard Susskind, who had also been developing holography, largely independently. Susskind argued that the oscillation of the horizon of a black hole is a complete description^{[note 2]} of both the infalling and outgoing matter, because the world-sheet theory of string theory was just such a holographic description. While short strings have zero entropy, he could identify long highly excited string states with ordinary black holes. This was a deep advance because it revealed that strings have a classical interpretation in terms of black holes.

This work showed that the black hole information paradox is resolved when quantum gravity is described in an unusual string-theoretic way assuming the string-theoretical description is complete, unambiguous and non-redundant.^[15] The space-time in quantum gravity would emerge as an effective description of the theory of oscillations of a lower-dimensional black-hole horizon, and suggest that any black hole with appropriate properties, not just strings, would serve as a basis for a description of string theory.

In 1995, Susskind, along with collaborators Tom Banks, Willy Fischler, and Stephen Shenker, presented a formulation of the new M-theory using a holographic description in terms of charged point black holes, the D0 branes of type IIA string theory. The matrix theory they proposed was first suggested as a description of two branes in 11-dimensional supergravity by Bernard de Wit, Jens Hoppe, and Hermann Nicolai. The later authors reinterpreted the same matrix models as a description of the dynamics of point black holes in particular limits. Holography allowed them to conclude that the dynamics of these black holes give a complete non-perturbative formulation of M-theory. In 1997, Juan Maldacena gave the first holographic descriptions of a higher-dimensional object, the 3+1-dimensional type IIB membrane, which resolved a long-standing problem of finding a string description which describes a gauge theory. These developments simultaneously explained how string theory is related to some forms of supersymmetric quantum field theories.

Information content is defined as the logarithm of the reciprocal of the probability that a system is in a specific microstate, and the information entropy of a system is the expected value of the system's information content. This definition of entropy is equivalent to the standard Gibbs entropy used in classical physics. Applying this definition to a physical system leads to the conclusion that, for a given energy in a given volume, there is an upper limit to the density of information (the Bekenstein bound) about the whereabouts of all the particles which compose matter in that volume. In particular, a given volume has an upper limit of information it can contain, at which it will collapse into a black hole.

This suggests that matter itself cannot be subdivided infinitely many times and there must be an ultimate level of fundamental particles. As the degrees of freedom of a particle are the product of all the degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles, the degrees of freedom of the original particle would be infinite, violating the maximal limit of entropy density. The holographic principle thus implies that the subdivisions must stop at some level.

The most rigorous realization of the holographic principle is the AdS/CFT correspondence by Juan Maldacena. However, J.D. Brown and Marc Henneaux had rigorously proved already in 1986, that the asymptotic symmetry of 2+1 dimensional gravity gives rise to a Virasoro algebra, whose corresponding quantum theory is a 2-dimensional conformal field theory.^[16]

Neutronium and the periodic table[edit]

The term "neutronium" was coined in 1926 by Andreas von Antropoff for a conjectured form of matter made up of neutrons with no protons or electrons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table.^[6] It was subsequently placed in the middle of several spiral representations of the periodic system for classifying the chemical elements, such as those of Charles Janet (1928), E. I. Emerson (1944), and John D. Clark (1950).

Although the term is not used in the scientific literature either for a condensed form of matter, or as an element, there have been reports that, besides the free neutron, there may exist two bound forms of neutrons without protons.^[7] If neutronium were considered to be an element, then these neutron clusters could be considered to be the isotopes of that element. However, these reports have not been further substantiated.

Mononeutron: An isolated neutron undergoes beta decay with a mean lifetime of approximately 15 minutes (half-life of approximately 10 minutes), becoming a proton (the nucleus of hydrogen), an electron and an antineutrino.
Dineutron: The dineutron, containing two neutrons, was unambiguously observed in 2012 in the decay of beryllium-16.^[8]^[9] It is not a bound particle, but had been proposed as an extremely short-lived resonance state produced by nuclear reactions involving tritium. It has been suggested to have a transitory existence in nuclear reactions produced by helions (helium 3 nuclei, completely ionised) that result in the formation of a proton and a nucleus having the same atomic number as the target nucleus but a mass number two units greater. The dineutron hypothesis had been used in nuclear reactions with exotic nuclei for a long time.^[10] Several applications of the dineutron in nuclear reactions can be found in review papers.^[11] Its existence has been proven to be relevant for nuclear structure of exotic nuclei.^[12] A system made up of only two neutrons is not bound, though the attraction between them is very nearly enough to make them so.^[13] This has some consequences on nucleosynthesis and the abundance of the chemical elements.^[11]^[14]
Trineutron: A trineutron state consisting of three bound neutrons has not been detected, and is not expected to exist^{[citation needed]}even for a short time.
Tetraneutron: A tetraneutron is a hypothetical particle consisting of four bound neutrons. Reports of its existence have not been replicated.^[15]
Pentaneutron: Calculations indicate that the hypothetical pentaneutron state, consisting of a cluster of five neutrons, would not be bound.^[16]

Although not called "neutronium", the National Nuclear Data Center's Nuclear Wallet Cards lists as its first "isotope" an "element" with the symbol n and atomic number Z = 0 and mass number A = 1. This isotope is described as decaying to element H with a half life of 10.24±0.2 min.^[17]

Properties[edit]

Neutron matter is equivalent to a chemical element with atomic number 0, which is to say that it is equivalent to a species of atoms having no protons in their atomic nuclei. It is extremely radioactive; its only legitimate equivalent isotope, the free neutron, has a half-life of 10 minutes, which is approximately half that of the most stable known isotope of francium. Neutron matter decays quickly into hydrogen. Neutron matter has no electronic structure on account of its total lack of electrons. As an equivalent element, however, it could be classified as a noble gas.

Bulk neutron matter has never been viewed. It is assumed that neutron matter would appear as a chemically inert gas, if enough could be collected together to be viewed as a bulk gas or liquid, because of the general appearance of the elements in the noble gas column of the periodic table.

While this lifetime is long enough to permit the study of neutronium's chemical properties, there are serious practical problems. Having no charge or electrons, neutronium would not interact strongly with ordinary low-energy photons (visible light) and would feel no electrostatic forces, so it would diffuse into the walls of most containers made of ordinary matter. Certain materials are able to resist diffusion or absorption of ultracold neutrons due to nuclear-quantum effects, specifically reflection caused by the strong interaction. At ambient temperature and in the presence of other elements, thermal neutrons readily undergo neutron capture to form heavier (and often radioactive) isotopes of that element.

Neutron matter at standard pressure and temperature is predicted by the ideal gas law to be less dense than even hydrogen, with a density of only 0.045 kg/m³ (roughly 27 times less dense than air and half as dense as hydrogen gas). Neutron matter is expected to remain gaseous down to absolute zero at normal pressures, as the zero-point energy of the system is too high to allow condensation. However, neutron matter should in theory form a degenerate gaseous superfluid at these temperatures, composed of transient neutron-pairs called dineutrons. Under extremely low pressure, this low temperature, gaseous superfluid should exhibit quantum coherence producing a Bose–Einstein condensate. At higher temperatures, neutron matter will only condense with sufficient pressure, and solidify with even greater pressure. Such pressures exist in neutron stars, where the extreme pressure causes the neutron matter to become degenerate. However, in the presence of atomic matter compressed to the state of electron degeneracy, β⁻ decay may be inhibited due to the Pauli exclusion principle, thus making free neutrons stable. Also, elevated pressures should make neutrons degenerate themselves.

Compared to ordinary elements, neutronium should be more compressible due to the absence of electrically charged protons and electrons. This makes neutronium more energetically favorable than (positive-Z) atomic nuclei and leads to their conversion to (degenerate) neutronium through electron capture, a process that is believed to occur in stellar cores in the final seconds of the lifetime of massive stars, where it is facilitated by cooling via
ν
e emission. As a result, degenerate neutronium can have a density of 4×10¹⁷ kg/m³^{[citation needed]}, roughly 14 orders of magnitude denser than the densest known ordinary substances. It was theorized that extreme pressures of order 100 MeV/fm³ might deform the neutrons into a cubic symmetry, allowing tighter packing of neutrons,^[18] or cause a strange matter formation.

Hydrostatic paradox[edit]

Wikimedia Commons has media related to Hydrostatic paradox.

Diagram illustrating the hydrostatic paradox

Any quantity of liquid, however small, may be made to support any weight, however large.

W={\frac {wA}{\alpha }}.

Demonstrations of the hydrostatic paradox are used in teaching the phenomenon.^[5]^[6]

https://en.wikipedia.org/wiki/Vertical_pressure_variation#Hydrostatic_paradox

https://en.wikipedia.org/wiki/Gravitational_singularity

https://en.wikipedia.org/wiki/Zero-point_energy

The quantum vacuum state or simply quantum vacuum refers to the quantum state with the lowest possible energy.

Quantum vacuum may also refer to:

Quantum chromodynamic vacuum (QCD vacuum), a non-perturbative vacuum
Quantum electrodynamic vacuum (QED vacuum), a field-theoretic vacuum
Quantum vacuum (ground state), the state of lowest energy of a quantum system
- Quantum vacuum collapse, a hypothetical vacuum metastability event
- Quantum vacuum expectation value, an operator's average, expected value in a quantum vacuum
Quantum vacuum energy, an underlying background energy that exists in space throughout the entire Universe
The Quantum Vacuum, a physics textbook by Peter W. Milonni

Chronology[edit]

Radiation-dominated era[edit]

After Inflation, and until about 47,000 years after the Big Bang, the dynamics of the early universe were set by radiation (referring generally to the constituents of the universe which moved relativistically, principally photons and neutrinos).^[9]

For a radiation-dominated universe the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is obtained solving the Friedmann equations:

a(t)\propto t^{{1/2}}.\,

^[10]

Matter-dominated era[edit]

Between about 47,000 years and 9.8 billion years after the Big Bang,^[11] the energy density of matter exceeded both the energy density of radiation and the vacuum energy density.^[12]

When the early universe was about 47,000 years old (redshift 3600), mass–energy density surpassed the radiation energy, although the universe remained optically thick to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time of recombination), at which the photons which compose the cosmic microwave background radiation were last scattered, is often mistaken^{[neutrality is disputed]} as marking the end of the radiation era.

For a matter-dominated universe the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is easily obtained solving the Friedmann equations:

a(t)\propto t^{2/3}

Dark-energy-dominated era[edit]

In physical cosmology, the dark-energy-dominated era is proposed as the last of the three phases of the known universe, the other two being the matter-dominated era and the radiation-dominated era. The dark-energy-dominated era began after the matter-dominated era, i.e. when the Universe was about 9.8 billion years old.^[13] In the era of cosmic inflation, the Hubble parameter is also thought to be constant, so the expansion law of the dark-energy-dominated era also holds for the inflationary prequel of the big bang.

The cosmological constant is given the symbol Λ, and, considered as a source term in the Einstein field equation, can be viewed as equivalent to a "mass" of empty space, or dark energy. Since this increases with the volume of the universe, the expansion pressure is effectively constant, independent of the scale of the universe, while the other terms decrease with time. Thus, as the density of other forms of matter – dust and radiation – drops to very low concentrations, the cosmological constant (or "dark energy") term will eventually dominate the energy density of the Universe. Recent measurements of the change in Hubble constant with time, based on observations of distant supernovae, show this acceleration in expansion rate,^[14] indicating the presence of such dark energy.

For a dark-energy-dominated universe, the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is easily obtained solving the Friedmann equations:

a(t)\propto \exp(H_{0}t)

Here, the coefficient $H_{0}$ in the exponential, the Hubble constant, is

H_{0}={\sqrt {8\pi G\rho _{\mathrm {full} }/3}}={\sqrt {\Lambda /3}}.

This exponential dependence on time makes the spacetime geometry identical to the de Sitter universe, and only holds for a positive sign of the cosmological constant, which is the case according to the currently accepted value of the cosmological constant, Λ, that is approximately 2 · 10⁻³⁵ s⁻². The current density of the observable universe is of the order of 9.44 · 10⁻²⁷ kg m⁻³ and the age of the universe is of the order of 13.8 billion years, or 4.358 · 10¹⁷ s. The Hubble constant, $H_{0}$ , is ≈70.88 km s⁻¹ Mpc⁻¹ (The Hubble time is 13.79 billion years).

History[edit]

A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by Stephen Smale, in 1961, of the Poincaré conjecture in five or more dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics. In 2002, Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton's Ricci flow, an idea belonging to the field of geometric analysis.

Overall, this progress has led to better integration of the field into the rest of mathematics.

Two dimensions[edit]

A surface is a two-dimensional, topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R³—for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.

Classification of surfaces[edit]

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:

the sphere;
the connected sum of g tori, for $g\geq 1$ ;
the connected sum of k real projective planes, for $k\geq 1$ .

The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 2g.

The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is 2 − k.

Teichmüller space[edit]

In mathematics, the Teichmüller space T_X of a (real) topological surface X, is a space that parameterizes complex structures on Xup to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in T_X may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space.

Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it was introduced by Oswald Teichmüller (1940).^[1]

Uniformization theorem[edit]

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature. This classifies Riemannian surfaces as elliptic (positively curved—rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their universal cover.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

Three dimensions[edit]

A topological space X is a 3-manifold if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space.

The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.

Knot and braid theory[edit]

Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³ (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R³ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knot complements are frequently-studied 3-manifolds. The knot complement of a tame knot K is the three-dimensional space surrounding the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N,

X_{K}=M-{\mbox{interior}}(N).

A related topic is braid theory. Braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit presentations, as was shown by Emil Artin (1947).^[2] For an elementary treatment along these lines, see the article on braid groups. Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.

Hyperbolic 3-manifolds[edit]

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously. See also Kleinian model.

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends that are the product of a Euclidean surface and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact. In this case, the ends are of the form torus cross the closed half-ray and are called cusps. Knot complements are the most commonly studied cusped manifolds.

Poincaré conjecture and geometrization[edit]

Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply-connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.^[3]

Four dimensions[edit]

A 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds that admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds that are homeomorphic but not diffeomorphic).

4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

Exotic R⁴[edit]

An exotic R⁴ is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R⁴. The first examples were found in the early 1980s by Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.^[4] There is a continuum of non-diffeomorphic differentiable structures of R⁴, as was shown first by Clifford Taubes.^[5]

Prior to this construction, non-diffeomorphic smooth structures on spheres—exotic spheres—were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2018). For any positive integer n other than 4, there are no exotic smooth structures on Rⁿ; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rⁿ is diffeomorphic to Rⁿ.^[6]

Other special phenomena in four dimensions[edit]

There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in four dimensions. Here are some examples:

In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H⁴(M,Z/2Z) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures.
Four is the only dimension n for which Rⁿ can have an exotic smooth structure. R⁴ has an uncountable number of exotic smooth structures; see exotic R⁴.
The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. In 2013, Ciprian Manolescu posted a preprint on the ArXiv showing that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.

A few typical theorems that distinguish low-dimensional topology[edit]

There are several theorems that in effect state that many of the most basic tools used to study high-dimensional manifolds do not apply to low-dimensional manifolds, such as:

Steenrod's theorem states that an orientable 3-manifold has a trivial tangent bundle. Stated another way, the only characteristic class of a 3-manifold is the obstruction to orientability.

Any closed 3-manifold is the boundary of a 4-manifold. This theorem is due independently to several people: it follows from the Dehn–Lickorish theorem via a Heegaard splitting of the 3-manifold. It also follows from René Thom's computation of the cobordismring of closed manifolds.

The existence of exotic smooth structures on R⁴. This was originally observed by Michael Freedman, based on the work of Simon Donaldson and Andrew Casson. It has since been elaborated by Freedman, Robert Gompf, Clifford Taubes and Laurence Taylor to show there exists a continuum of non-diffeomorphic smooth structures on R⁴. Meanwhile, Rⁿ is known to have exactly one smooth structure up to diffeomorphism provided n ≠ 4.

Results[edit]

The goal of the experiment was to measure the rate at which time passes in a higher gravitational potential, so to test this the maser in the probe was compared to a similar maser that remained on Earth.^{[citation needed]} Before the two clock rates could be compared, the Doppler shift was subtracted from the clock rate measured by the maser that was sent into space, to correct for the relative motion between the observers on Earth and the motion of the probe. The two clock rates were then compared and further compared against the theoretical predictions of how the two clock rates should differ. The stability of the maser permitted measurement of changes in the rate of the maser of 1 part in 10¹⁴ for a 100-second measurement.

The experiment was thus able to test the equivalence principle. Gravity Probe A confirmed the prediction that deeper in the gravity well time flows slower,^[7] and the observed effects matched the predicted effects to an accuracy of about 70 parts per million.

https://en.wikipedia.org/wiki/Gravity_Probe_A

https://en.wikipedia.org/wiki/Hydrogen_maser

https://en.wikipedia.org/wiki/Low-dimensional_topology

https://en.wikipedia.org/wiki/Category:Atomic_clocks

https://en.wikipedia.org/wiki/Dark_matter

https://en.wikipedia.org/wiki/Thermodynamics

https://en.wikipedia.org/wiki/Energy

https://en.wikipedia.org/wiki/Space_rock

https://en.wikipedia.org/wiki/Oldest_dated_rocks

https://en.wikipedia.org/wiki/Radiation

https://en.wikipedia.org/wiki/Black_hole

https://en.wikipedia.org/wiki/Magnum_opus_(alchemy)

https://en.wikipedia.org/wiki/Dimensional_transmutation

https://en.wikipedia.org/wiki/Nuclear_transmutation

https://en.wikipedia.org/wiki/Nuclear_reaction

https://en.wikipedia.org/wiki/Nuclear_fission

https://en.wikipedia.org/wiki/Fusion

https://en.wikipedia.org/wiki/Atmosphere

https://en.wikipedia.org/wiki/Hertz

Terahertz waves lie at the far end of the infrared band, just before the start of the microwave band.

https://en.wikipedia.org/wiki/Terahertz_radiation

https://en.wikipedia.org/wiki/Terahertz_gap

https://en.wikipedia.org/wiki/T-ray_(disambiguation)

https://en.wikipedia.org/wiki/Higgs_mechanism

https://en.wikipedia.org/wiki/Gauge_boson

https://en.wikipedia.org/wiki/W_and_Z_bosons

The
Z0
boson is electrically neutral and is its own antiparticle.

https://en.wikipedia.org/wiki/W_and_Z_bosons

https://en.wikipedia.org/wiki/Elementary_particle

https://en.wikipedia.org/wiki/Antimatter

https://en.wikipedia.org/wiki/List_of_particles#Composite_particles

https://en.wikipedia.org/wiki/Force_carrier

Pressuron

https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange

https://en.wikipedia.org/wiki/Virtual_particle

https://en.wikipedia.org/wiki/Casimir_effect

https://en.wikipedia.org/wiki/Scattering_theory

https://en.wikipedia.org/wiki/Pion

https://en.wikipedia.org/wiki/Path_integral_formulation

https://en.wikipedia.org/wiki/Nucleon

https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange#The_Coulomb_potential_in_a_vacuum

https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange#Electrostatics

https://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange#The_Yukawa_potential:_The_force_between_two_nucleons_in_an_atomic_nucleus

https://en.wikipedia.org/wiki/Holon_(physics)

https://en.wikipedia.org/wiki/Orbiton

https://en.wikipedia.org/wiki/Quasiparticle

https://en.wikipedia.org/wiki/Spinon

Spinons are one of three quasiparticles, along with holons and orbitons, that electrons in solids are able to split into during the process of spin–charge separation, when extremely tightly confined at temperatures close to absolute zero.^[1] The electron can always be theoretically considered as a bound state of the three, with the spinon carrying the spin of the electron, the orbitoncarrying the orbital location and the holon carrying the charge, but in certain conditions they can behave as independent quasiparticles.

The term spinon is frequently used in discussions of experimental facts within the framework of both quantum spin liquid and strongly correlated quantum spin liquid.^[2]

Electrons, being of like charge, repel each other. As a result, in order to move past each other in an extremely crowded environment, they are forced to modify their behavior. Research published in July 2009 by the University of Cambridge and the University of Birmingham in England showed that electrons could jump from the surface of the metal onto a closely located quantum wire by quantum tunneling, and upon doing so, will separate into two quasiparticles, named spinons and holons by the researchers.^[3]

The orbiton was predicted theoretically by van den Brink, Khomskii and Sawatzky in 1997-1998.^[4]^[5] Its experimental observation as a separate quasiparticle was reported in paper sent to publishers in September 2011.^[6]^[7] The research states that by firing a beam of X-ray photons at a single electron in a one-dimensional sample of strontium cuprate, this will excite the electron to a higher orbital, causing the beam to lose a fraction of its energy in the process. In doing so, the electron will be separated into a spinon and an orbiton. This can be traced by observing the energy and momentum of the X-rays before and after the collision.

References[edit]

^ Tomonaga, S.-i. (1950). "Remarks on Bloch's method of sound waves applied to many-fermion problems". Progress of Theoretical Physics. 5 (4): 544. Bibcode:1950PThPh...5..544T. doi:10.1143/ptp/5.4.544.
^ Luttinger, J. M. (1963). "An Exactly Soluble Model of a Many‐Fermion System". Journal of Mathematical Physics. 4 (9): 1154. Bibcode:1963JMP.....4.1154L. doi:10.1063/1.1704046.
^ Haldane, F. D. M. (1981). "'Luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas". Journal of Physics C: Solid State Physics. 14 (19): 2585. Bibcode:1981JPhC...14.2585H. doi:10.1088/0022-3719/14/19/010.
^ "Spin-charge separation optical lattices". Optical-lattice.com. Retrieved 2016-07-11.
^ "UK | England | Physicists 'make electrons split'". BBC News. 2009-08-28. Retrieved 2016-07-11.
^ Discovery About Behavior Of Building Block Of Nature Could Lead To Computer Revolution. Science Daily (July 31, 2009)
^ Yarris, Lynn (2006-07-13). "First Direct Observations of Spinons and Holons". Lbl.gov. Retrieved 2016-07-11.

External links[edit]

https://en.wikipedia.org/wiki/Spin–charge_separation

Ultracold atoms are atoms that are maintained at temperatures close to 0 kelvin (absolute zero), typically below several tens of microkelvin (µK). At these temperatures the atom's quantum-mechanical properties become important.

To reach such low temperatures, a combination of several techniques typically has to be used.^[1] First, atoms are usually trapped and pre-cooled via laser cooling in a magneto-optical trap. To reach the lowest possible temperature, further cooling is performed using evaporative cooling in a magnetic or optical trap. Several Nobel prizes in physics are related to the development of the techniques to manipulate quantum properties of individual atoms (e.g. 1995-1997, 2001, 2005, 2012, 2017).

Experiments with ultracold atoms study a variety of phenomena, including quantum phase transitions, Bose–Einstein condensation (BEC), bosonic superfluidity, quantum magnetism, many-body spin dynamics, Efimov states, Bardeen–Cooper–Schrieffer (BCS) superfluidity and the BEC–BCS crossover.^[2] Some of these research directions utilize ultracold atom systems as quantum simulators to study the physics of other systems, including the unitary Fermi gas and the Ising and Hubbard models.^[3] Ultracold atoms could also be used for realization of quantum computers.^[4]

https://en.wikipedia.org/wiki/Ultracold_atom

https://en.wikipedia.org/wiki/Nitrogen-vacancy_center

https://en.wikipedia.org/wiki/Kane_quantum_computer

https://en.wikipedia.org/wiki/Toric_code

https://en.wikipedia.org/wiki/Quantum_logic_gate

https://en.wikipedia.org/wiki/Stop_squark

https://en.wikipedia.org/wiki/Scalar_boson

https://en.wikipedia.org/wiki/Spin–charge_separation

Pionium is an exotic atom consisting of one
π+
and one
π−
meson. It can be created, for instance, by interaction of a proton beam accelerated by a particle accelerator and a target nucleus. Pionium has a short lifetime, predicted by chiral perturbation theory to be 2.89×10⁻¹⁵ s. It decays mainly into two
π0
mesons, and to a smaller extent into two photons.

https://en.wikipedia.org/wiki/Pionium

A mesonic molecule is a set of two or more mesons bound together by the strong force.^[1]^[2] Unlike baryonic molecules, which form the nuclei of all elements in nature save hydrogen-1, a mesonic molecule has yet to be definitively observed.^[3] The X(3872)discovered in 2003 and the Z(4430) discovered in 2007 by the Belle experiment are the best candidates for such an observation.

https://en.wikipedia.org/wiki/Mesonic_molecule

https://en.wikipedia.org/wiki/X(3872)

https://en.wikipedia.org/wiki/Dark_photon

Preons come in four varieties, plus, anti-plus, zero and anti-zero. W bosons have 6 preons and quarks have only 3.

https://en.wikipedia.org/wiki/Preon

https://en.wikipedia.org/wiki/X_and_Y_bosons

https://en.wikipedia.org/wiki/XYZ_particle

https://en.wikipedia.org/wiki/Grand_Unified_Theory

https://en.wikipedia.org/wiki/Elementary_particle

https://en.wikipedia.org/wiki/Chargino

https://en.wikipedia.org/wiki/Delta_baryon

https://en.wikipedia.org/wiki/Phi_meson

https://en.wikipedia.org/wiki/Quarkonium

https://en.wikipedia.org/wiki/Hexaquark

https://en.wikipedia.org/wiki/Pomeron

https://en.wikipedia.org/wiki/Exciton

https://en.wikipedia.org/wiki/Davydov_soliton

https://en.wikipedia.org/wiki/Magnon

https://en.wikipedia.org/wiki/Plasmaron

https://en.wikipedia.org/wiki/Trion_(physics)

https://en.wikipedia.org/wiki/List_of_mesons

https://en.wikipedia.org/wiki/Eightfold_way_(physics)

https://en.wikipedia.org/wiki/Polariton

Plane waves in vacuum[edit]

Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.

Electromagnetic waves in a vacuum[edit]

For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:

\omega =ck.\,

This is a linear dispersion relation. In this case, the phase velocity and the group velocity are the same:

v={\frac {\omega }{k}}={\frac {d\omega }{dk}}=c;

they are given by c, the speed of light in vacuum, a frequency-independent constant.

De Broglie dispersion relations[edit]

The free-space dispersion plot of kinetic energy versus momentum, for many objects of everyday life

Total energy, momentum, and mass of particles are connected through the relativistic dispersion relation^[1] established by Paul Dirac:

E^{2}=(mc^{2})^{2}+(pc)^{2},

which in the ultrarelativistic limit is

E=pc

and in the nonrelativistic limit is

E=mc^{2}+{\frac {p^{2}}{2m}},

where $m$ is the invariant mass. In the nonrelativistic limit, $mc^2$ is a constant, and $p^{2}/(2m)$ is the familiar kinetic energy expressed in terms of the momentum $p = mv$ .

The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from p to p² as shown in the log–log dispersion plot of E vs. p.

Elementary particles, atomic nuclei, atoms, and even molecules behave in some contexts as matter waves. According to the de Broglie relations, their kinetic energy E can be expressed as a frequency ω, and their momentum p as a wavenumber k, using the reduced Planck constant ħ:

E=\hbar \omega ,\quad p=\hbar k.

Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads

\omega ={\frac {\hbar k^{2}}{2m}}.

showAnimation: phase and group velocity of electrons

https://en.wikipedia.org/wiki/Dispersion_relation

https://en.wikipedia.org/wiki/Dispersion_(optics)

https://en.wikipedia.org/wiki/Dissipative_system

Pointwise relations[edit]

In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by f ≤ g if and only if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with pointwise order.^[1] Using the pointwise order on functions one can concisely define other important notions, for instance:^[2]

A closure operator c on a poset P is a monotone and idempotent self-map on P (i.e. a projection operator) with the additional property that id_A ≤ c, where id is the identity function.
Similarly, a projection operator k is called a kernel operator if and only if k ≤ id_A.

An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions

(f_{n})_{n=1}^{\infty }

with

f_{n}:X\longrightarrow Y

converges pointwise to a function $f$ if for each $x$ in $X$

\lim _{n\rightarrow \infty }f_{n}(x)=f(x).

https://en.wikipedia.org/wiki/Pointwise

https://en.wikipedia.org/wiki/Category:Mathematical_terminology

https://en.wikipedia.org/wiki/Glossary_of_order_theory#P

https://en.wikipedia.org/wiki/Partially_ordered_set

https://en.wikipedia.org/wiki/Monotonic_function

https://en.wikipedia.org/wiki/Idempotence

https://en.wikipedia.org/wiki/Identity_function

https://en.wikipedia.org/wiki/Closure_operator#Closure_operators_on_partially_ordered_sets

https://en.wikipedia.org/wiki/Lattice_(order)#Continuity_and_algebraicity

https://en.wikipedia.org/wiki/Finitary

https://en.wikipedia.org/wiki/Pointwise_convergence

https://en.wikipedia.org/wiki/Semiring

https://en.wikipedia.org/wiki/Zero_element#Additive_identities

In mathematics, the zero module is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name zero module. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.

https://en.wikipedia.org/wiki/Zero_element#Additive_identities

https://en.wikipedia.org/wiki/Category_of_groups

https://en.wikipedia.org/wiki/Greatest_element_and_least_element

https://en.wikipedia.org/wiki/Lattice_(order)

https://en.wikipedia.org/wiki/Partially_ordered_set

https://en.wikipedia.org/wiki/Zero_morphism

https://en.wikipedia.org/wiki/Function_composition

https://en.wikipedia.org/wiki/Ideal_(ring_theory)

https://en.wikipedia.org/wiki/Chinese_remainder_theorem

https://en.wikipedia.org/wiki/Ideal_number

https://en.wikipedia.org/wiki/Algebraic_number_field

https://en.wikipedia.org/wiki/Field_extension

https://en.wikipedia.org/wiki/Galois_theory

https://en.wikipedia.org/wiki/Angle_trisection

https://en.wikipedia.org/wiki/Neusis_construction

https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems

https://en.wikipedia.org/wiki/Entscheidungsproblem

https://en.wikipedia.org/wiki/Anomalous_cancellation

https://en.wikipedia.org/wiki/Mathematical_fallacy

https://en.wikipedia.org/wiki/0.999...

https://en.wikipedia.org/wiki/Computational_complexity_theory

https://en.wikipedia.org/wiki/Probability

https://en.wikipedia.org/wiki/Indeterminism

https://en.wikipedia.org/wiki/Analog

https://en.wikipedia.org/wiki/Error_detection_and_correction

https://en.wikipedia.org/wiki/Memorylessness

https://en.wikipedia.org/wiki/Markov_chain

https://en.wikipedia.org/wiki/Deterministic_system

https://en.wikipedia.org/wiki/Collectively_exhaustive_events

https://en.wikipedia.org/wiki/Conditional_probability

https://en.wikipedia.org/wiki/Probability_space

https://en.wikipedia.org/wiki/Random_variable

https://en.wikipedia.org/wiki/Event_(probability_theory)

https://en.wikipedia.org/wiki/Law_of_total_probability

https://en.wikipedia.org/wiki/Marginal_distribution

https://en.wikipedia.org/wiki/Mutual_exclusivity

https://en.wikipedia.org/wiki/Law_of_large_numbers

The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of nresults taken from the Cauchy distribution or some Pareto distributions (α<1) will not converge as n becomes larger; the reason is heavy tails. The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation,^[4] whereas the expectation of the Pareto distribution (α<1) is infinite.^[5] Another example is where the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as n goes to infinity.

https://en.wikipedia.org/wiki/Law_of_large_numbers

In computer programming, a rope, or cord, is a data structure composed of smaller strings that is used to efficiently store and manipulate a very long string. For example, a text editing program may use a rope to represent the text being edited, so that operations such as insertion, deletion, and random access can be done efficiently.^[1]

https://en.wikipedia.org/wiki/Rope_(data_structure)

https://en.wikipedia.org/wiki/Mathematical_joke

Multivalued functions[edit]

Many functions do not have a unique inverse. For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots.

https://en.wikipedia.org/wiki/Mathematical_fallacy

Positive and negative roots[edit]

Care must be taken when taking the square root of both sides of an equality. Failing to do so results in a "proof" of^[9] 5 = 4.

Proof:

Start from

-20=-20

Write this as

25-45=16-36

Rewrite as

5^{2}-5\times 9=4^{2}-4\times 9

Add 814 on both sides:

5^{2}-5\times 9+{\frac {81}{4}}=4^{2}-4\times 9+{\frac {81}{4}}

These are perfect squares:

\left(5-{\frac {9}{2}}\right)^{2}=\left(4-{\frac {9}{2}}\right)^{2}

Take the square root of both sides:

5-{\frac {9}{2}}=4-{\frac {9}{2}}

Add 92 on both sides:

5=4

Q.E.D.

The fallacy is in the second to last line, where the square root of both sides is taken: a² = b² only implies a = b if a and b have the same sign, which is not the case here. In this case, it implies that a = –b, so the equation should read

5-{\frac {9}{2}}=-\left(4-{\frac {9}{2}}\right)

which, by adding 9/2 on both sides, correctly reduces to 5 = 5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity^[10]

\cos ^{2}x=1-\sin ^{2}x

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

\cos x={\sqrt {1-\sin ^{2}x}}

Evaluating this when x = $π$ , we get that

-1={\sqrt {1-0}}

-1=1

which is incorrect.

The error in each of these examples fundamentally lies in the fact that any equation of the form

x^{2}=a^{2}

where $a\neq 0$ , has two solutions:

x=\pm a

and it is essential to check which of these solutions is relevant to the problem at hand.^[11] In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to $π$ , the second equation is rendered invalid.

Square roots of negative numbers[edit]

Invalid proofs utilizing powers and roots are often of the following kind:

1={\sqrt {1}}={\sqrt {(-1)(-1)}}={\sqrt {-1}}{\sqrt {-1}}=i\cdot i=-1.

The fallacy is that the rule ${\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}$ is generally valid only if at least one of $x$ and $y$ is non-negative (when dealing with real numbers), which is not the case here.^[12]

Alternatively, imaginary roots are obfuscated in the following:

i={\sqrt {-1}}=\left(-1\right)^{\frac {2}{4}}=\left(\left(-1\right)^{2}\right)^{\frac {1}{4}}=1^{\frac {1}{4}}=1

The error here lies in the third equality, as the rule $a^{bc}=(a^{b})^{c}$ only holds for positive real a and real b,c.

Complex exponents[edit]

When a number is raised to a complex power, the result is not uniquely defined (see Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

{\begin{aligned}e^{2\pi i}&=1\\\left(e^{2\pi i}\right)^{i}&=1^{i}\\e^{-2\pi }&=1\\\end{aligned}}

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being {e^{2 $π$ n} | n ∈ ℤ}.

https://en.wikipedia.org/wiki/Mathematical_fallacy

https://en.wikipedia.org/wiki/Negative_mass

https://en.wikipedia.org/wiki/Faster-than-light

Note. Error, losses, irremediable outcome cascade

A tachyon (/ˈtækiɒn/) or tachyonic particle is a hypothetical particle that always travels faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics.^[1]^[a] If such particles did exist, and could send signals faster than light, then according to the theory of relativity they would violate causality, leading to logical paradoxes such as the grandfather paradox.^[1] Tachyons would also exhibit the unusual property of increasing in speed as their energy decreases, and would require infinite energy to slow down to the speed of light. No experimental evidence for the existence of such particles has been found.

https://en.wikipedia.org/wiki/Tachyon

Traversable wormholes[edit]

The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary matter vacuum energy, and it has been shown theoretically that quantum field theory allows states where energy can be arbitrarily negative at a given point.^[26] Many physicists, such as Stephen Hawking,^[27] Kip Thorne,^[28] and others,^[29]^[30]^[31]argued that such effects might make it possible to stabilize a traversable wormhole.^[32]^[33] The only known natural process that is theoretically predicted to form a wormhole in the context of general relativity and quantum mechanics was put forth by Leonard Susskind in his ER=EPR conjecture. The quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,^[34]^{: 494–496}^[35] and stable versions of such wormholes have been suggested as dark matter candidates.^[36]^[37] It has also been proposed that, if a tiny wormhole held open by a negative mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.^[38]

Image of a simulated traversable wormhole that connects the square in front of the physical institutes of University of Tübingen with the sand dunes near Boulogne-sur-Mer in the north of France. The image is calculated with 4D raytracing in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.^{[note 1]}

Lorentzian traversable wormholes would allow travel in both directions from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated in a 1973 paper by Homer Ellis^[39] and independently in a 1973 paper by K. A. Bronnikov.^[40] Ellis analyzed the topology and the geodesics of the Ellis drainhole, showing it to be geodesically complete, horizonless, singularity-free, and fully traversable in both directions. The drainhole is a solution manifold of Einstein's field equations for a vacuum spacetime, modified by inclusion of a scalar field minimally coupled to the Ricci tensor with antiorthodox polarity (negative instead of positive). (Ellis specifically rejected referring to the scalar field as 'exotic' because of the antiorthodox coupling, finding arguments for doing so unpersuasive.) The solution depends on two parameters: m, which fixes the strength of its gravitational field, and n, which determines the curvature of its spatial cross sections. When m is set equal to 0, the drainhole's gravitational field vanishes. What is left is the Ellis wormhole, a nongravitating, purely geometric, traversable wormhole.

Kip Thorne and his graduate student Mike Morris, unaware of the 1973 papers by Ellis and Bronnikov, manufactured, and in 1988 published, a duplicate of the Ellis wormhole for use as a tool for teaching general relativity.^[41] For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, was from 1988 to 2015 referred to in the literature as a Morris–Thorne wormhole.

Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made where the traversing path does not pass through a region of exotic matter. However, in the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology) exotic matter is not needed in order for wormholes to exist—they can exist even with no matter.^[42] A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,^[38] in which it was proposed that such wormholes could have been naturally created in the early universe.

Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out how to convert a wormhole traversing space into one traversing time by accelerating one of its two mouths.^[28] However, according to general relativity, it would not be possible to use a wormhole to travel back to a time earlier than when the wormhole was first converted into a time "machine". Until this time it could not have been noticed or have been used.^[34]^: 504

^{https://en.wikipedia.org/wiki/Wormhole#Traversable_wormholes}

Thus the drainhole is 'traversable' by test particles in both directions. The same holds for photons.

A complete catalog of geodesics of the drainhole can be found in the Ellis paper.^[1]

Absence of horizons and singularities; geodesical completeness[edit]

For a metric of the general form of the drainhole metric, with $\partial _{t}+cf(\rho )\partial _{\rho }$ as the velocity field of a flowing ether, the coordinate velocities $d\rho /dt$ of radial null geodesics are found to be $c(f(\rho )+1)$ for light waves traveling against the ether flow, and $c(f(\rho )-1)$ for light waves traveling with the flow. Wherever $f(\rho )>-1$ , so that $c(f(\rho )+1)>0$ , light waves struggling against the ether flow can gain ground. On the other hand, at places where $f(\rho )\leq -1$ upstream light waves can at best hold their own (if $f(\rho )=-1$ ), or otherwise be swept downstream to wherever the ether is going (if $f(\rho )<-1$ ). (This situation is described in jest by: "People in light canoes should avoid ethereal rapids."^[1])

The latter situation is seen in the Schwarzschild metric, where $\textstyle f_{\text{S}}(\rho )=-{\sqrt {2M/\rho }}\,$ , which is $-1$ at the Schwarzschild event horizon where $r_{\text{S}}(\rho )=\rho =2M$ , and less than $-1$ inside the horizon where $\rho <2M$ .

By contrast, in the drainhole $\textstyle f^{2}(\rho )<1-e^{-2m\pi /a}<1$ and $\textstyle f(\rho )=-\left[f^{2}(\rho )\right]^{1/2}>-1$ , for every value of $\rho$ , so nowhere is there a horizon on one side of which light waves struggling against the ether flow cannot gain ground.

Because

$r$ and $f$ are defined on the whole real line, and
$r$ is bounded away from $0$ by $r(2m)$ ), and
$f^{2}$ is bounded away from $1$ (by ${\sqrt {1-e^{-2m\pi /a}}}$ ),

the drainhole metric encompasses neither a 'coordinate singularity' where $1-f^{2}(\rho )\to 0$ nor a 'geometric singularity' where $r(\rho )\to 0$ , not even asymptotic ones. For the same reasons, every geodesic with an unbound orbit, and with some additional argument every geodesic with a bound orbit, has an affine parametrization whose parameter extends from $-\infty$ to $\infty$ . The drainhole manifold is, therefore, geodesically complete.

https://en.wikipedia.org/wiki/Ellis_drainhole

https://en.wikipedia.org/wiki/Geodesic_manifold

https://en.wikipedia.org/wiki/Schwarzschild_metric

https://en.wikipedia.org/wiki/Angular_momentum

https://en.wikipedia.org/wiki/Pseudovector

https://en.wikipedia.org/wiki/Point_particle

https://en.wikipedia.org/wiki/Preon

https://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix

https://en.wikipedia.org/wiki/Vector_boson

https://en.wikipedia.org/wiki/Gauge_boson

https://en.wikipedia.org/wiki/Force_carrier

https://en.wikipedia.org/wiki/Pressuron

https://en.wikipedia.org/wiki/Scalar–tensor_theory

https://en.wikipedia.org/wiki/Dimensionless_quantity

https://en.wikipedia.org/wiki/Boundary_value_problem

https://en.wikipedia.org/wiki/Eigenfunction

https://en.wikipedia.org/wiki/Function_space

https://en.wikipedia.org/wiki/Vector_space

https://en.wikipedia.org/wiki/Topological_space

https://en.wikipedia.org/wiki/Linear_space_(geometry)

https://en.wikipedia.org/wiki/Incidence_geometry

https://en.wikipedia.org/wiki/Incidence_structure

https://en.wikipedia.org/wiki/Projective_plane

https://en.wikipedia.org/wiki/Square_matrix

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

https://en.wikipedia.org/wiki/Loop_(graph_theory)

https://en.wikipedia.org/wiki/Incidence_structure

https://en.wikipedia.org/wiki/Partial_linear_space

https://en.wikipedia.org/wiki/Multiple_edges

https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Graph

https://en.wikipedia.org/wiki/Glossary_of_graph_theory#infinite

https://en.wikipedia.org/wiki/Null_graph

https://en.wikipedia.org/wiki/Duality_(projective_geometry)

https://en.wikipedia.org/wiki/Isomorphism

https://en.wikipedia.org/wiki/Hypergraph

https://en.wikipedia.org/wiki/Block_design

https://en.wikipedia.org/wiki/Fano_plane

https://en.wikipedia.org/wiki/Nimber

https://en.wikipedia.org/wiki/GF(2)

https://en.wikipedia.org/wiki/Linear_subspace

https://en.wikipedia.org/wiki/Incidence_matrix

The order-zero graph, $K_{0}$ , is the unique graph having no vertices (hence its order is zero). It follows that $K_{0}$ also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude $K_{0}$ from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including $K_{0}$ as a valid graph is useful depends on context. On the positive side, $K_{0}$ follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures $K_{0}$ is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children). On the negative side, including $K_{0}$ as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph", or the definition of connected graphs has to be modified not to include K₀). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies "graph with at least one vertex" unless context suggests otherwise.^[1]^[2]

In category theory, the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category.

$K_{0}$ does fulfill (vacuously) most of the same basic graph properties as does $K_{1}$ (the graph with one vertex and no edges). As some examples, $K_{0}$ is of size zero, it is equal to its complement graph $\overline {K_{0}}$ , a forest, and a planar graph. It may be considered undirected, directed, or even both; when considered as directed, it is a directed acyclic graph. And it is both a complete graph and an edgeless graph. However, definitions for each of these graph properties will vary depending on whether context allows for $K_{0}$ .

Edgeless graph[edit]

Edgeless graph (empty graph, null graph)
Vertices	n
Edges	0
Radius	0
Diameter	0
Girth	$\infty$
Automorphisms	n!
Chromatic number	1
Chromatic index	0
Genus	0
Properties	Integral Symmetric
Notation	$\overline K_{n}$
Table of graphs and parameters

For each natural number n, the edgeless graph (or empty graph) $\overline K_{n}$ of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted.^[1]^[2]

It is a 0-regular graph. The notation $\overline K_{n}$ arises from the fact that the n-vertex edgeless graph is the complement of the complete graph $K_{n}$ .

List of forbidden characterizations for graphs and hypergraphs[edit]

This list is incomplete; you can help by adding missing items. (August 2008)

Family	Obstructions	Relation	Reference
Forests	loops, pairs of parallel edges, and cycles of all lengths	subgraph	Definition
Forests	a loop (for multigraphs) or a triangle K₃ (for simple graphs)	graph minor	Definition
Claw-free graphs	star K_1,3	induced subgraph	Definition
Comparability graphs		induced subgraph
Triangle-free graphs	triangle K₃	induced subgraph	Definition
Planar graphs	K₅ and K_3,3	homeomorphic subgraph	Kuratowski's theorem
Planar graphs	K₅ and K_3,3	graph minor	Wagner's theorem
Outerplanar graphs	K₄ and K_2,3	graph minor	Diestel (2000),^[1]p. 107
Outer 1-planar graphs	six forbidden minors	graph minor	Auer et al. (2013)^[2]
Graphs of fixed genus	a finite obstruction set	graph minor	Diestel (2000),^[1]p. 275
Apex graphs	a finite obstruction set	graph minor	^[3]
Linklessly embeddable graphs	The Petersen family	graph minor	^[4]
Bipartite graphs	odd cycles	subgraph	^[5]
Chordal graphs	cycles of length 4 or more	induced subgraph	^[6]
Perfect graphs	cycles of odd length 5 or more or their complements	induced subgraph	^[7]
Line graph of graphs	nine forbidden subgraphs (listed here)	induced subgraph	^[8]
Graph unions of cactus graphs	the four-vertex diamond graph formed by removing an edge from the complete graph K₄	graph minor	^[9]
Ladder graphs	K_2,3 and its dual graph	homeomorphic subgraph	^[10]
Split graphs	$C_{4},C_{5},{\bar {C}}_{4}\left(=K_{2}+K_{2}\right)$	induced subgraph	^[11]
2-connected series–parallel(treewidth ≤ 2, branchwidth ≤ 2)	K₄	graph minor	Diestel (2000),^[1]p. 327
Treewidth ≤ 3	K₅, octahedron, pentagonal prism, Wagner graph	graph minor	^[12]
Branchwidth ≤ 3	K₅, octahedron, cube, Wagner graph	graph minor	^[13]
Complement-reducible graphs (cographs)	4-vertex path P₄	induced subgraph	^[14]
Trivially perfect graphs	4-vertex path P₄ and 4-vertex cycle C₄	induced subgraph	^[15]
Threshold graphs	4-vertex path P₄, 4-vertex cycle C₄, and complement of C₄	induced subgraph	^[15]
Line graph of 3-uniform linear hypergraphs	a finite list of forbidden induced subgraphs with minimum degree at least 19	induced subgraph	^[16]
Line graph of k-uniform linear hypergraphs, k > 3	a finite list of forbidden induced subgraphs with minimum edge degree at least 2k² − 3k + 1	induced subgraph	^[17]^[18]
Graphs ΔY-reducible to a single vertex	a finite list of at least 68 billion distinct (1,2,3)-clique sums	graph minor	^[19]
General theorems
A family defined by an induced-hereditary property	a, possibly non-finite, obstruction set	induced subgraph
A family defined by a minor-hereditary property	a finite obstruction set	graph minor	Robertson–Seymour theorem

Additive identities[edit]

An additive identity is the identity element in an additive group. It corresponds to the element 0 such that for all x in the group, 0 + x = x + 0 = x. Some examples of additive identity include:

The zero vector under vector addition: the vector of length 0 and whose components are all 0. Often denoted as $\mathbf {0}$ or ${\vec {0}}$ .^[1]^[2]^[3]
The zero function or zero map defined by z(x) = 0, under pointwise addition (f + g)(x) = f(x) + g(x)
The empty set under set union
An empty sum or empty coproduct
An initial object in a category (an empty coproduct, and so an identity under coproducts)

Absorbing elements[edit]

An absorbing element in a multiplicative semigroup or semiring generalises the property 0 ⋅ x = 0. Examples include:

The empty set, which is an absorbing element under Cartesian product of sets, since { } × S = { }
The zero function or zero map defined by z(x) = 0 under pointwise multiplication (f ⋅ g)(x) = f(x) ⋅ g(x)

Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a field or ring, which is both the additive identity and the multiplicative absorbing element, and whose principal ideal is the smallest ideal.

Zero objects[edit]

A zero object in a category is both an initial and terminal object (and so an identity under both coproducts and products). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:

The trivial group, containing only the identity (a zero object in the category of groups)
The zero module, containing only the identity (a zero object in the category of modules over a ring)

Zero morphisms[edit]

A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0_XY : X → Y is the zero morphism among morphisms from X to Y, and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0_XY = 0_XB and 0_XY ∘ f = 0_AY.

If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0_XY : X → Y. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.

Least elements[edit]

A least element in a partially ordered set or lattice may sometimes be called a zero element, and written either as 0 or ⊥.

Zero module[edit]

Zero ideal[edit]

In mathematics, the zero ideal in a ring $R$ is the ideal $\{0\}$ consisting of only the additive identity (or zero element). The fact that this is an ideal follows directly from the definition.

Zero matrix[edit]

In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. It is alternately denoted by the symbol $O$ .^[1] Some examples of zero matrices are

0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\

The set of m × n matrices with entries in a ring K forms a module $K_{m,n}$ . The zero matrix $0_{{K_{{m,n}}}}$ in $K_{m,n}$ is the matrix with all entries equal to $0_{K}$ , where $0_{K}$ is the additive identity in K.

0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}

The zero matrix is the additive identity in $K_{m,n}$ . That is, for all $A\in K_{m,n}$ :

0_{K_{m,n}}+A=A+0_{K_{m,n}}=A

There is exactly one zero matrix of any given size m × n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix also represents the linear transformation which sends all vectors to the zero vector.

Zero tensor[edit]

In mathematics, the zero tensor is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector.

Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.

Narrow escape[edit]

The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.

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The Artin conjecture[edit]

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Big Rip mechanism[edit]

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Black hole information paradox[edit]

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Neutronium and the periodic table[edit]

Exotic R⁴[edit]