title: Philosophy of mathematics

text: Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus, humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered; triangles, for example, are real entities, not the creations of the human mind.


title: Philosophy of mathematics

text: Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the continuum hypothesis conjecture might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.


title: Philosophy of mathematics

text: Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them. Major forms of mathematical realism include Platonism and Aristotelianism.


title: Philosophy of mathematics

text: Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include formalism and fictionalism.


title: Philosophy of mathematics

text: The view that claims that mathematics is the aesthetic combination of assumptions, and then also claims that mathematics is an art. A famous mathematician who claims that is the British G. H. Hardy. For Hardy, in his book, "A Mathematician's Apology", the definition of mathematics was more like the aesthetic combination of concepts.


title: Philosophy of mathematics

text: Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term "Platonism" is used because such a view is seen to parallel Plato's Theory of Forms and a "World of Ideas" (Greek: "eidos" (εἶδος)) described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular "Pythagoreans" of ancient Greece, who believed that the world was, quite literally, generated by numbers.


title: Philosophy of mathematics

text: A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.


title: Philosophy of mathematics

text: Kurt Gödel's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic "a priori".) Davis and Hersh have suggested in their 1999 book "The Mathematical Experience" that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.


title: Philosophy of mathematics

text: Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the excluded middle, and the axiom of choice). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from a single consistent set of axioms.


title: Philosophy of mathematics

text: Set-theoretic realism (also set-theoretic Platonism) a position defended by Penelope Maddy, is the view that set theory is about a single universe of sets. This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of Paul Benacerraf's epistemological problem. A similar view, termed Platonized naturalism, was later defended by the Stanford–Edmonton School: according to this view, a more traditional kind of Platonism is consistent with naturalism; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of abstract objects.


title: Philosophy of mathematics

text: Max Tegmark's mathematical universe hypothesis (or mathematicism) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: "All structures that exist mathematically also exist physically". That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".


title: Philosophy of mathematics

text: Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. Logicists hold that mathematics can be known "a priori", but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.


title: Philosophy of mathematics

text: Gottlob Frege was the founder of logicism. In his seminal "Die Grundgesetze der Arithmetik" ("Basic Laws of Arithmetic") he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts "F" and "G", the extension of "F" equals the extension of "G" if and only if for all objects "a", "Fa" equals "Ga"), a principle that he took to be acceptable as part of logic.


title: Philosophy of mathematics

text: Frege's construction was flawed. Bertrand Russell discovered that Basic Law V is inconsistent (this is Russell's paradox). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop much of mathematics, such as the "axiom of reducibility". Even Russell said that this axiom did not really belong to logic.


title: Philosophy of mathematics

text: Modern logicists (like Bob Hale, Crispin Wright, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as Hume's principle (the number of objects falling under the concept "F" equals the number of objects falling under the concept "G" if and only if the extension of "F" and the extension of "G" can be put into one-to-one correspondence). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.


title: Philosophy of mathematics

text: Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all.


title: Philosophy of mathematics

text: Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: "if" one assigns meaning to the strings in such a way that the rules of the game become true (i.e., true statements are assigned to the axioms and the rules of inference are truth-preserving), "then" one must accept the theorem, or, rather, the interpretation one has given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism.) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.


title: Philosophy of mathematics

text: A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were seriously undermined by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.


title: Philosophy of mathematics

text: Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.


title: Philosophy of mathematics

text: Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists.


title: Philosophy of mathematics

text: Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.


title: Philosophy of mathematics

text: The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.


title: Philosophy of mathematics

text: Recently, some formalist mathematicians have proposed that all of our "formal" mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition—see QED project for a general overview.


title: Philosophy of mathematics

text: The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry should not be regarded as "a priori" truth. He held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.


title: Philosophy of mathematics

text: In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L. E. J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the "a priori" forms of the volitions that inform the perception of empirical objects.


title: Philosophy of mathematics

text: A major force behind intuitionism was L. E. J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted.


title: Philosophy of mathematics

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


title: Philosophy of mathematics

text: Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. Important work was done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis as constructive analysis in his 1967 "Foundations of Constructive Analysis."


title: Philosophy of mathematics

text: Finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. In her book "Philosophy of Set Theory", Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists.


title: Philosophy of mathematics

text: Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Another variant of finitism is Euclidean arithmetic, a system developed by John Penn Mayberry in his book "The Foundations of Mathematics in the Theory of Sets". Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that super-exponentiation is not a legitimate finitary function.


title: Philosophy of mathematics

text: Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their "places" in such structures, consequently having no intrinsic properties. For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.


title: Philosophy of mathematics

text: Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what "kind" of entity a mathematical object is, not to what kind of "existence" mathematical objects or structures have (not, in other words, to their ontology). The kind of existence mathematical objects have would clearly be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard.


title: Philosophy of mathematics

text: The "ante rem" structuralism ("before the thing") has a similar ontology to Platonism. Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem of explaining the interaction between such abstract structures and flesh-and-blood mathematicians (see Benacerraf's identification problem).


title: Philosophy of mathematics

text: The "in re" structuralism ("in the thing") is the equivalent of Aristotelian realism. Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures.


title: Philosophy of mathematics

text: The "post rem" structuralism ("after the thing") is anti-realist about structures in a way that parallels nominalism. Like nominalism, the "post rem" approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical "systems" exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.


title: Philosophy of mathematics

text: Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects (requiring the human senses such as sight for detecting the objects, touch; and signalling from the brain). It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.


title: Philosophy of mathematics

text: The cognitive processes of pattern-finding and distinguishing objects are also subject to neuroscience; if mathematics is considered to be relevant to a natural world (such as from realism or a degree of it, as opposed to pure solipsism).


title: Philosophy of mathematics

text: Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the evolution of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as illusion, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and hallucinations). As such, this may also raise questions for the modern scientific method for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by empiricism which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts ins science such as quantum nonlocality, and action at a distance).


title: Philosophy of mathematics

text: Another issue is that one numeral system may not necessarily be applicable to problem solving. Subjects such as complex numbers or imaginary numbers require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.


title: Philosophy of mathematics

text: Alternatively, computer programmers may use hexadecimal for its 'human-friendly' representation of binary-coded values, rather than decimal (convenient for counting because humans have ten fingers). The axioms or logical rules behind mathematics also vary through time (such as the adaption and invention of zero).


title: Philosophy of mathematics

text: As perceptions from the human brain are subject to illusions, assumptions, deceptions, (induced) hallucinations, cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: philosophy of being), and the nature of empiricism itself in relation to the universe and whether it is independent to the senses and the universe.


title: Philosophy of mathematics

text: The human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition.


title: Philosophy of mathematics

text: Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe.


title: Philosophy of mathematics

text: The most accessible, famous, and infamous treatment of this perspective is "Where Mathematics Comes From", by George Lakoff and Rafael E. Núñez. In addition, mathematician Keith Devlin has investigated similar concepts with his book "The Math Instinct", as has neuroscientist Stanislas Dehaene with his book "The Number Sense". For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics.


title: Philosophy of mathematics

text: Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots. Aristotelian realism is defended by James Franklin and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.


title: Philosophy of mathematics

text: The Euclidean arithmetic developed by John Penn Mayberry in his book "The Foundations of Mathematics in the Theory of Sets" also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.


title: Philosophy of mathematics

text: Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).


title: Philosophy of mathematics

text: John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists, past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his "The Foundations of Arithmetic", and many of his works and essays, including his review of Husserl's "Philosophy of Arithmetic". Edmund Husserl, in the first volume of his "Logical Investigations", called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by Charles Sanders Peirce and Maurice Merleau-Ponty.


title: Philosophy of mathematics

text: Mathematical empiricism is a form of realism that denies that mathematics can be known "a priori" at all. It says that we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill. Mill's view was widely criticized, because, according to critics, such as A.J. Ayer, it makes statements like come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.


title: Philosophy of mathematics

text: Karl Popper was another philosopher to point out empirical aspects of mathematics, observing that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience."


title: Philosophy of mathematics

text: Contemporary mathematical empiricism, formulated by W. V. O. Quine and Hilary Putnam, is primarily supported by the indispensability argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of being distinct from the other sciences.


title: Philosophy of mathematics

text: Putnam strongly rejected the term "Platonist" as implying an over-specific ontology that was not necessary to mathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist. It is also sometimes called "postmodernism in mathematics" although that term is considered overloaded by some and insulting by others. Quasi-empiricism argues that in doing their research, mathematicians test hypotheses as well as prove theorems. A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. Putnam has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics—at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in "New Directions". Quasi-empiricism was also developed by Imre Lakatos.


title: Philosophy of mathematics

text: The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. consilience after E.O. Wilson. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is extraordinarily central, and that it would be extremely difficult for us to revise it, though not impossible.


title: Philosophy of mathematics

text: For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's "Realism in Mathematics". Another example of a realist theory is the embodied mind theory.


title: Philosophy of mathematics

text: For experimental evidence suggesting that human infants can do elementary arithmetic, see Brian Butterworth.


title: Philosophy of mathematics

text: Mathematical fictionalism was brought to fame in 1980 when Hartry Field published "Science Without Numbers", which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.


title: Philosophy of mathematics

text: Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of useful fiction. He showed that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like is just as fictitious as "Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions.


title: Philosophy of mathematics

text: Another fictionalist, Mary Leng, expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way.


title: Philosophy of mathematics

text: By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions.


title: Philosophy of mathematics

text: Social constructivism sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historically-defined discipline.


title: Philosophy of mathematics

text: This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an overemphasis on axiomatic proof and peer review as practices.


title: Philosophy of mathematics

text: The social nature of mathematics is highlighted in its subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias, or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless.


title: Philosophy of mathematics

text: Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko, although it is not clear that either would endorse the title. More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the Erdős number. Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach, similar to but not quite the same as that associated with Alvin White; one of Hersh's co-authors, Philip J. Davis, has expressed sympathy for the social view as well.


title: Philosophy of mathematics

text: Rather than focus on narrow debates about the true nature of mathematical truth, or even on practices unique to mathematicians such as the proof, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.


title: Philosophy of mathematics

text: Realist and constructivist theories are normally taken to be contraries. However, Karl Popper argued that a number statement such as can be taken in two senses. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other on realist lines.


title: Philosophy of mathematics

text: Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, the "language" of science. Although some mathematicians and philosophers would accept the statement "mathematics is a language", linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.


title: Philosophy of mathematics

text: Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics. Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such as tense), but many of the same analytical tools can be used (such as context-free grammars). One important difference is that mathematical objects have clearly defined types, which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its part of speech in another; and this operation has no analogue in natural language."


title: Philosophy of mathematics

text: This argument, associated with Willard Quine and Hilary Putnam, is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The form of the argument is as follows.


title: Philosophy of mathematics

text: The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry, but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.


title: Philosophy of mathematics

text: The anti-realist "epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field. Platonism posits that mathematical objects are "abstract" entities. By general agreement, abstract entities cannot interact causally with concrete, physical entities ("the truth-values of our mathematical assertions depend on facts involving Platonic entities that reside in a realm outside of space-time"). Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. Another way of making the point is that if the Platonic world were to disappear, it would make no difference to the ability of mathematicians to generate proofs, etc., which is already fully accountable in terms of physical processes in their brains.


title: Philosophy of mathematics

text: Field developed his views into fictionalism. Benacerraf also developed the philosophy of mathematical structuralism, according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism.


title: Philosophy of mathematics

text: The argument hinges on the idea that a satisfactory naturalistic account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else. One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose.


title: Philosophy of mathematics

text: Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non-causal, and not analogous to perception. This argument is developed by Jerrold Katz in his 2000 book "Realistic Rationalism".


title: Philosophy of mathematics

text: A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.


title: Philosophy of mathematics

text: Many practicing mathematicians have been drawn to their subject because of a sense of beauty they perceive in it. One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics—where, presumably, the beauty lies.


title: Philosophy of mathematics

text: In his work on the divine proportion, H.E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art—the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as literature.


title: Philosophy of mathematics

text: Philip J. Davis and Reuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of . The first is the traditional proof by contradiction, ascribed to Euclid; the second is a more direct proof involving the fundamental theorem of arithmetic that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem.


title: Philosophy of mathematics

text: Paul Erdős was well known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof; Gregory Chaitin has argued against this idea.


title: Philosophy of mathematics

text: Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound.


title: Philosophy of mathematics

text: Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in G. H. Hardy's book "A Mathematician's Apology", in which Hardy argues that pure mathematics is superior in beauty to applied mathematics precisely because it cannot be used for war and similar ends.


title: Unified Diagnostic Services

text: Unified Diagnostic Services (UDS) is a diagnostic communication protocol used in electronic control units (ECUs) within automotive electronics, which is specified in the ISO 14229-1. It is derived from ISO 14230-3 (KWP2000) and the now obsolete ISO 15765-3 (Diagnostic Communication over Controller Area Network (DoCAN)). 'Unified' in this context means that it is an international and not a company-specific standard. By now this communication protocol is used in all new ECUs made by Tier 1 suppliers of Original Equipment Manufacturer (OEM), and is incorporated into other standards, such as AUTOSAR. The ECUs in modern vehicles control nearly all functions, including electronic fuel injection (EFI), engine control, the transmission, anti-lock braking system, door locks, braking, window operation, and more.


title: Unified Diagnostic Services

text: Diagnostic tools are able to contact all ECUs installed in a vehicle, which has UDS services enabled. In contrast to the CAN bus protocol, which only uses the first and second layers of the OSI model, UDS utilizes the fifth and seventh layers of the OSI model. The Service ID (SID) and the parameters associated with the services are contained in the payload of a message frame.gh


title: Unified Diagnostic Services

text: Modern vehicles have a diagnostic interface for off-board diagnostics, which makes it possible to connect a computer (client) or diagnostics tool, which is referred to as tester, to the communication system of the vehicle. Thus, UDS requests can be sent to the controllers which must provide a response (this may be positive or negative). This makes it possible to interrogate the fault memory of the individual control units, to update them with new firmware, have low-level interaction with their hardware (e.g. to turn a specific output on or off), or to make use of special functions (referred to as routines) to attempt to understand the environment and operating conditions of an ECU to be able to diagnose faulty or otherwise undesirable behavior.


title: Suckling pig

text: A suckling pig is a piglet fed on its mother's milk (i.e., a piglet which is still a "suckling"). In culinary contexts, a suckling pig is slaughtered between the ages of two and six weeks. It is traditionally cooked whole, often roasted, in various cuisines. It is usually prepared for special occasions and gatherings. The most popular preparation can be found in Spain and Portugal under the name lechón (Spanish) or leitão (Portuguese).


title: Suckling pig

text: The meat from suckling pig is pale and tender and the cooked skin is crisp and can be used for pork rinds. The texture of the meat can be somewhat gelatinous due to the amount of collagen in a young pig.


title: Suckling pig

text: There are many ancient recipes for suckling pig from Roman and Chinese cuisine. Since the pig is one of the first animals domesticated by human beings for slaughter, many references to pigs are found in human culture. The suckling pig, specifically, appears in early texts such as the sixth-century Salic law. As an example of a law governing the punishment for theft, Title 2, article 1, is, in Latin, . "If someone has stolen a suckling pig and this is proven against him, the guilty party will be sentenced to 120 denarii which adds up to three solidi (Latin coins)." The words are written in Frankish; (or in other manuscripts) is the gloss for "suckling pig"; . These glosses in Frankish, the so-called "Malbergse Glossen", are considered the earliest attested words in Old Dutch.


title: Suckling pig

text: It has been argued that the use of pigs for human consumption is unethical, especially in the case of young animals removed from their mothers earlier than weaning would happen in nature: natural weaning takes place at around 12 weeks of age, whereas suckling pigs are slaughtered at 2 to 6 weeks of age. Further, investigations by media and animal rights groups have uncovered animal cruelty and inhumane conditions related to the farming of suckling pigs. The sows, mother pigs, are in many countries kept in gestation crates or farrowing crates, which render them nearly immobile and unable to interact with their newborn offspring.


title: Suckling pig

text: Lechón (Spanish, ; from "leche" "milk" + -ón) or leitão (Portuguese; from "leite" "milk" + -ão) is a pork dish in several regions of the world, most specifically in Spain (in particular Segovia), Portugal (in particular Bairrada) and regions worldwide previously colonized by the Portuguese Empire or Spanish Empire. "Lechón/Leitão" is a word referring to a roasted baby pig (piglet) which was still fed by suckling its mother's milk (a suckling pig). Lechón/Leitão is a popular item in the cuisine in Los Angeles (in the United States), Spain, Cuba, Puerto Rico, Honduras, Argentina, Uruguay, Bolivia, Ecuador, Peru, Costa Rica, the Dominican Republic, and other Spanish-speaking nations in Latin America, as well as in Portugal, Cape Verde, Angola, Mozambique and other Portuguese-speaking nations. It is also present as cochon de lait in the French-Swiss and French cuisines (in particular in Metz), in Italy (in particular in Sardinian cuisine as su porcheddu) and Romania. The dish features a whole roasted suckling pig cooked over charcoal. It has been described as a national dish of Cuba, Puerto Rico, Spain, Portugal, as well as the Philippines whose pig-roasting traditions (similar to other Austronesian regions) have native pre-colonial origins. In the latter case, the meaning of the designation diverted in these regions from the original Spanish term to become a general term for "roasted pig", and is nowadays used in reference to adult roasted pigs more often than to lechones (milk suckling pigs), with Cebu being asserted by American chef Anthony Bourdain as having the best pigs.


title: Suckling pig

text: In most of these regions, lechón/leitão is prepared throughout the year for special occasions, during festivals. It is the centerpiece of the tradition Cuban Christmas feast "La Noche Buena". After seasoning, the piglet is cooked by skewering the entire animal, entrails removed, on a large stick and cooking it in a pit filled with charcoal. The piglet is placed over the charcoal, and the stick or rod it is attached to is turned in a rotisserie action.


title: Suckling pig

text: The dish has been described as a national dish of Puerto Rico. The name of the dish in Puerto Rico is "lechón asado". It is a traditional dish served at festivals and holidays.


title: Suckling pig

text: It is similar in style to many preparations made in other South American countries, consisting of a roasted pig stuffed with yellow peas, green onion, yellow rice and spices, cooked in an outdoor brick oven for several hours.


title: Suckling pig

text: It is mostly traditional to the Tolima Department in central Colombia and is usually accompanied by arepas, a corn-based bread.


title: Suckling pig

text: In most regions of the Philippines, lechón (spelled "lechon" without diacritic but also "litson" or "lichon") is traditionally prepared throughout the year for special occasions, festivals, and the holidays. Although it acquired the Spanish name, Philippine lechon has pre-Hispanic origins as pigs are one of the native domesticated animals of all Austronesian cultures and were carried throughout the Austronesian Expansion all the way to Polynesia. In the former Spanish colony of the Philippines, () is considered a national dish. As the usage of the term has evolved over the years, has now come to refer to roasted pig in general (including suckling pigs). Suckling pigs in the country are referred to as , which corresponds to the term in Spain.


title: Suckling pig

text: The native name of lechón is inihaw [na baboy] in Tagalog, a general term meaning "charcoal-roasted/barbecued [pig]". Native names were also preserved in other regions until recently, like in Cebu where it was previously more commonly known as "inasal" until Tagalog influence changed it to "lechon" in the 2000s. As the usage of the Spanish loanword evolved over the years in the languages of the Philippines, "lechon" has come to refer to roasted pig in general (including suckling pigs). Roasted suckling pigs are now referred to in the Philippines as "lechon de leche" (which in Spanish would be a linguistic redundancy, though corresponding to the term cochinillo in Spain).


title: Suckling pig

text: It is a national dish of the Philippines. There are two major methods of preparing lechon in the Philippines, the "Manila lechon" (or "Luzon lechon"), and the "Cebu lechon" (or "Visayas lechon").


title: Suckling pig

text: Visayan lechon is prepared stuffed with herbs which usually include scallions, bay leaves, black peppercorn, garlic, salt, and distinctively "tanglad" (lemongrass) or leaves from native "Citrus" trees or tamarind trees, among other spices. A variant among Hiligaynon people also stuffs the pig with the sour fruits of "batuan" or "binukaw" ("Garcinia binucao"). It is usually cooked over charcoal made from coconut husks. Since it is already flavored with spices, it is served with minimal dipping sauces, like salt and vinegar or silimansi (soy sauce, calamansi, and labuyo chili).


title: Suckling pig

text: Luzon lechon on the other hand, is typically not stuffed with herbs. When it is, it is usually just salt and pepper. Instead, the distinctiveness of Manila lechon comes from the liver-based sauce, known as the "lechon sauce". Lechon sauce is made from vinegar, brown sugar, salt, pepper, mashed liver (or liver spread), breadcrumbs, garlic and onion. Manila lechon is also typically cooked over woodfire.


title: Suckling pig

text: Most lechon can either be cooked based on the two main versions, or mix techniques from both. Both variants also rub salt or spices unto the skin to make it crispier, as well as continually baste the lechon as it cooks. Sometimes carbonated drinks may also be used. They are cooked on a bamboo spit over charcoal for a few hours with constant (traditionally manual) turning. The pig is roasted on all sides for several hours until done. The process of cooking and basting usually results in making the pork skin crisp and is a distinctive feature of the dish.


title: Suckling pig

text: Leftover parts from the lechon, such as the head and feet, are usually cooked into another popular dish, "lechon paksiw". Like lechon itself, "lechon paksiw" also differs based on whether it is prepared Luzon-style or Visayas-style, with the former using liver sauce as an essential ingredient, while the latter does not. In some cases, these parts or stale lechon can be repurposed into another dish, such as Sisig.


title: Suckling pig

text: There is also variant of suckling pig among the Indonesian non-Muslim ethnic groups, such as the Balinese, Batak, and Minahasa. Some pork dishes (e.g. in Singapore) are also influenced by ethnic Chinese. In Southeast Asia, roast suckling pig is eaten in Chinese or Vietnamese restaurants for important parties. It is also a popular dish at wedding dinners or a party for a baby's completion of its first month of life.


title: Suckling pig

text: The European cuisines of Austria, Croatia, Georgia, Germany, Macedonia, the Netherlands, Russia, Serbia, Slovenia, Switzerland and Sweden favor the dish highly as well. It accompanies goose as the traditional Christmas feast of families in Russia and Serbia, while the Russian Navy maintains a tradition of presenting a roast piglet (or several) to the crew of a ship returning from deployment.


title: Suckling pig

text: Suckling pig is known in German, Austrian and German-Swiss cuisines as Spanferkel and in the Dutch cuisine as speenvarken. It can be roasted in the oven or grilled, and is often served at festive occasions such as Oktoberfest.


title: Suckling pig

text: In Sweden suckling pig is called spädgris, it is usually cooked in the oven, or sometimes roasted directly over the fire. It is often stuffed with various fruits such as apples and plums, together with butter and breadcrumbs.


title: Suckling pig

text: The suckling pig is used in Cajun cuisine in the southern U.S., where the "Cochon de Lait" Festival is held annually in the small town of Mansura, Louisiana. During this festival, as its name implies, suckling pigs are roasted. Other uses for the suckling pig in the U.S. include slow roasting in an oven or (as in a Hawaiian-style pig roast) in a pit. The latter remains popular in the cuisine of the Southern United States.


title: Martin Dúbravka

text: Martin Dúbravka (; born 15 January 1989) is a Slovak professional footballer who plays as a goalkeeper for club Manchester United, on loan from Newcastle United, and the Slovakia national team.


title: Martin Dúbravka

text: After playing in the top flights of Slovak, Danish and Czech football, Dúbravka joined English club Newcastle United in January 2018, initially on loan. He won the club's Player of the Year award in the 2019–20 season.


title: Martin Dúbravka

text: Dúbravka represented Slovakia at under-21 level, before making his senior international debut in May 2014. He started all three of Slovakia's games at UEFA Euro 2020.


title: Martin Dúbravka

text: As a child, Dúbravka enjoyed playing ice hockey as well as football, and could have moved into the former sport were it not for a leg injury he sustained when he was five years old.


title: Martin Dúbravka

text: In the 2009–10 Slovak Superliga he played 26 Slovak Superliga games, finishing as a league champion. He became a first choice after the transfer of Dušan Perniš in January 2010. He qualified with Žilina to the 2010–11 UEFA Champions League and played all 6 group games. He kept a clean sheet in the play-off round against Sparta Prague. On 30 January 2014, he signed a 3½-year deal with the Danish Superliga club Esbjerg fB.


title: Martin Dúbravka

text: Dúbravka joined Slovan Liberec of the Czech First League in July 2016 on a one-year contract. He joined Sparta Prague in June 2017, signing a three-year contract.


title: Martin Dúbravka

text: In January 2018, Newcastle United showed interest in Dúbravka, initially offering Sparta Prague a loan for €500,000, with an option to sign in the summer for €4.5 million. He finally joined the Premier League side on 31 January 2018, the final day of the winter transfer window on a half-year loan deal, due to expire at the end of 2017–18 season. According to isport.cz, he joined the club for €2 million, with an option to sign in the summer for €4 million. He made his debut on 11 February, playing a pivotal role and keeping a clean sheet in a 1–0 win against Manchester United.


title: Martin Dúbravka

text: After a very successful loan spell, he signed permanently for Newcastle on 30 May 2018 for an undisclosed fee of around €5 million. On 17 February 2019 in Durham, Dúbravka was recognised as the 2018 Player of the Year by the North East Football Writers' Association, joining the likes of Alan Shearer, Kevin Keegan and Shay Given. He became the seventh goalkeeper to win the award, over its thirty-nine seasons. Dúbravka stated that he was happy to win the award because "amazing players have won this trophy before". "FourFourTwo" ranked Dúbravka as the twentieth best player of the Premier League, in the 2018–19 season. He was the highest ranked goalkeeper, ahead of highly valued goalkeepers such as Alisson (Liverpool) and David de Gea (Manchester United).


title: Martin Dúbravka

text: While Dúbravka was ever present for Newcastle for the next two seasons, he was mostly replaced in the FA Cup and EFL Cup, by either Karl Darlow or Freddie Woodman. His performances during the 2019–20 season, saw him win the club's Player of the Year award. In August 2020, Dúbravka injured his ankle in a training session prior to the start of the 2020–21 season. Darlow played in his absence and he returned to the bench for the club's last match of the year, a 0–0 draw with Liverpool. On 9 January 2021, Dúbravka started his first match of the season, in a FA Cup extra-time loss to Arsenal. On 27 February, Dúbravka started his first league match of the season in a 1–1 draw with Wolverhampton Wanderers. Afterwards, he went on to make twelve more appearances in goal as Newcastle, despite having been in a relegation battle for much of the season, finished in twelfth place.


title: Martin Dúbravka

text: On 1 September 2022, Dúbravka joined Manchester United on a season-long loan from Newcastle. On 3 September 2022, it was confirmed that he would wear the number 31 shirt last worn by Nemanja Matić. He made his debut in United's 4–2 win over Aston Villa in the EFL Cup third round on 10 November 2022.


title: Martin Dúbravka

text: Dúbravka made 10 appearances for the Slovakia under-21 team. On 23 May 2014, Dúbravka made his debut for the Slovakia national football team in a 2–0 win over Montenegro.


title: Martin Dúbravka

text: In January 2017, he played his first full game for the national team, having played 45 minutes in both of his previous matches. While he was benched against Uganda (3–1 loss), on 12 January he was fielded in an unofficial friendly match against Sweden in Abu Dhabi, United Arab Emirates. Slovakia lost the game 6–0, despite only trailing 1–0 at half time. This match also marked the first time Dúbravka conceded in the national team and he was also Slovak captain for the game, being the most experienced with international football.


title: Martin Dúbravka

text: He made his competitive debut in a 2018 FIFA World Cup qualifier, against Lithuania (2–1 win), on 10 June 2017, when the preferred goalkeeper in the qualification, Matúš Kozáčik, was unavailable due to injury. He however achieved his first competitive clean sheet on 1 September in a home qualifier against Slovenia. Slovakia won thanks to an 81st-minute goal by Adam Nemec. Just as in the previous cap, Dúbravka was chosen due to a prolonged injury of Kozáčik. After Kozáčik's return to the squad, Dúbravka remained the preferred choice for the remaining matches against England, Scotland and Malta. In the match against Scotland, Dúbravka made a series of saves before he was finally beaten by an own goal from Martin Škrtel.


title: Martin Dúbravka

text: Dúbravka missed all the national team fixtures of 2020 due to injuries ruling him out of play during the autumn of the year, he returned to the national team in March 2021, in a 0–0 draw with Cyprus. He made notable saves in the second half preventing and upset loss and received some praise for them, while the team received overall criticisms for the upset result and the performance labelled as "boring".


title: Martin Dúbravka

text: He was named in Slovakia's final squad for UEFA Euro 2020, where he played in all three group matches. Going into the match against Spain on 23 June, Slovakia had a slim chance to advance to the knockout stage as a third-placed team. Dúbravka started well, saving a penalty from Álvaro Morata in the twelfth minute. After a shot from Pablo Sarabia hit the crossbar, he tried to tip the ball away and instead punched the ball into his own net. In doing so, he became the first goalkeeper in European Championship history, to save a penalty and score an own goal in the same match. Dúbravka would go on to concede four goals, as Spain won the match 5–0.


title: Martin Dúbravka

text: Dúbravka was called up for a double fixture against Czech Republic and Sweden on 13 October and 16 October 2018, respectively. He played the entirety of both games, but was also involved in an incident, that had led to resignation of Ján Kozák, then Slovakia's longest-serving and most successful coach, who introduced him to the national team.


title: Martin Dúbravka

text: Dúbravka violated the code of conduct of national team players, along with Michal Šulla, Norbert Gyömbér, Ľubomír Šatka, Milan Škriniar, Vladimír Weiss and Stanislav Lobotka. On the night of 13 October, after a 2–1 loss in Slovakia's second UEFA Nations League fixture and a derby match against the Czech Republic, they left the hotel and went out, missed the bedtime and violated the wellness policy, regarding regeneration and rehabilitation. They were allegedly discovered by Kozák at about midnight, admitting to the incident and apologising during the next day. Kozák resigned from the national team during the afternoon of 14 October, revealing the details of his decision in a press conference on 18 October, to avoid distractions during the preparation for the fixture against Sweden, that was managed by his former assistant Štefan Tarkovič on a caretaker basis. Kozák cited his inability to work with such squad under these conditions as the primary reason, as almost a third of the team was concerned, including players, those Kozák described as crucial, with Dúbravka being a part of this group. President of SFZ, Ján Kováčik, said that the players will be punished by not receiving any financial rewards for future nominations and performances in the national team during the upcoming qualification cycle. During the press conference, Kozák also revealed that Dúbravka had claimed "difficult times at Newcastle" as one of the reasons for his conduct.


title: Martin Dúbravka

text: On 19 October, SFZ published Dúbravka's official statement. He acknowledged that missing the bedtime was a violation, but maintained that the rest was within norms. He accepted any sanctions and stressed that the conduct was not aimed against Kozák, commenting that he regrets that Kozák did not accept an apology from the trespassers. He also stated that Kozák's resignation surprised him, because of the good relations within the team.


title: Martin Dúbravka

text: Dúbravka is a third-generation goalkeeper: his father and grandfather also played in the position. Dúbravka has been dubbed the "quintessential sweeper-keeper" due to his "outstanding footwork" and he believes "in modern football the keeper's almost like a libero. You need to play with the ball, not just kick it long".


title: Martin Dúbravka

text: Louise Taylor of "The Guardian" christened Dúbravka the "reluctant keeper" after he admitted his favourite position to play is right wing; he sometimes plays as an outfielder in training and coaching staff have suggested that he switch to a midfield role.


title: Martin Dúbravka

text: Dúbravka lives with his girlfriend Lucia, and his dog. The couple had their first child in March 2021.


title: Whitesnake discography

text: The British-American hard rock band Whitesnake have released thirteen studio albums, nine live albums, twelve compilation albums, three box sets, two extended plays (EPs), 40 singles, nine video albums and 29 music videos. Formed in Middlesbrough in 1978 by vocalist David Coverdale, the band originally featured guitarists Micky Moody and Bernie Marsden, bassist Neil Murray, keyboardist Peter Solley and drummer Dave Dowle. The group's debut EP "Snakebite" was released in June 1978 and reached number 61 on the UK Singles Chart. After replacing Solley with Jon Lord, the band released their debut full-length album "Trouble" later in the year, which reached number 50 on the UK Albums Chart. 1979's "Lovehunter" reached number 29 on the chart. Lead single "Long Way from Home" charted at number 55.


title: Whitesnake discography

text: Dowle was replaced by Ian Paice after the release of "Lovehunter", and in 1980 Whitesnake reached the UK top ten for the first time with "Ready an' Willing", which peaked at number 6 and was certified gold by the British Phonographic Industry (BPI). The album was also the band's first to register on the US "Billboard" 200, reaching number 90. The album's lead single "Fool for Your Loving" reached number 13 on the UK Singles Chart. The group's first live release, "Live... in the Heart of the City", reached number 5 on the UK Albums Chart and was certified platinum by the BPI. 1981's "Come an' Get It" and 1982's "Saints & Sinners" also both reached the UK top ten, with the former reaching a peak of number 2 (the band's highest to date). Three singles from across the two albums reached the UK Singles Chart top 40.


title: Whitesnake discography

text: After a brief hiatus and several lineup changes, Whitesnake resurfaced in 1984 with "Slide It In", which was their first album to reach the US top 40. It was also the band's first release to be certified by the Recording Industry Association of America (RIAA), reaching double platinum status. 1987's "Whitesnake" was even more successful, reaching number 2 on the "Billboard" 200, number 5 on the Canadian Albums Chart, and the top ten in several other regions. The album was certified eight times platinum by the RIAA, five times platinum by Music Canada, and platinum by the BPI. The single "Here I Go Again" topped the "Billboard" Hot 100 and the Canadian Singles Chart, as well as reaching the top ten of the UK Singles Chart. Its follow-up "Is This Love" reached number 2 on the Hot 100 and number 9 in the UK. The band's 1989 release "Slip of the Tongue" was the band's only other album to reach the US top ten, and the last to be certified by the RIAA (reaching platinum status).


title: Whitesnake discography

text: Whitesnake broke up in 1991. Three years later, "Greatest Hits" was released as the band's first compilation album, reaching number 4 on the UK Albums Chart and being certified gold by the BPI. The band returned with "Restless Heart" in 1997, which peaked at number 34 in the UK. In 2002 the band returned on a permanent basis. The 2003 compilation "Best of Whitesnake" reached number 44 in the UK and was certified gold by the BPI, while the 2006 video release "Live... in the Still of the Night" reached number 2 on the UK Music Video Chart and was also certified gold. Whitesnake's tenth studio album "Good to Be Bad" was released in 2008, reaching number 7 in the UK and number 62 in the US. In 2011, the band reached number 33 with "Forevermore" and number 81 with "Live at Donington 1990", in 2013 they reached number 67 with "Made in Japan", and in 2015 they reached number 18 with "The Purple Album".


title: Unexplained wealth of the Marcos family

text: The Marcos family, a political family in the Philippines, owns various assets that Philippine courts have determined to have been acquired through ilicit means during the presidency of Ferdinand Marcos from 1965–1986. These assets are referred to using several terms, including "ill-gotten wealth" and "unexplained wealth," while some authors such as Philippine Senator Jovito Salonga and Belinda Aquino more bluntly refer to it as the "Marcos Plunder."


title: Unexplained wealth of the Marcos family

text: Legally, the Philippine Supreme Court defines this "ill-gotten wealth" as the assets the Marcoses acquired beyond the amount legally declared by Ferdinand and Imelda Marcos in the president's statements of assets and liabilities—only about US$13,500.00 worth from his salary as president. The court also deems such wealth to have been forfeited in favor of the government or of the human rights victims of Marcos's authoritarian regime. Estimates of the amount the Marcoses reportedly acquired in the last few years of the Marcos administration range from US$5 billion to $13 billion. No exact figures can be determined for the amount acquired through the entire 21 years of the Marcos regime. But prominent Marcos-era economist Jesus Estanislao has suggested that the amount could go up to as high as US$30 billion.


title: Unexplained wealth of the Marcos family

text: Among the sources of the Marcos wealth are alleged to be: diverted foreign economic aid, US Government military aid (including huge discretionary funds at Marcos disposal as a "reward" for sending some Filipino troops to Vietnam) and kickbacks from public works contracts over a two-decades-long rule.


title: Unexplained wealth of the Marcos family

text: This wealth includes: real estate assets both within the Philippines and in several other countries, notably the United States; collections of jewelry and artwork; shares and other financial instruments; bank accounts, both in the Philippines and overseas, notably Switzerland, the United States, Singapore, and the British Virgin Islands; and in some instances, actual cash assets.


title: Unexplained wealth of the Marcos family

text: Some of this wealth has been recovered as the result of various court cases, either as funds or properties returned to the Philippine government, or by being awarded as reparations to the victims of human rights abuses under Marcos's presidency. Some of it has also been recovered by the Philippine government through settlements and compromise deals, either with the Marcoses themselves or with cronies who said that certain properties had been entrusted to them by the Marcoses. Some of the recovery cases have been dismissed by the courts for reasons including improper case filing procedures and technical issues with documentary evidence. An unknown amount is not recoverable because the full extent of the Marcos wealth is unknown.


title: Unexplained wealth of the Marcos family

text: Estimates of the ill-gotten wealth of the Marcos family vary, with most sources accepting a figure of about US$5–10 billion for wealth acquired in the last years of the Marcos administration, but with rough extreme estimates of wealth acquired since the 1950s going as high as US$30 billion.


title: Unexplained wealth of the Marcos family

text: In a 1985 report to the United States Congress House Committee on Foreign Affairs, Ambassador Stephen Bosworth estimated that the Marcoses had stolen an accumulated wealth of US$10 billion "in recent years", in the context of the rapid decline of the Philppine economy in the early 1980s. The same figure was cited by the Philippines' Office of the Solicitor General soon after Marcos was deposed by the EDSA Revolution in 1986.


title: Unexplained wealth of the Marcos family

text: Bosworth's source, Dr. Bernardo Villegas of the Philippine think tank the Center for Research and Communication (CRC), noted that the rough 10 Billion figure ultimately cited by Bosworth was already below well below the conservative estimate which can be derived from an analysis of the Philippines' import and export figures. In an interview with the Catholic newspaper Veritas, Villegas cited economic data gathered by the CRC saying that the conservative figure for the Marcoses' unexplained wealth would be a minimum of $13.145 Billion, and could be as high as $30 billion. Villegas noted that his analysis was based on figures tabulated by the CRC from 1977 to 1985, and added that the hidden wealth was self-evident, saying "kitang kita ang ebidensya" (""the evidence is very visible"").


title: Unexplained wealth of the Marcos family

text: The PCGG's first chair Jovito Salonga later said that he estimated figure of US$5 billion–10 billion, based on the documentary trail left behind by the Marcoses in 1986. Internationally, Salonga's estimate has become the popularly cited estimate of the Marcoses' unexplained wealth, and it is this amount for which the Marcoses were cited by Guinness World Records as having perpetrated the "largest-ever theft from a government" in 1989—a record they are still holding in 2021.


title: Unexplained wealth of the Marcos family

text: However, Dr. Jesus Estanislao, another noted economist from the Center for Research and Communication, pointed out that this figure reflected amounts taken out of the country in the years immediately prior to the ouster of the Marcos administration, and that there was no way to accurately estimate the wealth acquired by the Marcoses since the 1950s when he was in the legislature, and the mid 1960s when he became president. Estanislao suggested that the figure could be closer to $30 billion.


title: Unexplained wealth of the Marcos family

text: Among the sources of the Marcos wealth are alleged to be diverted foreign economic aid, US Government military aid (including huge discretionary funds at Marcos disposal as a "reward" for sending some Filipino troops to Vietnam) and kickbacks from public works contracts over a 2-decades-long rule.


title: Unexplained wealth of the Marcos family

text: According to Jovito Salonga in his book "Presidential Plunder", which details Salonga's time as head of the Presidential Commission on Good Government, these cronies helped the Marcoses amass his wealth by aiding him in one or more of what Salonga called "Marcos' Techniques of Plunder".


title: Unexplained wealth of the Marcos family

text: Shortly after Marcos was removed through the 1986 People Power revolution, an quasi-judicial government agency named the Presidential Commission on Good Government was created, with the primary mandate of recovering ill-gotten wealth accumulated by the Marcoses, their relatives, subordinates, and close associates, whether located in the Philippines or abroad. It was also tasked with investigating other cases of graft and corruption; and instituting of corruption prevention measures.


title: Unexplained wealth of the Marcos family

text: Two years later in 1998, the Comprehensive Agrarian Reform Law mandated that the funds recovered by the PCGG be automatically appropriated to fund the Philippines' agrarian reform programs. Since then, assets recovered from the Marcoses by the PCGG has funded more than 80 percent of the Philippines' budget for agrarian reform. On January 28, 2013, the Philippine Congress enacted the Human Rights Victims Reparation and Recognition Act of 2013 (Republic Act No. 10368). Through this law, the Philippine government acknowledged its moral and legal obligation to recognize and/or provide reparation for the victims of human rights violations during the Marcos regime. For this purpose, Congress allocated ₱10 billion from funds transferred to the Philippine government by virtue of the December 10, 1997 Order of the Swiss Federal Supreme Court, adjudged by the Supreme Court of the Philippines as final and executory in Republic vs. Sandiganbayan on July 15, 2003, (G.R. No. 152154) as Marcos ill-gotten wealth and forfeited in favor of the Republic of the Philippines.


title: Unexplained wealth of the Marcos family

text: Some of the most tangible examples of the unexplained wealth of the Marcos family are the various overseas real estate properties that the Marcoses acquired while they were in power.


title: Unexplained wealth of the Marcos family

text: The best known of these properties are the Marcoses' multi-million dollar real estate investments in the United States, particularly Imelda's purchases of buildings and real estate in New York, the estates purchased in New Jersey for the use of the Marcos children, Jose Yao Campos's investments in Seattle, various properties in Hawaii including the Makiki Heights estate where they lived during their exile, and their ownership of the California Overseas Bank in Los Angeles. According to Ricardo Manapat's book "Some Are Smarter Than Others", which was one of the earliest to document details of the Marcos wealth, lesser known properties include gold and diamond investments in South Africa, banks and hotels in Israel, and various landholdings in Austria, London, and Rome.


title: Unexplained wealth of the Marcos family

text: Many of these properties are to have been acquired under the name of several Marcos cronies. One of them, Jose Yao Campos, cooperated with the Philippine Government and made an immunity deal, revealing how he fronted Marcos’ investments both locally and abroad via numerous interlocking shell corporations.


title: Unexplained wealth of the Marcos family

text: Aside from the overseas properties, there are fifty-or-so "Marcos mansions" acquired by the Marcos family within the Philippines itself. Locations of these "Marcos mansions" include properties in Philippines' "Summer Capital" of Baguio, in the Ilocos region where the Marcoses trace their ancestry, Leyte where Imelda Marcos's family came from, and throughout the Greater Manila Area and its outskirts.


title: Unexplained wealth of the Marcos family

text: Some of these properties are titled in the name of Marcos family members, but others are titled in the name of identified “Marcos cronies” while being reserved for the use of the Marcos family. In some cases, several such mansions were located close together, with specific mansions meant for individual members of the family, as was the case of the Marcos mansions on Outlook Drive in Baguio. Many of the Marcos mansions were sequestered by the Philippine government when the Marcoses were expelled from the country as a result of the 1986 EDSA Revolution.


title: Unexplained wealth of the Marcos family

text: Also sequestered by the PCGG in 1986 were three sets of Imelda Marcos's jewelry, which have collectively come to be known as the "Marcos jewels" or "Imelda jewels."


title: Unexplained wealth of the Marcos family

text: Individually, the three collections of jewels sequestered by the PCGG have been come to be called the "Hawaii collection," the "Malacanang collection", and the "Roumeliotes collection", respectively. The "Hawaii collection" refers to a group of jewels seized by the US Bureau of Customs from the Marcoses when went into exile in Hawaii in 1986. The "Malacanang collection" refers to a group of jewels that were discovered in Malacañang Palace after the Marcoses fled the Philippines. The "Roumeliotes collection" refers to a group of jewels that were confiscated from Demetriou Roumeliotes, said to have been a close associate of Imelda Marcos, after he was caught trying to smuggle them out of the Philippines at Manila International Airport.


title: Unexplained wealth of the Marcos family

text: In February 2016, the government of the Philippines announced that the three collections had been appraised as valuing , and that they would eventually be auctioned off after having been kept unsold by the government for three decades.


title: Unexplained wealth of the Marcos family

text: Much of the Marcos wealth that the PCGG has been trying to recover is kept in various overseas bank accounts, which the PCGG was able to identify based on documents left by the Marcoses in Malacañang in February 1986. Marcos opened his first bank account by depositing $215,000.00 in Chase Manhattan Bank in New York City on July 7, 1967.


title: Unexplained wealth of the Marcos family

text: The most famous of these overseas accounts of the Marcoses are the four so-called William Saunders and Jane Ryan accounts, opened with Credit Suisse in Zurich in March 1968. Marcos famously used the alias "William Saunders" for the name of the account, while Imelda Marcos choose the alias "Jane Ryan." Four checks, totaling US$950,000.00 were used to make the initial deposit. These were later moved into other accounts under various dummy foundations, but when records of them were discovered by the new Philippine government after the 1986 EDSA revolution, the Swiss Federal Council froze them. On December 21, 1990, the Swiss Federal Supreme Court ruled that these accounts could be turned over to the Philippine government, on the condition that there be a concurring "final and absolute judgment" by a Philippine court. In 1997, the Swiss Federal Supreme Court established the funds to have been "of criminal provenance" and permitted their transfer to an escrow account in Manila, pending a ruling from a Philippine court that came in the form of a confiscation ruling by the Philippine Supreme court on July 15, 2003. Switzerland finally released a total of $683 million in Marcos funds to the Philippines Treasury in 2004.


title: Unexplained wealth of the Marcos family

text: Aside from the Saunders account, another well known overseas account of the Marcoses is known as the Arelma account, opened in 1972 at the brokerage firm of Merrill, Lynch, Pierce, Fenner & Smith, Inc. in New York under the name of the Arelma Foundation, a Panamanian corporation. The initial deposit had been for only $2 million in 1972, but the account had grown to approximately $35 million by 2000, and $42 million by 2014.


title: Unexplained wealth of the Marcos family

text: Among the early successes of the PCGG were achieved through compromise deals with Imelda Marcos or various Marcos cronies. Banks involved included Japan's Sanwa Bank, and in the United States, Redwood Bank and California Overseas Bank.


title: Unexplained wealth of the Marcos family

text: As the Marcos family fled to Hawaii after the EDSA Revolution, the opposition organized themselves and Pres. Corazon Aquino released Executive Order 1 on 28 February 1986, which was the creation of the Presidential Commission on Good Government (PCGG), with Jovito Salonga as its chair. The task of the PCGG was to recover all assets and moneys amassed by the Marcoses, relatives, and cronies.


title: Unexplained wealth of the Marcos family

text: The first attempt to recover was organized by Salonga, Sedfrey Ordoñez, Charlie Avila, and BGen. Jose Almonte, and was called Operation Big Bird. Armed with documents recovered in Manila after the revolution, the committee worked with the European banking systems and Swiss government to recover the money hidden at Swiss banks. Operations Big Bird as formulated by Gen. Almonte was to use Filipino bankers in Europe to double-cross the Marcoses in issuing a special power of attorney (SPA) to move their money from possible investigation by Philippine, and European governments. While $7 billion of accounts were identified, this mission was only successful to transfer $356 million from Marcos connected accounts to the Philippine government accounts. Sensing the double-cross, the Marcos withdrew their SPA, and the Swiss Attorney-General has ordered the freezing of the rest of the accounts until the Philippine government can prove that these accounts belong to the Filipino people.


title: Unexplained wealth of the Marcos family

text: The recovered money has ballooned to $570 million by 1998 when it was transferred from Switzerland to a Philippine National Bank escrow account. By 2003 the Philippine Supreme Court released a decision with finality that the said money was considered public fund. President Gloria Macapagal Arroyo in accordance to the law declared that the said money was to be used for the agricultural sector and agrarian reform. However, as 2004 elections are in the corner, the said funds were allegedly used for funding the campaign of Arroyo and her party-mates and allies. This was to become known as the Fertilizer Fund scam.


title: Unexplained wealth of the Marcos family

text: By 2019, the PCGG had recovered more than ₱171 billion of ill-gotten wealth from the Marcoses and from Marcos cronies since its creation in 1986. Some of this came from money sequestered by the PCGG or surrendered under various compromise agreements, and some of it came from the sale of various surrendered or sequestered properties. By 2020 it became ₱174 billion.


title: 5-HT2A receptor

text: The 5-HT receptor is a subtype of the 5-HT receptor that belongs to the serotonin receptor family and is a G protein-coupled receptor (GPCR). The 5-HT receptor is a cell surface receptor, but has several intracellular locations. 5-HT is short for 5-hydroxy-tryptamine or serotonin. This is the main excitatory receptor subtype among the GPCRs for serotonin, although 5-HT may also have an inhibitory effect on certain areas such as the visual cortex and the orbitofrontal cortex. This receptor was first noted for its importance as a target of serotonergic psychedelic drugs such as LSD and psilocybin mushrooms. Later it came back to prominence because it was also found to be mediating, at least partly, the action of many antipsychotic drugs, especially the atypical ones.


title: 5-HT2A receptor

text: Downregulation of post-synaptic 5-HT receptor is an adaptive process provoked by chronic administration of selective serotonin reuptake inhibitors (SSRIs) and atypical antipsychotics. Suicidal and otherwise depressed patients have had more 5-HT receptors than normal patients. These findings suggest that post-synaptic 5-HT overdensity is involved in the pathogenesis of depression.


title: 5-HT2A receptor

text: Paradoxical down-regulation of 5-HT receptors can be observed with several 5-HT antagonists. Thus, instead of tolerance, reverse-tolerance would be expected from 5-HT antagonists. However, there is at least one antagonist at this site which has been shown to up-regulate 5-HT receptors. Additionally, a couple of other antagonists may have no effect on 5-HT receptor number. Nevertheless, upregulation is the exception rather than the rule. Neither tolerance nor rebound is observed in humans with regard to the slow-wave sleep(SWS) promoting effects of 5-HT antagonists.


title: 5-HT2A receptor

text: 5-HT receptors were split into two classes by John Gaddum and Picarelli when it was discovered that some of the serotonin-induced changes in the gut could be blocked by morphine, while the remainder of the response was inhibited by dibenzyline, leading to the naming of M and D receptors, respectively. 5-HT is thought to correspond to what was originally described as D subtype of 5-HT receptors by Gaddum and Picarelli. In the era before molecular cloning, when radioligand binding and displacement was the only major tool, spiperone and LSD were shown to label two different 5-HT receptors, and neither of them displaced morphine, leading to naming of the 5-HT, 5-HT and 5-HT receptors, corresponding to high affinity sites from LSD, spiperone and morphine, respectively. Later it was shown that the 5-HT was very close to 5-HT and thus were grouped together, renaming the 5-HT into 5-HT. Thus, the 5-HT receptor family is composed of three separate molecular entities: the 5-HT (formerly known as 5-HT or D), the 5-HT (formerly known as 5-HT) and the 5-HT (formerly known as 5-HT) receptors.


title: 5-HT2A receptor

text: It is expressed near most of the serotonergic terminal rich areas, including neocortex (mainly prefrontal, parietal, and somatosensory cortex) and the olfactory tubercle. Especially high concentrations of this receptor on the apical dendrites of pyramidal cells in layer V of the cortex may modulate cognitive processes, working memory, and attention by enhancing glutamate release followed by a complex range of interactions with the 5-HT, GABA, adenosine A, AMPA, mGluR, mGlu5, and OX receptors. In the rat cerebellum, the protein has also been found in the Golgi cells of the granular layer, and in the Purkinje cells.


title: 5-HT2A receptor

text: In the periphery, it is highly expressed in platelets and many cell types of the cardiovascular system, in fibroblasts, and in neurons of the peripheral nervous system. Additionally, 5-HT mRNA expression has been observed in human monocytes. Whole-body distribution of the 5-HT2A/2C receptor agonist, [11C]Cimbi-36 show uptake in several internal organs and brown adipose tissue (BAT), but it is not clear if this represents specific 5-HT2A receptor binding.


title: 5-HT2A receptor

text: The 5-HT receptor is known primarily to couple to the Gα signal transduction pathway. Upon receptor stimulation with agonist, Gα and β-γ subunits dissociate to initiate downstream effector pathways. Gα stimulates phospholipase C (PLC) activity, which subsequently promotes the release of diacylglycerol (DAG) and inositol triphosphate (IP3), which in turn stimulate protein kinase C (PKC) activity and Ca release.


title: 5-HT2A receptor

text: Activation of the 5-HT receptor is necessary for the effects of the "classic" psychedelics like LSD, psilocin and mescaline, which act as full or partial agonists at this receptor, and represent the three main classes of 5-HT agonists, the ergolines, tryptamines and phenethylamines, respectively. A very large family of derivatives from these three classes has been developed, and their structure-activity relationships have been extensively researched. Agonists acting at 5-HT receptors located on the apical dendrites of pyramidal cells within regions of the prefrontal cortex are believed to mediate hallucinogenic activity. Some findings reveal that psychoactive effects of classic psychedelics are mediated by the receptor heterodimer 5-HT–mGlu2 and not by monomeric 5-HT receptors. However, newer research suggests that 5HT and mGlu2 receptors do not physically associate with each other, so the former findings have questionable relevance. Agonists enhance dopamine in PFC, enhance memory and play an active role in attention and learning.


title: 5-HT2A receptor

text: One effect of 5-HT receptor activation is a reduction in intraocular pressure, and so 5-HT agonists can be useful for the treatment of glaucoma. This has led to the development of compounds such as AL-34662 that are hoped to reduce pressure inside the eyes but without crossing the blood–brain barrier and producing hallucinogenic side effects. Animal studies with this compound showed it to be free of hallucinogenic effects at doses up to 30 mg/kg, although several of its more lipophilic analogues did produce the head-twitch response known to be characteristic of hallucinogenic effects in rodents.


title: 5-HT2A receptor

text: Increased 5-HT2A expression is observed in patients with coronary thrombosis, and the receptor has been associated with processes that influence atherosclerosis. As the receptor is present in coronary arteries and capable of mediating vasoconstriction, 5-HT2A has also been linked to coronary artery spasms. 5HT antagonism, therefore, has potential in the prevention of cardiovascular disease, however, no studies have been published so far.


title: 5-HT2A receptor

text: 5-HT-receptor ligands may differentially activate the transductional pathways (see above). Studies evaluated the activation of two effectors, PLC and PLA2, by means of their second messengers. Compounds displaying more pronounced functional selectivity are 2,5-DMA and 2C-N. The former induces IP accumulation without activating the PLA2 mediated response, while the latter elicits AA release without activating the PLC mediated response.


title: 5-HT2A receptor

text: Recent research has suggested potential signaling differences within the somatosensory cortex between 5-HT agonists that produce headshakes in the mouse and those that do not, such as lisuride, as these agents are also non-hallucinogenic in humans despite being active 5-HT agonists.


title: 5-HT2A receptor

text: One known example of differences in signal transduction is between the two 5-HT agonists serotonin and DOI that involves differential recruitment of intracellular proteins called β-arrestins, more specifically arrestin beta 2. Cyclopropylmethanamine derivatives such as (-)-19 have also been shown to act as 5-HT agonists with functional selectivity for Gq-mediated signaling compared with β-arrestin recruitment.


title: 5-HT2A receptor

text: The gene has previously been called just HTR2 until the description of two related genes "HTR2B" and "HTR2C".


title: 5-HT2A receptor

text: Probable role in fibromyalgia as the T102C polymorphisms of the gene 5HT2A were common in fibromyalgia patients.


title: 5-HT2A receptor

text: Human HTR2A gene is thought to consist of 3 introns and 4 exons and to overlap with human gene HTR2A-AS1 which consists of 18 exons. There are over 200 organisms that have orthologs with the human HTR2A. Currently, the best documented orthologs for "HTR2A" gene are the mouse, and zebrafish. There are 8 paralogs for the HTR2A gene. The "HTR2A" gene is known to interact and activate G-protein genes such as GNA14, GNAI1, GNAI3, GNAQ , and GNAZ. These interactions are critical for cell signaling and homeostasis in many organisms.


title: 5-HT2A receptor

text: In human brain tissue, regulation of "HTR2A" varies depending on the region: frontal cortex, amygdala, thalamus, brain stem and cerebellum. In a paper from 2016, they found that "HTR2A" undergoes a variety of different splicing events, including utilization of alternative splice acceptor sites, exon skipping, rare exon usage, and intron retention.


title: 5-HT2A receptor

text: There are a few mechanisms of regulation for "HTR2A" gene such regulated by DNA methylation at particular transcript binding sites. Another mechanism for the correct regulation of gene expression is achieved through alternative splicing. This is a co-transcriptional process, which allows the generation of multiple forms of mRNA transcript from a single coding unit and is emerging as an important control point for gene expression. In this process, exons or introns can be either included or excluded from precursor-mRNA resulting in multiple mature mRNA variants. These mRNA variants result in different isoforms which may have antagonistic functions or differential expression patterns, yielding plasticity and adaptability to the cells. One study found that the common genetic variant rs6311 regulates expression of "HTR2A" transcripts containing the extended 5' UTR.


title: 5-HT2A receptor

text: Several studies have seen links between the -1438G/A polymorphism and mood disorders, such as bipolar disorder