folk mathematics

The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. Specifically, it is the search for foundations of mathematics in human cognition. This approach was long preceded by the study, in cognitive sciences proper, of human cognitive bias, especially in statistical thinking, most notably by Amos Tversky and Daniel Kahneman, including theories of measurement, risk and behavioral finance from these and other authors. These studies suggested that mathematical practice and p ...

Including:

• Cognitive science of mathematics - Topics

Read more here: » Cognitive science of mathematics: Encyclopedia - Cognitive science of mathematics

Integrity comprises the personal inner sense of "wholeness" deriving from honesty and consistent uprightness of character. The etymology of the word relates it to the Latin adjective integer (whole, complete). Evaluators, of course, usually assess integrity from some point of view, such as that of a given ethical tradition or in the context of an ethical relationship. Integrity - Popular views of Integrity. Many people appear to use the word "integrity" in a vague manner as an alternative to the perc ...

Including:

• Integrity - Popular views of Integrity
• Integrity - Mensuration
• Integrity - Integrity in the Religious Society of Friends
• Integrity - Integrity in modern ethics
• Integrity - The Law
• Integrity - Mathematics
• Integrity - Cryptography
• Integrity - Science
• Integrity - Other Integrities

Read more here: » Integrity: Encyclopedia - Integrity

There exists however a more formal study of the term integrity and its meaning in modern ethics. It is often understood not only as a refusal to engage in behavior that evades responsibility, but as an understanding of different modes or styles in which some discourse takes place, and which aims at the discovery of some truth. Integrity - The Law. An adversarial process, for instance, has a certain type of integrity, in which those engaged in it commit not only to advance the case for "their own" si ...

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Integrity, Integrity - Popular views of Integrity, Integrity - Mensuration, Integrity - Integrity in the Religious Society of Friends, Integrity - Integrity in modern ethics, Integrity - The Law, Integrity - Mathematics, Integrity - Cryptography, Integrity - Science, Integrity - Other Integrities

Read more here: » Integrity: Encyclopedia II - Integrity - Integrity in modern ethics

Mathematical realism 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. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built u ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Mathematical realism or Platonism

There exists however a more formal study of the term integrity and its meaning in modern ethics. It is often understood not only as a refusal to engage in behavior that evades responsibility, but as an understanding of different modes or styles in which some discourse takes place, and which aims at the discovery of some truth. Integrity - The Law. An adversarial process, for instance, has a certain type of integrity, in which those engaged in it commit not only to advance the case for "their own" si ...

See also:

Integrity, Integrity - Popular views of Integrity, Integrity - Mensuration, Integrity - Integrity in the Religious Society of Friends, Integrity - Integrity in modern ethics, Integrity - The Law, Integrity - Mathematics, Integrity - Cryptography, Integrity - Science, Integrity - Other Integrities

Read more here: » Integrity: Encyclopedia II - Integrity in modern ethics

English-speakers often measure such integrity on a one-dimensional vertical scale dominated by two reference points: those of the highest integrity and no integrity (also known as a total lack of integrity). Some prescriptive dualistic schemas of ethics divide human activity into two fields and speak of behaviour as "in integrity" (approved) or as "out of integrity" (despised). ...

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Integrity, Integrity - Popular views of Integrity, Integrity - Mensuration, Integrity - Integrity in the Religious Society of Friends, Integrity - Integrity in modern ethics, Integrity - The Law, Integrity - Mathematics, Integrity - Cryptography, Integrity - Science, Integrity - Other Integrities

Read more here: » Integrity: Encyclopedia II - Mensuration

This theory 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 compared to 'reality' and may be discarded if they don't agree with observation or prove pointless. 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 mathema ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Social constructivism or social realism

English-speakers often measure such integrity on a one-dimensional vertical scale dominated by two reference points: those of the highest integrity and no integrity (also known as a total lack of integrity). Some prescriptive dualistic schemas of ethics divide human activity into two fields and speak of behaviour as "in integrity" (approved) or as "out of integrity" (despised). ...

See also:

Integrity, Integrity - Popular views of Integrity, Integrity - Mensuration, Integrity - Integrity in the Religious Society of Friends, Integrity - Integrity in modern ethics, Integrity - The Law, Integrity - Mathematics, Integrity - Cryptography, Integrity - Science, Integrity - Other Integrities

Read more here: » Integrity: Encyclopedia II - Integrity - Mensuration

These 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. 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. The physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Embodied mind theories

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, you can generat ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Formalism

Some philosophers of mathematics view their task as giving an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism. Criticisms can, however, have important ramifications for mathematical practice, so the philosophy of mathematics can be of direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs is not firmly established, raising the probability of an undetected error. Such errors can thus only be reduced by knowing where they are ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Relation to philosophy proper

The philosophy of mathematics has seen several different schools or strains, which primarily focus on questions of metaphysics, e.g., "Why does it work?" and the related--but logically separate-- "Why does mathematics explain the physical world as we see it so well?" Three schools, intuitionism, logicism and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigour wi ...

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Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Why does it work?

Logicism holds that logic is the proper foundation of mathematics, and that all mathematical statements are necessary logical truths. For instance, the statement "If Socrates is a human, and every human is mortal, then Socrates is mortal" is a necessary logical truth. To the logicist, all mathematical statements are precisely of the same type; they are analytic truths, or tautologies. Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up ar ...

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Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Logicism

These schools maintain that only mathematical entities which can be explicitly constructed have a claim to existence and should be admitted in mathematical discourse. A typical quote comes from Leopold Kronecker: "The natural numbers come from God, everything else is man's work." A major force behind Intuitionism was L.E.J. Brouwer, who postulated a new logic different from the classical Aristotelian logic; this intuitionistic logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The ...

See also:

Philosophy of mathematics, Philosophy of mathematics - Relation to philosophy proper, Philosophy of mathematics - Why does it work?, Philosophy of mathematics - Mathematical realism or Platonism, Philosophy of mathematics - Formalism, Philosophy of mathematics - Logicism, Philosophy of mathematics - Constructivism and Intuitionism, Philosophy of mathematics - Embodied mind theories, Philosophy of mathematics - Social constructivism or social realism, Philosophy of mathematics - Beyond the schools, Philosophy of mathematics - Quasi-empiricism, Philosophy of mathematics - Action, Philosophy of mathematics - Unification, Philosophy of mathematics - Ethics, Philosophy of mathematics - Aesthetics, Philosophy of mathematics - Language

Read more here: » Philosophy of mathematics: Encyclopedia II - Philosophy of mathematics - Constructivism and Intuitionism