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In the mathematical field of topology, a homeomorphism or topological isomorphism (from the Greek words homoios = similar and μορφή (morphē) = shape = form (Latin deformation of morphe)) is a special isomorphism between topological spaces which respects topological properties. Two spaces with a homeomorphism between them are called homeomorphic. From a topological viewpoint they are the same.

Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Intuitively, a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.

Reference:
http://en.wikipedia.org/wiki/Homeomorphic

In structural proof theory, an analytical proof is a proof whose structure is simple in a special way. The term does not admit an uncontroversial definition, but for several proof calculi there is an accepted notion of analytic proof. For example:

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

However it is possible to extend both calculi so that there are proofs that satisfy the condition but are not analytic: a particularly tricky example of this is the analytic cut rule: this is a special case of the cut rule where the cut formula is a subformula of side formulae of the cut rule; a proof that contains an analytic cut is by virtue of that rule not analytic.

Reference:
http://en.wikipedia.org/wiki/Analytic_proof

Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972. Martin-Löf has modified his proposal a few times; his early, impredicative formulations were inconsistent as demonstrated by Girard's paradox, and later formulations were predicative. He also proposed extensional and then intensional variants of intuitionistic type theory.

Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs. This identification is usually called the Curry–Howard isomorphism, which was originally formulated for propositional logic and simply typed lambda calculus. Type Theory extends this identification to predicate logic by introducing dependent types, that is types which contain values. Type Theory internalizes the interpretation of intuitionistic logic proposed by Brouwer, Heyting and Kolmogorov, the so called BHK interpretation. The types of Type Theory play a similar role as sets in set theory but functions definable in Type Theory are always computable.

Reference:
http://en.wikipedia.org/wiki/Intuitionistic_type_theory

In proof theory and mathematical logic, the sequent calculus is a widely known proof calculus for first-order logic (and propositional logic as a special case of it). The term "sequent calculus" applies both to a family of formal systems sharing a certain style of formal inference, and to its individual members, of which the first, and best known, is known under the name LK, distinguishing it from other systems in the family, such as LJ. Another term for such systems in general is Gentzen systems.

The sequent calculus LK was introduced by Gerhard Gentzen as a tool for studying natural deduction in 1934. It has turned out to be a very useful calculus for constructing logical derivations. The name itself is derived from the German term Logischer Kalkül, meaning "logical calculus." Sequent calculi and the general concepts relating to them are used widely in the whole field of proof theory and mathematical logic.

Reference:
http://en.wikipedia.org/wiki/Sequent_calculus

In computer science, anti-patterns are specific repeated practices that appear initially to be beneficial, but ultimately result in bad consequences that outweigh the hoped-for advantages.

According to the authors of the book, AntiPatterns, there must be at least two key elements present to formally distinguish an actual anti-pattern:

1. Some repeated pattern of action, process or structure that initially appears to be beneficial, but ultimately produces more bad consequences than beneficial results, and
2. A refactored solution that is clearly documented, proven in actual practice and repeatable.

Often pejoratively named with clever oxymoronic neologisms, many anti-pattern ideas amount to little more than mere mistakes, rants, unsolvable problems, or just plain bad practices to be avoided if possible. Sometimes called pitfalls or dark patterns the general, informal use of the term has however come to refer to classes of commonly reinvented bad solutions to problems. Thus, there are always many more candidate anti-patterns under debate which could never be formally considered as anti-patterns.

The term anti-pattern originates in computer science, inspired by the Gang of Four's book Design Patterns, which displayed examples of high-quality programming methods. The authors named these methods design patterns, by analogy with the same term used in architecture. The book AntiPatterns (by William Brown, Raphael Malveau, Skip McCormick and Tom Mowbray, and more recently Scott Thomas) describes anti-patterns as natural counterparts or follow-ons to the study of design patterns. By formally describing repeated mistakes, one can recognize the forces that lead to their repetition and learn how others have refactored themselves out of these broken patterns. Anti-patterns are not mentioned in the first edition of Design Patterns, which predated the term anti-pattern; however, one of the original Gang of Four authors, the late John Vlissides, offers an endorsement on the back cover.

Avoiding anti-patterns requires diagnosing them as early as possible in the software life-cycle. The concept of anti-patterns is readily applied to engineering in general.

If a software application is garnished with the unfortunate combination or mix of too many anti-patterns, then it may be known under the expression "full monty".

Reference:
http://en.wikipedia.org/wiki/Anti-pattern

Antinomy (Greek αντι-, against, plus νομος, law) literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology.

The term acquired a special significance in the philosophy of Immanuel Kant, who used it to describe the equally rational but contradictory results of applying to the universe of pure thought the categories or criteria of reason proper to the universe of sensible perception or experience (phenomena). Reason cannot here play the role of establishing rational truths because it goes beyond possible experience and is applied to the sphere of that which transcends it.

For Kant there are four antinomies connected with

1. the limitation of the universe in respect of space and time,
2. the theory that the whole consists of indivisible atoms (whereas, in fact, none such exist),
3. the problem of freedom in relation to universal causality
4. the existence of a necessary being

about each of which pure reason contradicts the empirical, as thesis and antithesis. This was part of Kant's critical program of determining limits to science and philosophical inquiry. Kant claimed to solve these contradictions by saying, that in no case is the contradiction real, however really it has been intended by the opposing partisans, or must appear to the mind without critical enlightenment. It is wrong, therefore, to impute to Kant, as is often done, the view that human reason is, on ultimate subjects, at war with itself, in the sense of being impelled by equally strong arguments towards alternatives contradictory of each other. The difficulty arises from a confusion between the spheres of phenomena and noumena. In fact no rational cosmology is possible.

It can also be argued that antinomies do not highlight limitations in the power of logical reasoning. This is because the conclusion that there is a limitation is (supposedly) derived from the antinomy by logical reasoning; therefore any limitation in the validity of logical reasoning imposes a limitation on the conclusion that there is a limitation on logical reasoning. (This is an argument by self-reference.) In short, in terms of the validity of logical reasoning as a whole, antinomies are self-isolating: they are like scattered discontinuities within the field of logic, incapable of casting doubt on anything else but themselves.

This carefree position is incompatible with the principle of explosion. In mathematical logic, antinomies are patently not self-isolating, and are usually seen as disasters for the formal system in which they arise (as Russell's paradox in Frege's work).

Reference:
http://en.wikipedia.org/wiki/Antinomy

The noumenon (plural: noumena) classically refers to an object of human inquiry, understanding or cognition. The term is generally used in contrast with, or in relation to, "phenomenon" (plural: phenomena), which refers to appearances, or objects of the senses. A phenomenon is that which is perceived; A noumenon is the actual object that emits the phenomenon in question.

The philosopher Immanuel Kant used the term noumenon synonymously with the phrase thing in itself (German: Ding an sich).

Reference:
http://en.wikipedia.org/wiki/Noumena

Epistemology or theory of knowledge is a branch of philosophy concerned with the nature and scope of knowledge. The term was introduced into English by the Scottish philosopher James Frederick Ferrier (1808-1864).

Much of the debate in this field has focused on analyzing the nature of knowledge and how it relates to similar notions such as truth, belief, and justification. It also deals with the means of production of knowledge, as well as skepticism about different knowledge claims. In other words, epistemology primarily addresses the following questions: "What is knowledge?", "How is knowledge acquired?", and "What do people know?"

Reference:
http://en.wikipedia.org/wiki/Epistemology