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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists. See constructive proof. Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will...
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ...
Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Constructivism does not, and is entirely consonant with an objective view of mathematics. In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ...
Constructivist mathematics
Constructivist mathematics use constructivist logic, which is essentially a removal of the law of the excluded middle from classical logic. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable as theorems. It is just that the law is not assumed as an axiom. (The law of non-contradiction, on the other hand, is still valid.) Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...
The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it...
For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers,  is a theorem (where x,y,z... are the free variables in the proposition p). In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence is not assumed to extend to those which talk about infinite collections. Heyting arithmetic is the basic arithmetic of intuitionism (not to be confused with Heyting algebra). ...
In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...
In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In logic, the laws of bivalence, excluded middle, and non-contradiction are related, but not the same. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In fact, L.E.J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as something which was abstracted from finite experience, and which was then applied by mathematicians to the infinite, without justification. For instance, Goldbach's conjecture is the assertion that every even number (greater than 2) is the sum of two prime numbers. It is possible to test for any particular even number whether or not it is the sum of two primes (for instance by exhaustive search), so it is fair to say of any one of them that it is either the sum of two primes, or it is not. And so far, every one thus tested has in fact been the sum of two primes. Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ...
Theory of justification is that part of epistemology that attempts to understand the justification of statements and beliefs. ...
In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...
In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...
But there is no known proof that all of them are so, nor any known proof that not all of them are so. Thus to Brouwer, one cannot say "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems; to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution.
By doing away with the law of the excluded middle as an axiom, the remaining logical system has an existence property which classical logic does not: whenever  is proven constructively, then in fact P(a) is proven constructively for (at least) one particular  . Thus the proof of the existence of a mathematical object is tied to the possibility of its construction. In mathematical logic, the disjunction property is satisfied by a logic if whenever is a theorem, then either is a theorem, or is a theorem. ...
Example from real analysis In classical real analysis, one way to define a real number is as a Cauchy sequence of rational numbers. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
In mathematics, there are a number of ways of defining the real number system as an ordered field. ...
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In constructive mathematics, one way to construct a real number is as a function f that takes a positive integer n and outputs a rational f(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that In mathematics, a function returns a unique output for a given input. ...
 so that as n increases, the values of f(n) get closer and closer together. We can use f and g together to compute as close a rational approximation as we like to the real number they represent.
Under this definition, a simple representation of the real number e is: The mathematical constant e (occasionally called Eulers number after the Swiss mathematician Leonhard Euler, or Napiers constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. ...
 This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists (in a classical sense) a member in the sequence after which all members are closer together than that distance. In the constructive version, it is required that, for any given distance, it is possible to actually specify the point in the sequence where this happens (this required specification is often called the modulus of convergence). In fact, the standard constructive interpretation of the mathematical statement In mathematics, an existence theorem is a theorem with a statement beginning there exist(s) .., or more generally for all x, y, ... there exist(s) .... That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. ...
In mathematical logic, the BHK interpretation of intuitionistic predicate logic was proposed by L. E. J. Brouwer, Arendt Heyting and independently by Kolmogorov. ...
 is precisely the existence of the function computing the modulus of convergence. Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement "for all... there exists..."
This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different versions of constructivism diverge on this point. Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the recursive functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers. In mathematics, a function returns a unique output for a given input. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are computable in some intuitive sense. ...
In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ...
Cardinality To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet Cantor's diagonal argument shows that real numbers have higher cardinality. Furthermore the diagonal argument seems perfectly constructive. To identify the real numbers with the computable numbers would then be a contradiction. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...
And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number which doesn't fit, and thereby proves a contradiction. We can indeed enumerate algorithms to construct a function T from the natural numbers onto the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (T is a partial function), so this fails to produce the required bijection. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable. They are, therefore exactly countable. However this reasoning is not constructive, as it still does not construct the required bijection. In fact the cardinality of sets fails to be totally ordered (see Cantor–Bernstein–Schroeder theorem). In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ...
In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of...
Attitude of mathematicians Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because of the limitations that it poses for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists" [1]. (The law of excluded middle is not valid in constructivist logic.) Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework. Nevertheless, not every mathematician accepts that Bishop did so successfully, since his book is necessarily more complicated than a classical analysis text would be. In any case, most mathematicians see no need to restrict themselves to constructivist methods, even if this can be done. David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
1928 (MCMXXVIII) was a leap year starting on Sunday (link will take you to calendar). ...
In logic, the law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P ∨ ¬P). ...
Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...
Errett Bishop (1928-1983) was an American mathematician who managed to prove versions of the most important theorems in real analysis within the constructivist framework. ...
1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar. ...
[1] Translation from the Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/mathematics-constructive/. The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ...
Mathematicians who have contributed to constructivism Errett Bishop (1928-1983) was an American mathematician who managed to prove versions of the most important theorems in real analysis within the constructivist framework. ...
Paul Lorenzen (born March 24, 1915 in Kiel, Germany) is a philosopher and mathematician. ...
Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...
Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ...
Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician. ...
Branches Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...
Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ...
In mathematics, constructive analysis is mathematical analysis done according to the principles of constructivist mathematics. ...
Computability logic is a formal theory of computability, introduced by Giorgi Japaridze in 2003. ...
See also In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ...
Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ...
In the philosophy of mathematics, 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. ...
Game semantics (German: dialogische Logik) is an approach to the semantics of logic that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player. ...
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ...
The Critique of Pure Reason (Kritik der reinen Vernunft), first published in 1781 with a second edition in 1787, is widely regarded as the most influential and widely read work of the German philosopher Immanuel Kant and one of the most influential and important in the entire history of Western...
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