Foundations of mathematics is a term sometimes used for certain fields of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, such as mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the, axiomatic set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, proof theory Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the, model theory In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or, type theory In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general. In programming language theory, a branch of computer science, type theory can refer to the design, analysis and study of type systems, although some computer and recursion theory Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive. The search for foundations of mathematics is also a central question of the philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives. The logical and structural: On what ultimate basis can mathematical statements A proposition is a sentence expressing something true or false. In philosophy, particularly in logic, a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers be called true In logic and mathematics, a logical value, also called a truth value, is a value indicating the relation of a proposition to truth?

Philosophical foundations of mathematics

Platonism

“Platonists, such as Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, (1906–1978), hold that numbers are abstract, necessarily existing objects, independent of the human mind”^[1]

The foundational philosophy of Platonist mathematical realism The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives. The logical and structural, as exemplified by mathematician Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness In 1960, the physicist Eugene Wigner published an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". In it, he observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and (2) argued that this is not just a ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world? ^[2]

Formalism

“Formalists, such as David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of (1862–1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” ^[1]

The foundational philosophy of formalism, as exemplified by David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of, is based on axiomatic set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics and formal logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the. Virtually all mathematical theorems In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic. ^[2]

Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental) appear to be true, and so on. In some cases these may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. The method can briefly be described as "going backwards from the theorems to the axioms." This contrasts with the ordinary mathematical practice of deriving theorems from axioms and computational complexity theory Computational complexity theory is a branch of the theory of computation in computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved by a computer. Informally, a. Formal logical systems also run the risk of inconsistency In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term; in Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental, this arguably has already been settled with several proofs of consistency In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term, but there is debate over whether or not they are sufficiently finitary 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. In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and to be meaningful. Gödel's second incompleteness theorem Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that establishes that logical systems of arithmetic can never contain a valid proof of their own consistency In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S!

Intuitionism

“Intuitionists, such as L. E. J. Brouwer Luitzen Egbertus Jan Brouwer [ˈlœyt.sən ɛx.ˈbɛʁ.təs jɑn ˈbʁʌu.əʁ] , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.”^[1]

The foundational philosophy of intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent or constructivism In the philosophy of mathematics, constructivism asserts that it is necessary to find 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, as exemplified in the extreme by Brouwer Luitzen Egbertus Jan Brouwer [ˈlœyt.sən ɛx.ˈbɛʁ.təs jɑn ˈbʁʌu.əʁ] , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis and more coherently by Stephen Kleene Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum Reductio ad absurdum is a form of argument in which a proposition is disproven by following its implications to a logical but absurd consequence. A particular kind of reductio ad absurdum, in its strictest sense, is proof by contradiction (also called indirect proof) where an assumption is proven false because it leads to a contradiction (for is suspect. ^[2]

Some modern theories In philosophy, theory refers to contemplation or speculation, as opposed to action. Theory is especially often contrasted to "practice" (Greek praxis, πρᾶξις) a concept that in its original Aristotelian context referred to actions done for their own sake, but can also refer to "technical" actions instrumental to some in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems, and aim to describe and analyze the actual working of mathematicians as a social group In the social sciences a group can be defined as two or more humans who interact with one another, accept expectations and obligations as members of the group, and share a common identity. By this definition, society can be viewed as a large group, though most social groups are considerably smaller. Others try to create a cognitive science of mathematics The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. It proposes to ground the foundations of mathematics in the empirical study of human cognition and metaphor, and to analyze mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.

Logicism

Logicism Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind and Gottlob Frege. Dedekind's path to logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, socialist, pacifist, and social critic. He spent most of his life in England; he was born in Wales where he also died, aged 97 and Alfred North Whitehead Alfred North Whitehead, OM was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education. He co-authored the epochal Principia Mathematica with Bertrand Russell championed this theory fathered by Gottlob Frege Friedrich Ludwig Gottlob Frege was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and.

Foundational crisis

The foundational crisis of mathematics (in German German (Deutsch, [ˈdɔʏtʃ] ) is a West Germanic language, thus related to and classified alongside English and Dutch. It is one of the world's major languages and the most widely spoken first language in the European Union. Globally, German is spoken by approximately 120 million native speakers and also by about 80 million non-native speakers: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives. The logical and structural ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes A paradox is a true statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth . Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or (such as Russell's paradox In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other) and to be inconsistent In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols (van Dalen, 2008). The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic – a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove). Meanwhile, the intuitionistic school had not attracted many adherents among working mathematicians, due to difficulties of constructive mathematics.

In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided.

A working perspective

To give an example, in number theory there is a huge body of doctrine, a tiny fraction of which has been developed in a particular axiomatic system, say Peano arithmetic (PA). Most of this work could be developed in PA; as a famous example, the prime number theorem is provable in PRA (Sudac (2001)), a much weaker theory than PA. But the working number theorist is concerned with proving theorems from initial assumptions which are obviously true using proof methods which are obviously correct, not with any particular logical system. In fact, the "crisis"-causing assertions discovered by Gödel are assertions about Diophantine equations, one of the main avenues in number theory. It may or may not be the case that there is a fundamental limit to what humans can understand about numbers (i.e., there may be true number-theoretical principles which cannot be perceived as being true by any human), but Gödel's theorem does not tell us which of these is the case, and we have no way of knowing. It may or may not be that we are required to introduce principles which are not expressible in the language of first order arithmetic in order to decide questions which are (e.g. the consistency of PA), but Gödel's theorem does not tell us which of these is the case, and again we have no way of knowing. It is often asserted that in light of Gödel's theorem one must introduce set-theoretical principles in order to decide certain number theoretical questions, but this assertion is unjustified. Gödel's theorem does not put any such constraints on the nature of the principles involved (i.e. the language in which they must be expressed). The attitude of the working number theorist is thus a reasonable one: one does not spend time thinking about such things, as there is simply no way to know. Instead one continues to prove theorems, and true principles which may be outside this or that logical system will be appealed to as required. Such principles will be introduced by people thinking about and solving actual problems, on the front-line. The problems (assuming there is no limit to what humans can understand about numbers) will be solved by people carrying on in the same way as they did before.

Attempt at a definition

Since the time of Pythagoras, mathematicians have wondered about the nature of mathematical truth, the ontology of mathematical entities and the reasons for the validity of proof and, more generally, mathematical knowledge. From the Enlightenment until the middle of the 19th century, the prevailing scientific ideology saw mathematics as the only way of reaching a truth that is final, absolute and totally independent of the human mind's capacity to understand it. The basic notions of mathematics were thought to reflect essential properties of the cosmos and the theorems to be the truths of a higher reality. This absolute faith in mathematics is reflected in the crowning of the discipline as the "Queen of the Sciences", a title whose previous holder, significantly, was theology. This view is usually termed mathematical Platonism, having its roots in the views of Plato - and, at least partly, Pythagoras before him - on the transcendent Ideas. Yet, in the 19th century this traditional belief was undermined in the minds of some people and eventually led to a serious foundational crisis in mathematics. The first of the discoveries which caused the loss of faith, dating from the time of the Renaissance, was that of the imaginary numbers (i.e. those involving the square root of minus one).

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