Philosophy of mathematics
The philosophy of mathematics is an area of theoretical philosophy, which seeks, conditions, subject matter, method and nature of mathematics to understand and explain.
Starting point
Systematically fundamentally are questions about
The starting point is almost constantly the view that mathematical propositions are apodictic certainly timeless and exactly and their accuracy depends neither on empirical evidence, nor by personal views. Task is to determine the conditions of the possibility of such knowledge as to question even this starting point.
Realism, Platonism
A widespread among mathematicians position is the realism, including representatives of by Kurt Godel and Paul Erdős. Mathematical objects (numbers, geometric shapes, structures) and laws are not concepts that arise in the mind of the mathematician, but it is attributed to them an independent existence by human thought. Mathematics is therefore not invented, but discovered. With this view of the objective, ie interpersonal character of mathematics are met. This ontological realism is incompatible with all varieties of materialistic philosophy.
The classic form of realism is Platonism, according to which mathematical objects and sentences detached from the material world and independent of space and time exist, " Divine " along with the other ideas such as the " good guys", the " beauty", or the. The main problem of Platonism in the philosophy of mathematics is the question of how we can recognize as a limited being the mathematical objects and truths if they are located in this " heaven of ideas ." According to Gödel, this makes a mathematical intuition, which, similar to a sense organ, makes us humans perceive parts of this other world. Such rational intuitions are also used by most of the classics and rationalism in recent debates about justification or knowledge a priori, inter alia, defended by Laurence Bonjour.
Aristotle treats his philosophy of mathematics in the books XIII and XIV of metaphysics. He criticizes Platonism here and in many places.
Logicism
Logicism was founded, among others, Gottlob Frege, Bertrand Russell and Rudolf Carnap. According to this thesis, the mathematics can be completely attributed to the formal logic and is therefore to be understood as a part of logic. Logicists take the view that mathematical knowledge is a priori valid. Mathematical concepts are derived from logical concepts, mathematical theorems follow directly from the axioms of pure logic.
Gottlob Frege, who is considered one of the great thinkers of the 20th century, resulted in its Basic Laws of Arithmetic, the laws of building number arithmetic on logical principles back. But Frege's construction proved before its full publication as brittle after Russell showed with his famous antinomy that circular arguments and contradictions he founded on formal logic mathematical building took the foundation. Russell pointed this out in a letter to Frege, whereupon he fell into a deep personal crisis. Later could be avoided with more complicated axiom systems, the contradictions, so that the set theory and in particular the theory of natural numbers could be established without contradiction.
Is criticized on logicism especially that it does not solve the basic problems of mathematics, but merely pushes on basic problems of logic and thus are no satisfactory answers.
Formalism, deductivism
The formalism understands the math similar to a game based on a certain set of rules, with the strings (English strings) are manipulated. For example, in the game " Euclidean geometry " of the Pythagorean theorem is obtained by certain strings ( the axioms ) with certain rules ( which logical reasoning ) as building blocks are joined together. Mathematical statements thus lose the character of truths (about about geometric figures or numbers), they are ultimately no more statements "about anything ".
As deductivism is often a variant of the formalism referred to, in the example, the Pythagorean theorem is not an absolute truth more, but only a relative: If you assign the strings in a way meanings, so that the axioms and rules of inference are true, then you have the conclusions, such as the Pythagorean theorem, as being true. Seen in the formalism must not remain a meaningless symbolic play. The mathematician may hope that there is an interpretation of strings, for example, pretending it physics or other natural sciences, so that the rules lead to true statements rather. A deductivist mathematician so can keep clear both from the responsibility for the interpretations and of the ontological problems of philosophers.
David Hilbert is considered an important early proponent of formalism. It aims to achieve a consistent axiomatic construction of the entire mathematics, which he in turn selects the natural numbers as a starting point, assuming thus a complete and consistent system without having. This view has a short time later Kurt Gödel with his incompleteness theorem revoked the floor. This made for each axiom system that includes the natural numbers, proved that it is either incomplete or contradictory in itself.
Structuralism
Structuralism considers the mathematics primarily as a science that deals with general structures, ie with the relations of elements in a system. To illustrate this, one can consider as an example of "system " as the management of a sports club. The various offices (such as board of directors, auditor, treasurer, etc.) can differ from the people who take on these tasks. If one considers only the skeleton of the offices (and thus the actual persons to fill, " omit " ), then you get the general structure of an association. The club itself with the persons who have taken over the offices, illustrates this structure.
Also exemplified each system whose elements have a clear successor, the structure of the natural numbers: The same applies for other mathematical objects. Since such numbers are not separated from its whole structure or structuralism objects considered, but " in a structure places " sees more than he evades the question of the existence of mathematical objects or clears them as a category mistake. For instance, the two no longer be considered in isolation as a natural number from the structure of the natural numbers, but an identifier for the " second place in the structure of natural numbers ": it has neither internal properties or its own structure. Accordingly, there are both variants of structuralism, take the mathematical objects to exist, as well as those who reject its existence.
Problems arise with this particular flow from the question of the properties and the existence of the structures. Similar to the universals apparently it is for structures to be something that can play many systems simultaneously. Thus, the structure of a football team is certainly exemplified by thousands of teams. This raises the question of whether and how structures exist, whether they exist as independent systems. Other open questions concerning access to structures; how can we learn about structures?
Current representatives of structuralism are Stewart Shapiro, Michael Resnik and Geoffrey Hellman.
Other theories
The Luitzen founded by Brouwer intuitionism denies the existence of mathematical concepts outside of the human mind, so used constructive proofs and not those that existential statements ohn make a design, which is why not used in the used intuitionistic formal logic of the law of excluded middle is acknowledged. A generalization of intuitionism is constructivism.
The conventionalism was developed by Henri Poincaré and partly by logical empiricists (Rudolf Carnap, Alfred Jules Ayer, Carl Hempel ) further developed.
Another important issue is the justification of a mathematical theory. Since mathematics (unlike the natural sciences) can not be tested experimentally, one looks for reasons, a certain mathematical theory to hold true (see also epistemology ).
From the perspective of the mathematician starting and at the same time resorting to the epistemology of Immanuel Kant, the question is after the categorical constitution of man, from which the mathematical disciplines can be derived ( cf. Ernst Kleinert ).
Even in popular scientific literature questions the philosophy of mathematics are presented. Thus, inter alia, discussed by John D. Barrow and Roger Penrose, why mathematics is useful at all, and why it fits so well on the world.