# Philosophy of mathematics

The philosophy of mathematics is an area of theoretical philosophy that seeks to understand and explain the premises, subject matter, method and nature of mathematics .

## starting point

1. the way of being of the mathematical objects: do they exist “really” and independently of a specific use, and if so, in what sense? What does it even mean to relate to a mathematical object? What is the character of mathematical theorems? What are the relationships between logic and mathematics? - These are ontological questions .
2. the origin of mathematical knowledge : what is the source and essence of mathematical truth ? What are the conditions of mathematical science? What are your research methods in principle? What role does human nature play in this? - These are epistemological questions .
3. the relationship between mathematics and reality : what is the relationship between the abstract world of mathematics and the material universe? Is math anchored in experience , and if so, how? How is it that mathematics “fits so perfectly with the objects of reality” ( Albert Einstein ) ? In what way do concepts such as number , point , and infinity acquire their significance that extends beyond the internal mathematical realm?

The starting point is almost always the view that mathematical propositions are apodictically certain, timeless and exact and that their correctness does not depend on empirical results or personal views. The task is to determine the conditions of the possibility of such knowledge, as well as to question this starting point.

## Realism, Platonism, Materialism

A position widespread among mathematicians is realism , represented u. a. by Kurt Gödel and Paul Erdős . Mathematical objects (numbers, geometrical figures , structures) and laws are not concepts that arise in the mathematician's head, but are given an existence independent of human thought , as Friedrich Engels emphasizes in Anti-Dühring . Mathematics is therefore not invented , but discovered. This conception corresponds to the objective, i.e. interpersonal, character of mathematics. This ontological realism is materialistic philosophy.

The classic form of realism is Platonism , according to which the mathematical objects and propositions exist separately from the material world and independent of space and time, together with other ideas such as the "good", the " beautiful " or the "divine" . The main problem of Platonism in the philosophy of mathematics is the question of how we as limited beings can recognize mathematical objects and truths when they are at home in this “heaven of ideas”. According to Gödel, this is achieved through a mathematical intuition that, similar to a sense organ , allows us humans to perceive parts of this other world. Such rational intuitions are also used by most of the classics of rationalism and in more recent debates about justification or knowledge a priori and the like. a. defended by Laurence Bonjour .

Aristotle covers his philosophy of mathematics in Books XIII and XIV of Metaphysics . He criticizes Platonism here and in many places .

## Logicism

The logicism was, among others, Gottlob Frege , Bertrand Russell , and Rudolf Carnap founded. He pursued a program to reduce mathematics completely to formal logic and consequently to understand it as part of logic . Logicists hold the view that mathematical knowledge is valid a priori . Mathematical concepts are derived from or constructed from logical concepts, mathematical propositions follow directly from the axioms of pure logic.

Gottlob Frege , who is considered one of the great thinkers of the 20th century, traced the legal structure of numerical calculation back to logical principles in his basic laws of arithmetic . Frege's construction, however, proved to be fragile even before its full publication, after Russell showed with his famous antinomy that contradictions in Frege's mathematical can be deduced. Russell informed Frege of this in a letter, whereupon he fell into a deep personal crisis. Later, the contradictions could be avoided with more complicated axiom systems, so that set theory and especially the theory of natural numbers could be justified without contradiction. However, these axioms cannot be justified purely logically in the sense of Frege's fundamental laws.

The main criticism of logicism is that it does not solve the basic problems of mathematics, but only pushes it to the basic problems of logic and thus does not provide any satisfactory answers.

## Formalism, deductivism

The formalism refers to the mathematics similar to a game based on a certain set of rules, with the strings (engl. Strings) to be manipulated. For example, in the game " Euclidean Geometry ", the Pythagorean Theorem is won by putting certain strings of characters (the axioms ) together like building blocks with certain rules (those of logical reasoning). Mathematical statements lose the character of truths (about geometric figures or numbers), they are ultimately no longer statements “about anything”.

A variant of the formalism is often referred to as deductivism . For example, the Pythagorean theorem no longer represents an absolute truth, but only a relative one : If one assigns meanings to the character strings in such a way that the axioms and the rules of inference are true, then one must follow the conclusions, B. regard the Pythagorean theorem as true. Seen in this way, formalism need not remain a meaningless symbolic game. Rather, the mathematician can hope that there is an interpretation of the character strings that B. prescribe physics or other natural sciences so that the rules lead to true statements. A deductivist mathematician can thus keep himself free from responsibility for the interpretations as well as from the ontological difficulties of the philosophers.

David Hilbert is considered to be an important early exponent of formalism. He strives for a consistent axiomatic structure of all mathematics, again choosing the natural numbers as a starting point, assuming that he has a complete and consistent system with them. A short time later Kurt Gödel defied this view with his incompleteness theorem . With this it was proven for every axiom system which includes the natural numbers that it is either incomplete or contradicting itself.

## structuralism

The structuralism considers the mathematics primarily as a science that deals with general structures, d. H. with the relations of elements within a system. To illustrate this, one can consider the administration of a sports club as an example system. The various offices (such as the board of directors, auditor, treasurer, etc.) can be distinguished from the people who take on these tasks. If you only look at the framework of the offices (and thus omit the specific people who fill them), you get the general structure of an association. The association itself with the people who have taken over the offices exemplifies this structure.

Likewise, every system whose elements have a clear successor exemplifies the structure of natural numbers: the same applies to other mathematical objects. Since structuralism does not consider objects such as numbers detached from their totality or structure, but rather sees them as places in a structure, it evades the question of the existence of mathematical objects or clarifies them as category errors . For example, two as a natural number can no longer be viewed separately from the structure of natural numbers, but rather an identifier for second place in the structure of natural numbers: it has neither internal properties nor its own structure. Accordingly, there are variants of structuralism that accept mathematical objects as existent, as well as those that reject their existence.

Problems arise with this current in particular from the question of the properties and the nature of the structures. Similar to the universality dispute, structures are obviously something that can apply to many systems at the same time. The structure of a soccer team is surely exemplified by thousands of teams . The question arises as to whether and how structures exist, whether they exist independently of systems. Other open questions concern access to structures; how can we learn about structures?

Current representatives of structuralism are Stewart Shapiro , Michael Resnik and Geoffrey Hellman .

## Other theories

The intuitionism founded by Luitzen Brouwer denies the existence of mathematical concepts outside the human mind, therefore uses constructive proofs and not those that make statements about existence without specifying a construction, which is why the proposition of the excluded third party is not used in the intuitionist formal logic used. A generalization of intuitionism is constructivism .

The Conventionalism was of Henri Poincaré developed and partly of logical Empiricists ( Rudolf Carnap , A. J. Ayer , Carl Hempel developed).

Starting from the perspective of the mathematician and at the same time going back to the epistemological criticism of Immanuel Kant , the question arises as to the categorical constitution of the human being, from which the mathematical disciplines can be derived (cf. Ernst Kleinert ).

Questions of the philosophy of mathematics are also presented in popular scientific literature. So u. a. by John D. Barrow and Roger Penrose discuss why mathematics is useful in the first place and why it fits so well in the world.

## Individual evidence

1. ^ Karl Marx / Friedrich Engels - Works. (Karl) Dietz Verlag, Berlin. Volume 20. Berlin / GDR. 1962. "Herr Eugen Dlassung's upheaval in science", III. Classification. Apriorism
2. mlwerke.de
3. Cf. In Defense of Pure Reason, A Rationalist Account of A Priori Justification, 1998, ISBN 978-0-521-59236-9 and with direct reference to the philosophy of mathematics, for example, Hartry Field: Recent Debates About the A Priori ( Memento des Originals from September 3, 2006 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (with further literature; PDF; 128 kB).
4. Stewart Shapiro, “Thinking About Mathematics,” Oxford 2000, p. 263

## literature

Introductory notes for laypeople
Specialist literature
More special
• Hermann Weyl : Philosophy of mathematics and natural science , 6th edition, Oldenbourg Verlag 1990 (English Princeton University Press 1949) (from the manual of philosophy 1927).
• Eugene Wigner : The Unreasonable Effectiveness of Mathematics in the Natural Sciences , in: Communications on Pure and Applied Mathematics, vol. 13, No. I (1960), doi : 10.1002 / cpa.3160130102 .
• Christian Thiel : Philosophy and mathematics: an introduction to their interactions and to the philosophy of mathematics , Darmstadt: Wissenschaftliche Buchgesellschaft, 1995
• John R. Lucas : The Conceptual Roots of Mathematics . Routledge London / New York (2000). ISBN 0-415-20738-X .
• Saunders Mac Lane : Mathematics: Form and Function . Springer, New York (1986). ISBN 0-387-96217-4 .