Fundamental crisis in mathematics

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The fundamental crisis in mathematics was a phase of uncertainty among the mathematical public at the beginning of the 20th century, beginning with the publication of Russell's Antinomy in 1903 and ending around 1930. In the 1920s, the crisis culminated in the fundamental dispute in mathematics , which was essentially held by Main exponent of formalism , David Hilbert , and of that of intuitionism , LEJ Brouwer , was carried out. At the end of this controversy, the impression has prevailed that classical mathematics has overcome its basic problems and can expand to modern mathematics without any cuts in its inventory (which is triumphant , for example in computer networks , weather forecasting, in plant control including space travel and materials research has competed without equal). In particular, it has been shown that evidence by contradiction using the principle of tertium non datur remains a viable and successful evidence procedure.

Prehistory and trigger of the crisis

The discovery of irrational numbers and thus incommensurability by the Pythagorean Hippasus of Metaponton used to be described as the first fundamental crisis in mathematics . It was assumed that a previously prevailing fundamental conviction had been eliminated, according to which all phenomena could be expressed as integer numerical ratios and thus there could be no incommensurability. In fact, however, the existence of such a belief has not been established among the early Pythagoreans. Therefore there is no reason to assume a mathematical or philosophical crisis due to the discovery, rather there are clear indications against it.

Occasionally the uncertainty of mathematicians in the 18th and early 19th centuries when calculating with infinitesimal quantities is seen as a fundamental crisis. However , this forerunner does not deserve the term “ fundamental crisis” insofar as the mathematical public did not yet have any fundamental awareness . This awareness only developed in the second half of the 19th century following the discovery of non-Euclidean geometries . This made it clear for the first time that there can be not just one math, but several different maths, that certain sentences can be true in one mathematical system and false in another. As a result, people began to pay more attention to the system in which they were currently moving, or to try harder to make the respective system tangible, for example by axiomatizing it according to the Euclidean model .

While Giuseppe Peano had put the arithmetic of natural numbers and Moritz Pasch and Hilbert the geometry on a contemporary axiomatic basis, Gottlob Frege tried a big hit: He wrote his "Basic Laws of Arithmetic ", those of all mathematics, among other things the Analysis and Cantor's set theory , should provide a basis, namely a purely logical, no mathematical symbolism basis containing. Frege was thus the founder of logicism . But even the axioms given in the first volume of the Basic Laws of 1893 were inconsistent ; Russell's famous antinomy could be derived from them, as its discoverer Bertrand Russell Frege pointed out in a letter in 1902.

Not all mathematicians paid enough attention to this antinomy that one could speak of a continuous awareness of crises, but its discovery was decisive for all those who dealt with questions of foundation and axiomatics. Not only was Frege's attempt to be regarded as a failure, but the antinomy particularly concerned the so-called naive set theory , which Georg Cantor had developed into the basis of all mathematics around 1880. Cantor himself was an avowed Platonist , so he was of the opinion that the sets and all other mathematical objects that could be formed from them are ontological realities that objectively exist in their own, purely spiritual sphere of being, independent of the viewing subject. This view was now deeply shaken, because from the point of view of naive set theory there was nothing wrong with that set that Russell proved to be contradictory, but because of its contradictions it could not be ascribed to any ontological reality. This resulted in the question of drawing the line between consistent and contradicting sets - another motive for the axiomatic program - and the insight that the philosophical consideration of mathematical objects as objective spiritual realities should not be presupposed without reflection.

While Frege then stopped pursuing his program, others tried alternative foundations for mathematics. These experiments are usually divided into three schools:

Logicism

Russell and AN Whitehead published the Principia Mathematica (PM) between 1910 and 1913, a three-volume work in which, like Frege, they tried to reduce all mathematical concepts to logical ones and to prove the fundamental theorems on the basis of axioms in a strictly logical manner. They avoided antinomies like Russell's through a gradual structure, the so-called type theory . This gradation does not appear to be logically compelling, but rather as an ontological assertion about the world of logical objects. Such ontological moments became particularly clear in the PM in the axiom of infinity (“There are infinite sets”) and in the axiom of reducibility . These axioms have often been criticized as not logically evident and therefore not fitting into the program of logicism. The failure of logicism for mathematics became explicit in them.

In the further course of the fundamental crisis, the PM nevertheless formed an important point of reference because, compared to Frege's terminology, they introduced a simpler logical notation and were exemplary in terms of the rigor of the formal argumentation. Insofar as this fundamental attempt could hardly be mastered by a single reader due to the scope and in addition it appeared implausible in some places in the sense of logicism, one can see in its position as a standard work of the time a symptom of the fundamental crisis: The PM was accepted despite all difficulties at hand, for there was nothing better.

formalism

Compared to logicism, formalism emphasized the independence of mathematics, and thus from the outset did not seek to trace it back to logic. In view of the antinomies, formalism did not want to address the extent to which the objects of mathematics form their own sphere of being, rather it was anti-ontological and withdrew to the point of view that mathematical objects only existed as signs (on paper, on the blackboard, ... ). The free operation with signs according to predetermined mathematical-logical rules and axioms is justified by the success of mathematics and can only be limited in one respect: the operation must not generate any contradictions. The formalism therefore demanded that axiom systems always be proven to be free of contradictions. Such proofs could only be given in the context of a new mathematical discipline, metamathematics , which takes the axiom systems as its object. The main exponent of formalism and the metamathematic program was David Hilbert , who was able to see his demand for proofs of freedom from contradiction formulated as early as 1900 at the mathematics congress in Paris confirmed by Russell's antinomy, and who formulated his principles of metamathematics as a so-called " Hilbert program " in various articles. Hilbert is - also due to his work in other areas of mathematics - as the most influential mathematician of the first third of the 20th century.

In addition to Hilbert's own axiomatization of geometry (1899), the axiomatization of set theory by Hilbert's pupil Ernst Zermelo in 1907 was important, but initially it was hardly perceived as an attempt at the foundation of all mathematics - especially since the proof of consistency of his axioms required according to Zermelo was a long way off - and was only able to assert itself after the additions by AA Fraenkel in 1921 and Thoralf Skolem in 1929 to the so-called Zermelo-Fraenkel set theory as the standard foundation of mathematics and thus superseded the PM.

Intuitionism

Compared to formalism, intuitionism asserted that mathematical objects were more than mere signs. However, he did not locate the objects in a sphere of being independent of humans like Platonism, but said they existed exclusively in the human mind, in "intuition" as soon as it generates them. The intellectual production is by no means a linguistic one, and therefore not one that can be reduced to logic. The linguistic signs and the logical symbolism are only justified as representatives of the intellectual objects. Intuitionism criticized the formalistic view as an empty play of symbols, which often did not represent any mathematical objects or any mental operations. In particular, the “objects” of transfinite set theory, that is, infinite sets, are considered suspicious to him; with regard to the infinite, he is more inclined to understand the potential than the actual infinity . While this is a more philosophical question, the criticism of the logical theorem of the excluded third party , which, according to intuitionism, has no counterpart in mental operation and is therefore unjustified, had drastic effects on mathematics, as it is an important principle of proof. The main exponent of intuitionism LEJ Brouwer therefore developed a set theory independent of the theorem of the excluded third party, and presented it to the public in several essays from 1918 onwards. Since these elaborations of the intuitionistic fundamental attempt were kept very technical and therefore had practically no effect, the charismatic and polarizing personality of Brouwer and the fact that in the first few years after the, the charismatic and polarizing personality of Brouwer must be used as the reasons that a fundamental dispute broke out over intuitionism World War II the majority of the mathematical public was opposed to a formalistic view of mathematics.

The fundamental dispute between intuitionism and formalism

Weyl's appearance - Hilbert's reaction

While Brouwer's publications themselves met with little response, the essay "On the new fundamental crisis in mathematics" by Hermann Weyl from 1921 is considered to be the trigger for the fundamental dispute. After a meeting, Weyl was deeply impressed by Brouwer's personality as well as by his intuitionism and made himself his representative in the partly polemical essay by declaring: “Brouwer - this is the revolution!” Weyl's teacher Hilbert certainly responded to him too as a personal attack in the following year just as violent; he accused Weyl and Brouwer of a "coup attempt". In order to save “Cantor's Paradise” from this, he resumed his metamathematic program, on which he had not published anything further in the 1910s.

The following years were marked by a steadily growing number of articles by more and more authors in more and more languages, who spread the argument between intuitionism and formalism in the mathematical public. Hesseling counted over 250 scientific papers that responded to the dispute in the 1920s and early 1930s. There were several difficulties in getting to the heart of the two positions:

  • There was no more comprehensive, understandable account of intuitionism; Brouwer's philosophical position was only published in Dutch, his criticism of classical mathematics was spread over various, sometimes difficult-to-read writings, and Weyl's presentation in his essay reflected Brouwer's views only to a limited extent.
  • Hilbert's formalistic-metamathematic program was just emerging. To what extent proofs of freedom from contradiction were even possible for arithmetic and set theory was difficult to assess.

Due to the active publication activity of the Hilbertkreis - u. a. When his pupil Wilhelm Ackermann provided proof of the absence of contradictions for the principle of the excluded third party in 1925 - trust gradually grew in Hilbert's attempt at rescue. At the same time, the initial sympathy for intuitionism suffered from the gradually becoming clearer consequences that the circumcisions that it demanded would have had on the overall structure of mathematics and thus also its applicability. As early as 1924, Weyl stated that the application aspect of mathematics spoke for Hilbert.

Brouwers expulsion from the editorial board of the Mathematische Annalen

A preliminary decision was made in 1928. For the first time since the World War, mathematicians from Germany were invited to the international mathematicians' congress in Bologna, but they were still not entitled to vote. The Dutchman Brouwer expressed his solidarity with the Germans and called on them to boycott the congress in advance. However, this appeal only met with a positive response from a few nationalist-minded German mathematicians; the majority decided to participate, which was also clearly supported by Hilbert. In Bologna, for example, the Germans formed the second largest group after the Italians, and Hilbert gave a lecture on his formalistic basic program, without Brouwer being able to counter anything - he had not even come.

Hilbert had to see Brouwer's call for a boycott as an attempt to use political means to dispute his leading role among mathematicians. He reacted to this a few days after the Bologna Congress with a measure that is probably unique to this day in terms of science policy. Hilbert was one of the three main editors of the Mathematische Annalen , the most important specialist journal at the time, Brouwer was one of several co-editors. Without coordinating this with the other two main editors, Hilbert Brouwer informed him in a letter that he would be excluded from the co-editorship. Although this caused greater irritation, so that thereupon the entire editorship of the Annals had to be formed anew, but Hilbert's will to exclude Brouwer was complied with. The fact that Brouwer did not publish anything on intuitionism for many years after 1928 can probably be attributed to his frustration with the disembarkation.

Brouwer's retreat

After Brouwer's withdrawal, the intuitionist basic program faded more and more into the background in the public discussion, the question remained as to what extent the formalism offered a sufficient justification for current mathematical practice. Gödel's incompleteness sentences put a damper on the increasing optimism on this issue , but they in no way signified the end of the Hilbert program, but merely required a modification of the program. The proof theory based in the Hilbert program developed into a very fruitful part of basic mathematical research after the Second World War ; today it represents an important interface between philosophy and mathematics, insofar as its rigorous method fully meets mathematical requirements, but its main issue is a philosophical-scientific theoretical one : Which logical assumptions and which axioms do I need at least to be able to prove this and that theorem? Today's evidence theorists, however, care less whether these assumptions - as Hilbert had in mind - can be justified from the intuitionist or, more generally, from a non-Platonic standpoint; They mostly leave these questions to the “real” philosophers.

End of the dispute

Gödel already mentioned his incompleteness results, which he published in 1931, at a conference on the philosophy of science in Königsberg in autumn 1930, which can perhaps be seen as a kind of conclusion to the fundamental crisis. There Rudolf Carnap spoke about logicism, Arend Heyting about intuitionism and Johann von Neumann about formalism, and all three speakers chose an emphatically conciliatory style. This coming together was made possible by the insight that all parties had made their contribution in the previous decades to first become clear about the problem, then to look for solutions, and ultimately to be sufficiently sure about the fundamentals of mathematics and thus to end the mood of crisis: Logicism had successfully fought for the insight that every mathematical reasoning can ultimately be traced back to logical reasoning, which greatly increased the transparency of mathematical reasoning; The formalism had taken up this, but distanced itself from the failed attempt to reduce mathematical objects (numbers, quantities, ...) to purely logical objects. Intuitionism then criticized the early formalistic axiomatics by pointing out that there was an absolutely secure core of mathematics that was beyond any doubt, namely finite arithmetic, which every mathematician had at his disposal without any axiomatic or otherwise linguistic-logical processing is necessary. Hilbert took up this insight in his program, which consisted in proving the consistency of infinite mathematics exclusively with the means of finite arithmetic. Even if this has not yet succeeded in a satisfactory manner, the metamathematic reflection on the fundamentals in its multitude of very differentiated results has led to the fear of new antinomies being low today. But Gödel's incompleteness theorems remain valid. Complete certainty about the consistency of the axiom systems of essential parts of mathematics (such as arithmetic) cannot be obtained.

literature

classic
  • David Hilbert (1922): New foundations of mathematics: First communication. Treatises from the seminar at Hamburg University 1, 157–177.
  • David Hilbert (1928): The basics of mathematics . Treatises from the seminar at Hamburg University 6, 65–85.
  • Hermann Weyl (1921): About the new fundamental crisis in mathematics. Mathematical Journal 10, 39-79.
  • Hermann Weyl (1924): Marginal remarks on main problems in mathematics. Mathematical Journal 20, 131–150.
Secondary literature
  • Paul Benacerraf / Hilary Putnam (eds.): Philosophy of Mathematics , Cambridge University Press 2nd A. 1983.
  • M. Detlefsen: Hilbert's Program , Dordrecht 1986.
  • Dirk van Dalen (1990): The War of the Frogs and the Mice, or the Crisis of the 'Mathematische Annalen'. The Mathematical Intelligencer 12, 17-31.
  • Hesseling, Dennis E. (2003): Gnomes in the fog: the reception of Brouwer's intuitionism in the 1920s. Birkhäuser, Basel 2003.
  • Christian Thiel (1972): Fundamental crisis and fundamental dispute. Study of the normative foundation of the sciences using the example of mathematics and social science. Hain, Meisenheim am Glan 1972.
  • Herbert Mehrtens (1990): Modern Language Mathematics: A History of the Controversy about the Fundamentals of the Discipline and the Subject of Formal Systems. Frankfurt a. M .: Suhrkamp 1990.
  • Stuart Shapiro : Thinking About Mathematics , Oxford: OUP 2000, v. a. Cape. 6 (Formalism) and 7 (Intuitionism).

Web links

Individual evidence

  1. Walter Burkert contradicts the assumption of an ancient fundamental crisis in mathematics or the philosophy of mathematics : Wisdom and Science. Studies on Pythagoras, Philolaos and Plato , Nuremberg 1962, pp. 431–440. The same conclusion come Leonid Zhmud : science, philosophy and religion in early Pythagoreanism , Berlin 1997, pp 170-175, David H. Fowler: The Mathematics of Plato's Academy , Oxford 1987, pp 302-308 and Hans-Joachim Waschkies : Beginnings of arithmetic in the ancient Orient and among the Greeks , Amsterdam 1989, p. 311 and note 23. The hypothesis of a crisis or even a fundamental crisis is unanimously rejected in today's specialist literature on ancient mathematics.
  2. Weyl 1921, p. 56
  3. Hilbert 1922, p. 160
  4. Hesseling 2003, p. 346
  5. Weyl 1924, pp. 149f.
  6. The course of the affair is in v. Dalen traced in detail in 1990.
  7. s. Carnap 1931, Heyting 1931, v. Neumann 1931