The extremes of intuitionism and the failure of logicism forced the most famous and influential mathematician of the era, D. Hilbert, formulate your own view on the foundations of mathematics.
However, Hilbert by that time already acted as the legislator of the modern understanding of the axiomatic method in mathematics, which we continue to use so far and which has the most direct relation to the formalist program justification of mathematics. The Hilbertian understanding of the axiomatic method found a classical expression in his work "The Foundations of Geometry", published in 1899 . Reasoning about such a visual object, like Euclidean geometry, reaches Hilbert's unprecedented formalization. By the time R. Dedekind and D. Peano already constructed an axiomatics for the arithmetic of natural numbers (1888-1889), consisting of five axioms and based on three primary concepts - "natural number", "unit", "the following natural number & quot ;. Gilbert did the same for geometry. Gilbert obtained a system of 20 axioms, with three primary types of objects - points & quot ;, direct & quot ;, planes and a few primary relationships - membership & quot ;, between & quot ;, congruence & quot ;. Memories preserved the famous phrase that Gilbert threw in one of the conversations in 1891, which maximally capaciously and simultaneously visually reveals the essence of the Hilbertian understanding of the axiomatic method: "It is necessary that words such as" point, "" straight line, "" plane, "" in all sentences of geometry it was possible to replace, for example, with the words "table", "chair", "beer mug" [6, p. 237] (according to O. Blumenthal's recollections, the interlocutors were at the time at the railway station in Berlin [37, p. 79]). The meaning of these words is not difficult to understand: everything you need to know about primary objects and their relations for the deployment of the entire system of geometry must be explicitly spelled out in the axioms. This is a formal understanding of axiomatics, when all questions connected with the truth or, at least, the psychological persuasiveness of each of the axioms are taken beyond the scope of consideration. Instead, the system of axioms as a whole must satisfy the requirements 1) completeness - of axioms sufficient to derive any theorem of this theory; 2) independence - there is nothing superfluous in this system, removing any of the axioms will inevitably lead to the inability to prove any theorems; 3) no inconsistency - it is impossible to derive logically mutually exclusive results from each other.
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The proof of the consistency of a formal axiomatic system serves as a necessary compensation for the loss of axioms of visual meaning. Consistency at that time was proved by the method of reduction. Hilbert showed in his 1899 work that the system described by axioms proposed by him for Euclidean geometry is also realizable in the set of real numbers. Consequently, if there was a contradiction in the system of axioms proposed by him, it would exist in the set of real numbers. Axiomatics for a set of real numbers Hilbert constructed in the article "About the concept of a number [11, p. 315-3211 in the same year 1899, but the question of its consistency remained open.
In the list of famous "Hilbert problems", formulated by him in the report "Mathematical problems" [1, p. 11-64] at the Second International Congress of Mathematicians in Paris in 1900, there were several important points on the foundations of mathematics. The second problem was: "Explore the consistency of the axioms of arithmetic," suggesting that the next be done after the "Geometry Foundations" step. The sixth proposed to axiomatize the physical disciplines in which mathematics plays an important role. Hilbert was convinced that the axiomatization of theories is the main path not only for pure mathematics, but for all other areas of human knowledge as they mathematize. He later made a report on this "Axiomatic Thinking" (1917) [10, p. 409-417].
Hilbert formulated his approach to the paradox problem of set theory in the report "On the foundations of logic and arithmetic" at the Third International Congress of Mathematicians in Heidelberg in 1904. It was easy to guess that he was starting from his understanding of the axiomatic method: an explicit formulation of axioms and a proof of the consistency of the resulting axiomatic system are needed. It is the consistency of the corresponding mathematical theory that decides the question of the existence of certain mathematical objects. For the latter, it is impossible to use the method of information again and again; at some level, we need to get a direct proof of consistency. What is this level? On the one hand, the logicians believed that the last level is logic, it does not require proof of consistency, it is enough to carry out the reduction to logical axioms. On the other hand, Henri Poincare saw in the constructions of the logicians the vicious circle, for him arithmetic is already hidden in the logic that the logicians want to rely on.
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According to Hilbert, we should not reduce arithmetic to logic, as suggested by logicians, and do not reduce logic to arithmetic, as intuitionists will later assume. It is necessary to construct a direct proof of consistency for the joint logical-arithmetic axiomatics [10, p. 400]. Gilbert was very optimistic in this respect, he hoped that direct evidence of consistency could be given for natural numbers, and then in the same way for real and Cantor transfinite numbers.
However, at that time (1904-1905) from detailed implementation of these plans, Hilbert was distracted by other interests (primarily physics), and he returned to the problems of the foundations of mathematics only during the First World War. By this moment the situation has changed. An attempt at a detailed implementation of the logistic program by Russell and Whitehead made both strong and weak sides clear. Zermelo constructed an axiomatics for set theory. But the main thing is that Brouwer's intuitionism appeared and began to gain popularity during this time. It was precisely the danger of the wide dissemination of intuitionistic ideas among mathematicians (among his advocates of intuitionism at the time was his pupil, H. Weyl!) And in many ways forced Hilbert to take up seriously the practical implementation of his own program of substantiating classical mathematics. The main ideas of his new approach Hilbert set out in a series of reports made during the 1920s. "Everything that in the former sense is mathematics," Hilbert said in his report "The Logical Foundations of Mathematics" (1922), is subject to strict formalization in order to transform mathematics, or mathematics in a narrow sense, into a set of formulas. & lt; ... & gt; Formulas that serve as bricks from which a formal math building is built are called axioms. Proof is a figure that, as such, must visibly appear before us; & lt; ... & gt; A formula is said to be provable if it is either an axiom (or obtained by substitution from some axiom), or the final formula of any proof [10, p. 419].
Next to the familiar (informative) mathematics, Gilbert proposes to build its strictly formal analogue. After this, it is necessary to prove the consistency of the formal analogue for every substantive mathematical theory. But what means is it permissible to do?Along with the actual mathematics formalized in the above manner, Hilbert went on to say, "a new mathematics arises in a certain measure, a metamathematics necessary to ensure the reliability of mathematics proper, in which (in contrast to purely formal conclusions of mathematics proper) conclusions, but only to prove the consistency of the axioms [10, p. 419].
The third level in the Hilbert scheme is, therefore, metamathematics - a meaningful way of thinking about formal systems. He wants to limit himself to the meta-level with a minimal set of so-called limited means. That is, such that their use does not cause doubts among representatives of any point of view on mathematics, even among intuitionists, and the results obtained with their help have the status of "absolute truths". The motto of the meta-level reasoning is finiteness and visual visibility, which leave no reason for doubt.
But mathematics can not be limited to finite (finite), it constantly uses transfinite (infinite). The use of quantifiers of universality and existence in application to infinite subject areas puts us in the face of this problem. Here, his method of ideal elements comes to the aid of Hilbert. In my theory of proofs, transfinite axioms and formulas are added to finite axioms, just as in the theory of complex numbers to real elements attached imaginary, and in geometry to the real images are added the ideal. The motivations for this and the success of the method in my theory of proof are the same as there, namely: the additional inclusion of transfinite axioms occurs in the name of simplicity and completeness of the theory of [10, p. 426].
If finite mega-level tools can prove consistency of the cumulative system of axioms (combining both finite and transfinite axioms), the problem will be solved. This topic Hilbert discussed in detail in one of his most famous reports - a speech in Münster, dedicated to the memory of K. Weierstrass, "On the Infinite" (1925).
As a pupil of Koenigsberg, the city of Kant, Hilbert, in his mature philosophy of mathematics, gave clear preference to Kant before Leibniz. In a report of 1925, he unequivocally testified to this: "Kant already taught - and this constitutes an integral part of his teaching - that mathematics possesses an absolutely immanent content of logic, and therefore can never be justified by logic alone , why, by the way, the efforts of Dedekind and Frege and should have been wrecked. Moreover, we already have something in our view as a precondition for applying logical reasoning and performing logical operations: certain, extralogical concrete objects that are present in contemplation as immediate experiences before all thinking. For logical reasoning to be reliable, these objects must be fully observable in all parts, and the presentation of these objects, their differentiation, following one after another, or how one of them is relative to others, all this must be given directly visually together with by the objects themselves as something that can not be reduced to anything else and does not need such a reduction. This is the basic philosophical premise that I consider necessary for both mathematics and, in general, for all scientific thinking, understanding and communication. And in particular, in mathematics, the subject of our consideration are these specific signs themselves, whose appearance, according to our definition, is directly clear and can subsequently be recognized again and again. " [10, p. 439-440].It is easy to see that the Hilbert's intention of direct proof of the consistency of arithmetic with metamathematical means is a direct development of the Kantian conception of the "symbolic construction", which affirmed the preservation of the constructive nature of mathematical thinking in the transition from geometry, with its "ostensivnym design", to arithmetic and algebra [ 18, p. 530-531]. And his own method of ideal elements, Hilbert, in the same report of 1925, connects with the Kantian regulative ideas of pure reason (see paragraph 3.6) [10, p. 448].
Despite the fact that Brouwer's intuitionism also came from the philosophy of Kant's mathematics, the ways in which this philosophy developed in Brouwer and Hilbert turned out to be different. Moreover, Hilbert and Brower turned out to be antagonistic figures, and not allies in the struggle against logicism. Gilbert was sharply against putting classical mathematics on the Procrustean bed of intuitionistic rigid constraints. "No one will be able to drive us out of the paradise that Cantor created for us," these often quoted words of Hubert were voiced in a 1925 report [10, p. 439]. In particular, Hilbert did not agree with the intuitionistic criticism of the law of the excluded third and classical logic in general. In a report of 1927, he expressed his attitude with these words: "Taking away the law of the excluded third from mathematicians is like taking a telescope from astronomers or prohibiting a boxer from using his fists. The prohibition of existence theorems and the law of the excluded third is almost equivalent to a complete rejection of the mathematical science " [11, p. 383].Speaking about formalism in the philosophy of mathematics, it is important to clearly distinguish formalism as the position of David Hilbert and radical formalism in its popular understanding, which is also (though not entirely fair) sometimes associated with the name of Hilbert.
I will refer to Frank Ramsey's article "Foundations of mathematics" as an example. (1925). According to Ramsey, mathematical propositions, such as "2 + 2 = 4", the formalists proclaim "not meaningful formulas that are treated according to some arbitrary rules; they believe that mathematical knowledge consists in knowing which formulas can be derived from other formulas in accordance with certain rules [34, p. 17-18]. Ramsay directly points to Hilbert.
Such a radical understanding of formalism differs significantly from the position of Hilbert, who never identified mathematics with a set of formalisms. Moreover, the formalized theory was built by Hilbert not in order to replace with the relevant content theory, but in order to confirm the legitimacy meaningful way of reasoning. It is noteworthy in this respect that, simultaneously with developments in the theory of evidence, Gilbert read lectures on visual geometry in Göttingen (1920-1921) as far from any formalization. Foreword to their edition of 1932 he began with the words: "In mathematics, as in general in scientific research, there are two tendencies: the tendency to abstraction - it tries to develop a logical point of view on the basis of different material and bring all this material into a systematic connection - and a different tendency, a tendency towards clarity, which, in contrast, tends to a living understanding of objects and their internal relations [14, p. 5]. Gilbert never offered to abandon the second trend in the name of the undivided domination of the first, he sought to maintain their balance. The three-level structure (the content level - the formal level - the level of the metalanguage) that Hilbert created in the framework of his project of substantiating mathematics is of a nature and has the main aim to justify the first (informative) level.
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