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Chwistek, Leon (1884–1944)


Leon Chwistek, a Polish mathematical logician, philosopher, aesthetician, essayist, and painter, was a lecturer at the University of Kraków and from 1930 a professor of mathematical logic at the University of Lvov.

Theory of Realities

The central problem of Chwistek's philosophy was a criticism of the idea of a uniform reality. It had been shown by Bertrand Russell that in logic admission of the totality of all functions of x produces contradictions; Chwistek claimed that in philosophy, likewise, many obscure and misleading thoughts result from the assumption of a single all-inclusive reality.

The results of this criticism led Chwistek to the thesis of a plurality of realities. Out of many possible realities four are particularly important to philosophy. The first, the reality of natural objects, is assumed by common sense; natural objects are of a given form regardless of our perception. Chwistek's defense of natural reality and our knowledge of it is reminiscent of the British commonsense philosophy of the nineteenth century. The objects studied in physics are not natural; the telescopic and microscopic worlds, matter, and the particles upon which the forces are supposed to act form a second reality. They are constructions, not something naturally given. The third reality, that of impressions, the elements of sensation, as studied by David Hume or Ernst Mach, forms the world of appearances. The fourth reality is that of images, produced by us and dependent on our will, fantasy, and creative processes.

All four of these realities are necessary to account for our knowledge. In addition, when we reflect that we speak about a reality, we cannot include ourselves or our reflection in this reality. Such a reflection must be a part of a higher reality. Otherwise confusions and contradictions arise. The act of discourse cannot be a part of the universe of discourse.


Chwistek applied the doctrine of plurality of realities to investigations in many areasaesthetics, for example. Natural reality is dealt with by primitive art. In primitive art each object is given one color only, and perspective is not obeyed. The primitivist paints not as he sees but as things are supposed to be by themselves. He uses his vision, but mainly he uses his knowledge about the world. Realism in art depicts the physical reality as it is conceived at a given time. Impressionism is the art of the reality of impressions; it flourished in a society that had developed psychological research and made psychologism its fundamental scientific method. Futurism is the art of free images, of an actively created reality of fantasy and mental constructions.

In each style of art the artist tries to give a perfect form to his creation independent of the kind of reality he is working with. The form is the common feature of all works of art. Thus, Chwistek justified all styles by relating them to different realities, and he advocated formism: evaluation of form, not of reality, is the proper aesthetic evaluation.

Mathematics and Semantics

Chwistek extended his pluralism to mathematics. There is no one system of mathematics, but there are many mutually exclusive systems. Various geometries coincide only in part. When we build analysis based on logic, we can accept, reject, or accept the negations of some extralogical existence axioms, such as the axiom of choice, the axiom of infinity, and the assumption of the existence of transfinite cardinal numbers. Logic itself should not decide any existence problem.

This restrained program for logic was paired with the requirement that logic be understandable in a nominalistic manner and deal with expressions in a constructive, mechanically computable way. Among principles often accepted as logical are some propositions questionable from the constructivist point of viewfor example, the axiom of reducibility and the axiom of extensionality. The axiom of reducibility has to do with the distinction between predicative and impredicative concepts. An impredicative concept is a concept definable only by a definiens containing a quantifier that accepts as one of its values the very concept being defined. Russell and Chwistek ruled out such definitions as involving a vicious circle.

As was incisively pointed out by Kurt Gödel (in The Philosophy of Bertrand Russell, P. A. Schilpp, ed. [Evanston, IL, 1946], pp. 135138), impredicative definitions involve a vicious circle only if one takes, as Chwistek did and Russell did not, a nominalistic attitude toward logic. Only if the quantifier is understood as a summary reference (infinite conjunction) to all of its values that are expressions and if one of the values of a quantifier that occurs in the definiens is the expression that is the definiendum do we presuppose what we want to define. Russell was not a nominalist. His exclusion of impredicative definitions was a way of avoiding antinomies. By differentiating between ranges of values of variables according to the way the quantifier binding a variable occurs, Russell constructed the ramified theory of logical types. This is a somewhat awkward theory. In analysis we want to speak about, for example, the real number that is the least upper bound of a set of real numbers that has a bound. To introduce this concept we must quantify over real numbers greater than all real numbers of a class that includes the least of them. Russell's theory avoids this impredicativeness by setting the least upper bound in a different logical type from the starting real numbers. But then the least upper bound and the real numbers involved cannot be values of the same variables, and several statements about particular sets of real numbers (for example, that a given function is continuous) are impossible.

To overcome this difficulty Russell accepted the axiom of reducibility, which says that every propositional function is coextensive with a predicative one. In many cases we cannot construct such a predicative function, and therefore constructivists, such as Chwistek, cannot accept this axiom. Moreover, for a nominalist, that two propositional functions are coextensive is not a sufficient guarantee of their identity. Thus, Chwistek attempted the task, which Russell called "heroic," of forming a purely constructivist system of the foundations of mathematics without impredicative definitions, the axiom of reducibility, or the axiom of extensionality. He observed, as F. P. Ramsey did, that results similar to Russell's can be obtained by the simple theory of types (where one distinguishes only between variables ranging over individuals, properties of individuals, properties of such properties, etc.) instead of the more complicated ramified theory. But simple type theory is inconsistent with the axiom of intensionality, which Chwistek wanted to be free to accept and which asserts the nonidentity of the concepts defined by two different propositional functions (even if they are coextensive).

The systems Chwistek constructed for the foundations of mathematics were such that they answered the philosophical needs of their author. They were admittedly more complicated than Russell's. "But it may be erroneous to think that clear ideas are never complicated; while we must agree that many simple ideas are, as a matter of fact, very obscure." Chwistek presented several formulations of his attempts at a constructivist theory, all of them too sketchy to be judged definitive. The relation to other constructivist systems is hard to establish. The last few versions were called "rational metamathematics." This theory deals with expressions, some of which are theorems.

A principal part of rational metamathematics, the fundamental system of semantics, uses two specific primitive signs, c and *, about which we stipulate that c is an expression and that if E and F are expressions, then *EF is an expression. These formation rules assign a definite tree (or grouping) structure to each finite expression as well as to any two expressions written one after the other. Some of the allowed combinations of c and * may have no meaningin this Chwistek was a formalist. To some other expressions we assign meaning, and in accordance with this assignment we accept proper axioms. We take 0 to be an abbreviation of *cc. The fundamental substitution pattern (EFGH )which is read "H is the result of substituting G for every occurrence of F in E "is taken to be an abbreviation of ****EE *FF *GG *HH. The Sheffer stroke function, |EF, is regarded as an abbreviation of ***EE **EE *EE ***FF **FE *FF. Identity = EF stands for (EOOF ).

See also Aesthetics, History of; Aesthetics, Problems of; Gödel, Kurt; Russell, Bertrand Arthur William; Semantics.


selected works by chwistek

Wielość Rzeczywistości (Plurality of realities). Kraków, 1921.

"The Theory of Constructive Types." Annales de la Société Polonaise de Mathématique 2 (1924, for 1923): 948, and 3 (1925, for 1924): 91141.

"Fondements de la métamathématique rationnelle." Bulletin de l'Académie Polonaise des Sciences et des Lettres, Series A (1933): 253264. Written with W. Hetper and J. Herzberg.

"Remarques sur la méthode de la construction des notions fondamentales de la métamathématique rationnelle." Bulletin de l'Académie Polonaise des Sciences et des Lettres, Series A (1933): 265275. Written with W. Hetper and J. Herzberg.

Granice Nauki. Lvov and Warsaw, 1935. Rev. ed. translated by H. C. Brodie and A. P. Coleman as The Limits of Science. London: K. Paul, Trench, Trubner, 1948.

Wielość Rzeczywistości w Sztuce (Plurality of realities in art). Edited by K. Estreicher. Warsaw, 1960.

Pisma Filozoficzne i Logiczne (Philosophical and logical writings). Edited by K. Pasenkiewicz, 2 vols. Warsaw: Naukwoe, 19611963.

A reformulation of Chwistek's system of semantics by John Myhill appears in the Journal of Symbolic Logic 14 (1949): 119125, and 16 (1951): 3542.

Palace Boga. Próba rekonstrukcji. Warsaw: Panstwowy Instytut Wydawniczy, 1968.

Niejedna rzeczywistosc: racjonalizm krytyczny Leon Chwistka. Sens i rzeczywistosc. Crakow: Inter esse, 2004.

H. Hiż (1967)

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Chwistek, Leon

Chwistek, Leon

(b. Zakopane, Poland, 13 January 1884; d. Berwisza, near Moscow, U. S. S.R., 20 August 1944),

philosophy, logic, aesthetics.

An acute thinker who was strongly opposed to metaphysics and idealistic philosophy, Chwistek was professor of logic at the University of Lvov from 1930 to 1940, when he took refuge in the Soviet Union. In 1921 he published his theory of the plurality of realities. Rejecting the idea of one reality, he distinguished four main concepts of reality: natural reality, physical reality, reality of sensation, and reality of images. These concepts should not be confused, and each has its proper sphere of application. He used this theory to classify movements and styles in art. He maintained that aesthetic evaluation should be based not on reality but on form. From 1919 to 1920 he was coeditor of the periodical Formiści.

Under the influence of Poincare, Chwistek developed a strictly nominalistic attitude toward science, particularly logic and mathematics. In 1921 he observed that for the removal of Russell’s paradox in the theory of classes, the simplified theory of types suffices. Dissatisfied with Russell’s foundation of mathematics, in which he rejected such idealistic elements as the axiom of reducibility in the theory of types, Chwistek proposed his theory of constructive types in 1924. His main contribution was the foundation of logic and mathematics on his system of rational semantics.

Rational semantics is a system of expressions constructed from the symbols * and c according to two rules: (1) c is an expression and (2) if E and F are expressions, then * EF is an expression. The role of everyday language is restricted to the use of (a) E is an expression, (b) E is a theorem, (c) If X, then Y, in cases where E denotes an expression and X and Y denote propositions of form (a), (b), or (c).

Integers .0L, .1L, .2L… of type L (an expression) occur as expressions *LL, **LL*L, ***LL*LL**LL*LL,…. The expression *cc is abbreviated as 0. With the help of the expressions *.0L. 1L and *.1L.0L, abbreviated as IL. and IIL, respectively, the fundamental pattern of Sheffer’s stroke/EF is introduced as *.IE. IIF. Essential is the fundamental pattern of substitution (EFGH)[L], an expression of the form ****.IL.0E.0F.0G.0H. It reads: H is the result of substituting G for F in E. The type is indicated by L.

From the expressions theorems are singled out by certain rules, e.g., by (0*EF G0) [c] is a theorem and (EEFF) [c] is a theorem. Variables, quantification, and the axioms of logic are introduced by patterns. Quantification is always over constructed expressions. There results a collection of systems [MN] described by = [MN]E, short for **.IM.0E. IIN, Where M is an integer greater than N and E an expression. For instance, the fact that .01 is an expression of type I in system [20] is reflected by the definite expression = [20] Expr [1] .01.

Chwistek succeeded in constructing a theory of classes based on types. The members of a class are of a higher type than the class, and there is no highest type. Chwistek and Hetper reconstructed the arithmetic of natural and rational numbers and considered the possibility of developing analysis. For a detailed and overall picture of Chwistek’s scientific activity see his posthumous Limits of Science (1948). His ideas have been taken up and carried further by J. Myhill.


I. Original Works. Chwistek’s writings are “Antynomej logiki formalnej” (“Antinomies of Formal Logic”), in Przeglad filozoficzny, 24 (1921), 164–171; Wielośé rceczywistości (“The Plurality of Realities”; Cracow, 1921); “Über die Antinomien der Prinzipien der Mathematik.” in Mathematische Zeitschrif, 14 (1922), 236–243;” Zsatowanie metody konstrukcyjnej do teorji poznania” (“The Application of the Constructive Method in the Theory of Knowledge”), in Przegląd filozoficzny, 26 (1923), 175–187, and 27 (1927),296–298; “The Theory of Constructive Types. Principles of Logic and Mathematics,” in Annales de la Société polonaise de mathématique, 2 (1924), 9–48, and 3 (1925), 92–141; “Pluralité des réalités,” in Atti del V Congresso internazionale di filosofia (Naples, 1925), pp. 19–24; “Neue Grundlagen der Logik and Mathematik,” in Mathematische Zeitschrift, 34 (1932), 527–534; “Die nominalistische Grundlegung der Mathematik,” in Erkenntnis, 3 (1932/1933), 367–388; “Fondements de la métamathématique rationnelle,” in Bulletin de l’Académie polonaise des sciences et des lettres, Classe des sciences mathématiques et naturelles, ser. A (1933), 253–264, written with W. Hetper and J. Herzberg; Granice nauki (“The Limits of Science”; Lvov-Warsaw, 1935); “New Foundations of Formal Metamathematics,” in Journal of Symbolic Logic, 3 (1938), 1–36, written with W. Hetper; “A Formal Proof of Gödel’s Theorem,” ibid., 4 (1939), 61–68; and The Limits of Science, Helen C. Brodie, ed. (London, 1948).

II. Secondary Literature. On Chwistek or his work, see A. A. Fraenkel, Abstract Set Theory (Amsterdam, 1953); A. A. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory (Amsterdam, 1958); and J. Myhill, review of The Limits of Science, in Journal of Symbolic Logic, 14 (1949), 119–125; “Report on Some Investigations Concerning the Consistency of the Axiom of Reducibility,” ibid., 16 (1951), 35–42; and “Towards a Consistent Set-Theory,” ibid., 130–136.

B. van Rootselaar

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