László Kalmár
Encyclopedia
László Kalmár was a Hungarian
mathematician
and Professor at the University of Szeged
. Kalmár is considered the founder of mathematical logic
and theoretical Computer Science
in Hungary.
Kalmár's brilliance manifested itself while in Budapest schools. At the University of Budapest, his teachers included Kürschák
and Fejér
. His fellow students included the future logician Rózsa Péter
. Kalmár graduated in 1927. He discovered mathematical logic, his chosen field, while visiting Göttingen in 1929.
Upon completing his doctorate at Budapest, he took up a position at the University of Szeged. That university was mostly made up of staff from the former University of Kolozsvár, a major Hungarian university before World War I
that found itself after the War in Romania. Kolozsvár was renamed Cluj. The Hungarian university moved to Szeged in 1920, where there had previously been no university. The appointment of Haar
and Riesz
turned Szeged into a major research center for mathematics. Kalmár began his career as a research assistant to Haar and Riesz. Kalmár was appointed a full professor at Szeged in 1947. He was the inaugural holder of Szeged's chair for the Foundations of Mathematics and Computer Science. He also founded Szeged's Cybernetic Laboratory and the Research Group for Mathematical Logic and Automata Theory.
In mathematical logic
, Kalmár proved that certain classes of formulas of the first order predicate calculus were decidable
. In 1936, he proved that the predicate calculus could be formulated using a single binary predicate, if the recursive
definition of a term
was sufficiently rich. (This result is commonly attributed to a 1954 paper of Quine
's.) He discovered an alternative form of primitive recursive arithmetic, known as elementary recursive arithmetic, based on primitive functions that differ from the usual kind. He did his utmost to promote computers and computer science in Hungary. He wrote on theoretical computer science, including programming languages, automatic error correction, non-numerical applications of computers, and the connection between computer science
and mathematical logic.
Kalmar was elected to the Hungarian Academy of Sciences
in 1949, and was awarded the Kossuth Prize in 1950 and the Hungarian State Prize in 1975.
In 1933 Kalmár married Erzsébet Arvay; they had four children.
and variables, the constants 0 and 1, repeated addition + of the constants, proper subtraction ∸, bounded summation and bounded product (Kleene 1952:526). Elimination of the bounded product from this list yields the subelementary or lower elementary functions. By use of the abstract computational model called a register machine
Schwichtenberg provides a demonstration that "all elementary functions are computable and totally defined" (Schwichtenberg 58).
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and Professor at the University of Szeged
University of Szeged
The University of Szeged is one of Hungary's most distinguished universities, and is among the most prominent higher education institutions in Central Europe...
. Kalmár is considered the founder of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and theoretical Computer Science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
in Hungary.
Biography
His early life mixed promise and tragedy. His father died when he was young, and his mother died when he was 17, the year he entered the University of Budapest, making him essentially an orphan.Kalmár's brilliance manifested itself while in Budapest schools. At the University of Budapest, his teachers included Kürschák
József Kürschák
József Kürschák was a Hungarian mathematician noted for his work on trigonometry and for his creation of the theory of valuations. He proved that every valued field can be embedded into a complete valued field which is algebraically closed. In 1918 he proved that the sum of reciprocals of...
and Fejér
Lipót Fejér
Lipót Fejér , was a Hungarian mathematician. Fejér was born Leopold Weiss, and changed to the Hungarian name Fejér around 1900....
. His fellow students included the future logician Rózsa Péter
Rózsa Péter
Rózsa Péter , Hungarian name Péter Rózsa, was a Hungarian mathematician. She is best known for her work with recursion theory....
. Kalmár graduated in 1927. He discovered mathematical logic, his chosen field, while visiting Göttingen in 1929.
Upon completing his doctorate at Budapest, he took up a position at the University of Szeged. That university was mostly made up of staff from the former University of Kolozsvár, a major Hungarian university before World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
that found itself after the War in Romania. Kolozsvár was renamed Cluj. The Hungarian university moved to Szeged in 1920, where there had previously been no university. The appointment of Haar
Alfréd Haar
Alfréd Haar was a Jewish Hungarian mathematician. In 1904 he began to study at the University of Göttingen. His doctorate was supervised by David Hilbert. The Haar measure, Haar wavelet, and Haar transform are named in his honor....
and Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
turned Szeged into a major research center for mathematics. Kalmár began his career as a research assistant to Haar and Riesz. Kalmár was appointed a full professor at Szeged in 1947. He was the inaugural holder of Szeged's chair for the Foundations of Mathematics and Computer Science. He also founded Szeged's Cybernetic Laboratory and the Research Group for Mathematical Logic and Automata Theory.
In mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, Kalmár proved that certain classes of formulas of the first order predicate calculus were decidable
Decidable
The word decidable may refer to:* Decidable language*Decidability for the equivalent in mathematical logic*Gödel's incompleteness theorem, a theorem on the indecidability of languages consisting of "true statements" in mathematical logic....
. In 1936, he proved that the predicate calculus could be formulated using a single binary predicate, if the recursive
Recursive
Recursive may refer to:*Recursion, the technique of functions calling themselves*Recursive function, a total computable function*Recursive language, a language which is decidable...
definition of a term
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
was sufficiently rich. (This result is commonly attributed to a 1954 paper of Quine
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
's.) He discovered an alternative form of primitive recursive arithmetic, known as elementary recursive arithmetic, based on primitive functions that differ from the usual kind. He did his utmost to promote computers and computer science in Hungary. He wrote on theoretical computer science, including programming languages, automatic error correction, non-numerical applications of computers, and the connection between computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
and mathematical logic.
Kalmar was elected to the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
in 1949, and was awarded the Kossuth Prize in 1950 and the Hungarian State Prize in 1975.
In 1933 Kalmár married Erzsébet Arvay; they had four children.
Elementary functions
Kalmar defined what are known as elementary functions, number-theoretic functions (i.e. those based on the natural numbers) built up from the notions of compositionComposition (number theory)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Any integer...
and variables, the constants 0 and 1, repeated addition + of the constants, proper subtraction ∸, bounded summation and bounded product (Kleene 1952:526). Elimination of the bounded product from this list yields the subelementary or lower elementary functions. By use of the abstract computational model called a register machine
Register machine
In mathematical logic and theoretical computer science a register machine is a generic class of abstract machines used in a manner similar to a Turing machine...
Schwichtenberg provides a demonstration that "all elementary functions are computable and totally defined" (Schwichtenberg 58).
External links
- MacTutor The source for most of this entry.