Leon Chwistek was a Polish avant-garde painter, theoretician of modern art, literary critic, logician and mathematician. Starting in 1929 Chwistek was a Professor of Logic at the University of Lwów in a position for which Alfred Tarski had applied, his interests in the 1930s were in a general system of philosophy of science, published in a book translated in English 1948 as The Limits of Science. In the 1920s-30s, many European philosophers attempted to reform traditional philosophy by means of mathematical logic. Leon Chwistek did not believe, he thought that reality could not be described in one homogeneous system, based on the principles of formal logic, because there was not one reality but many. Chwistek argued against the axiomatic method by demonstrating that the extant axiomatic systems are inconsistent. Chwistek developed his theory of the multiplicity of realities first with regard to the arts, he distinguished four basic types of realities matched them with four basic types of painting.
The four types of realities were: 1. Popular reality 2. Physical reality 3. Phenomenal reality 4. Visionary/intuitive reality; the types of painting corresponding to the above were: 1. Primitivism 2. Realism 3. Impressionism 4. FuturismChwistek never intended his views to constitute a new metaphysical theory, he was a defender of "common sense" against irrational feeling. His theory of plural reality was an attempt to specify the various ways in which the term, “real,” is used. Chwistek's fellow-artist and closest friend, Stanislaw Ignacy Witkiewicz, harshly criticized his philosophical views. Witkiewicz’s own philosophy was based on a monadic character to the individual's existence, embracing a multiplicity of existences, with the world being made up of a multiplicity of Particular Existences; the limits of science. Outline of logic and of the methodology of the exact sciences. Translated from the Polish by Helen Charlotte Brodie and Arthur P. Coleman. New York: Harcourt, Brace, 1948 History of philosophy in Poland Polish Philosophy Page: Leon Chwistek at the Wayback Machine Profile of Leon Chwistek at Culture.pl Instituto Polaco de Cultura: Artola, Inés R. Formiści: la síntesis de la modernidad.
Conexiones y protagonistas, Granada: Libargo, ISBN 978-84-938812-7-6
Alfred North Whitehead
Alfred North Whitehead was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found application to a wide variety of disciplines, including ecology, education, biology and psychology, among other areas. In his early career Whitehead wrote on mathematics and physics, his most notable work in these fields is the three-volume Principia Mathematica, which he wrote with former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library. Beginning in the late 1910s and early 1920s, Whitehead turned his attention from mathematics to philosophy of science, to metaphysics, he developed a comprehensive metaphysical system which radically departed from most of western philosophy. Whitehead argued that reality consists of processes rather than material objects, that processes are best defined by their relations with other processes, thus rejecting the theory that reality is fundamentally constructed by bits of matter that exist independently of one another.
Today Whitehead's philosophical works – Process and Reality – are regarded as the foundational texts of process philosophy. Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us." For this reason, one of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization and environmental ethics pioneered by John B. Cobb Jr. Alfred North Whitehead was born in Ramsgate, England, in 1861, his father, Alfred Whitehead, was a minister and schoolmaster of Chatham House Academy, a school for boys established by Thomas Whitehead, Alfred North's grandfather. Whitehead himself recalled both of them as being successful schools, but that his grandfather was the more extraordinary man. Whitehead's mother was Maria Sarah Whitehead Maria Sarah Buckmaster. Whitehead was not close with his mother, as he never mentioned her in any of his writings, there is evidence that Whitehead's wife, had a low opinion of her.
Whitehead was educated at Sherborne School, one of the best public schools in the country. His childhood was described as over-protected, but when at school he excelled in sports and mathematics and was head prefect of his class. In 1880, Whitehead began attending Trinity College and studied mathematics, his academic advisor was Edward John Routh. He earned his BA from Trinity in 1884, graduated as fourth wrangler. Elected a fellow of Trinity in 1884, Whitehead would teach and write on mathematics and physics at the college until 1910, spending the 1890s writing his Treatise on Universal Algebra, the 1900s collaborating with his former pupil, Bertrand Russell, on the first edition of Principia Mathematica, he was a Cambridge Apostle. In 1890, Whitehead married an Irish woman raised in France. Eric Whitehead died in action at the age of 19, while serving in the Royal Flying Corps during World War I. Alfred's brother Henry became Bishop of Madras, wrote a observed ethnographic account of the Village Gods of South-India, still of value today.
In 1910, Whitehead resigned his senior lectureship in mathematics at Trinity and moved to London without first lining up another job. After being unemployed for a year, Whitehead accepted a position as lecturer in applied mathematics and mechanics at University College London, but was passed over a year for the Goldsmid Chair of Applied Mathematics and Mechanics, a position for which he had hoped to be considered. In 1914 Whitehead accepted a position as professor of applied mathematics at the newly chartered Imperial College London, where his old friend Andrew Forsyth had been appointed chief professor of mathematics. In 1918 Whitehead's academic responsibilities began to expand as he accepted a number of high administrative positions within the University of London system, of which Imperial College London was a member at the time, he was elected dean of the Faculty of Science at the University of London in late 1918, a member of the University of London's Senate in 1919, chairman of the Senate's Academic Council in 1920, a post which he held until he departed for America in 1924.
Whitehead was able to exert his newfound influence to lobby for a new history of science department, help establish a Bachelor of Science degree, make the school more accessible to less wealthy students. Toward the end of his time in England, Whitehead turned his attention to philosophy. Though he had no advanced training in philosophy, his philosophical work soon became regarded. After publishing The Concept of Nature in 1920, he served as president of the Aristotelian Society from 1922 to 1923. In 1924, Henry Osborn Taylor invited the 63-year-old Whitehead to join the faculty at Harvard University as a professor of philosophy. During his time at Harvard, Whitehead produced his most important philosophical contributions. In 1925, he wrote Science and the Modern World, hailed as an alternative to the Cartesian dualism that plagued popular scien
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology, the Church–Rosser theorem, he worked on philosophy of language. Alonzo Church was born on June 14, 1903, in Washington, D. C. where his father, Samuel Robbins Church, was the judge of the Municipal Court for the District of Columbia. The family moved to Virginia after his father lost this position because of failing eyesight. With help from his uncle named Alonzo Church, the son attended the private Ridgefield School for Boys in Ridgefield, Connecticut. After graduating from Ridgefield in 1920, Church attended Princeton University, where he was an exceptional student, he published his first paper on Lorentz transformations and graduated in 1924 with a degree in mathematics. He stayed at Princeton for graduate work, earning a Ph.
D. in mathematics in three years under Oswald Veblen. He married Mary Julia Kuczinski in 1925; the couple had three children, Alonzo Church, Jr. Mary Ann and Mildred. After receiving his Ph. D. he taught as an instructor at the University of Chicago. He received a two-year National Research Fellowship that enabled him to attend Harvard University in 1927–1928, the University of Göttingen and University of Amsterdam the following year, he taught philosophy and mathematics at Princeton for nearly four decades, 1929–1967. He taught at the University of California, Los Angeles, 1967–1990, he was a Plenary Speaker at the ICM in 1962 in Stockholm. He received honorary Doctor of Science degrees from Case Western Reserve University in 1969, Princeton University in 1985, the University at Buffalo, The State University of New York in 1990 in connection with an international symposium in his honor organized by John Corcoran. A religious person, Church was a lifelong member of the Presbyterian church, he was buried in Princeton Cemetery.
Church is known for the following significant accomplishments: His proof that the Entscheidungsproblem, which asks for a decision procedure to determine the truth of arbitrary propositions in a first-order mathematical theory, is undecidable. This is known as Church's theorem, his proof that Peano arithmetic is undecidable. His articulation of what has come to be known as the Church–Turing thesis, he was the founding editor of the Journal of Symbolic Logic, editing its reviews section until 1979. His creation of the lambda calculus; the lambda calculus emerged in his 1936 paper showing the unsolvability of the Entscheidungsproblem. This result preceded Alan Turing's work on the halting problem, which demonstrated the existence of a problem unsolvable by mechanical means. Church and Turing showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, subsequently demonstrated a variety of alternative "mechanical processes for computation."
This resulted in the Church–Turing thesis. The efforts for automatically generating a controller implementation from specifications originates from his ideas; the lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor. In his honor the Alonzo Church Award for Outstanding Contributions to Logic and Computation was established in 2015 by the Association for Computing Machinery Special Interest Group for Logic and Computation, the European Association for Theoretical Computer Science, the European Association for Computer Science Logic, the Kurt Gödel Society; the award is for an outstanding contribution to the field published within the past 25 years and must not yet have received recognition via another major award, such as the Turing Award, the Paris Kanellakis Award, or the Gödel Prize. Church’s elaboration of a methodology involving the logistic method, his philosophical criticisms of nominalism and his defense of realism, his argumentation leading to conclusions about the theory of meaning, the detailed construction of the Fregean and Russellian intensional logics, are more than sufficient to place him high up among the most important philosophers of this century.
Many of Church's doctoral students have led distinguished careers, including C. Anthony Anderson, Peter B. Andrews, George A. Barnard, David Berlinski, William W. Boone, Martin Davis, Alfred L. Foster, Leon Henkin, John G. Kemeny, Stephen C. Kleene, Simon B. Kochen, Maurice L'Abbé, Isaac Malitz, Gary R. Mar, Michael O. Rabin, Nicholas Rescher, Hartley Rogers, Jr. J. Barkley Rosser, Dana Scott, Raymond Smullyan, Alan Turing. A more complete list of Church's students is available via Mathematics Genealogy Project. Alonzo Church, Introduction to Mathematical Logic Alonzo Church, The Calculi of Lambda-Conversion Alonzo Church, A Bibliography of Symbolic Logic, 1666–1935 C. Anthony Anderson and Michael Zelëny, Logic and Computation: Essays in Memory of Alonzo Church Church–Turing–Deutsch principle Higher-order logic List of pioneers in computer science Modern Platonism Universal set Enderton, Herbert B. Alonzo Church: Life and Work. Introduction to the Collected Works of Alonzo Church, MIT Press, not yet published.
Enderton, Herbert B. In memoriam: Alonzo Church, The Bulletin of Symbolic Logic, vol. 1, no. 4, pp. 486–488. Wade, Alonzo Church, 92, Theorist of the Limits of Mathematics, The New York Times, September 5, 1995, p. B6. Hodges, Wilf
Peter B. Andrews
Peter Bruce Andrews is an American mathematician and Professor of Mathematics, Emeritus at Carnegie Mellon University in Pittsburgh and the creator of the mathematical logic Q0. He received his Ph. D. from Princeton University in 1964 under the tutelage of Alonzo Church. He received the Herbrand Award in 2003, his research group designed the TPS automated theorem prover. A subsystem ETPS of TPS is used to help students learn logic by interactively constructing natural deduction proofs. Andrews, Peter B.. "Resolution in type theory". Journal of Symbolic Logic 36, 414–432. Andrews, Peter B.. "Theorem proving via general matings". J. Assoc. Comput. March. 28, no. 2, 193–214. Andrews, Peter B.. An introduction to mathematical logic and type theory: to truth through proof. Computer Science and Applied Mathematics. ISBN 978-0-1205-8535-9. Academic Press, Inc. Orlando, FL. Andrews, Peter B.. "On connections and higher-order logic". J. Automat. Reason. 5, no. 3, 257–291. Andrews, Peter B.. "TPS: a theorem-proving system for classical type theory".
J. Automat. Reason. 16, no. 3, 321–353. Andrews, Peter B.. An introduction to mathematical logic and type theory: to truth through proof. Second edition. Applied Logic Series, 27. ISBN 978-1-4020-0763-7. Kluwer Academic Publishers, Dordrecht. Peter B. Andrews
Joachim "Jim" Lambek was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph. D. degree in 1950 with Hans Zassenhaus as advisor. Lambek was born in Leipzig, where he attended a Gymnasium, he came to England in 1938 as a refugee on the Kindertransport. From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger interned, wrote the McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an M. Sc. A year later. In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first Ph. D. in mathematics. Lambek became assistant professor at McGill, he spent his sabbatical year 1965–66 in at the Institute for Mathematical Research at ETH Zurich, where Beno Eckmann had gathered together a group of researchers interested in algebraic topology and category theory, including Bill Lawvere.
There Lambek reoriented his research into category theory. Lambek continued his involvement at McGill's mathematics department. In 2000 a festschrift celebrating Lambek's contributions to mathematical structures in computer science was published. On the occasion of Lambek's 90th birthday, a collection Categories and Types in Logic and Physics was produced in tribute to him. Lambek's PhD thesis investigated vector fields using the biquaternion algebra and in Minkowski space, as well as semigroup immersion in a group; the second component was published by the Canadian Journal of Mathematics. He returned to biquaternions when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the Riemann-Silberstein bivector to express the free-space electromagnetic equations. Lambek supervised 16 doctoral students, has 51 doctoral descendants, he has over 100 publications listed including 6 books. His earlier work was in module theory torsion theories, non-commutative localization, injective modules.
One of his earliest papers, Lambek & Moser, proved the Lambek-Moser theorem about integer sequences. His more recent work is in formal languages, he is noted, among other things, for the Lambek calculus, an effort to capture mathematical aspects of natural language syntax in logical form and a work, influential in computational linguistics, as well as for developing the connections between typed lambda calculus and cartesian closed categories. His last works were on pregroup grammar. Fine, N. J.. Rings of quotients of rings of functions. McGill University Press. MR 0200747. Lambek, Joachim. Lectures on rings and modules. Blaisdell Publishing. ISBN 9780821849002. MR 0206032. Lambek, Joachim. Completions of categories. Seminar lectures given in 1966 in Zürich. Lecture Notes in Mathematics, No. 24. Berlin, New York: Springer-Verlag. MR 0209330. Lambek, Joachim. Torsion theories, additive semantics, rings of quotients. Lecture Notes in Mathematics. 177. Berlin, New York: Springer-Verlag. MR 0284459. Lambek, J.. J..
Introduction to Higher Order Categorical Logic. Cambridge University Press. ISBN 978-0-521-35653-4. MR 0856915. Anglin, W. S.. The heritage of Thales. Undergraduate Texts in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94544-6. MR 1369087. Casadio, Claudia. Computational Algebraic Approaches to Natural Language. Polimetrica. ISBN 978-88-7699-125-7. Lambek, J.. From word to sentence: a computational algebraic approach to grammar. Polimetrica. ISBN 978-88-7699-117-2. Lambek, J. "The immersibility of a semigroup into a group", Canadian Journal of Mathematics, 3: 34–43, doi:10.4153/CJM-1951-005-8 Lambek, Joachim. "The Mathematics of Sentence Structure", The American Mathematical Monthly, 65: 154–170, doi:10.2307/2310058, ISSN 0002-9890, JSTOR 1480361 Lambek, Joachim, "How to program an infinite abacus", Canadian Mathematical Bulletin, 4: 295–302, doi:10.4153/CMB-1961-032-6 Lambek, Joachim. "Deductive systems and categories II. Standard constructions and closed categories". Lecture Notes in Mathematics.
86. Berlin, Heidelberg: Springer Berlin Heidelberg. Pp. 76–122. Doi:10.1007/bfb0079385. ISBN 978-3-540-04605-9. ISSN 0075-8434. Lambek, Joachim, "Bicommutators of nice injectives", Journal of Algebra, 21: 60–73, doi:10.1016/0021-869390034-8, ISSN 0021-8693, MR 0301052 Lambek, Joachim, "Localization and completion", Journal of Pure and Applied Algebra, 2: 343–370, doi:10.1016/0022-404990011-4, ISSN 0022-4049, MR 0320047 Lambek, Joachim, "A mathematician looks at Latin conjugation", Theoretical Linguistics, 6: 221–234, doi:10.1515/thli.1979.6.1-3.221, ISSN 0301-4428, MR 0589163 J. Lambek "If Hamilton had prevailed: Quaternions and Physics", Mathematical Intelligencer 17: 7–15, reprinted in Mathematical Conversations Robin Wilson & Jeremy Gray editors, Springer books, ISBN 978-1-4613-0195-0 J Lambek & Michael Barr In Praise of Quaternions from McGill University Joachim Lambek at the Mathematics Genealogy Project Faculty profile of Joachim Lambek at McGi
The Principia Mathematica is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✸9 and all-new Appendix B and Appendix C. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM was conceived as a sequel volume to Russell's 1903 Principles, but as PM states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became evident that the subject is a much larger one than we had supposed. PM, according to its introduction, had three aims: to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions and axioms, inference rules; this third aim motivated the adoption of the theory of types in PM.
The theory of types adopts grammatical restrictions on formulas that rules out the unrestricted comprehension of classes and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of PM. There is no doubt that PM is of great importance in the history of mathematics and philosophy: as Irvine has noted, it sparked interest in symbolic logic and advanced the subject by popularizing it. Indeed, PM was in part brought about by an interest in Logicism, the view on which all mathematical truths are logical truths, it was in part thanks to the advances made in PM that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems. For all that, PM is not used today: the foremost reason for this is its reputation for typographical complexity. Somewhat infamously, several hundred pages of PM precede the proof of the validity of the proposition 1+1=2.
Contemporary mathematicians tend to use a modernized form of the system of Zermelo–Fraenkel set theory. Nonetheless, the scholarly and philosophical interest in PM is great and ongoing: for example, the Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century; the Principia covered only set theory, cardinal numbers, ordinal numbers, real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism, it was clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third; as noted in the criticism of the theory by Kurt Gödel, unlike a formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that immediately in the theory, interpretations are presented in terms of truth-values for the behaviour of the symbols "⊢", "~", "V".
Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be arbitrary and unfamiliar; the theory would specify only. By assignment of "values", a model would specify an interpretation of what the formulas are saying, thus in the formal Kleene symbol set below, the "interpretation" of what the symbols mean, by implication how they end up being used, is given in parentheses, e.g. "¬". But this is not a pure Formalist theory; the following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows: Symbols used: This set is the starting set, other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols: "→", "&", "V", "¬", "∀", "∃". Symbol strings: The theory will build "strings" of these symbols by concatenation.
Formation rules: The theory specifies the rules of syntax as a recursive definition that starts with "0" and specifies how to build acceptable strings or "well-formed formulas". This includes a rule for "substitution" of strings for the symbols called "variables". Transformation rule: The axioms that specify the behaviours of the symbols and