Sorin Istrail Links
Friday, August 19, 2011
2:00 pm, CIT 241
FROM THE EUCLIDEAN TO THE NON-EUCLIDEAN PARADIGM
Starting as a statement concerning an apparently very particular, narrow question in geometry, the Fifth Postulate of Euclid's Elements increased step by step in importance and impact. In the long list of attempts and failuresto prove it, we focus our attention on John Wallis' s result (1663), stating that the Parallel Postulate is equivalent to the assertion that there exist similar figures that are not congruent. We argue: Why Euclidean geometry is the source of aesthetic ideal of simplicity, regularity and symmetry associated with Antiquity and Renaissance; how is it in agreement with our intuitive perception of the world and with the Galileo-Newtonian paradigm; why Euclidean geometry fits with the macroscopic Universe. Hyperbolic and absolute geometries invented towards the end of the thirties in the 19-th century are in the focus of interest of philosophers, artists and writers, while with Riemann and Felix Klein non-Euclidean geometries get a general mathematical status; they gain the interest of logicians, of physicists (with special and general theory of relativity), then, along the 20-th century, they become involved in the quasi-totality of the fields of mathematics. In the second half of the past century, non-Euclidean geometries are involved in psychology, biology and computer science, while Patrick Suppes raises the question whether the visual space is Euclidean and some authors wonder about the possible non-Euclidean approach to reality of children, immediately after birth.
A comparative analysis is proposed between Choice Axiom and the Continuum Hypothesis, on the one hand, and Bolyai's Absolute Geometry, on the other hand. We also call attention on the similarity between the mathematical eventsrelated to the years 1831 and 1931 and on the non-Euclidean nature of Mandelbrot's Fractal Geometry.
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Friday, August 19, 2011
3:30 pm, CIT 241
CORRECTNESS AND MEANING: COOPERATION OR CONFLICT?
Correctness is related to explicit rules and binary predicates; meaning is a matter of typology and of degree; no rule. Correctness has in view the syntactic dimension of a text, while meaning is the object of semantics. Thereis also the pragmatic dimension of a text, concerning its interactions with its users.
In a formal system S (for instance, as Hilbert defined it), syntax refers to what happens in the interior of S. Semantics refers to some mappings of S into other (formal or non-formal) systems; these mappings are said to be interpretations of S. Pragmatics refers to the relations between S and its users.
At the syntactic level, the main problem is the correctness in the way various explicit rules are used. Semantics is related to the meaning of primitive terms, of axioms, of problems, theorems, conjectures, hypotheses, proofs etc. Pragmatics refers mainly to the cost , the complexity of various entities, questions and procedures.
In a usual (and especially in a scientific, mathematical text), syntactic problems are mainly related to the validity of various types of inferences, to the use of various logical operations, to the linguistic cohesion of the discourse and to the consistency in the iterative use of various terms and symbols. To syntax belongs also the accuracy in using previous concepts and results. All these aspects are usually labeled by the term 'correctness'.
Semantic problems are mainly related to the choice of primitive terms and of axioms, to definitions, to the choice of problems to be investigated, to the choice of tools and methods to be used, to the way we formulate questions, guesses, hypotheses and conjectures, to the way we build hypothetical explanatory models and metaphors. Meaning is to a large extent located at the interface with other fields: physics, computer science, biology, economics, linguistics, music etc. History of science and its cultural environment are a rich source of meaning. Questions of adequacy, of relevance and of interpretation are at home in semantics; but they cannot be approachedby binary predicates.
Syntax and contextual behavior may generate meaning, so there is no sharp border between semantics and syntax.
Semantics and syntax need each other. In order to answer the question 'What does it mean a 'mathematical proof', Hilbert used syntactic means, visible in the structure of a formal system. In order to develop deductive inferences, we need some starting primitive terms and axioms, whose choice is a semantic job. Meaning may co-exist with some benign variants of incorrectness, while correctness may co-exist with absence of meaning (like in Chomsky's example 'colorless green ideas sleep furiously').
Meaning needs correctness and rigor, because in absence of the latter we risk to fall in confusion in the way we understand some notions and some statements. The role of syntax is in this respect crucial. But a formal definition of a notion x is not able to exhaust the meaning of x; the former is finite, while the latter is practically infinite. The syntactic-contextual behavior of a concept or of a symbol may always add something to its 'genus-differentia' description. In elementary mathematics, 'zero' is a typical example in this respect. Most notions, terms and symbols in logic, in linguistics and in computer science, to a large extent in algebra too, are defined and explained by indicating their syntactic-contextual behavior. This is the cooperative aspect of the syntax-semantics (correctness-meaning) interaction.
However, there are other aspects pointing out for the couples syntax-semantics, correctness-meaning the status of a conjugate pair (i.e., a pair where each term can be improved only at the expense of the other term). For instance:
a) Precision and exactness are often obtained at the expense of truth (see the French saying: ''Presque' et 'quasiment' empêchent de mentir'). A photograph is more exact than a portrait, but the latter may reveal more (artistic) truth than the former;
b) Einstein: 'In so far as the propositions of math are certain they do not refer to reality, and in so far as they refer to reality they are not certain'.
c) Socrates: Rigor needs to replace the real world by a fictional one. Math is related to the real world only by mediation of a fictional one.
d) RenéThom: Rigor is obtained at the expense of meaning; more rigor, less meaning.
Too much stress on formal aspects works at the expense of meaning at all levels: learning, teaching, and research. We do desire more from computation than simple symbol manipulation. Errett Bishop warns already in 1973: 'Do not ask whether a statement is true until you know what it means'. William Thurston (1994) proposes to change, in math education, the usual scenario 'definitions-theorem-proof-examples-applications' by another one, with accenton history, motivations, examples, explications, questions etc.'
'Math is not a deductive science', points out Paul R. Halmos.
Chomsky's syntactic mistakes were detected and deleted only after 17 years, while his semantic fallacies are still under examination.
In calculus, the move from Newton, Leibniz, Euler and Lagrange to Cauchy, Riemann and Weierstrass gained in rigor, but in some respect, at the expense of meaning, because it was a move from dynamical and intuitive to static and formal. Formal rigor replaces the natural, historical order, by an artificial one.
Today mentality in education, learning, teaching and research has created a gap between syntax and semantics; meaning is marginalized at the expense of correctness, syntactic procedures and rules. Pragmatics tries to bridge this gap. All these matters are our concern.
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Tuesday, August 23, 2011
3:00 pm, CIT 241
SYMMETRY IN MATHEMATICS AND BEYOND
Questions of the following types are considered:
1) To what extent is symmetry optional and to what extent is symmetry unavoidable?
2) To what extent is symmetry visible and to what extent is symmetry only intelligible?
3) Is arbitrariness compatible with symmetry?
4) How is symmetry possible in an infinite word, when periodicity is no longer valid?
5) Is chaos compatible with symmetry?
6) How is symmetry involved in randomness?
7) What is the relation between local and global symmetry?
8) How is symmetry involved in our choices?
9) How is symmetry involved in our mistakes?
10) Is total absence of symmetry possible?
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Tuesday, August 23, 2011
4:30 pm, CIT 241
MATHEMATICS, BETWEEN SEMIOSIS AND COGNITION
In March 2011, an International Workshop took place at the FIELDS Institute of Mathematics, in Toronto, about the interaction between Mathematics, Cognitive Science and Semiotics. The need of such a Workshop followed a long period of development of research concerning the nature of mathematical cognition; done by people working in cognitive science, anthropology, psychology, linguistics, semiotics and computer science. The culminating moment of this adventure was the publication, in 2000, of the book Where mathematics comes from?How the embodied mind brings mathematics into being, by George Lakoff (linguist) and Rafael Núñez (anthropologist).
The present lecture is the development of my invited presentation at thisWorkshop.
Here are some ideas I am considering.
Why is the nature of the cognitive function of mathematics so controversial? Because the fundamental question MATHEMATICS IS ABOUT WHAT? remains with no generally accepted answer. Mathematics is equally involved in the understanding of the architecture of the universe and in the understanding of the structure of the human mind. What do we really know by mathematics, taking into account that it is not concerned with no clear specific slide ofthe surrounding world, as it happens with physics, chemistry or biology?
Mathematical knowledge is predominantly of a global nature. The historical development of the mathematical sign system is presented and a typology of 34 species of the mathematical way of thinking is proposed. A special attention is directed towards the metaphorical nature of mathematical creativity, but our approach is completely different from that used by Lakoff and Núñez, leading however to the same idea: mathematical knowledge is of a metaphorical nature. Our approach is based on the analysis of the historical process of development taking place within the internal life of mathematics as a research enterprise, while Lakoff and Núñez refer to the cerebral (brain) mechanisms involved in the respective processes. But just the fact that, by two completely different approaches, the same idea is reached shows the solidity of the hypothesis claiming that mathematical concepts and results are obtained by metaphorical processes.
We analyze the metaphorical itinerary from natural numbers to integers, from integers to rational and from rational to real numbers, showing that in all these cases we have a self-referential metaphorical process. We point outthe similarity and the difference with respect to the metaphorical itinerary leading to the idea of Divinity. We show the existence of a very rich common denominator between mathematics and art, concomitant with a long listofdifferences between them.
About Professor Solomon Marcus
Born March 1, 1925, Bacau, Romania. Mother Sima , father Alter, both taylors. The merits of his father were acknowledged by the Ministry of War: 'The title of war veteran is assigned to the solder Alter Marcus for participation in the independence and in the completing wars'.
Elementary school and high school in Bacau, Romania. First four classes of high school in the state school; the last four classes in a private Jewish school, according to the laws valid in Romania between 1940 and 1944. First classified at 'Bacalaureat' (school-leaving examination) September 1944. Faculty of Science, Mathematics, University of Bucuresti, Romania, 1944-1949 with Diploma of Merit. Assistant Professor 1950, Lecturer 1955, Associate Professor 1964, Professor 1966, Emeritus Professor 1991, Faculty of Mathematics, University of Bucuresti. PhD in Mathematics 1956, State Doctor in Sciences 1968, University of Bucuresti, Romania.
Corresponding Member of the Romanian Academy, April 1993. Full Member of the Romanian Academy, December 2001.
Research and teaching in the fields of Mathematical Analysis, Measure Theory, General Topology, Linguistics, Theoretical Computer Science, Poetics, Theory of Literature, Semiotics, Cultural Anthropology, History and Philosophyof Science (all of them seen through the glasses of the mathematical way of thinking), Applications of Mathematics in Natural and Social Sciences. In these fields, he published about 50 books in Romanian, English, French, Italian, German, Spanish, Russian, Hungarian, Czech, Serbo-Croatian, Greek and more than 400 papers in research journals of almost all European countries, of United States, Canada, Brazil, Argentine, Mexic, Japan, China, India, New Zealand. He also published several hundreds of articles in cultural magazines, about problems of general interest, but mainly related to educational issues and to the need to connect science and the humanities and to develop a new attitude, making possible to capture the unity of culture, the metabolism of disciplines of all kinds, be they scientific or artistic, philosophical or technological, science and the humanities, natural sciences or social sciences. Marcus fights for a new vision in education and in culture, proposing an integrative approach against a fragmentary one, still dominant.
He is recognized as one of the initiators of the new fields called Mathematical Linguistics and Mathematical Poetics, as it follows from citations in Brockhaus, Encyclopaedia Universalis, Enciclopedia Italiana, Great Soviet Encyclopedia etc.
Here are the titles of some books published by Solomon Marcus: Algebraic Linguistics; Analytical Models. Academic Press, New York and London, 1967; Introduction Mathématique à la Linguistique Structurale. Dunod, Paris, 1967; Introduzione alla Linguistica Matematica (in collab. with Edmond Nicolau and Sorin Stati), Casa Editrice Riccardo Patron, Bologna, Italia, 1971; Mathematische Poetik. Athenäum Verlag, Frankfurt/Main, 1973; co-author and editor of La Sémiotique Formelle du Folklore. Klincksieck, Paris, 1978; co-author and editor of Contextual Ambiguities in Natural and in Artificial Languages. Communication and Cognition, Ghent, Belgium, vol. 1, 1981; vol.2, 1983; The Paradox (in Romanian), Ed. Albatros, Bucuresti, 1984; The Meeting of Extremities (in Romanian), Editura Paralela 45, Bucuresti, 2005; Universal Paradigms (in Romanian), 4 volumes, Editura Paralela 45, Bucuresti, 2005-2010; Words and Languages Everywhere. Polimetrica, Milano, 2007; The Education in Spectacle (in Romanian). Spandugino Publishing House, Bucuresti, 2010.
A monograph published by Gheorghe Paun, Marcus Contextual Grammars, Kluwer Academic Pubkisher, Dordrecht-Boston-London, 1997, is devoted to a class of grammars introduced by Solomon Marcus. The book Meetings with Solomon Marcus, eds. Lavinia Spandonide, Gheorghe Paun, Spandugino Publ. House, Bucuresti, 2010, collects reactions from about 450 authors about their meetings with Solomon Marcus's works or/and personality.
About 30 international scientific journals included his name in their Editorial Boards, for some large periods of time. Here are some of them: Revue Roumaine de Mathématiques Pures et Appliquée (Bucuresti); Poetics (Amsterdam), Zeitschrift für Literaturwissenschaft und Linguistik (Siegen), International Journal of Computer Mathematics (London), Poetics Today (Tel Aviv), Fundamenta Informaticae (Warsaw), Foundations of Computing and Decision Sciences (Poznan), Galaxia (Sao Paulo); Symmetry: Culture and Science (Budaspest); Semiotica (Berlin and Toronto) .
Solomon Marcus was an invited plenary speaker of about hundred international scientific meetings. He gave invited lectures in more than hundred universities from Europe, North and South America, Israel, Japan, China, Sri Lankaand New Zealand.
More than 1000 authors cited in their works Solomon Marcus's ideas and results. Tens of scholars, today university professors or researchers in Europe, America, Israel, Oceania recognized Marcus as one of their mentors.
