One is a science, the other an art; one useful, the other seemingly decorative, but mathematics and music share common origins in cult and mystery and have been linked throughout history. Emblems of Mind is Edward Rothstein’s classic exploration of their profound similarities, a journey into their “inner life.” Along the way, Rothstein explains how mathematics makes sense of space, how music tells a story, how theories are constructed, how melody is shaped. He invokes the poetry (...) of Wordsworth, the anthropology of Le;vi-Strauss, the imagery of Plato, and the philosophy of Kant. Math and music, Rothstein shows, apply comparable methods as they create their abstractions, display similar concerns with ratio and proportion, and depend on metaphors and analogies to create their meanings. Ultimately, Rothstein argues, they reveal the ways in which we come to understand the world. They are images of the mind at work and play; indeed, they are emblems of Mind itself. Jacques Barzun called this book “splendid.” Martin Gardner said it was “beautifully written, marvelous and entertaining.” It will provoke all serious readers to think in new ways about the grand patterns in art and life. “Lovely, wistful. . . . Rothstein is a wonderful guide to the architecture of musical space, its tensions and relations, its resonances and proportions. . . . His account of what is going on in the music is unfailingly felicitous.”— New Yorker “Provocative and exciting. . . . Rothstein writes this book as a foreign correspondent, sending dispatches from a remote and mysterious locale as a guide for the intellectually adventurous. The remarkable fact about his work is not that it is profound, as much of the writing is, but that it is so accessible.”— Christian Science Monitor. (shrink)
The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical musictheory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification (...)theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical musictheory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)
Music theorists of almost all ages employ a concept of "Nature" to justify observations or statements about music. The understanding of what "Nature" is, however, is subject to cultural and historical differences. In tracing these explanatory strategies and their changes in music theories between c. 1600 and 1900, these essays explore (for the first time in a book-length study) how the multifarious conceptions of nature, located variously between scientific reason and divine power, are brought to bear on (...)musictheory and how they affect our understanding of music. (shrink)
Recent encounters with structuralist and poststructuralist critical theory, linguistics, and cognitive sciences have brought the theory and analysis of music into the orbit of important developments in present-day intellectual history. Without seeking to impose an explicit redefinition of either theory or analysis, this book explores the limits of both. Essays on decidability, ambiguity, metaphor, music as text, and music analysis as cognitive theory are complemented by studies of works by Debussy, Schoenberg, Birtwistle and (...) Boulez. (shrink)
A green philosopher's peripeteia.--Physics and metaphysics of music.--The roots of arithmetic.--Critique of new geometrical abstractions.--The philosophical value of science.
Serial music was one of the most important aesthetic movements to emerge in post-war Europe, but its uncompromising music and modernist aesthetic has often been misunderstood. This book focuses on the controversial journal die Reihe, whose major contributors included Stockhausen, Eimert, Pousseur, Dieter Schnebel and G. M. Koenig, and discusses it in connection with many lesser-known sources in German musicology. It traces serialism's debt to the theories of Klee and Mondrian, and its relationship to developments in concrete art, (...) modern poetry and the information aesthetics and semiotics of Max Bense and Umberto Eco. M. J. Grant sketches an aesthetic theory of serialism as experimental music, arguing that serial theory's embrace of both rigorous intellectualism and aleatoric processes is not, as many have suggested, a paradox, but the key to serial thought and to its relevance for contemporary theory. (shrink)
A theory of musical narrative. An introduction to narrative analysis : Chopin's prelude in G major, op. 28, no. 3 ; Perspectives and critiques ; A theory of musical narrative : conceptual considerations ; A theory of musical narrative : analytical considerations ; Narrative and topic -- Archetypal narratives and phases. Romance narratives and Micznik's degrees of narrativity ; Tragic narratives : an extended analysis of Schubert, piano sonata in B flat major, D. 960, first movement ; (...) Ironic narratives : subtypes and phases ; Comic narratives and discursive strategies ; Summary and conclusion. (shrink)
The Consolations of Philosophy by Boethius, whose English translators include King Alfred, Geoffrey Chaucer, and Queen Elizabeth I, ranks among the most remarkable books to be written by a prisoner awaiting the execution of a tyrannical death sentence. Its interpretation is bound up with his other writings on mathematics and music, on Aristotelian and propositional logic, and on central themes of Christian dogma. -/- Chadwick begins by tracing the career of Boethius, a Roman rising to high office under (...) the Gothic King Theoderic the Great, and suggests that his death may be seen as a cruel by-product of Byzantine ambitions to restore Roman imperial rule after its elimination in the West in AD 476. Subsequent chapters examine in detail his educational programme in the liberal arts designed to avert a threatened collapse of culture and his ambition to translate into Latin everything he could find on Plato and Aristotle. -/- Boethius has been called `last of the Romans, first of the scholastics'. This book is the first major study in English of a writer who was of critical importance in the history of thought. (shrink)
The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is (...) divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)
A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.
The ten contributions in this volume range widely over topics in the philosophy of mathematics. The four papers in Part I (entitled "Set Theory, Inconsistency, and Measuring Theories") take up topics ranging from proposed resolutions to the paradoxes of naïve set theory, paraconsistent logics as applied to the early infinitesimal calculus, the notion of "purity of method" in the proof of mathematical results, and a reconstruction of Peano's axiom that no two distinct numbers have the same successor. (...) Papers in the second part ("The Challenge of Nominalism") concern the nominalistic thesis that there are no abstract objects. The two contributions in Part III ("Historical Background") consider the contributions of Mill, Frege, and Descartes to the philosophy of mathematics. (shrink)
In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, (...) as the structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism. (shrink)
Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of (...) geometry was sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’ (16–20 April 2007), organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)
Julian Cole argues that mathematical domains are the products of social construction. This view has an initial appeal in that it seems to salvage much that is good about traditional platonistic realism without taking on the ontological baggage. However, it also has problems. After a brief sketch of social constructivist theories and Cole’s philosophy of mathematics, I evaluate the arguments in favor of social constructivism. I also discuss two substantial problems with the theory. I argue that unless and (...) until social constructivists can address the two concerns, we have reason to be skeptical about social constructivism in the philosophy of mathematics. (shrink)
This paper begins by distinguishing intrinsic and extrinsic contributions of mathematics to scientific representation. This leads to two investigations into how these different sorts of contributions relate to confirmation. I present a way of accommodating both contributions that complicates the traditional assumptions of confirmation theory. In particular, I argue that subjective Bayesianism does best accounting for extrinsic contributions, while objective Bayesianism is more promising for intrinsic contributions.
The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). it gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-L6f 's theory of types as a formal system for BISH.
We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to (...) the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively. (shrink)
We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in (...) which we use work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods. (shrink)
The purpose of this paper is to present a bio-physical basis of mathematics. The essence of the theory is that function in the nervous system is mathematical. The mathematics arises as a result of the interaction of energy (a wave with a precise curvature in space and time) and matter (a molecular or ionic structure with a precise form in space and time). In this interaction, both energy and matter play an active role. That is, the interaction (...) results in a change in form of both energy and matter. There are at least six mathematical operations in a simple synaptic region. It is believed the form of both energy and matter are specific, and their interaction is specific, that is, function in most of the nervous system is stereotyped. It is suggested that mathematics be taken out of the mind and placed where it belongs — in nature and the synaptic regions of the nervous system; it results in both places from a precise interaction between energy (in a precise form) and matter (in a precise structure). (shrink)
In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.
David Hilbert famously remarked, “No one will drive us from the paradise that Cantor has created.” This volume offers a guided tour of modern mathematics’ Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor’s transfinite paradise; axiomatic set theory; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. (...) Philosophers and mathematicians will find an abundance of intriguing topics in this text, which is appropriate for undergraduate- and graduate-level courses. 1989 ed. 32 figures. (shrink)
Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James (...) H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. (shrink)
We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known (...) class='Hi'>mathematics must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound. (shrink)
These are notes designed to bring the beginning student of the philosophy of quantum mechanics 'up to scratch' on the mathematical background needed to understand elementary finite-dimensional quantum theory. There are just three chapters: Ch. 1 'Vector Spaces'; Ch. 2 'Inner Product Spaces'; and Ch. 3 'Operators on Finite-Dimensional Complex Inner Product Spaces'. The notes are entirely self-contained and presuppose knowledge of only high school level algebra.
Method and system Francois- Bernard Mdche I want to raise the issue of the possible significance of the use of digital machines for a composer whose work is ...
Presented here are translations of two essays of the Austrian logician, philosopher and experimental psychologist Ernst Mally, originally delivered at the Third International Congress of Philosophy in Heidelberg, Germany. Both essays conclude with discussion between Mally and Kurt Grelling. Mally was a student of Alexius Meinong and a contributor to logical investigations in the field of object theory (Gegenstandstheorie). In these essays, Mally introduces a vital distinction between formal and extra-formal ?determinations? (Bestimmungen), and he argues that formal determinations are (...) not part of the identity conditions for intended objects, but provide the basis for a theory of pure logical and mathematical relations. Mally then proceeds to develop a formal logic of formal and extra-formal determinations, whose interrelations of ontic and modal predications provide an analysis of fundamental object theory concepts. (shrink)
A historically feminized profession, education in North America remains remarkably unaffected by feminism, with the notable exception of pedagogy and its impact on curriculum. The purpose of this paper is to describe characteristics of feminism that render it particularly useful and appropriate for developing potentialities in education and music education. As a set of flexible methodological tools informed by Gilles Deleuze's notions of philosophy and art, I argue feminism may contribute to education's becoming more efficacious, reflexive, and reflective of (...) the values of its participants. Its impetus involves ‘feminist imperative(s)’ to help in the sense articulated by Elizabeth Grosz: to provoke thought, challenge, and problematize. (shrink)
What is music for? How does it work? What can it teach us? Intuitively, we feel there must be answers to such questions, but they tend to be scattered throughout a wide range of different areas of study, from acoustics to music history, from psychology to composition. In this brilliant and thought-provoking book Maconie seeks the answers to these and other fundamental questions about music, integrating music and appropriate scientific research in a new evaluation of his (...) topic. In so doing, he argues passionately for a reappraisal of music, not as mere entertainment, but as something basic to our experience of listening and communicating in sound, and an art which has exerted a profound influence on society. (shrink)
The article offers a comparative analysis of the uses of set theory in Castoriadis's "The Imaginary Institution of Society" and Badiou's "Being and Event".
Music's ability to express and arouse emotions is a mystery that has fascinated both experts and laymen at least since ancient Greece. The predecessor to this book 'Music and Emotion' (OUP, 2001) was critically and commercially successful and stimulated much further work in this area. In the years since publication of that book, empirical research in this area has blossomed, and the successor to 'Music and Emotion' reflects the considerable activity in this area. The Handbook of (...) class='Hi'>Music and Emotion offers an 'up-to-date' account of this vibrant domain. It provides comprehensive coverage of the many approaches that may be said to define the field of music and emotion, in all its breadth and depth. The first section offers multi-disciplinary perspectives on musical emotions from philosophy, musicology, psychology, neurobiology, anthropology, and sociology. The second section features methodologically-oriented chapters on the measurement of emotions via different channels (e.g., self report, psychophysiology, neuroimaging). Sections three and four address how emotion enters into different aspects of musical behavior, both the making of music and its consumption. Section five covers developmental, personality, and social factors. Section six describes the most important applications involving the relationship between music and emotion. In a final commentary, the editors comment on the history of the field, summarize the current state of affairs, as well as propose future directions for the field. The only book of its kind, The Handbook of Music and Emotion will fascinate music psychologists, musicologists, music educators, philosophers, and others with an interest in music and emotion (e.g., in marketing, health, engineering, film, and the game industry). It will be a valuable resource for established researchers in the field, a developmental aid for early-career researchers and postgraduate research students, and a compendium to assist students at various levels. In addition, as with its predecessor, it will also attract interest from practising musicians and lay readers fascinated by music and emotion. (shrink)
This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates (...) proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
The volumes of G¨ odel’s collected papers under review consist almost entirely of a rich selection of his philosophical/scientific correspondence, including English translations face-to-face with the originals when the latter are in German. The residue consists of correspondence with editors (more amusing than of any scientific value) and five letters from G¨ odel to his mother, in which explains to her his religious views. The term “selection” is strongly operative here: The editors state the total number of items of personal (...) and scientific correspondence in G¨ odel’s Nachlass to be around thirty-five hundred. The correspondence selected involves fifty correspondents, and the editors list the most prominent of these: Paul Bernays, William Boone, Rudolph Carnap. Paul Cohen, Burton Dreben, Jacques Herbrand, Arend Heyting, Karl Menger, Ernest Nagel, Emil Post, Abraham Robinson, Alfred Tarski, Stanislaw Ulam, John von Neumann, Hao Wang, and Ernest Zermelo. The correspondence is arranged alphebetically, with A-G in Volume IV. The imbalance results from the disproportionate size of the Bernays correrspondence: 85 letters are included (almost all of them), spanning 234 pages) including the face-to-face originals and translations). Each volume contains a calendar of all the items included in the volume together with separate calendars listing all known correspondence (whether included or not) with the major correspondents (seven in Volume IV and ten in Volume V). Let me recommend to the reader the review of these same volumes by Paolo Mancosu in the Notre Dame Journal of Formal Logic 45 (2004):109- 125. This essay very nicely describes much of the correspondence in terms of broad themes relating, especially, to the incompleteness theorems—their origins in G¨ odel’s thought, their reception, their impact on Hilbert’s program. (shrink)
Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic ...
Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including (...) an essay on the philosophy of logic first published in 1971. (shrink)
Robertson's earlier work, The New Renaissance projected the likely future impact of computers in changing our culture. Phase Change builds on and deepens his assessment of the role of the computer as a tool driving profound change by examining the role of computers in changing the face of the sciences and mathematics. He shows that paradigm shifts in understanding in science have generally been triggered by the availability of new tools, allowing the investigator a new way of seeing into (...) questions that had not earlier been amenable to scientific probing. (shrink)
In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice (...) principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn—Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. (shrink)
