Embed or link this publication
Popular Pages
p. 1
alexandre v borovik mathematics under the microscope notes on cognitive aspects of mathematical practice august 20 2008 creative commons[close]
p. 3
a lexandre v b orovik m athematics under the m icroscope the book can be downloaded for free from http www.maths.manchester.ac.uk/avb/micromathematics/downloads unless otherwise expressly stated all original content of this book is created and copyrighted c 2006 by alexandre v borovik and is licensed for non-commercial use under a c reative c ommons a ttribution -n on c ommercial -n o d erivs 2.0 l icense you are free · to copy distribute display and perform the work under the following conditions · · · attribution you must give the original author credit non-commercial you may not use this work for commercial purposes no derivative works you may not alter transform or build upon this work for any reuse or distribution you must make clear to others the licence terms of this work any of these conditions can be waived if you get permission from the copyright holder your fair use and other rights are in no way affected by the above this is a human-readable summary of the legal code the full licence see http creativecommons.org/licenses/by-nc-nd/2.0/uk/legalcode.[close]
p. 5
preface skvoz volxebnyi pribor levenguka nikolai zabolockii the portrayal of human thought has rarely been more powerful and convincing than in vermeer s astronomer the painting creates the illusion that you see the movement of thought itself as an embodied action as a physical process taking place in real space and time i use astronomer as a visual metaphor for the principal aim of the present book i attempt to write about mathematical thinking as an objective real-world process something which is actually moving and happening in our brains when we do mathematics of course it is a challenging task inevitably i have to concentrate on the simplest atomic activities involved in mathematical practice hence the microscope of the title among other things · i look at simple minute activities like placing brackets in the sum a b c d e i analyze everyday observations so routine and self-evident that their mathematical nature usually remains unnoticed for example when you fold a sheet of paper the crease for some reason happens to be a perfect straight line i use palindromes like m adam i m a dam to illustrate how mathematics deals with words composed of symbols and how it relates the word symmetry of palindromes to the geometric symmetry of solid bodies i even discuss the problem of dividing 10 apples among 5 people · · · why am i earnestly concerned with such ridiculously simple questions why do i believe that the answers are important for our understanding of mathematics as a whole?[close]
p. 6
vi preface we cannot seriously discuss mathematical thinking without taking into account the limitations of our brain in this book i argue that we cannot seriously discuss mathematical thinking without taking into account the limitations of the informationprocessing capacity of our brain in our conscious and totally controlled reasoning we can process about 16 bits per second in activities related to mathematics this miserable bit rate is further reduced to 12 bits per second in addition of decimal numbers and to 3 bits in counting individual objects meanwhile the visual processing module of our brain easily handles 10,000,000 bits per second [207 pp 138 and 143 we can handle complex mathematical constructions only because we repeatedly compress them until we reduce a whole theory to a few symbols which we can then treat as something simple also because we encapsulate potentially infinite mathematical processes turning them into finite objects which we then manipulate on a par with other much simpler objects on the other hand we are lucky to have some mathematical capacities directly wired in the powerful subconscious modules of our brain responsible for visual and speech processing and powered by these enormous machines as you will see i pay special attention to order symmetry and parsing that is bracketing of a string of symbols as prominent examples of atomic mathematical concepts or processes i put such atomic particles of mathematics at the focus of the study my position is diametrically opposite to that of martin krieger who said in his recent book doing mathematics [59 that he aimed at a description of some of the the work that mathematicians do employing modern and sophisticated examples unlike krieger i write about simple things however i freely use examples from modern mathematical research and my understanding of simple is not confined to the elementary-school classroom i hope that a professional mathematician will find in the book sufficient non-trivial mathematical material the book inevitably asks the question how does the mathematical brain work i try to reflect on the explosive development of mathematical cognition an emerging branch of neurophysiology which purports to locate structures and processes in the human brain responsible for mathematical thinking [155 167 however i am not a cognitive psychologist i write about the cognitive mechanisms of mathematical thinking from the position of a practicing mathematician who is trying to take a very close look through the magnifying glass at his own everyday work i write not so much about discoveries of cognitive science as of their implications for our understanding of mathematical practice i do not even insist on the ultimate correctness of my interpretations of findings of cognitive psychologists and neurophysiologists with science developing at its present pace the current understanding of the internal work m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 7
preface vii ing of the brain is no more than a preliminary sketch it is likely to be overwritten in the future by deeper works instead i attempt something much more speculative and risky i take as a working hypothesis the assumption that mathematics is produced by our brains and therefore bears imprints of some of the intrinsic structural patterns of our mind if this is true then a close look at mathematics might reveal some of these imprints not unlike the microscope revealing the cellular structure of living tissue i am trying to bridge the gap between mathematics and mathematical cognition by pointing to structures and processes of mathematics which are sufficiently non-trivial to be interesting to a mathematician while being deeply integrated into certain basic structures of our mind and which may lie within reach of cognitive science for example i pay special attention to coxeter theory this theory lies in the very heart of modern mathematics and could be informally described as an algebraic expression of the concept of symmetry it is named after h s m coxeter who laid its foundations in his seminal works [331 332 coxeter theory provides an example of a mathematical theory where we occasionally have a glimpse of the inner working of our mind i suggest that coxeter theory is so natural and intuitive because its underlying cognitive mechanisms are deeply rooted in both the visual and verbal processing modules of our mind moreover coxeter theory itself has clearly defined geometric visual and algebraic verbal components which perfectly match the great visual verbal divide of mathematical cognition however in paying attention to the microcosm of mathematics i try not to lose the large-scale view of mathematics one of the principal mathematics is the study of mental objects with reproducible properties points of the book is the essential vertical unity of mathematics the natural integration of its simplest objects and concepts into the complex hierarchy of mathematics as a whole the astronomer is again a useful metaphor the celestial globe the focal point of the painting boldly places it into a cosmological perspec one of the principal points of the book is the essential vertical unity of mathemattive the astronomer is reaching out ics to the universe but according to the widely held attribution of the painting he is vermeer s neighbor and friend antonij van leeuwenhoek the inventor of the microscope and the discoverer of the microcosm a beautiful world of tiny creatures which no-one had ever seen before van leeuwenhoek also discovered the cellular structure of living organisms the basis of the unity of life m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 8
viii preface microstructure of nerve fibers a drawing by antonij van leeuwenhoek circa 1718 public domain the next principal feature of the book is that i center my discussion of mathematics as a whole in all its astonishing unity around the thesis due to davis and hersh [21 that mathematics is the study of mental objects with reproducible properties in the book the davishersh thesis works at three levels firstly it allows us to place mathematics in the wider context of the evolution of human culture chapter 11 of the book is a brief diversion into memetics an emerging interdisciplinary area of research concerned with the mechanisms of evolution of human culture the term meme an analogue of gene was made popular by richard dawkins [163 and was introduced into mainstream philosophy and cultural studies by daniel dennett [25 it refers to elementary units of cultural transmission i discuss the nature and role of mathematical memes in detail sufficient i hope for making the claim that mathematical memes play a crucial role in many meme complexes of human culture they increase the precision of reproduction of the complex thus giving it an evolutionary advantage remarkably the memes may remain invisible unnoticed for centuries and not recognized as rightly belonging to mathematics in this book i argue that this is a characteristic property of mathematical memes if a meme has the intrinsic property that it increases the precision of reproduction and error correction of the meme complexes it belongs to and if it does that without resorting to external social or cultural restraints then it is likely to be an object or construction of mathematics so far research efforts in mathematical cognition have been concentrated mostly on brain processes during quantification and counting i refer the reader to the book the number sense how the mind creates mathematics by stanislas dehaene [167 for a m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 9
preface ix first-hand account of the study of number sense and numerosity important as they are these activities occupy a very low level in the hierarchy of mathematics not surprisingly the remarkable achievements of cognitive scientists and neurophysiologists are mostly ignored by the mathematical community this situation may change fairly soon since conclusions drawn from neurophysiological research could be very attractive to policymakers in mathematics education especially since neurophysiologists themselves do not shy away from making direct recommendations i believe that hi-tech brain scan cognitive psychology and neurophysiology will more and more influence policies in mathematics education if mathematicians do not pay attention now it may very soon be too late we need a dialogue with the neurophysiological community the development of neurophysiology and cognitive psychology has reached the point where mathematicians should start some initial dis cognitive psychology and neurophysiology will more and more influence policussion of the issues involved furcies in mathematics education if maththermore the already impressive ematicians do not pay attention now body of literature on mathematical it may very soon be too late we need cognition might benefit from a criti a dialogue with the neurophysiological cal assessment by mathematicians community secondly the davishersh thesis puts the underlying cognitive mechanisms of mathematics into the focus of the study finally the davishersh thesis is useful for understanding the mechanisms of learning and teaching mathematics it forces us to analyze the underlying processes of interiorization and reproduction of the mental objects of mathematics in my book i am trying to respond to a sudden surge of interest in mathematics education which can be seen in the mathematical research community it appears that it has finally dawned on us that we are a dying breed that the very reproduction of mathematics as a social institution and a professional community is under threat i approach the problems of mathematical education from this viewpoint which should not be easily set aside what kind of mathematics teaching allows the production of future professional mathematicians what is it that makes a mathematician what are the specific traits which need to be encouraged in a student if we want him or her to be capable of a rewarding career in mathematics i hope that my observations and questions might be interesting to all practitioners and theorists of general mathematical education but i refrain from any critique of or recommendations for school mathematics teaching the unity of mathematics means that there are no boundaries between recreational elementary undergraduate and research mathematics in my book i freely move throughout the whole range nevertheless i am trying to keep the book as non m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 10
x preface technical as possible i hope that the book will find readers among school teachers as well as students in a few instances the mathematics used appears to be more technical this usually happens when i have to resort to metamathematics a mathematical description of the structure and role of mathematical theories but even in such cases mathematical concepts are no more than a presentation tool for a very informal description of my observations occasionally i could not resist the temptation to include some comments on matters of my own professional interest however such comments are indicated in the text by smaller print photographs in this book i come from childhood as from a homeland antoine de saint-exup´ ry pilot de guerre e i tried to place on the margins of the book a photograph of every living mathematician computer scientist historian of mathematics /philosopher of mathematics scholar of mathematics mentioned or quoted in the book the catch is i am using childhood photographs in my book i write a lot about children and early mathematical education and i wish my book to bear a powerful reminder that we all were children once i hope that the reader agrees that the photographs make a fascinating gallery and my warmest thanks go to everyone who contributed his or her photograph i tried to place a photograph of a particular person in those section of the book where his/her views had some impact on my writing the responsibility for my writings is my own and a photograph a person should not be construed as his or her tacit endorsement of my views alexandre borovik aged 11 apologies this book may need more than one preface and in the end there would still remain room for doubt whether anyone who had never lived through similar experiences could be brought closer to the experience of this book by means of prefaces friedrich nietzsche i hope that the reader will forgive me that the book reflects my personal outlook on mathematics to preempt criticism of my sweeping generalizations and of the even greater sin of using introspection as a source of empirical data i quote sholom aleichem m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 11
preface xi man s life is full of mystery and everyone tries to compare it to something simple and easier to grasp i knew a carpenter and he used to say a man he is like a carpenter look at the carpenter the carpenter lives lives and then dies and so does a man and to ward off another sort of criticism i should state clearly that i understand that by writing about mathematics instead of doing mathematics i am breaking a kind of taboo as g h hardy famously put it in his book a mathematician s apology [43 p 61 the function of a mathematician is to do something to prove new theorems to add to mathematics and not to talk about what he or other mathematicians have done statesmen despise publicists painters despise art-critics and physiologists physicists mathematicians have similar feelings there is no scorn more profound or on the whole justifiable than that of the men who make for the men who explain exposition criticism appreciation is work for second-rate minds having broken a formidable taboo of my own tribe i can only apologize in advance if i have disregarded inadvertently or through ignorance any sacred beliefs of other disciplines and professions to reduce the level of offence i ask the discerning reader to treat my book not so much as a statement of my beliefs but as a list of questions which have puzzled me throughout my professional career in mathematics and which continue to puzzle me perhaps my questions are naive however i worked on the book for several years and it is several months now as i keep the text on the web occasionally returning to it to put some extra polish or correct the errors so far the changes in the book were limited to expanding and refining the list of questions not inserting answers i cannot find any in the existing literature this is one the reasons why i believe that perhaps at least some of my questions deserve a thorough discussion in the mathematical educational and cognitive-science communities my last apology concerns the use of terminology some terms and expressions which attained a specialized meaning in certain mathematics-related disciplines are used in this book in their original wider and vaguer sense and therefore are more reader-friendly to fend off a potential criticism from nit-picking specialists i quote a fable which i heard from one of the great mathematicians of our time israel gelfand a student corrected an old professor in his lecture by pointing out that a formula on the blackboard should contain cotangent instead of tangent the professor thanked the student corrected the formula and then added young man i am old and no longer see much difference between tangent and cotangent and i do not advise you to do so either m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 12
xii preface indeed when mathematicians informally discuss their work they tend to use a very flexible language exactly because the principal technical language of their profession is exceptionally precise i follow this practice in my book i hope it allows me to be friendly towards all my readers and not only my fellow mathematicians acknowledgements inspiration and help the gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle hermann weyl i thank my children sergey and maria who read a much earlier version of the book and corrected my english further errors introduced by me are not their responsibility and who introduced me to the philosophical writings of terry pratchett i am grateful to my wife anna the harshest critic of my book this book would never have appeared without her she also provided a number of illustrations as the reader may notice israel gelfand is the person who most influenced my outlook on mathematics i am most grateful to him for generously sharing with me his ideas and incisive observations i am indebted to gregory cherlin reuben hersh and to my old friend owl for most stimulating conversations and many comments on the book some of the topics in the book were included on their advice almost everyday chats with hovik khudaverdyan about mathematics and teaching of mathematics seriously contributed to my desire to proceed with this project during our conversation in paris the late paul moszkowski put forcefully the case for the development of the theory of coxeter groups without reference to geometry and pointed me toward his remarkable paper [384 jeff burdges gregory cherlin david corfield chandler davis ed dubinsky eric ellers tony gardiner ray hill chris hobbs david pierce john stillwell robert thomas ijon tichy and neil white carefully read and corrected the whole or parts of the book my thanks are due to a number of people for their advice and comments on the specific areas touched on in the book to david corfield on philosophy of mathematics to susan blackmore on memetics to vladimir radzivilovsky for explaining to me the details of his teaching method to satyan devadoss on diagrams and drawings used in this book to ray hill on the history of ´ ´ coding theory to p´ ter pal palfy on universal algebra to sergey e utyuzhnikov on chess turbulence and dimensional analysis to alexander jones and jeremy gray on history of euclidean geometry to victor goryunov on multivalued analytic functions to m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 13
preface xiii thomas hull on history of origami to gordon royle on sudoku to alexander kuzminykh and igor pak on convex geometry to dennis lomas-on visual thinking to semen kutateldaze on philosophy and convex geometry and finally to paul ernest and inna korchagina for general encouraging comments jody azzouni barbara sarnecka and robert thomas sent me the texts of their papers [5 6 [159 221 [89 david petty provided diagrammatic instructions for the origami chinese junk figures 11.6 and 11.7 dougald dunham allowed me to use his studies of hyperbolic tesselations in m c escher s engravings figures 5.4 5.5 bruno berenguer allowed me to use one of his chess diagrams figure 7.6 ali nesin made illustrations for chapter 10 john baez provided photograph of figure 2.3 simon thomas provided me with diagrams used in section 12.8 i am lucky that my university colleagues david broomhead paul glendinning bill lionheart and mark muldoon are involved in research into mathematical imaging and/or mathematical models of neural activity and perception their advice has been invaluable paul glendinning gave me a permission to quote large fragments of his papers [179 181 my work on genetic algorithms shaped my understanding of the evolution of algorithms i am grateful to my collaborator rick booth who shared with me the burden of the project also the very first seed which grew into this book can be found in our joint paper [105 finally my thanks go to the blogging community i have picked in the blogosphere some ideas and quite a number of references and especially to numerous anonymous commentators on my blog anonymous age unknown acknowledgements hospitality i developed some of the ideas of section 7.1 in a conversation with ´ maria do rosario pinto i thank her and maria leonor moreira for their hospitality in porto parts of the book were written during my visits to university paris vi in january 2004 and june 2005 on invitation from michel las vergnas and i use this opportunity to tell janette and michel las vergnas how enchanted i was by their hospitality section 10.5 of the book is a direct result of a mathematical tour of cappadocia in january 2006 organized by my turkish colleagues ayse berkman david pierce and sukru yalcinkaya my ¸ ¸¨ ¨ ¸ warmest thanks to them for their hospitality in turkey on that and many other occasions acknowledgements institutional an invitation to the conference the coxeter legacy reflections and projections at the university of toronto had a considerable influ m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 14
xiv preface ence on my work on this book and i am most grateful to its organizers my work on genetic algorithms was funded by epsrc grant gr/r29451 while working on the book i used on several occasions facilities of mathematisches forschungsinstitut oberwolfach the fields institute for research in mathematical sciences and the isaac newton institute for mathematical sciences chapter 7 of this book was much influenced by the discussion meeting where will the next generation of uk mathematicians come from held in march 2005 in manchester the meeting was supported by the manchester institute for mathematical sciences by the london mathematical society by the institute of mathematics and applications and by the uk mathematics foundation it was during the modnet conference on model theory in antalya 211 november 2006 that i had made the final decision to distribute the book under a creative commons license and placed the fist chapter of the book on the internet modnet marie curie research training network in model theory and applications is funded by the european commission under contract no mrtnct-2004-512234 in july 2007 i enjoyed the hospitality of mathematical village in sirince turkey built and run by ali nesin ¸ i started writing this book in caf´ de flore paris an extreme e case of vanity publishing since then i continued my work in many fine establishments among them airbrau das brauhaus im ¨ flughafen in munich cafe del turco in antalya l authre bistro on rue des ecoles and caf´ des arts on place de la contrescarpe in e paris i thank them all alexandre borovik 19 september 2007 durham manchester m athematics under the m icroscope v er 0.93 20-aug -2008/17:33 c a lexandre v b orovik[close]
p. 15
contents part i simple things how structures of human cognition reveal themselves in mathematics 1 a taste of things to come 3 1.1 simplest possible example 3 1.2 switches and flows some questions for cognitive psychologists 5 1.3 choiceless computation 7 1.3.1 polynomial time complexity 7 1.3.2 choiceless algorithms 8 1.4 analytic functions and the inevitability of choice 9 1.5 you name it we have it 11 1.6 why are certain repetitive activities more pleasurable than others 14 1.7 what lies ahead 17 what you see is what you get 2.1 the starting point mirrors and reflections 2.2 image processing in humans 2.3 a small triumph of visualisation coxeter s proof of euler s theorem 2.4 mathematics interiorization and reproduction 2.5 how to draw an icosahedron on a blackboard 2.6 self-explanatory diagrams the wing of the hummingbird 3.1 parsing 3.2 number sense and grammar 3.3 what about music 3.4 palindromes and mirrors 3.5 parsing continued do brackets matter 3.6 the mathematics of bracketing and catalan numbers 3.7 the mystery of hipparchus 21 21 23 26 29 31 36 43 43 46 47 49 52 53 56 2 3[close]
Comments
no comments yet