Readings and Resources


2008-08-28 -17:26 -0700

If there is any hint of mathematical logic to a book or article, I automatically  classify it as a Reading in Logic.  Perhaps it is because I am not entirely at home considering myself to be a mathematician, yet I say that mathematical logic is very much a subject for me.  

I propose to reform and place more materials on the philosophy of mathematics and the foundations of mathematics here.  Nevertheless, much about mathematical logic and that great achievement, set theory, will continue to be viewed as part of logic.  I don't think of myself as a logicist, though I am closer to that than at risk of becoming a formalist or intuitionist.  In any event, here is much about the overall tapestry of mathematics and its foundations, beyond the focused treatment of logic, computation, and language that centers around mathematical logic.

-- Dennis E. Hamilton
Seattle, Washington
2002 September 2

see also
Readings in Logic
Readings in Theory of Computation (Miser Project)
Readings in Philosophy
Readings in Science
Beck, Anatole., Bleicher, Michael N., Crowe, Donald Warren.  Excursion into Mathematics: The Millennium Edition.  With a foreword by Martin Gardner.  A. K. Peters, Ltd. (Natick, MA: 1969, 2000).  ISBN 1-56881-115-2 pbk: alk. paper.
     "Ours is a mathematical age.  Not only are we ever more dependent on the fruits of the physical sciences, not only is the art of data processing being developed to free man of much of his routine work, but also the ideas of mathematics are beginning to permeate our sociological, philosophical, linguistic, and artistic world.  The Chinese compare an illiterate to a blind man.  As our age develops, we may soon feel the same way of the man ignorant of mathematical thinking."
     "Excursions into Mathematics is designed to acquaint the general reader with some of the flavor of mathematics.  By the general reader we mean someone who has had two or three years of high school mathematics and is either a high school graduate or senior."
     "Each excursion deals with a single area of mathematical interest and builds around it a body of theory.  The initial problems are elementary, but as one penetrates deeper, new questions arise, new approaches suggest themselves. ...
     "The conceptual level of the work is intended to be about the same as calculus, and the student should be prepared for some hard thinking.  However, he can make a fresh start with each new chapter, since the six chapters are almost completely independent.  Any one of the first three can be take first.  Each of these three chapters will introduce the student to the principle of mathematical induction.  Each of the last three chapters presupposes that the student has mastered this principle."
     "The oldest material in the book antedates written history in Europe, the newest is being published here for the first time."  -- From the 1969 Preface, pp. ix-x.
     2001-11-26: I was reluctant to pick up this book.  I don't have need for one more unread mathematics book.  The winning attraction is Chapter 6, which addresses matters of use in a consideration of number theory for the Miser Project.  There are also potential applications in Chapter 4's work on Finite Fields, Chapter 3's introduction of Measure Theory, and Chapter 2's treatment of fundamental number theory.  Chapter 1's section 5 treats Hamilton circuits and the distinct Euler circuit. -- dh.
     Preface to the Millennium Edition
     Note to the Instructors

     Chapter 1.  Euler's Formula for Polyhedra, and Related Topics (Donald W. Crowe)
          1. Introduction
          2. Regular Polyhedra
          3. Deltahedra
          4. Polyhedra without Diagonals
          5. The n-dimensional Cube and the Tower of Hanoi
          6. The Four-color Problem and the Five-color Theorem
          7. The Conquest of Saturn, the Scramble for Africa, and Other Problems
     Chapter 2.  The Search for Perfect Numbers (Michale N. Bleicher)
          1. Introduction
          2. Prime Numbers and Factorization
          3. Euclidean Perfect Numbers
          4. Primes and Their Distribution
          5. Factorization Techniques
          6. Non-Euclidean Perfect Numbers
          7. Extensions and Generalizations
     Chapter 3.  What is Area?  (Anatole Beck)
          1. Introduction
          2. Rectangles and Grid Figures
          3. Triangles
          4. Polygons
          5. Polygonal Regions
          6. Area in General
          7. Pathology
     Chapter 4.  Some Exotic Geometries (Donald W. Crowe)
          1. Historical Background
          2. Spherical Geometry
          3. Absolute Geometry
          4. More History -- Saccheri, Bolyai, and Lobachevsky
          5. Hyperbolic Geometry
          6. New Beginnings
          7. Analytic Geometry -- A Reminder
          8. Finite Arithmetics
          9. Finite Geometries
          10. Application
          11. Circles and Quadratic Equations
          12. Finite Affine Planes
          13. Finite Projective Planes
          14. Ovals in a Finite Plane
          15. A Finite Version of Poincaré's Universe
          Appendix.  Excerpts from Euclid
     Chapter 5.  Games (Anatole Beck)
          1. Introduction
          2. Some Tree Games
          3. The Game of Hex
          4. The Game of Nim
          5. Games of Chance
          6. Matrix Games
          7. Applications of Matrix Games
          8. Positive-sum Games
          9. Sharing
          10. Cooperative Games
     Chapter 6. What's in a Name?  (Michael N. Bleicher)
          1. Introduction
          2. Historical Background
          3. Place Notation for the Base b
          4. Some Properties of Natural Numbers Related to Notation
          5. Fractions: First Comments
          6. Farey Fractions
          7. Egyptian Fractions
          8. The Euclidean Algorithm
          9. Continued Fractions
          10. Decimal Fractions
          11. Concluding Remarks
     Glossary of Symbols
     Appendix 2000

Bell, Eric Temple. Men of Mathematics.  Simon & Schuster (New York: 1937).  Touchstone edition ISBN 0-671-62818-6 pbk.
     I don't recall how many times I have owned this book.  At least twice before.  As a short-term entering freshman at Caltech, I had already read John Taine's Ralph 124C41+, one of Bell's science fiction novels.   At the end of the hallway in the mathematics department there was a closed door with Bell's name and "Professor Emeritus" carefully painted on the glass.  I did not know what "emeritus" meant other than he must be really old (indeed, a mere 11 years older than I am now).  I never saw the door be open, though others said that they had seen him come and go.  Later, I would have this book and make his acquaintance thereby.
     A number of things have stayed with me.  First, that the proof of the fundamental theorem of algebra took quite a long time to "get right," although the validity of the theorem was never much in doubt, though now I can't put my finger on where that is told.  And that George Boole "fully realized that he had done great work."  I am always touched by that.  A new honor for this book is that Julia Robinson was inspired to become a mathematician after reading it as a college student [Henderson1996: p.105].  -- dh:2002-09-05
     "The lives of mathematicians presented here are addressed to the general reader and to others who may wish to see what sort of human beings the men were who created modern mathematics.  Our object is to lead up to some of the dominating ideas governing vast tracts of mathematics as it exists today and to do this through the lives of the men responsible for those ideas." -- from the Introduction, p. 3.
     "It must not be imagined that the sole function of mathematics -- 'the handmaiden of the sciences' -- is to serve science. ... Mathematics has a light and wisdom of its own, above any possible application to science, and it will richly reward any intelligent human being to catch a glimpse of what mathematics means to itself.  This is not the old doctrine of art for art's sake; it is art for humanity's sake.  After all, the whole purpose of science is not technology -- God knows we have gadgets enough already; science also explores depths of a universe that will never, by any stretch of the imagination, be visited by human beings or affect our material existence.  So we shall attend also to some of the things which the great mathematicians have considered worthy of loving understanding for their intrinsic beauty."  -- from the Introduction, p. 4.
     "Looking back over the long struggle to make the concepts of real number, continuity, limit, and infinity precise and consistently usable in mathematics, we see that Zeno and Eudoxus were not so far in time from Weierstrass, Dedekind, and Cantor as the twenty four or twenty five centuries which separate modern Germany from ancient Greece might seem to imply.  There is no doubt that we have a clearer conception of the nature of the difficulties involved than our predecessors had, because we see the same unsolved problems cropping up in new guises and in fields the ancients never dreamed of, but to say that we have disposed of those hoary old difficulties is a gross mis-statement of fact.  Nevertheless the net score records a greater gain than any which our predecessors could rightfully claim. ... Cantor's revolutionary work gave our present activity its initial impulse.   But it was soon discovered -- twenty one years before Cantor's death -- that his revolution was either too revolutionary or not revolutionary enough.  The latter now appears to be the case. ...
     "What will mathematics be like a generation hence when -- we hope -- these difficulties will have been cleared up?  ... If there is any continuity at all in the evolution of mathematics -- and the majority of dispassionate observers believe that there is -- we shall find that the mathematics which is to come will be broader, firmer, and richer in content than that which we or our predecessors have known.  ...
     "If we  may rashly venture a prediction, what is to come will be fresher, younger in every respect, and closer to human thought and human needs -- freer of appeal for its justification to extra-human 'existences' -- than what is now being vigorously refashioned."  -- from Chapter 29, pp. 575, 579.
     1. Introduction
     2. Modern Minds in Ancient Bodies [Zeno, Eudoxus, Archimedes]
     3. Gentleman, Soldier, and Mathematician [Descartes (1596-1650)]
     4. The Prince of Amateurs [Fermat (1601-1665)]
     5. "Greatness and Misery of Man" [Pascal (1623-1662)]
     6. On the Seashore [Newton (1642-1727)]
     7. Master of All Trades [Liebniz (1646-1716)]
     8. Nature or Nurture? [The Bernoullis (17th & 18th centuries)]
     9. Analysis Incarnate [Euler (1707-1783)]
     10. A Lofty Pyramid [Lagrange (1736-1813)]
     11. From Peasant to Snob [Paplace (1749-1827)]
     12. Frinds of an Emperor [Monge (1746-1818), Fourier (1768-1830)]
     13. The Day of Glory [Poncelet (1788-1867)]
     14. The Prince of Mathematicians [Gauss (1777-1855)]
     15. Mathematics and Windmills [Cauchy (1789-1857)]
     16. The Copernicus of Geometry [Lobatchewsky (1793-1856)]
     17. Genius and Poverty [Abel (1802-1829)]
     18. The Great Algorist [Jacobi (1804-1851)]
     19. An Irish Tragedy [Hamilton (1805-1865)]
     20. Genius and Stupidity [Galois (1811-1832)]
     21. Invariant Twins [Sylvester (1814-1897), Cayley (1821-1895)]
     22. Master and Pupil [Weierstrass (1815-1897) and Sonja Kowalewski (1850-1891)]
     23. Complete Independence [Boole (1815-1864)]
     24. The Man, Not the Method [Hermite (1822-1901)]
     25. The Doubter [Kronecker (1823-1891)]
     26. Anima Candida [Riemann (1826-1866]
     27. Arithmetic the Second [Kummer (1810-1893) and Dedekind (1831-1916)]
     28. The Last Universalist [Poincaré (1854-1912)]
     29. Paradise Lost? [Cantor (1845-1918)]
Berlinski, David.  A Tour of the CalculusRandom House Pantheon Books (New York: 1995).  Random House Vintage edition ISBN 0-679-74788-5.
     Based on two samples thus far (cf. [Berlinski2000]), there is something poetic, metaphoric and lyrical in Berlinski's writings that captures me.  I find myself thinking like he writes and wanting to engage in literary expansions far beyond my ordinary inclination to write too much.   I am possessed.
     If this book were "merely" about the calculus, I would not have bothered.  I didn't bother on other occasions where I chanced across the book.   This time, I looked more closely, having Cantor and Einstein on my mind at the time, and saw that there is much here about the entire development of mathematics that made the calculus possible and, also, essential.  It is that treatment of the calculus tied to the very foundation and fabric of what mathematics is up to that has me commend the material now.   I have not completed my first reading and there is already a rich collage of Post-Its sticking out here and there in a variety of day-glo hues. -- dh:2002-09-02
     "The sense of intellectual discomfort by which the calculus was provoked into consciousness in the seventeenth century lies deep within the memory.  It arises from an unsettling contrast, a division of experience.  Words and numbers are, like the human beings that employ them, isolated and discrete; but the slow and measured movement of the stars across the night sky, the rising and the setting of the sun, ... -- these are, all of them, continuous and smoothly flowing processes.  Their parts are inseparable.  How can language account for what is not discrete, and numbers for what is not divisible?
     "Space and time are the great imponderables of human experience, the continuum within which every life is lived and every river flows.  In its largest, its most architectural aspect, the calculus is a great, even spectacular theory of space and time, a demonstration that in the real numbers there is an instrument adequate to their representation.  ...
     "It is sometimes said and said sometimes by mathematicians that the usefulness of the calculus resides in its applications.  This is an incoherent, if innocent, view of things.  However much the mathematician may figure in myth, absently applying stray symbols to an alien physical world, mathematical theories apply only to mathematical facts, and mathematics can no more be applied to facts that are not mathematical than shapes may be applied to liquids.  ... It is in the world of things and places, times and troubles and dense turbid processes, that mathematics is not so much applied as illustrated.  -- from the Introduction, pp. xi-xii.
I am quite taken with this Berlinski offering. The chapter title, though charming, do not easily yield up their themes to my scrutiny and I have provided additional annotation to tempt the reader to plunge into this story's embrace.  You can tell how far I have gotten thus far.  -- dh:2002-09-02
     A Note to the Reader
     The Frame of the Book

     1. Masters of the Symbols [Zeno, Newton, Liebnitz]
     2. Symbols of the Masters [mathematical representation, Euclidean geometry, natural numbers, rationals]
     3. The Black Blossoms of Geometry [distance, measure, zero, negative numbers]
     4. Cartesian Coordinates [equations of lines, Descartes]
     5. The Unbearable Smoothness of Motion [continuity, density, and incommensurable magnitudes]
     6. Yo [proof that square root of 2 is not a (rational) number, acceptance of irrationals]
     7. Thirteen Ways of Looking at a Line [severability and the Dedekind cut]
     8. The Doctor of Discovery [Richard Dedekind, the irrational numbers made real]
     9. Real World Rising [Leopold Kronecker, commitment to the existence of reals]
     10. Forever Familiar, Forever Unknown [mathematical functions, existence of functions as relations]
     11. Some Famous Functions [algebraic functions, transcendental functions, Leonhard Euler]
     12. Speed of Sorts
     13. Speed, Strange Speed
     14. Paris Days
     15. Prague Interlude
     16. Memory of Motion
     17. The Dimpled Shoulder
     18. Wrong Way Rolle
     19. The Mean Value Theorem
     20. The Song of Igor
     21. Area
     22. Those Legos Vanish
     23. The Integral Wishes to Compute an Area
     24. The Integral Wishes to Become a Function
     25. Between the Living and the Dead
     26. A Farewell to Continuity

Beck, Anatole., Bleicher, Michael N., Crowe, Donald Warren.  Excursion into Mathematics: The Millennium Edition.  With a foreword by Martin Gardner.  A. K. Peters, Ltd. (Natick, MA: 2000).  ISBN 1-56881-115-2 pbk: alk. paper.  See [Beck2000].
Courant, Richard., Robbins, Herbert.  What is Mathematics, ed. 2.  Revised by Ian Stewart.  Oxford University Press (New York: 1941, 1996).  ISBN 0-19-510519-2 pbk.
     "What Is Mathematics? is one of those great classics, a sparkling collection of mathematical gems, one of whose aims was to counter the idea that 'mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician.'  In short, it wanted to put the meaning back in mathematics.  But it was meaning of a very different kind from physical reality, for the meaning of mathematical objects states 'only the relationships between mathematically `undefined objects´ and the rules governing operations with them.'  It doesn't matter what mathematical things are: it's what they do that counts.  Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible.   This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics--what I have elsewhere called its 'unreal reality.'  Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either."  -- from the Preface to the Second Edition, unnumbered p. vii.
     "For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person.  Today the traditional place of mathematics in education is in grave danger.  Unfortunately, professional representatives of mathematics share in the responsibility.  The teaching of mathematics has sometimes degenerated into empty drill in problem solving, which may develop formal ability but does not lead to real understanding or to greater intellectual independence.  ... Teachers, students, and the educated public demand constructive reform ... .  The goal is genuine comprehension of mathematics as an organic whole and as a basis for scientific thinking and acting.
     "Some splendid books on biography and history and some provocative popular writings have stimulated the latent general interest.  But knowledge cannot be attained by indirect means alone.  Understanding of mathematics cannot be transmitted by painless entertainment any more than education in music can be brought by the most brilliant journalism to those who never have listened intensively.  Actual contact with the content of mathematics is necessary.  Nevertheless technicalities and detours should be avoided, and the presentation of mathematics should be just as free from emphasis on routine as from forbidding dogmatism which refuses to disclose motives or goal and which is an unfair obstacle to honest effort.  It is possible to proceed on a straight road from the very elements to vantage points from which the substance and driving forces of modern mathematics can be surveyed.
     "The present book is an attempt in this direction. ..."  -- from the Preface to the First Edition, unnumbered p. x.
     Foreword (Ernest D. Courant, 1995)
     Preface to the Second Edition (Ian Stewart, 1995)
     Preface to the Revised Editions (Richard Courant, 1947)
     Preface to the First Edition (Richard Courant, 1941)
     How to Use the Book
     What Is Mathematics?
     I.  The Natural Numbers
     II. The Number System of Mathematics
     III. Geometrical Constructions.  The Algebra of Number Fields
     IV. Projective Geometry.  Axiomatics.  Non-Euclidean Geometries
     V. Topology
     VI.  Functions and Limits
     VII.  Maxima and Minima
     VIII.  The Calculus
     IX.  Recent Developments
     Appendix: Supplemental Remarks, Problems, and Exercises
     Suggestions for Further Reading
     Suggestions for Additional Reading
Beck, Anatole., Bleicher, Michael N., Crowe, Donald Warren.  Excursion into Mathematics: The Millennium Edition.  With a foreword by Martin Gardner.  A. K. Peters, Ltd. (Natick, MA: 2000).  ISBN 1-56881-115-2 pbk: alk. paper.  See [Beck2000].
Devlin, Keith JThe Language of Mathematics: Making the Invisible VisibleW. H. Freeman (New York: 1998, 2000).  ISBN 0-7167-3967-4 pbk.
     "This book tries to convey the essence of mathematics, both its historical development and its current breadth.  It is not a 'how to' book; it is an 'about' book, which sets out to describe mathematics as a rich and living part of humankind's culture.  It is intended for the general reader, and does not assume any mathematical knowledge or ability." -- from the Preface, p.vii.
     "It was only within the last thirty years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns.  What the mathematician does is examine abstract 'patterns' ... .  Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, purely utilitarian or of little more than recreational interest.  They can arise from the world around us, from the depths of space and time, or from the inner workings of the human mind." -- from the Prologue, p.3.
     I ordered this book, sight unseen, because of the title.  Struggling with how mathematics frames theories, and how the theoretical can be made real in mathematics (and "present" in computers), I thought that here I would find freedom from my plodding theorizing about theories and be able to move on, pointing others to these pages for what I find so difficult to articulate.  Making the invisible visible.  That's what I see (mathematical) theories providing.  When I found the very book in the neighborhood public library, I had this sudden dread of wasted money.  It is about patterns.   Oh no.  That can't be it.  
     Plugging along on how theories expressed mathematically provide access to the abstract, I made some headway and then the book arrived.   Oh, yes, that is it.  Not how I want to say it, and it's there.
     "To convey something of this modern conception of mathematics, this book takes eight general themes, covering patterns of counting, patterns of reasoning and communicating, patterns of motion and change, patterns of shape, patterns of symmetry and regularity, patterns of position, patterns of chance, and the fundamental patterns of the universe.  Though this selection leaves out a number of major areas of mathematics, it should provide a good overall sense of what contemporary mathematics is about.  The treatment of each theme, while at a purely descriptive level, is not superficial."  -- from the Prologue, pp.3-4.
     I have expanded the content to include all subheadings used through the end of Chapter 3, also identifying some of the content areas to suggest the appealing reach of this material.  I mean to suggest how the foundational themes of mathematics are portrayed, and where to look for more in the coverage.  I stopped with mathematical analysis because that is enough for one complete turn through the foundation, establishing the persistent demand to accommodate the infinite, something those of us who deal with mathematics in terms of discrete computations too-readily neglect.
     Read this book. -- dh:2002-07-23.
     Prologue: What is Mathematics?
          It's not just numbers
          Mathematics in motion
          The science of patterns
          Symbols of progress
          When seeing is discovering
          The hidden beauty in the symbols
          Making the invisible visible
          The invisible universe
     Chapter 1: Why Numbers Count
          You can count on them 
          These days, children do it before they're five [counting]
          A token advance [physical numbering systems]
          Symbolic progress [written numbering systems]
          For a long time it was all Greek
          A fatal flaw is discovered [irrational numbers]
          Here's looking at Euclid [the mathematical method]
          Numbers in prime condition [fundamental theorem of arithmetic]
          Prime order [the density of primes]
          The child genius [Gauss]
          Gauss's clock arithmetic [modular arithmetic]
          The great amateur [Fermat]
          Taking the prime test [ARCLP test for primality]
          Keeping secrets [factorization complexity, Diffie, Hellman, Rivest, Shamir, Adleman]
          Easy to guess, hard to prove [Goldbach, Mersenne]
          Fermat's last theorem 
          The Fermat saga begins [expanding proofs of limited cases]
          The domino effect [mathematical induction]
     Chapter 2: Patterns of the Mind
          Proof beyond doubt
          The logical patterns of Aristotle [valid syllogisms]
          How Euler circled the syllogism
          An algebra of thought [Boole]
          The atomic approach to logic [propositional logic]
          The patterns of reason [propositional inference]
          Splitting the logical atom [Peano, Frege, predicate logic]
          The dawn of the modern age [mathematical truth and proof]
          The power of abstraction [axiomatic method]
          The versatile concept of sets [Cantor]
          Numbers from nothing [sets for everything]
          Cracks in the foundations [Predicative sets, Russell, Zermelo, Fraenkel]
          The rise and fall of Hilbert's program [Gödel]
          The golden age of logic [mathematical logic, model theory, computation theory]
          Patterns of language [Chomsky]
          The fingerprint hidden in our words [individual styles]
     Chapter 3: Mathematics in Motion
          A world in motion
          The two men who invented calculus [Liebniz and Newton]
          The paradox of motion [Zeno]
          Taming infinity [convergent series]
          Infinity bites back [non-convergent series]
          Functions provide the key [functions as objects of manipulation]
          How to compute slopes [derivatives]
          Ghosts of departed quantities [fluxions, infinitesimals]
          Chasing sound intuitions [limits of sequences]
          The differential calculus
          Is there a danger from the radiation? [differential equations]
          Waves that drive the pop music industry [Fourier analysis]
          Making sure it all adds up [integral calculus]
          The real numbers [real analysis - Cauchy, Wierstrass, Dedekind]
          Complex numbers
          Where all equations can be solved [fundamental theorem of algebra - d'Alambert, Euler, Gauss]
          Euler's amazing formula
          Uncovering the hidden patterns of numbers [Riemann]
     Chapter 4: Mathematics Gets into Shape [Geometries]
     Chapter 5: The Mathematics of Beauty [symmetries, transformations, packing, coloring, tiling]
     Chapter 6: What Happens When Mathematics Gets into Position [topology]
     Chapter 7: How Mathematicians Figure the Odds [probability]
     Chapter 8: Uncovering the Hidden Patterns of the Universe [mathematical physics]

Michalewicz, Zbigniew., Fogel, David B.  How to Solve It: Modern Heuristics.  Corrected Second printing.  Springer-Verlag (Berlin: 2000).  ISBN 3-540-66061-5 alk.paper.   See [Michalewicz2000]
Beck, Anatole., Bleicher, Michael N., Crowe, Donald Warren.  Excursion into Mathematics: The Millennium Edition.  With a foreword by Martin Gardner.  A. K. Peters, Ltd. (Natick, MA: 2000).  ISBN 1-56881-115-2 pbk: alk. paper.  See [Beck2000].
Hart, Wilbur Dyre (ed.).  The Philosophy of Mathematics.  Oxford University Press (Oxford: 1996).  ISBN 0-19-875120-6 pbk.
     I. Mathematical Proof.  Paul Benacerraf
     II. Two Dogmas of Empiricism.  W. V. Quine
     III. Access and Inference.  W. D. Hart
     IV. The Philosophical Basis of Intuitionist Logic.  Michael Dummett
     V. Mathematical Intuition.  Charles Parsons
     VI. Perception and Mathematical Intuition.  Penelope Maddy
     VII. Truth and Proof: The Platonism of Mathematics.  W. W. Tait
     VIII. Mathematics without Foundations.  Hilary Putnum
     IX. The Consistency of Frege's Foundations of ArithmeticGeorge Boolos
     X. Arithmetical Truth and Hidden Higher-Order Concepts.  Daniel Isaacson
     XI.  Conservatism and Incompleteness.  Stewart Shapiro
     XII. Is Mathematical Knowledge Just Logical Knowledge?  Hartry Field
     XIII. The Structuralist View of Mathematical Objects.  Charles Parsons
     Notes on the Contributors
     Suggestions for Further Reading
Henderson, Harry.  Modern MathematiciansFacts on File (New York: 1996).  ISBN 0-8160-3235-1.
     "The title of this book is Modern Mathematicians because the people in this book lived and worked in the last two centuries ... .   It would take many more books like this one to survey all the remarkable lives and important achievements that make up modern mathematics.  But the stories of the 13 mathematicians profiled in the book's 12 chapters will give you a good taste of the diversity of modern mathematics." -- from the Introduction, pp. xi-xii.
     These are all brief sketches, but they provide some insight into the different times and interests of the selected mathematicians. -- dh:2002-09-02
          Charles Babbage (1792-1871) and Ada Lovelace (1815-1852)
          George Boole (1815-1864)
     Georg Cantor (1845-1918)
          Sofia Kovalevskaia (1850-1891)
          Emmy Noether (1882-1935)
     Srinivasa Ramanujan (1887-1920)
     Stanislaw Ulam (1909-1984)
     Shiing-Shen Chern (1911- )
          Alan Turing (1912-1954)
          Julia Bowman Robinson (1919-1985)
          Benoit Mandelbrot (1924- )
          John H. Conway (1937- )
Körner, Stephan.  The Philosophy of Mathematics: An Introductory Essay.  Hutchinson & Co (London: 1960, 1968).  Unabridged and unaltered republication by Dover Publications (New York: 1968).  ISBN 0-486-25048-2 pbk.
     "... Questions we have asked about the apparently isolated statement ['1+1=2'] will be seen immediately to extend to the system or systems to which it belongs.  In a similar way we shall be forced to inquire into the pure system or systems of geometry and class-algebra, and into the structure of applied arithmetic, geometry and class-algebra.  And this inquiry in turn will raise the question of the structure and function of pure and applied mathematical theories in general.
     "The full implications of a philosopher's answer to this last and central question will, of course, become clearer by considering the manner in which he deals with more specific problems, in particular controversial ones.  One of these--one of the most important--concerns the proper analysis of the notion of infinity.  The problem arises at an early stage of our reflection upon the apparently unlimited possibilities of continuing the sequence of natural numbers, and of subdividing the distance between two points; and it re-emerges at all later and more subtle stages of philosophizing about discrete and continuous quantities.  If in the history of mathematics a new epoch can sometimes be marked by a new conception of infinite quantities and sets, then this is even more true of the history of the philosophy of mathematics.
     "We are now in a position to indicate in a preliminary way the topics of our present discussion.  They are first the general structure and function of the propositions and theories belonging to pure mathematics, secondly the general structure and function of the propositions and theories belonging to applied mathematics, and thirdly questions about the role of the notion of infinity in the various systems in which it occurs."  -- from the Introduction, pp.10-11.

     I. Some Older Views
     II. Mathematics as Logic: Exposition
     III. Mathematics as Logic: Criticism
     IV. Mathematics as the Science of Formal Systems: Exposition
     V. Mathematics as the Science of Formal Systems: Criticism
     VI. Mathematics as the Activity of Intuitive Constructions: Expositions
     VII. Mathematics as the Activity of Intuitive Constructions: Criticism
     VIII. The Nature of Pure and Applied Mathematics
     Appendix A.  On the classical theory of real numbers
     Appendix B. Some suggestions for further reading

Lakoff, George., Núñez, Rafael E. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.  Basic Books (New York: 2000).  ISBN 0-465-03771-2 pbk.
     Introduction: Why Cognitive Science Matters to Mathematics
     Part I: The Embodiment of Basic Arithmetic
          1. The Brain's Innate Arithmetic
          2. A Brief Introduction to the Cognitive Science of the Embodied Mind
          3. Embodied Arithmetic: The Grounding Metaphors
          4. Where Do the Laws of Arithmetic Come From?
     Part II: Algebra, Logic, and Sets
          5. Essence and Algebra
          6. Boole's Metaphor: Classes and Symbolic Logic
          7. Sets and Hypersets
     Part III: The Embodiment of Infinity
          8. The Basic Metaphor of Infinity
          9. Real Numbers and Limits
          10. Transfinite Numbers
          11. Infinitesimals
     Part IV: Banning Space and Motion: The Discretization Program that Shaped Modern Mathematics
          12. Points and the Continuum
          13. Continuity for Numbers: The Triumph of Dedekind's Metaphors
          14. Calculus Without Space or Motion: Weierstrass's Metaphorical Masterpiece
     Le trou normand: A Classic Paradox of Infinity
     Part V: Implications for the Philosophy of Mathematics
          15. The Theory of Embodied Mathematics
          16. The Philosophy of Embodied Mathematics
     Part VI: e
πi + 1 = 0 A Case Study of the Cognitive Structure of Classical Mathematics
          Case Study 1.  Analytic Geometry and Trigonometry
          Case Study 2.  What is e?
          Case Study 3.  What is i?
          Case Study 4. 
eπi + 1 = 0 -- How the Fundamental Ideas of Classical Mathematics Fit Together

Michalewicz, Zbigniew., Fogel, David B.  How to Solve It: Modern Heuristics.  Corrected Second printing.  Springer-Verlag (Berlin: 2000).  ISBN 3-540-66061-5 alk.paper.

     I. What Are the Ages of My Three Sons?
     1. Why Are Some Problems Difficult to Solve?
     II. How Important Is a Model?
     2. Basic Concepts
     III. What Are the Prices in 7-11?
     3. Traditional Methods -- Part 1
     IV. What Are the Numbers?
     4. Traditional Methods - Part 2
     V. What's the Color of the Bear?
     5. Escaping Local Optima
     VI. How Good Is Your Intuition?
     6. An Evolutionary Approach
     VII. One of These Things Is Not Like the Others
     7. Designing Evolutionary Algorithms
     VIII. What Is the Shortest Way?
     8. The Traveling Salesman Problem
     IX. Who Owns the Zebra?
     9. Constraint-Handling Techniques
     X. Can You Tune to the Problem?
     10. Tuning the Algorithm to the Problem
     XI. Can You Mate in Two Moves?
     11. Time-Varying Environments and Noise
     XII.  Day of the Week of January 1st
     12. Neural Networks
     XIII. What Was the Length of the Rope?
     13. Fuzzy Systems
     XIV. Do You Like Simple Solutions?
     14. Hybrid Systems
     15. Summary
     Appendix A: Probability and Statistics
     Appendix B: Problems and Projects

Lakoff, George., Núñez, Rafael E. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.  Basic Books (New York: 2000).  ISBN 0-465-03771-2 pbk.  See [Lakoff2000]
Pólya, George.  How to Solve It.  ed.2.  Princeton University Press (Princeton, NJ: 1945, 1957).  ISBN 0-691-08097-6.
     I was put off by this book when I first saw it as a young college student.  It looked too hard, and there was too much geometry.  I tossed in judgments like that.
    Yet, I am mindful that there is almost no context in which Donald Knuth doesn't pay homage to Pólya's work and tutelage.  (My evidence is in the indexes of Concrete Mathematics«Literate Programming», Selected Papers on Computer Science, and The Art of Computer Programming.)
     Wandering through an used-book shop near my wife's pottery studio, I was shocked to see principles that I had been emphasizing while tutoring a high-school student last summer, and my 11-year-old grand-nephew this school year: "Can you check the result?  Can you check the argument?  Can you derive the result differently?  Can you see it at a glance?  Can you use the result, or the method, for some other problem?  (p. xvii)"  We do these things to have that mastery in our private and public affairs and to some degree over nature itself, that provides for reliability in an inherently chancy world.  That we can do so by mastering theoretical abstractions and validly applying them in the world is one of the marvels of human existence.  What a gift it is that computation works at all!
     It is time I learned more with the guidance of this master.  [dh:2001-02-04]  
     From the Preface to the First Printing
     From the Preface to the Seventh Printing
     Preface to the Second Edition
     "How to Solve It" list

     I. In the Classroom
     II. How to Solve It
     III. Short Dictionary of Heuristic
     IV. Problems, Hints, Solutions
Courant, Richard., Robbins, Herbert.  What is Mathematics, ed. 2.  Revised by Ian Stewart.  Oxford University Press (New York: 1941, 1996).  ISBN 0-19-510519-2 pbk.  See [Courant1996]
Robinson, Abraham.  Non-Standard Analysis.  ed.2.  Princeton University Press (Princeton, NJ: 1965, 1973, 1996).  ISBN 0-691-04490-2 pbk.  Re-issue of the 1973 second edition with a 1996 foreword by Wilhelmus A. J. Luxemburg.
     This book is about mathematics and in particular analysis, based on a non-standard model of numbers in which infinitesimal and infinitely large elements have first-class standing as individuals of the theory.  A highly-productive approach grounded in mathematical logic, this book provides an application of model-theoretic approaches.
     Foreword (1996)
     Preface to the Second Edition

     I. General Introduction
     II. Tools from Logic
          2.1. The Lower Predicate Calculus
          2.2. Interpretation
          2.3. Ultraproducts
          2.4. Prenex normal form
          2.5. The finiteness principle
          2.6 Higher order structures and corresponding languages
          2.7 Type symbols
          2.8 Finiteness principle for higher order theories
          2.9 Enlargements
          2.10 Examples of enlargements
          2.11 General properties of enlargements
          2.12 Remarks and references
     III. Differential and Integral Calculus
     IV. General Topology
     V. Functions of a Real Variable
     VI. Functions of a Complex Variable
     VII. Linear Spaces
     VIII. Topological Groups and Lie Groups
     IX. Selected Topics
     X. Concerning the History of the Calculus
     Index of Authors
     Subject Index
Stewart, Ian.  Concepts of Modern Mathematics.  Dover Publications (New York: 1975, 1981, 1995).  An unabridged, slightly corrected republication of the 1981 edition of the work first published by Penguin Books, Harmondsworth, Middlesex, England, 1975.  ISBN 0-486-28424-7 pbk.
     Preface to the Dover Edition
     Preface to the First Edition

     1.  Mathematics in General
     2.  Motion without Movement
     3.  Short Cuts in the Higher Arithmetic
     4.  The Language of Sets
     5.  What is a Function?
     6.  The Beginnings of Abstract Algebra
     7.  Symmetry: The Group Concept
     8.  Axiomatics
     9.  Counting: Finite and Infinite
     10. Topology
     11. The Power of Indirect Thinking
     12. Topological Invariants
     13. Algebraic Topology
     14. Into Hyperspace
     15. Linear Algebra
     16. Real Analysis
     17. The Theory of Probability
     18. Computers and Their Use
     19. Applications of Modern Mathematics
     20. Foundations
     Glossary of Symbols

Courant, Richard., Robbins, Herbert.  What is Mathematics, ed. 2.  Revised by Ian Stewart.  Oxford University Press (New York: 1941, 1996).  ISBN 0-19-510519-2 pbk.  See [Courant1996]
Tymoczko, Thomas (ed.).  New Directions in the Philosophy of Mathematics: An Anthology.  ed. 2.  Princeton University Press (Princeton, NJ: 1986, 1998).  ISBN 0-691-03498-2 pbk.

     I. Challenging Foundations
          Reuben Hersh.  Some Proposals for Reviving the Philosophy of Mathematics
          Imre Lakatos.  A Renaissance of Empiricism in the Recent Philosophy of Mathematics?
          Hilary Putnum.  What Is Mathematical Truth?
          René Thom.  "Modern" Mathematics: An Educational and Philosophical Error?
          Nicholas D. Goodman.  Mathematics as an Objective Science
          George Polya. From the Preface of Induction and Analogy in Mathematics.
          George Polya.  Generalization, Specialization, Analogy.
     II. Mathematical Practice
          Hao Wang.  Theory and Practice in Mathematics
          Imre Lakatos.  What Does a Mathematical Proof Prove?
          Philip J. Davis.  Fidelity in Mathematical Discourse: Is One and One Really Two?
          Philip J. Davis and Reuben Hersh.  The Ideal Mathematician
          Robert L. Wilder.  The Cultural Basis of Mathematics
          Judith V. Grabiner.  Is Mathematical Truth Time-Dependent?
          Philip Kitcher.  Mathematical Change and Scientific Change
          Thomas Tymoczko.  The Four-Color Problem and Its Philosophical Significance
          Richard A. De Millo, Richard J. Lipton, and Alan J. Perlis.  Social Processes and Proofs of Theorems and Programs
          Gregory Chaitin.  Information-Theoretic Computational Complexity
          Gregory Chaitin.  Godel's Theorem and Information.
     III. Current Concerns
          Michael D. Resnik.  Proof as a Source of Truth
          William P. Thurston.  On Proof and Progress in Mathematics
          Penelope Maddy.  Does V Equal L?
     Supplemental Bibliography of Recent Work
Whitehead, Alfred North.  An Introduction to Mathematics.  Oxford University Press (New York: 1911, 1948).  12th printing of paperback edition issued 1958.  ISBN 0-19-500211-3.
     "Alfred North Whitehead, British mathematician and philosopher, died in 1947.  He is the author of many books, and is the co-author, with Bertrand Russell, of the monumental Principia Mathematica."  -- from the back cover.
     "The object of the following chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena.  All allusion in what follows to detailed deductions in any part of the science will be inserted merely for the purpose of example, and care will be taken to make the general argument comprehensible, even if here and there some technical process or symbol which the reader does not understand is cited for the purpose of illustration." p.2
     "... All science as it grows toward perfection becomes mathematical in its ideas." p.6
     I find this material of interest not so much for its mathematical content as for its effort to address just what mathematics is that it is important to us, and how casting abstractions in mathematical terms provides a way to form a general idea of the course of natural events, "to see what is general in what is particular and what is permanent in what is transitory [p.4]."  That is, to abstract the perceived orderliness of nature into mathematical theories that are interpreted to predict the course of events.  I am interested to learn how the very idea of mathematics as a device for marshalling abstract theories separated from particular interpretations was perceived by someone active in the period where this point of view was strengthened so much by revolutionary developments in logic and the mathematical language of science. -- dh:2002-07-05.
     1. The Abstract Nature of Mathematics
     2. Variables
     3. Methods of Application
     4. Dynamics
     5. The Symbolism of Mathematics
     6. Generalizations of Number
     7. Imaginary Numbers
     8. Imaginary Numbers (continued)
     9. Co-ordinate Geometry
     10. Conic Sections
     11. Functions
     12. Periodicity in Nature
     13. Trigonometry
     14. Series
     15. The Differential Calculus
     16. Geometry
     17. Quantity

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