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
- [Beck2000]
- 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.
Content
Preface to the Millennium Edition
Preface
Foreword
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
Index
Appendix 2000
- [Bell1937]
- 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.
Content
Acknowledgments
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)]
Index
- [Berlinski1995]
- Berlinski, David. A Tour of the Calculus. Random
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
Content
Introduction
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
Epilogue
Acknowledgments
Index
- [Bleicher2000]
- 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].
- [Courant1996]
- 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.
Content
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
Index
-
- [Crowe2000]
- 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].
- [Devlin2000]
- Devlin, Keith J. The Language of Mathematics: Making the
Invisible Visible. W. 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.
Content
Preface
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]
Postscript
Index
- [Fogel2000]
- 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]
-
- [Gardner2000]
- 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].
- [Hart1996]
- Hart, Wilbur Dyre (ed.). The Philosophy of Mathematics.
Oxford University Press (Oxford: 1996). ISBN 0-19-875120-6 pbk.
Contents
Introduction
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 Arithmetic. George 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
Index
- [Henderson1996]
- Henderson, Harry. Modern Mathematicians. Facts
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
Content
Acknowledgments
Introduction
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- )
Index
- [Körner1968]
- 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.
Content
Preface
Introduction
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
Index
- [Lakoff2000]
- 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.
Content
Acknowledgments
Preface
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: eπi
+ 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
References
Index
- [Michalewicz2000]
- 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.
Content
Preface
Introduction
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
References
Index
- [Núñez2000]
- 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]
- [Pólya1957]
- 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]
Content
From the Preface to the First Printing
From the Preface to the Seventh Printing
Preface to the Second Edition
"How to Solve It" list
Introduction
I. In the Classroom
II. How to Solve It
III. Short Dictionary of Heuristic
IV. Problems, Hints, Solutions
- [Robbins1996]
- 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]
- [Robinson1996]
- 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.
Contents
Foreword (1996)
Preface
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
Bibliography
Index of Authors
Subject Index
- [Stewart1995]
- 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.
Content
Preface to the Dover Edition
Preface to the First Edition
Acknowledgments
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
Appendix
Notes
Glossary of Symbols
Index
- [Stewart1996]
- 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]
- [Tymoczko1998]
- 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.
Contents
Preface
Introduction
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
Interlude
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?
Afterword
Bibliography
Supplemental Bibliography of Recent Work
- [Whitehead1948]
- 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.
Content
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
Bibliography
Index
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