Keith Douglas' Web Page

About me Find out who I am and what I do.
My resumé A copy of my resumé and other documentation about my education and work experience for employers and the curious.
Reviews, theses, articles, presentations A collection of papers from my work, categorized and annotated.
Current research projects What I am currently working on, including some non-research material.
Interesting people People professionally "connected" to me in some way.
Interesting organizations Organizations I am "connected" to. (Some rather loosely.)
Intellectual/professional influences Influences on my work, including an organization chart. Here you can also buy many good books on philosophy and other subjects via I have included brief reviews of hundreds of books.
Professional resources Research sources, associates programs, etc.
What is the philosophy of computing? A brief introduction to my primary professional interest.
My intellectual heroes A partial list of important people. Limited to the dead.
My educational philosophy As a sometime teacher I've developed one. Includes book resources.

Book Influences - Mathematics

Purchase / Enjoy Cover
A Modern Analytic Geometry Edgett (et al) A textbook in senior high school analytic geometry. Nothing particularly noteworthy either way as to its contents other than a reasonably large number of exercises of varying difficulty. Presupposes a small amount of trignometry and (of course) elementary algebra.
A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering (2e) James A slim book on a very important "analytical" tool. Like Div, Grad, Curl (below) it is a math book written by a physicist. So, although mathematicians may wince at the lack of rigor, everyone else might find it a bit easier to use. My only complaint is that it has less computational implementation discussion than I would have liked. The FFT implementation shown is in a 70s-early 80s BASIC dialect (i.e. with GOTO and line numbers - ugh) and is about the only such mention. On the other hand, one of the author's goals is to avoid messy long integrations, so maybe the symbolic approaches can be done (in part) with symbolic manipulators these days.
Automata and Computability Kozen Nicholas Pippenger assigned this in his introductory theory of computation class at UBC. I like its presentation of automata - very "visualizable." It also got me thinking more about metaphysics of computation.
Axiomatic Set Theory Suppes An introductory short text on the obvious subject. Many theorems proved, but also many are exercises. This reprint from Dover has some typographic problems from time to time (faded/cut off pages) but nothing that makes comprehension impossble. Covers general matters, relations and functions, equipollence, finite sets, ordinals and cardinals, the reals (via Cauchy sequences), transfinite induction and the axiom of choice. The latter chapter proves a helpful list of ten equivalents to the latter postulate.
Bayesian Networks and Decision Graphs Jensen Very hard to read, and examples are very unclearly motivated.
Calculus of Several Variables Adams Some of the exercises are ridiculously simple relative to what one would expect. Some are a bit too challenging. I have not used its sequences and series material either, and so have no idea how good that part is. Not sure if the link is to the right item.
Calculus the Easy Way Downing A simple calculus book centered around a strange story. The discussion of how the integral sign was invented is fun, too, though of course, fiction. (I had heard something similar from a physics teacher years before I saw this book ...)
Contributions to the Founding of the Theory of Transfinite Numbers Cantor Cantor's two pioneering papers are here translated. Also included is a detailed introduction by noted historian of mathematics P. Jourdain places Cantor's intellectual context as being the theory of functions and analysis. Two minor complaints about these two components to the volume are: (1) the translation (and the English of the introduction) is archaic - no reference to "set theory"; (2) Jourdain makes a historical mistake of minor importance. He attributes the insight that an infinite set can have a proper subset of the same cardinality (to speak anchronistically) to Bolzano. As far as I know, the earliest recognition of this somewhat "paradoxical" fact is actually in Galileo - and it was also noted by Leibniz, who mentions Galileo by name.
Counterfactuals and Causal Inference: Methods and Principles for Social Research Morgan and Winship This is primarily a clear and well worked out book on relatively new statistical methods. The innovations center around inferring actual causal relationships. Corrolation does not imply causation, but corrolation plus other assumptions, if justified, do. This book is about exploring the other assumptions, both in the formalism and without. It is in the latter case where the techniques touch philosophy. (The authors even suggest a realist attitude is needed in order to properly discover causation.)
Data Quality and Record Linkage Techniques Herzog, Scheuren, Winkler This is sort of a literature (and to a lesser extent, software) review on the subjects indicated in the title. Both algorithmic and statistical features are discussed, though primarily the latter. (I would have liked to see more discussion of computational complexity, having experienced how slow even preparing data for record linkage can be.) On a slightly more personal level, it is nice to see Statistics Canada mentioned favourably repeatedly.
Deductive Geometry and Introduction to Trigonometry Petrie   High school deductive geometry book that also covers trigonometry and the logarithm and some applications.
Dialogues Concerning Two New Sciences Galileo Galileo's famous physics book also treats some mathematical topics.
Discrete Mathematics with Applications Epp A pretty good discrete math book geared towards computing students. Makes the usual mistakes concerning computability that many do: the premiss that the putative halt() program must be that, a program as ordinarily understood. Also hardcover, which inflates the price.
Div, Grad, Curl and All That Schey An introductory vector calculus book motivated by problems in electricity and magnetism.
Elementary Algebra Rich As is usual for Schaum's, the strength of this book is its solved problems. This is a high school level book.
Elementary Linear Algebra Abridged Version Anton   A small linear algebra book. I have not found the abridged edition on
Elementary Mathematics Of Sets Brandt   A small book of set theory at roughly the high school level. I have not found it on
Elementary Numerical Analysis: An Algorithmic Approach Conte and de Boor How to represent numbers, solve equations (exactly and approximately), approximate functions, and so on. As this book (in my case bought used) dates from more than 40 years ago, it also includes computational implementation (as it should), but with examples in FORTRAN IV. The age also makes the discussion of floating point considerations somewhat out of date, being prior to the IEEE floating point standard and all.
Encyclopedia of Ignorance Duncan As this book is now decades old, an interesting project would be to go through it and see which areas in it are no longer areas of ignorance as well as to produce a new edition with new topics. This might illustrate how ignorance in a sense increases or at least does not decrease simply with the process of inquiry. (It is good to know we shall not run out of work!) I bought it used because the title seemed fascinating and the contents scientifically respectable.
Essays on the Theory of Numbers Dedekind Some translations of Dedekind's famous papers on number theory. Includes the appearance of what are sometimes called the Dedekind-Peano axioms, though this name is somewhat misleading.
Facts From Figures Moroney An elementary but dense introduction to statistics. My used copy is unfortunately in pieces, and I did not notice prior to buying that the first 5 chapters are missing. They must, given what follows, be extremely elementary indeed (means, etc. are no doubt covered). But the remainder which I have read is quite well written and not above a little humour, including a funny but critical (debatably too critical) analysis of time series and their uses/abuses. (Nowhere is there any discussion of laws, etc. which might render the "curve fitting" exercise that the author rightfully complains about a little less futile.)
Foundations and Fundamental Concepts of Mathematics Eves This survey of mathematics covers all the major branches of mathematics (including foundational matters like logic and set theory) except number theory. Its treatment of analysis is also very brief. These omissions are made up for by historical remarks and many exercises, including some quite challenging (to me, a non-mathmatician) ones. Especially interesting as exercises are a collection of fallacious proofs (of paradoxical conclusions) that one must figure out the error in.
Fundamental Concepts in the Design of Experiments Hicks (Note: link to later edition.) This, despite its title, is only about statistical processing of experiments and statistical analysis of same. Nothing about hypothesis selection or actually deciding what is feasable to measure, etc. to be found here. Hence it is basically a mathematics book and a relatively dry one at that. Various examples, but poorly motivated and very incompletely discussed. Not a book for beginners to statistics either.
Games and Decisions: Introduction and Critical Survey Luce and Raiffa An introduction at a relatively nontechnical (skippable proofs, little calculus and matrix algebra) level to game theory. What is missing in this otherwise erudite and reasonably readable tome is more about how to decide to make use of a game theoretic model of a real social situation. Of course, being a math book such applications are sort of not on topic, but still ...
Geometry for High Schools Lougheed High school geometry textbook that I got as a gift to improve my handwriting. Unfortunately, that aspect of it doesn't seem to have helped me ...
Group Theory And Its Application To Physical Problems Hamermesh Despite the promise of application to physical problems, this book is short on the latter. Nevertheless, the physicist interested in group theoretic approaches to quantum mechanics has likely come to the right place - many proofs are more informal than mathematicians would ordinarily tolerate, etc. The motivation behind using these techniques is also poorly justified. Nevertheless, the book covers a lot of material briskly and the value (as is typical for Dover) is quite high.
How to Solve It: A New Aspect of Mathematical Method Polya Polya's little book is an analysis of mathematical reasoning.
Innumeracy: Mathematical Illiteracy and Its Consequences Paulos Paulos explains why we need to know about numbers, logic, etc. Unfortunately a lot of it is preaching to the choir.
Introduction to Algebra Perlis Elementary algebra, as understood by mathematicians. Group theory, field theory, etc. are all introduced here. Claims that the entire book is to be understood as being based on set theory, though does not take that approach too literally.
Introduction to Mathematical Philosophy Russell If only all prisoners could write such lucid and interesting books.
Introduction to Probability Models Ross (Note: I only have the 2nd edition.) Very cursory introduction to everything from the Kolmogorov axioms for probability to queuing theory and related topics. Topics are not really very well motivated and seems to almost to consist of endless partial examples. Not easy or rigorous (for those who expect that sort of thing).
Introduction to Real Analysis Goffman I have hardly touched this book.
Introduction to Topology (3e) Mendelson A (weakly) structuralist (even includes a brief discussion of category theory) undergraduate textbook. Hardly motivates the field at all, no answers to exercises. It also presupposes some "calculus with proofs" though does rehearse "ε and δ style" proofs briefly. Not a bad book, just not a terribly inspired or interesting one. Dirt cheap, though.
Introductory Graph Theory Chartrand A curiously uneven book; many of the non-graphy theory examples of mathematical modelling are fairly complex (involving differential equations, for example). Yet this book has an appendix which explains what "function" means in the mathematical context. Some of the proofs are clear; some a bit sketchy. Strengths of the book include exercises (with selected hints and answers) as well as many "applied" problems, both worked and exercise-type.
Journey Through Genius: the great theorems of mathematics Dunham A neat little history of math book, rich with both the math and stories about the mathematicians.
Kurt Gödel Collected Works, Volume III Gödel and a whole team of editors 3rd of 5 volumes of the works of Gödel and translations of the same by a masterful team of scholars. Includes unpublished works and letters. I happen to just have vol. 3 because Wilfried Sieg kindly gave this particular volume to me as a gift.
Limits: A Transition to Calculus Buchanan This curious book is all about the notion of the limit in analysis, though at a rather basic level.
Mathematics for the Nonmathematican Kline All matter of interesting mathematics written sort of with a "liberal arts" audience in mind. For example, it has a discussion of projective geometry as it pertains to the history of painting.
Mathematics: Its Content, Methods and Meaning Aleksandrov, Kolmogorov, and Lavrent'ev (eds) This is a Dover reprint/translation of a 3 volume Soviet mathematics handbook. It is designed to illustrate the scope and usefulness of mathematics to the educated but unscientifically trained public. Includes chapters on virtually every general area of mathematics except foundations. Also of potential interest are the occaisonal philosophical remarks. For example, it includes what might well be a statement of what was the official Soviet philosophy of space, amongst other things.
Modèles mathématiques de la morphogenèse Thom This fascinating little book at the intersection of mathematics and exact metaphysics is unfortunately above my head mathematically. The ostensible subject is how novel forms arise in almost any domain.
New Directions in the the Philosophy of Mathematics Tymoczko This philosophy book also has some more directly mathematical papers in it, though ones with a foundational flavour.
Ordinary Differential Equations Tenenbaum and Pollard The title of this book makes the subject matter obvious. What isn't so obvious are the sheer numbers of applications to the natural sciences and engineering, the worked examples, etc. Also unusual is a disclaimer that the problems and such selected for the book are probably simpler than the ones one will get "from the real world". This book is clearly geared towards nonmathematicians in part, which is all to the good in such a "practical" branch of higher mathematics. As is usual for Dover, the price is certainly right, too.
Partial Differential Equations for Scientists and Engineers Farlow Much lighter on problems and examples than the ODEs book above. However, it is written in a much looser style, and is furthermore motivated by problems in physics, chemistry, biology, logistics, etc.
Probability and Random Processeses for Electrical Engineering (2e) Leon-Garcia A rather undistinguished textbook going from basic probability to computational and electrical design applications and queuing theory. Examples are often very elliptical and engineering applications are just enough to qualify as an engineering textbook. To be fair, it is not sold as one per se, rather than as a math book for engineers.
Probability and Statistics for Engineers and Scientists Walpole Having a reference on probability and statistics proves handy from time to time, so I got an edition of this book used.
Reflections on Kurt Gödel Wang Wang discusses the mathematics and philosophy of his sort-of teacher, Kurt Gödel.
Set Theory, Logic, and their Limitations Machover The text I used at McGill for my second course in logic. Quite mathematically advanced for some philosophers, particularly the set theory material we did not have time to cover.
Sets, Logic and Categories Cameron Another reasonably advanced textbook in logic.
Set Theory and the Continuum Hypothesis Cohen This is sort of a commemoration of a great mathematician and logician as it includes not just an account of forcing from the guy who invented it, but also an introduction by Davis and some remarks by Cohen on Gödel. The latter played an important role on the former's intellectual development and it is interesting to see how they saw one another. Cohen is not nearly as philosophical as Gödel and so the philosophical remarks are almost nil in this little book. However, there are a few, especially after the principle theorems about the continuum hypothesis are proved. Also in the book is a brief survey of the material necessary to understand most of the aforementioned proof.
Set Theory and Logic Stoll A somewhat haphazard (repetitive) text with plenty of exercises of varying difficulties. Probably suitable for a 300 level course or two, depending on how many chapters are selected. Topics include Cantor's concept of set, the natural number sequence, extensions of the natural numbers, propositional (herein: "sentence") and predicate calculus, introduction to axiomatization, boolean algebras, some axiomatic set theory, axiomatization of some algebra, brief metamathematical topics (completeness, etc.). In a way it is very similar to Machover's book (see above) but longer and more "chatty" which is all to the good - in this case -, as far as I am concerned.
Single Variable Calculus: Early Transcendtals Stewart Just a typical single variable calculus textbook. Does include a "lies my calculator or computer told me" appendix which explores some of the pitfalls of floating point arithmetic in some of its potentially hazardous (to begining students) formats.
State Space and Linear Systems Wiberg Another decent Schaum's book, though I have hardly ever used it.
Statistics as Principled Argument Abelson A useful corrective to "recipe" style math books; instead it shows how to integrate mathematics (specifically, as might be guessed) into a larger picture. Books like this would be of some merit in many areas: how to use calculus as part of a larger argument in physics, etc. would be neat. There's some of that, of course, in subject matter books, but a detailed discussion and attempts to elucidate the tacit knowledge involved would be profitable for students. In this case, the volume is primarily about how to fit statistics into arguments in psychology papers.
Symmetry Weyl A slim book introducing symmetry as studied in group theory, via lots of artistic, crystallographic and biological examples. I regret that Weyl does not consider the sort of dynamic symmetry that characterizes interesting chemical cases, though this is not surprising as a precise characterization of this is difficult. However, I was also pleasantly surprised to find that Weyl considers symmetry to be a philosophical concept as well, thus illustrating the Bungeian point about certain very general concepts being philosophical rather than belonging to any specific science. If Weyl had developed that throwaway remark a bit further, this book would count as a work in exact philosophy (metaphysics, specifically).
The Continuum Weyl A slim book about nonstandard analysis. Weyl's principle starting point is a desire to use only predicative definitions. This results in a revisionary set theory which then "percolates up" to the theory of real numbers and so on. It is interesting to notice also his emphasis on intuition and grasping of parts of proofs, etc. Yet he is not a Brouwer. Useful fodder against the Platonists if only because the understanding of the set theoretic "universe" and the usual one are incompatible and yet both could be claimed to have autonomous existence. Weyl does (explictly) skirt matters of ontology, but it is clear that he is a Platonist to some extent.
The Geometry Descartes A book that changed the world. Descartes (with Fermat) is often credited with inventing analytic geometry. This is the branch of geometry that so to speak "arithmetizes" geometry and so renders it also open to algebra. This particular edition of the book reproduces the French original and includes a suitable (though not thoroughly checked by me) English translation.
The Principia Newton Newton's physics and mathematics work of genius. I have no idea why Prometheus Books used this version of the title. Warning to the modern reader: following Newton's purely geometric (for the most part) proofs is extremely difficult to those of us raised on calculus.
The Visual Display of Quantitative Information Tufte Odd to classify as a mathematics book, this little gem nevertheless belongs here, as it is essentially a guide to better use of statistics. That said, it has important lessons for multimedia authoring, which is why I purchased it. For example, human interfaces should forgo "cutesy" but instead be functional. Depth cues should be useful, not flashy, and so on.
The Works of Archimedes Heath (ed.) This book is a partial (some semiredundant proofs removed) collection of the mathematical and physical works of Archimedes. Heath adds many (~150 pages) notes, commentary and introduction. Unfortunately this "Rough Draft Printing" edition is pretty shoddy: markings in the original show up on the pages, etc. I know the work is public domain, but really, that ought not to be an excuse.
Theory and Problems of Group Theory Baumslag and Chandler Another good Schaum's series book. I only bought this because there has been some interest in using group theory in mereology.
Van Norstrand's Scientific Encyclopedia Aroiam (editor)   I bought a used, ancient (1958) version of this because it contains many articles that are still useful. Still discusses computers that were humans!
Variables Complexes: Cours et Problemes Spiegel A Schaum's Outline, in French, on, needless to say, complex-variable calculus. Starts from kinds of numbers to rapidly introduce complex ones and ends with applications to physics. Like all Schaum's volumes, the strength lies in the exercises and solved problems, not in the exposition. In particular, not many of the key theorems are proved; an emphasis on calculation is instead the focus.


Finished with this section? Go back to the list of book subjects here.