From: Mok-Kong Shen Subject: About Principia Mathematica (long) Date: Tue, 19 Jun 2001 00:07:41 +0200 Newsgroups: sci.crypt,sci.math Summary: Just how important is "Principia Mathematica"? In connection with a recent discussion in sci.crypt, I obtained some seemingly radically different opinions or facts on the readability of Whitehead and Russell's Principia Mathematica, a book which I till the present have only heard talking about but never even actually seen. On the one extreme there was a regular in sci.crypt reporting that he had read most of that book while yet in high school. On the other extreme there was an acquaintance of mine claiming that most graduate students in math attempting to read that book would be coming up against a stone wall ('beissen auf Granit'). Fascinated thus by this huge disparity of opinions/facts, I undertook to collect certain matters concerning the book which appear to be of some general interest: (1) Availability. Currently the Cambridge University Press offers the full version at $595.00 and an abridged version at $52.95. Big public libraries are likely to have the full version (e.g. the library of Deutsches Museum in Munich). A company selling rare books offers on the internet the first edition (666+772+491 pages) for $45,000.00, while another offers the second edition (674+742+491 pages) for 3500 pounds. (2) Contents of the book. (Source: http://www.illc.uva.nl/~seop/archives/fall2000/ entries/principia-mathematica/) Principia Mathematica appeared in three volumes which together are divided into six parts. Volume 1 begins with a lengthy Introduction containing sections entitled "Preliminary Explanations of Ideas and Notations", "The Theory of Logical Types" and "Incomplete Symbols". It also contains Part I, entitled "Mathematical Logic", which contains sections on "The Theory of Deduction", "Theory of Apparent Variables", "Classes and Relations", "Logic of Relations", and "Products and Sums of Classes"; and Part II, entitled "Prolegomena to Cardinal Arithmetic", which contains sections on "Unit Classes and Couples", "Sub- Classes, Sub-Relations, and Relative Types", "One-Many, Many-One and One-One Relations", "Selections", and "Inductive Relations". Volume 2 begins with a "Prefatory Statement of Symbolic Conventions". It then continues with Part III, entitled "Cardinal Arithmetic", which itself contains sections on "Definition and Logical Properties of Cardinal Numbers", "Addition, Multiplication and Exponentiation", and "Finite and Infinite"; Part IV, entitled Relation-Arithmetic", which contains sections on "Ordinal Similarity and Relation- Numbers", "Addition of Relations, and the Product of Two Relations", "The Principle of First Differences, and the Multiplication and Exponentiation of Relations", and "Arithmetic of Relation-Numbers"; and the first half of Part V, entitled "Series", which contains sections on "General Theory of Series", "On Sections, Segments, Stretches, and Derivatives", and "On Convergence, and the Limits of Functions". Volume 3 continues Part V with sections on "Well-Ordered Series", "Finite and Infinite Series and Ordinals", and "Compact Series, Rational Series, and Continuous Series". It also contains Part VI, entitled "Quantity", which itself contains sections on "Generalization of Number", "Vector- Families", "Measurement", and "Cyclic Families". A fourth volume, on geometry, was planned but never completed. Even so, the book remains one of the great scientific documents of the twentieth century. (3) Excerpts from diverse web pages about the book. A. (Source: http://www.andrews.edu/~calkins/math/biograph/ biowhite.htm) This landmark work on mathematical logic and the foundations ofmathematics was written by Bertrand Russell and Alfred North Whitehead and was published in three volumes, in 1910, 1912 and 1913. Written as a defense of logicism, the book promoted wide acceptance of modern mathematical logic. Next to Aristotle's Organon, it is the most influential book on logic ever written. B. (Source: www.stephenwolfram.com/publications/talks/) I think Whitehead and Russell probably win the prize for the most notation-intensive non-machine-generated piece of work that's ever been done. C. (Source: http://www.cafecancun.com/bookarts/selfhits.htm) Cambridge University Press, is the publisher of record but refused to print it unless the authors paid part of the publication costs. Russell remarked, "We thus earned minus fifty pounds each for ten years' work." A two-thousand-page philosophical tome on mathematics published early in the century that, by the 1950s, had been read by only about six people. It costs $565.00 through Amazon.com. A few dozen have been sold. This famous book has just been included on a Modern Library list of the century's hundred greatest nonfiction book, reports The New Yorker, May 31, 1999. So the authors lost money on one of the most important books of the century. D. (Source: http://www.arts.ubc.ca/philos/irvine/ABtpm.htm) How is it that we have the mathematical knowledge that we do? ... One answer which immediately comes to mind is that we know that 2 + 2 = 4 because there exists a mathematical proof that 2 + 2 = 4, and a proof, of course, is just the kind of thing which is capable of justifying belief. ... However, despite all this, how many of us could actually provide the requisite justification by way of proof? ... To provide a comparison, it was not until proposition *110.643, on page 83 of the second volume of Principia Mathematica that Whitehead and Russell were able to prove that 1 + 1 = 2, let alone that 2 + 2 = 4. (In a book not otherwise noted for its humour, it is interesting to note that immediately below the proof is the following remarkable understatement: "The above proposition is occasionally useful. It is used at least three times ...") E. (Source: http://www.ams.org/new-in-math/06-1999-media.html) Unreadable but irresistible. Sales of Principia Mathematica have soared recently after it was voted one (number 23 to be exact) of the 100 greatest nonfiction books of the 20th century. This is reported in the May 31 New Yorker, in a ``Talk of the Town'' item by John Cassidy. And unnecessary: ``People don't need to read it because the important things in it have been done more clearly elsewhere,'' according to NYU mathematical philosopher Hartry Field, quoted by Cassidy. A good reference on Russell and Whitehead and the context of their work is the lecture notes by Stanley Burris at the University of Waterloo. Cassidy concludes, referring to Gödel's incompleteness theorems (which came some 20 years later and showed that Russell and Whitehead's goal of a complete axiomatic derivation of mathematics was impossible), ``Logic, even in the hands of figures as brilliant as Whitehead and Russell, has its limits. Book marketing, it appears, doesn't.'' (The lecture notes by Stanley Burris is available at http://www.thoralf.uwaterloo.ca/htdocs/scav/principia/ principia.html) M. K. Shen