Two new theorems in number theory

History of Science · Philosophy of Mathematics ·· More from December 2012

In grade school we learn how to divide one whole number by another. Sometimes nothing is left over, but often the division leaves a “remainder”. One learns to say “Eleven divided by five is two remainder one”. Numbers that always leave a remainder when divided by another number other than themselves or 1 are called prime. All other numbers are called composite (except 1, which is neither prime nor composite).

Friday nights at the Library

Books · Philosophy of Mathematics · Science ·· More from December 2005

On foundations and folklore

Maria Cristina Pedicchio (also ) and Walter Tholen, eds. Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Cambridge: Cambridge University Press, 2004. (Encyclopedia of Mathematics and its Applications, 97) See also the item “Atlantis” in Tholen’s publication list.

In Scholastic philosophy, a local assumption—that is, a premise that is not the conclusion of some prior argument—is often called a fundamentum, a “foundation”. To be a fundamentum is a rhetorical matter; distinctions treated as fundamenta in one context may be argued for at length elsewhere. A fundamentum functions as a point granted by both sides in a disputatio, because it is obvious, perhaps, or a matter of faith. Of course what is obvious to a well-trained Jesuit may not be obvious to you or me.

Mathematical argument is replete with fundamenta in the Scholastic sense. For example, I may start a proof by saying “let f : A → B be a map and let Im f be the image of f ”. I’m assuming of course that you know how to interpret the symbols; I’m also assuming a lemma of the form “Every map has an image”.

One fundamentum that turned out not to be easily proved is the Jordan Curve Theorem. The Theorem states that every simple closed curve in the plane divides the plane into exactly two pieces separated from one another by the curve. Everyone assumed, most often tacitly, its truth; but even Camille Jordan, after whom it is named, did not succeed in proving it; Oswald Veblen finally proved it in 1905. A rigorous proof (a formal proof, that is, checkable by a computer) was completed only in 2005 after fourteen years of effort. See Yakusa Nakamura, “Culmination of a complete proof of the Jordan Curve Theorem” (Sep 2005); the proof itself is accessible from the “abstract” at the Mizar Home Page.

It is my expectation that either you will have seen a proof of the lemma or that you will grant it for the sake of argument. In every branch of mathematics, as it develops, there is a collection of fundamenta, theorems that everyone takes as proved—the “folklore” of that branch. Some have in fact been proved; but some, taken as obvious, may not be explicitly proved until someone writes an introduction to the subject for outsiders. You won’t, after all, make your mark by proving what everyone “knows”—even if in fact they don’t. Though fundamenta are supposed to be obvious, and (so you’d think) easy to prove, sometimes the obvious and the easy-to-prove part ways. The worst case is when the obvious turns out to be false; but that, I think, is not so common. More often it turns out that what was taken to be obvious is not—that is, it depends on hitherto unstated conditions.

Real-time philosophy of real mathematics

Philosophy of Mathematics · Web/Tech ·· More from December 2005

David Corfield’s weblog Philosophy of real mathematics is a potpourri of items pertaining to mathematics and philosophy. Collingwood and MacIntyre show up alongside categories and 6j-symbols. I’ve mentioned his book before. Toward a philosophy of real mathematics is a manifesto on behalf of a philosophy of mathematics that, like the philosophies of physics and biology, takes mathematical practice seriously, rather than confining itself to the formalized versions that logicians have constructed to serve their ends. In my view the book says all that need be said on that point; but I was convinced beforehand. If you have any interest in the kind of program Corfield is proposing, you’ll find his weblog of interest too.

Mathematics: items of interest

Bibliography · Philosophy of Mathematics ·· More from June 2005

A few items on mathematics.

• Mathematical structures
  Peter Jipsen. A wiki-ish list of mathematical structures, mostly algebraic. Quite a menagerie. People who know about structures and have competence in LaTeX can contribute. Instances include tense algebras (pdf) (there don’t seem to be any relaxed algebras), modal algebras (pdf), and logic algebras (pdf). More algebraic structures are listed here than at Mathworld, but at the moment some entries are just stubs.
• Mathematics in Isabelle
  Jeremy Avigad. “The project goal [is] to formalize, and develop tools to assist in the formalization of, portions of mathematics in Isabelle’s higher-order logic”. Last year a proof of the Prime Number theorem (a theorem on the number of primes less than a given natural number n) was formalized;

  Research on the Prime Number Theorem has yielded one of the largest numbers with a proper name: Skewes’ number: 10^(10^(10^34))). For some time mathematicians believed that π(n), the number of primes less than n, was strictly less than Li(n), a function devised by Gauss as an approximation of π(n). J. E. Littlewood proved in 1914 that Li(n) > π(n) for infinitely many sufficiently large n. S. Skewes showed that the first n for which Li(n) > π(n) is less than the number later named after him. This bound has since been reduced to 1.39822 · 10^316 by Bays and Hudson (Mathematical computing 69(2000):1285–1296). In 1955 Skewes showed that if the Riemann Hypothesis is false, then Li(n) ≥ π(n) for at least one n less than 10^(10^(10^(10^3))), which is much larger than Skewes’ number; this is called the second Skewes number. An even larger number is Graham’s number. I wonder what Kant would have thought of them (see the Critique of judgment §26, Ak. 254).

  Avigad has written an informative note on the proof, with some brief remarks on its philosophical significance. Other projects with similar aims include the Logosphere Project, Mathscheme (“an integrated framework for computer algebra and computer theorem proving”), and Mathweb, which includes a database of definitions and theorems, along with various projects for coding and storing mathematical information. Of these projects, the Logosphere project and Avigad’s seem to be the most active.
• Lewis Carroll puzzles and games
  Cathy Dean. Carroll/Dodgson was both a logician and an entertainer of children. The result of putting the two together was puzzles of permanent interest. Only Richard Smullyan comes to mind as a rival in that department. Has any philosopher tried to explain how logic can be funny? Jokes involving the use of logic—e.g., the syllogism—seem to depend on incongruity between subject matter and method, or between method and results. But you won’t find me offering Yet Another Failed Theory of Humor. Not here.

Mathematical tidbits

Philosophy of Mathematics ·· More from March 2005

I’ve been flat on my back with the flu for the last three days, without an original thought in my head. So no philosophy. Instead a few more sites on mathematics:

• Carrés multimagies
  Christian Boyer. Carrés multimagiques (multimagic squares, multimagischen Quadrate) are magic squares such that if every number in the square is replaced its square (cube, etc.) the result is also a magic square.

  8	1	6
  3	5	7
  4	9	2

  A magic square is an n × n array of natural numbers (in a “normal” magic square, the consecutive numbers from 1 to n2) for which every row, column, and diagonal adds up to the same number. There are also magic cubes, for which every line of numbers between two faces or vertices (and, in a perfect magic cube, every line of numbers between two edges)—every “row, column, pillar, and space diagonal”—adds up to the same number. In 2003 Boyer and Walter Trump discovered a 5×5 perfect magic cube (the smallest possible). There is an animated version of it at Mathworld. Boyer’s site is a labor of love. It belongs to the Anneau des Mathématiques Francophones, which has 87, mostly pedagogical, sites.
• Mathcurve
  

  Conoid of Plücker, n=3.
  Source: Mathcurve
  Robert Ferréol. Une “encyclopédie des formes mathématiques remarquables”. All mathematical objects are abstract, I suppose. But some are more abstract than others. (Is there such a thing as a particular abstractum?) The study of curves and surfaces is rich in what a mathematician would call “concrete” examples. The Mathcurve site includes plane curves, space curves, surfaces, fractals, and polyhedra, including some in animation. There is also a fairly extensive bibliography. Philosophers have had almost nothing to say about what you might call the “natural history” side of mathematics: the gathering and classification of examples, some of which may be no more than curiosities (like the bimagic squares above), some of which turn out to be crucial (like Torricelli’s surface, which, though infinite in extent, is finite in volume). A good cheap book on plane curves is J. Dennis Lawrence, A catalog of special plane curves (Dover, 1972) · 0486602885 (find it at ABEBooks).
• New Proof of the Four-Color Theorem
  Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas. Twenty years ago, the first proof of the Theorem aroused some interest among philosophers because its proof, part of which was an enormous proof by cases, was carried out on computers. (See Thomas Tymoczko, “The Four-Color Problem and Its Philosophical Significance”, in New directions in the philosophy of mathematics, rev. ed., Princeton, 1998 · 0691034982.) Mathematicians accepted the result, but were, on the whole, disappointed that the proof of what would seem to be a simple property of the plane (“every map can be colored with no more than four colors”) did not have a conceptually more revealing proof. (I seem to remember Jean Dieudonné dismissing it as unfruitful.) You might call it an unwanted particular, a quirky fact that wasn’t supposed to be quirky. Robertson et al. have simplified the proof and produced a faster algorithm for proving that a configuration is four-colorable. They acknowledge that the proof is “not a proof in the traditional sense”, and cannot be made into one:

  We should mention that both our programs use only integer arithmetic, and so we need not be concerned with round-off errors and similar dangers of floating point arithmetic. However, an argument can be made that our ‘proof’ is not a proof in the traditional sense, because it contains steps that can never be verified by humans. In particular, we have not proved the correctness of the compiler we compiled our programs on, nor have we proved the infallibility of the hardware we ran our programs on. These have to be taken on faith, and are conceivably a source of error. However, from a practical point of view, the chance of a computer error that appears consistently in exactly the same way on all runs of our programs on all the compilers under all the operating systems that our programs run on is infinitesimally small compared to the chance of a human error during the same amount of case-checking. Apart from this hypothetical possibility of a computer consistently giving an incorrect answer, the rest of our proof can be verified in the same way as traditional mathematical proofs. We concede, however, that verifying a computer program is much more difficult than checking a mathematical proof of the same length.

  K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429–490.

  K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491—567.

  K. Appel and W. Haken, Every planar map is four colorable, Contemporary Math. 98 (1989).

  N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, The four colour theorem, J. Combin. Theory Ser. B. 70 (1997), 2–44.

  N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, A new proof of the four colour theorem, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 17–25 (electronic).

Mathematics sources online

History of Science · Philosophy of Mathematics ·· More from November 2004

Many of the canonical works in the history of mathematics can be found online. Let’s hope that more libraries receive funding to scan their collections. All the sites below are free.

• Digital Mathematics Library:
  Author & title listing of works online, together with journals.
• World Digital Mathematics Library:
  List of catalogues.
• Bibliothèque Nationale:
  Among the thousands of texts now online are a number of texts in mathematics, mostly French.
• Cornell Historical Mathematics Monographs:
  512 titles. These are 600 dpi bitonal scans made in the early 90s.
• Digitised European Periodicals:
  Includes the Monatshefte fü Mathematik und Physik, v. 1 to 51.
• Das Göttinger Digitalisierungszentrum (GDZ):
  Several hundred monographs and a thousand articles, mostly in German.
• Jahrbuch über die Fortschritte der Mathematik:
  An important German journal; includes most of the volumes published between 1868 and 1931.
• University of Michigan Historical Mathematics Collection:
  A good collection of old textbooks and important monographs. It was slow when I tried it.

Modal Glitch

Philosophy of Mathematics ·· More from November 2004

See the update…

It’s not mistaken claims generally that bother me—a historian of philosophy needs a high tolerance for error—but those that, being put forward with every appearance of confidence, and yet in my view visibly false, put me in a situation of doubt. Why would someone who seems to know their stuff say that?

With respect to some matters I can live with, even thrive on, doubt; in others, I want the itch scratched soon. Here’s a case in point. Writing on progress in philosophy(↓), Timothy Williamson says at one point that

The principle that every truth is possibly necessary can now be shown to entail that every truth is necessary by a chain of elementary inferences in a perspicuous notation unavailable to Hegel.

Grattan-Guinness’s Search for mathematical roots

Books · History of Philosophy · Philosophy of Mathematics ·· More from May 2004

I. Grattan-Guinness. The search for mathematical roots, 1870–1940. Logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel. Princeton & Oxford: Princeton University Press, 2000.

A richly documented study that takes issue with received views of the origins and content of logicism. GG begins with Lagrange’s “algebraization” of the study of differential equations; this, he argues, was one of the inspirations for Boole’s and De Morgan’s algebraic logic, which in turn fed into the algebraic logics of Peirce and Schröder near the end of the 19th century. I would regard that portion of the history as belonging to the history of the extension and generalization of algebraic notions, a history that also includes the emergence of “abstract” groups, vector algebra, and the beginnings of commutative algebra. (Whitehead’s Universal algebra (1898) belongs here too.)

What philosophers think of as “foundations”—the logics of Frege and Russell—was a largely separate line of development whose sources included the “arithmetization” of analysis, Cantor’s Mengenlehre (with Zermelo’s work on the axioms of set theory), and Peano’s efforts to create a symbolic language for mathematics. Whitehead and Russell’s Principia (1910–1913), not Frege’s earlier work, is the centerpiece of GG’s history. GG distinguishes the historical Frege from Frege´, the father (with Russell) of analytic philosophy. Frege´, whose influence since World War II has been enormous, is an invention, “an analytic philosopher of language writing in English about meaning and its meaning(s), and putting forward some attendant philosophy of mathematics”. The historical Frege, “a mathematician who wrote in German, in a markedly Platonic spirit, principally on the foundations of arithmetic and on a formal calculus in which it could be expressed”, has only a modest place in the history of logicism, though (as GG argues) he was read more widely than Russell implies in an appendix to the Principles of mathematics in 1903.