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The Mathematical Theory of Information Hecke, Oswald TeichmOller, Ernst Witt, are used for quantitative problems, Richard Courant, Edmund Landau, Fe- The Mathematical while hits (the number of correct clas- lix Hausdorff, Ernst Peschl, Paul Riebe- sifications), reliability, and nuts (von sell, Helmut Ulm, Alfred Stohr, Ernst Neumann's utility) appear in more qual- Zermelo, Gerhard Gentzen, Hans Pe- Theory of itative analyses. tersson, Erich K~ihler, and Wilhelm SOss. Although the author's ambition is to The names on this list range from the Information develop a (if not the) "mathematical the- committed Nazis, through non-Nazi by Jan Kahre ory of information," the embodiment is right wing nationalists, to naive, other- pre-mathematical. It is neither a naive worldly men who seemingly didn't BOSTON, KLUWER. 2002. 520 PP., US$50.O0 mathematical theory (as in naive set the- ISBN: 1-4020-7064-0 know much of what was going on, to ory) nor is it abused mathematics (in the those who stumbled into situations that REVIEWED BY CRISTIAN S. CALUDE sense of mathematics applied in mean- were beyond their control. The stories ingless ways). However, the mathemat- are gripping. In assessing guilt the ndoubtedly, the title of the book ical formalism is too rudimentary for a reader is hard pressed to decide with was well chosen: it is provoca- theory; I illustrate this point with two any certainty which people fall into U tive, promising, and full of infor- examples, the definitions of probability which categories. Teichm/_iller comes mation. Syntactically, the title can be and algorithmic complexity. The proba- off very badly, for example. Segal sug- viewed as a variation on the titles of bility P(a) is a real number that satisfies gests that he in fact came up with the both the seminal paper [11] Ca" is re- the following three axioms (pages theory that Aryan mathematics was dif- placed by "the") and the book [12] 25-26): probability cannot be a negative ferent and superior to "Jewish and ("Communication" is replaced by "In- number, the probability of something French" mathematics, a notion usually formation"). It provocatively questions that must occur is 1, and the probabil- credited to Bieberbach. TeichmOller Shannon's theory; according to [1] (page ity that a or b will occur is the sum of claimed the student revolt against 215), "no prophet remains unchallenged their probabilities provided that a and courses taught by Landau was not anti- for ever". And it promises "a new math- b cannot both occur. 1 Kolmogorov com- Semitic but pro-German. He was a bril- ematical theory of information, built on plexity is defined (page 234) as the liant mathematician but a Nazi fanatic. a single powerful postulate: The Law of length l(aO of the shortest algorithm He volunteered to go fight on the Russ- Diminishing Information." generating a given bj. In both cases the ian front and was killed there at the age The book was praised--"a bold new intuition is correct; even if some facts of 30. One should probably keep in approach to classical information the- can be deduced from those definitions, mind that because of his early death, ory"--by von Baeyer [1], who dedicated there is still a long way to a satisfactory he, unlike many others, never had a a special chapter of his book to it; de- mathematical presentation. chance to try to redeem his reputation tails about Kfthre and the fascinating au- The book is rather firmly based during the denazification period after tonomous islands ~,land on which he on Shannon's probabilistic view of in- the war. lives are presented too. formation and entropy; the standard In this last chapter, the reader can- Do we need a new information the- books [12, 6] are frequently used and not help regretting that certain other ory? Unsurprisingly, there is no one sin- cited. The information measure used in mathematicians were not included for gle theory of information, but several the book is defined by inf(B@A) = the this more expansive treatment: Hans theories: semantic theories [2], algorith- information B gives about A (author's Zassenhaus, Gustav Doetsch, Georg mic information theory [5,4], logic of in- notation). Here inf(B@A) is a real func- Hamel, Helmuth Kneser, or Erhard formation [7], information algebra [9], tion satisfying the Law of Diminishing Schmidt, for example. Perhaps Segal felt philosophy of information [8], informa- Information (or, simply, the Law, as it is that their involvement was adequately tion flow [3], quantum information the- referred in the book): Compared to di- described elsewhere in the text. ory [10], evolutionary information [13], rect reception, an intermediary can only Of those treated, some behaved well, to name just a few" (a workshop de- decrease the amount of information. If some badly. All were competent math- voted to various theories of information A--+ B--+ C denotes a transmission ematicians; some were giants. Their was recently held in Mtinchenwiler). chain, then the Law reads: inf(C@A) --< mathematics, however, did not save Each theory focuses on some specific inf(B@A). The "theory of information" some of them from being monsters. aspect of information, and overlaps are developed in the book is based on prob- minimal. There is little evidence that the ability (as defined below) plus the Law existing theories will converge towards (page 14): Department of Mathematics and a single, unified theory of information, [The Law] will be used as the fun- Computer Science so, indeed, there is ample room for damental axiom of the mathematical Santa Clara University (even a partial) unification. theory of information. The Law is Santa Clara, CA 95053-0290 The book discusses information from the pruning knife of information USA various angles, with interesting ideas theory: we will argue that the Law e-mail: [email protected] and many examples. Bits and entropy is the necessary and sufficient con- lit takes no fewer than 235 pages to realise that probabilities, defined in this way, apply only to finite sets (see section 8.4). 64 THE MATHEMATICALINTELLIGENCER dition for a mathenmtical function to tion physics and quantum information Evolutionary Perspective, Springer, Heidel- be accepted as an information mea- and applications. Some topics are bet- berg, 1997. sure, i.e. qualify as inf(B@A). ter presented than others. The chapter The Law is easy to understand and on algorithmic information, which is Department of Computer Science informally seems correct (for example, close to my expertise, is far from satis- The University of Auckland to anyone who has played the Tele- factory, as one can see by browsing the Private Bag 92019 phone game in which one person paragraphs 2, 3, and 4 on page 238. The Auckland chooses a sentence, whispers it into the main aim, a grand unification theory of New Zealand ear of her left neighbour, who in turn information, is certainly not achieved. e-mail: [email protected] whispers it into the ear of her left Despite this, the book, written by an neighbour, and so on down the line). original thinker, contains a number of It ties in well with other principles such interesting ideas which may inspire as the second law of thermodynamics, mathematically oriented readers to con- Negative Math: the data processing inequality [6], and tinue the project. the invariance of algorithmic complex- How Mathematical ity under computable tmnsfommtions [4]. REFERENCES Moreover, it is not difficult to see that 1. H. C. von Baeyer. Information: The New Rules Can Be there are infinitely many functions sat- Language of Science, Harvard University isfying the Law (trivially, each constant Press, Cambridge, Mass., 2003 (paper- Positively Bent function satisfies the inequality). Fi- back edition 2004). by Alberto A. Martinez nally, a weaker form of the Law has al- 2. Y. Bar Hillel (ed.). Language and Informa- ready been discussed in [6] (page 32) PRINCETON UNIVERSITY PRESS, 2006, 267 PP., tion: Selected Essays on Their Theory and as a consequence of the data process- ISBN-13:978-0-691-12309-7 Application, Addison-Wesley, Reading, ing inequality. Mass, 1964. REVIEWED BY ERIC GRUNWALD According to the book, the Law' is 3. J. Barwise and J. Seligman. information ubiquitous. It makes physics possible: Flow: The Logic of Distributed Systems, "systems are forgetting their past as they xtricating the book from its pack- Cambridge University Press, Cambridge, reach equilibrium, or rather, the initial aging, I was greeted by a picture 1997. conditions can be eliminated from their E of a large spoon on the dust wrap- description. Otherwise, physics would 4. C. S. Calude. Information and Random- per. What could this mean? Some sort be complicated beyond comprehen- ness: An Algorithmic Perspective, 2nd Edi- of reference to spoon-bending in the sion". It also explains evolution. It ap- tion, Springer-Verlag, Berlin, 2002. last word of the subtitle? Is the reader plies to information technology, game 5. G. J. Chaitin. Algorithmic Information The- going to be spoon fed? Surely the book theory, legislation, logic of research, al- ory, Cambridge University Press, Cam- isn't written in spoonerisms. Immedi- gorithmic information, chaos theory, bridge, 1987. ately after the title page, came the fol- control engineering, medical tests. It can 6. T. M. Cover and J. A. Thomas. Elements lowing: even be used as a legitimacy test: any of Information Theory, Wiley, New York, You can use a spoon to drive a acceptable in~brmation measure must 1991. screw into a wall. With practice, you satisfy the Law. 7. K. J. Devlin. Logic and Information, Cam- can become skillful at it. You can Is the Law true and should it be bridge University Press, Cambridge, 1991.
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