May 18, 2006
`I don't believe in natural science.'
From Incompleteness: The Proof and Paradox of Kurt Gödel, by Rebecca Goldstein. 2006. Norton paperback, ISBN 0-393-32760-4.
Though Princeton's population is well accustomed to eccentricity, trained not to look askance at rumpled speciments staring vacantly (or seeminly vacantly) off into space-time, Kurt Gödel struck almost everyone as seriously strange, presenting a formidable challenge to conversational exchange. A reticent person, Gödel, when he did speak, was more than likely to say something to which no possible response seemed forthcoming:
John Bahcall was a promising young astrophysicist when he was introduced to Gödel at a small Institute dinner. He identified himself as a physicist, to which Gödel's curt response was `I don't believe in natural science.'
The philosopher Thomas Nagel recalled also being seated next to Gödel at a small gathering for dinner at the Institute and discussing the mind-body problem with him, a philosophical chestnut that both men had tried to crack. Nagel pointed out to Gödel that Gödel's extreme dualist view (according to which souls and bodies have quite separate existences, linking up with one another at birth to conjoin in a sort of partnership that is severed upon death) seems hard to reconcile with the theory of evolution. Gödel professed himself a nonbeliever in evolution and topped this off by pointing out, as if this were additional corroboration for his own rejection of Darwinism: `You know Stalin didn't believe in evolution either, and he was a very intelligent man.'
`After that,' Nagel told me with a small laugh, `I just gave up.'
The linguist Noam Chomsky, too, reported being stopped dead in his linguistic tracks by the logician. Chomsky asked him what he was currently working on, and received an answer that probably nobody since the seventeenth-century's Leibniz had given: `I am trying to prove that the laws of nature are a priori.'
Three magnificent minds, as at home in the world of pure ideas as anyone on this planet, yet they (and there are more) reported hitting an insurmountable impasse in discussing ideas with Gödel.
Leave aside the comment about natural selection, and consider the other two anecdotal quotes attributed to Gödel. They are entirely consistent with Gödel's version of Leibniz's principle of sufficient reason; Gödel's so-called `interesting axiom' which is talked about earlier in Goldstein's book:
All of his thinking is governed by an `interesting axiom,' as Ernst Gabor Straus, Einstein's assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact. This conviction amounts to the assertion that there is no brute contingency in this world, no givens that need not have been given. In other words, the world will never, not even once, speak to us in the way that an exasperated parent will speak to her fractious adolescent: `Why, I'll tell you why. Because I said so!' The world always has an explanation for itself, or as (Gödel) puts it, Die Welt ist vernünftig, the world is intelligible.
About Gödel's comments on natural selection, I find it hard to say anything remotely reasonable. Rebecca Goldstein cites an explanation about Gödel's intuitions about Darwinism by Steven Pinker, which essentially states that being a logician, Gödel disliked the non-determinism inherent in the Darwinian explanation. It comes across more as an apology than an explanation. But we do not need any explanation, of course. There is little doubt about Gödel's accomplishments, but like Einstein, the public, and even other scientists, expected these geniuses to provide deep insights (purely intuitive or a priori, by their very nature) on topics outside their expertise. The fault is not with them, but with us in taking everything they said all too seriously.
Update 5/29/2006: This update is meant to clarify one point that might be misunderstood. Unlike natural selection, Gödel's interests did in fact extend quite clearly into physics and even astrophysics. For a Festschrift in Einstein's honor, Gödel reluctantly published a paper that laid out a completely new model for the famous Einstein equations of General Relativity. In Gödel's interpretation so-called "Closed Timelike Curves" could exist, in which time can have cycles and you could revisit the past cyclically (perhaps this is also related to Gödel's interest in the existence of non-standard models in logic). For reasons not entirely clear to me, this interpretation has some link to observations of galaxies where if a significant number of them had a strong preference for spinning in one direction vs. another this would be a relevant finding. In a story that appears much later in Goldstein's book, some astrophysicists who were involved in such observations were asked to confer with Gödel and were taken aback at the sharp penetrating questions he had for them. `I wish we had talked to Gödel before doing our work.' was their comment after this conversation. It still does not explain to me his mysterious statement to John Bahcall above, but I suspect it does not mean what it appears to at first glance.
November 22, 2005
The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity by Amir D. Aczel
This book is a short popular introduction to the theory of sets, the notion of an infinite set of countable numbers, and the existence of infinities greater than the infinity of the countable numbers. In addition, it also serves as a concise history of the life of Georg Cantor and his work on the continuum hypothesis, a statement that was later shown to be undecidable by Kurt Gödel.
The book also draws some parallels between ancient conceptions of infinity by the Greek mathematicians (particular the discovery of irrational numbers by the Pythagoreans) and modern ideas of mathematics. Where the comparisons become a bit frayed and sensationalistic are in comparisons of the Reimann Sphere which maps each point on a hemisphere to points on a plane with the concentric spheres that are frequently drawn in Kabbalistic texts and also the spheres of Dante in the Divine Comedy.
It seemed to me that the direct comparisons of Cantor's mental illess, and also Gödel's paranoid delusions with the madness of the Rabbis that contemplated the infinite nature of God using Kabbalistic methods are suggestive melodrama on Aczel's part, creating parallels where little evidence exists to show that such parallels should even be made. Such ideas belong to the realm of science-fiction: a similar idea was used in the movie "π" where a mathematician tries to find a single equation that describes all natural phenomena and goes mad in the process. π also picks up on the Kabbalistic idea of going mad by contemplating the word of God. ``Is it a coincidence that priests turned cryptographers and mathematicians both go mad?'', both π and Aczel seem to be indicating a deeper connection, where a superficial one will suffice.
Some mathematical ideas are expressed eloquently, while others are casually addressed, without enough information to make them sensible to an audience that is not already familiar with the ideas. In the central aspects of explaning the ideas on countable numbers, Aczel does an excellent job, while in peripheral ideas such as explaining the Reimann sphere he does displays a less then exemplary effort.
On the whole, for a layman's description of the nature of infinite sets in mathematics you can hardly find a better introduction than this book.
%T The Mystery of the Aleph %T :Mathematics, the Kabbalah, and the Search for Infinity %A Amir D. Aczel %I Pocket Books %D 2000 :trade paperback edition 2001 %G ISBN: 0743422996 (pb) %P 258 %K mathematics, religion
Review written: 2002/06/18
October 11, 2005
The Man Who Loved Only Numbers: the Story of Paul Erdös and the Search for Mathematical Truth by Paul Hoffmann
Paul Erdös was easily the most prolific mathematician of the 20th century. Part of this reason was that he did almost nothing else: he spent all his time engrossed in mathematics. He ingested the strongest coffee and addictive levels of amphetamines to ensure that he was continually alert to the possibility of a new theorem even with only three hours of sleep every day. He never married or even had to the best of anybody's knowledge any romantic feelings for anything other than prime numbers and graph theory. In a field of strange geniuses, Erdös was stranger and more of a genius than most.
He wrote or co-authored 1,475 academic papers, many of them monumental, and all of them substantial.
While Erdös' life and his work are fascinating, only the former gets adequate attention in this book. His curious behaviour and his homeland Hungary get a lot of attention by Hoffmann. The history of Hungary in the early 20th century provides a backdrop to the story of Erdös and his family who were Hungarian Jews during a time (as everyone knows) when it not so convenient to be Jewish in Europe.
While Hoffmann selects prime number theory and goes into its history and Erdös' contributions to this field in great depth, Hoffmann all but ignores the tremendous contributions of Erdös to random graphs and Ramsey theory. The latter is mentioned and discussed in the book, but is not covered in the depth that it deserved.
Instead, Hoffmann gets distracted by the colorful personalities of many of Erdös' academic collaborators. Even historical figures in mathematics that formed some of the foundations of the field are covered even though they have but tangential connections to the work of Erdös (their work might have deep connections, but they do so to the work of all mathematicians, and so their selection here is simply opportunistic). The work of Gauss is covered suitably for his proof of the Prime Number Theorem (a theorem about the log-like distribution of prime numbers). This was a theorem for which Erdös and Selberg gave a simpler proof than the original by Gauss. Other mathematicians such as Cantor get a lot of coverage in this book, sometimes not for very good reasons.
Despite these shortcomings, Hoffmann does a good job in introducing the intuitions behind the practice of number theory in mathematics.
It should be noted that the reviewer has an Erdös number of 4.
%T The Man Who Loved Only Numbers %T :the Story of Paul Erdös and the Search for Mathematical Truth %A Paul Hoffmann %I Fourth Estate, London %D 1998 %G ISBN: 1857028112 (pb) %P 302 %K mathematics, biography
Review written: 2002/02/22
September 28, 2005
The Code Book: The Evolution of Secrecy from Mary, Queen of Scots to Quantum Cryptography by Simon Singh
There have been many books that have explored the history, the scientific and engineering contributions and the mathematics behind cryptography. One of the most comprehensive early books on the topic was "Secret and Urgent, the story of codes and ciphers" by Fletcher Pratt (Blue Ribbon Books, 1942). More recently, "The Codebreakers; The Comprehensive History of Secret Communication from Ancient Times to the Internet" by David Kahn attempted a detailed history of cryptography. The emphasis in Kahn's book was on the historical details, more or less assuming a familiarity with the mathematics behind the algorithms.
Of late, popular writing about mathematics is better than ever. Many subjects previously thought too obscure have had skilled writers tackle the exposition of these subjects for a layperson audience. As a result, Simon Singh can attempt to explain in somewhat more detail the development of crypographic methods including all the math and the details of the ingenuity behind these methods. But one should keep in mind that this is a book for popular consumption -- those deeply interested in cryptography should pick up one of the many textbooks on the matter and read David Kahn's book for the history. This book is for those who want to read a compelling and important story.
Singh's book is a welcome addition to the selection of books about cryptography. You'll find some stories in Fletcher Pratt's book which don't make it into "The Code Book". And while David Kahn's book attempts to be far more comprehensive by being more than a 1000 pages long, Singh's narrative is trimmer and will inform you about the techniques and the history in a smaller dose.
Rather than concentrating on telling all the stories in the history of cryptography, Singh chooses a few important ones and tells them with their context and without sacrificing details (at least until the last chapters of the book). Many details are relegated to appendices which should not be skipped.
Singh describes the development of cryptographic methods as a result of co-evolutionary behaviour of those who encrypt messages to secure them (usually in a war with lives at stake) and those who are equally compelled to decrypt these messages. At some point, encryption methods seem to be unbreakable and induces a sense of security for the codemakers who think that their ciphers are unbreakable, until new methods are invented for their decipherment rendering the old encryption methods extinct and making way for a search for new forms of encryption and cycle repeats itself.
Singh also includes a chapter about the decipherment of ancient scripts which is unusual for a book on cryptography. But as the name suggests, such methods of de-cipher-ment have a lot in common with the methods of cryptography. Singh picks the examples of the decipherment of the ancient Egyptian script and that of the Linear B script from Crete. Singh avoids the uncomfortable example of the Mayan script to which various cryptographic techniques were applied to no avail, until the linguistic insight of the Russian philologist Knorosov provided success in decipherment.
The narrative also includes the story of the Navajo code talkers. And to drive home the wartime nature of cryptography the narrative includes stories like the following:
If you so much as held up your head six inches you were gone, the fire was so intense. And then in the wee hours, with no relief on our side or theirs, there was a dead standstill. It must have gotten so that this one Japanese couldn't take it anymore. He got up and yelled and screamed at the top of his voice and dashed over our trench, swinging a long samurai sword. I imagine he was shot from 25 to 40 times before he fell.
There was a buddy with me in the trench. But that Japanese had cut him across the throat, clear through to the cords on the back of his neck. He was still gasping through his windpipe. And the sound of him trying to breathe was horrible. He died, of course. When the Jap struck, warm blood spattered all over my hand that was holding the microphone. I was calling in code for help. They tell me that in spite of what happened, every syllable of my message came through.
From Doris Paul's book, The Navajo Code Talkers
The book also includes a detailed chapter on quantum cryptography. Since it is still in its budding stages, it is mostly a description of the pioneers in the field. The descriptions of quantum decryption are admirably done, but the explanations that accompany the story of quantum encryption methods are somewhat rushed and should've been presented as carefully as the earlier methods were.
The most admirable thing about Singh's writing is the attention to detail in the explanation of the various methods. He does not hurry through explanations of the Enigma or even the RSA algorithm. He provides several examples, works through them, and also provides metaphors and analogues to help understanding the algorithm. With this three-fold presentation, almost everyone should find something to keep them interested in the presentation.
There are some missteps in the writing. The descriptions of cryptographic techniques earlier in the book are longer and more lucid while those towards the end of the book seem rushed. Many historical details, especially about Mary, Queen of Scots are included at length, while details of events towards the end of the book from the near future are shortened drastically. There are several typographic errors to do with missing fonts and a few cases of lousy editing -- unforgivable lapses in a book that costs US $25. Also, a quick search on amazon provides several editions of "The Code Book" all of them written by Simon Singh but each with a different subtitle, some of them with 'Mary, Queen of Scots' and other citing 'Ancient Egypt'.
%T The Code Book %T :The Evolution of Secrecy from Mary, Queen of Scots to Quantum Cryptography %A Simon Singh %I Doubleday %D 1999 %G ISBN: 0385495315 (hc) %P 402 %K science, computer-science, cryptography, linguistics
Review written: 2001/10/16
June 30, 2004
Axis of EvilA gem of geek humor stolen from a post on Ernie's 3D Pancakes:
From "Shape Fitting with Outliers" by Sariel Har-Peled and Yusu Wang, SIAM J. Computing 33(2): 269–285, 2004.
DEFINITION 3.2. A set of hyperplanes I is a δ-sheaf if there exists a vertical segment s of length δ such that all the hyperplanes in I stab s, The vertical segment s is the axis of I. [Footnote: We will refer to it as the axis of evil when appropriate.]
So what does the axis of evil look like?
This shows a δ-sheaf in RxR with pq as the axis of evil.
June 25, 2004
The Man who knew Infinity: a life of the genius Ramanujan by Robert Kanigel
You might think: what would a science writer living in Baltimore know about how to present the life of a South Indian mathematical genius who traveled to England from India in the early 1900s.
You might think: how would anyone understand the psyche and the drives behind a person who was born into a demon-haunted late 19th century and was so enamoured with mathematics that he left to go to a completely alien place where even the food was not palatable.
You might think: perhaps this biography of Ramanujan will ignore his own story to concentrate on the more accessible lives of the famous Cambridge mathematicians like Hardy.
You might think: how can anyone make us understand why Ramanujan eventually died at an early age succumbing to tuberculosis; remaining a vegetarian in war-torn England even when he was consumed by malnutrition.
Well, think again.
This book is a tour-de-force in science writing. It is amazingly detailed in every aspect and Kanigel could not have done a better job if he was channeling Ramanujan himself. Kanigel is obviously fond of Ramanujan having spent so much time documenting his life, but he also has the necessary external point of view in many places which makes you thankful that this is not a mere hagiographic survey.
The math is dumbed down a bit as is necessary for a mass market book like this. However, the explanations of Ramanujan's math exploits are usually done well. At least interesting stories are not eliminated altogether because the math was be too hard to explain.
Here is one (non-mathematical) story from the book:
Even the prevalence of body odours among the English mystified him -- until, the story goes, one day he was enlightened about it at a tea party. A woman was complaining that the problem with the working classes was that they failed to bathe enough, sometimes not even once a week. Seeing disgust writ large on Ramanujan's face, she moved to reassure him that the Englishmen he met were sure to bathe daily. "You mean," he asked, "you bathe only once a day?"
%T The Man who knew Infinity %T :a life of the genius Ramanujan %A Robert Kanigel %I Washington Square Press %D 1991 %G ISBN: 0671750615 %P 438 %K mathematics, biography
Review written: 1999/08/04
March 19, 2004
The Night is Large: Collected Essays, 1938-1995 by Martin Gardner
Martin Gardner, of course, is famous for his mathematical recreation columns in Scientific American. He is also famous for the Annotated Alice books.
This is a collection of 47 essays spanning almost 70 years. The essays have some recurring themes which is either good or bad depending on whether you agree with Gardner's treatment of these themes.
For instance, he defends Platonic mathematical realism in many of his essays ("Mathematics and the Folkways", "How not to talk about Mathematics", "Computers near the Threshold ?", among others), defending it from the view that mathematics or any science as a purely solipsist human construct without any reality outside human brains. It is hard not to sympathize with this view for any person remotely connected with scientific activity. However, the controversy was still alive and well in the late 1990s, see The Sokal Hoax for more on this debate. But in none of his essays that deal with this issue does Gardner address any of the arguments that are put forth by other kinds of mathematicians like intuitionists or formalists. In fact, in defending the Lucas-Penrose arguments he also defends a narrow Platonist view that a mathematician can see, without proof, that certain statements of mathematics are true, which can then be used to see that certain artifacts like thinking machines are impossible. For more on this issue from another point of view read the essay by Edward Nelson called "Mathematics and the Mind" (available from Edward Nelson's web page).
Another common theme is the debunking of the fringe pseudosciences or other untenable positions. Since none of these were particularly controversial, these essays were not so interesting to me. Many of arguments used I had seen before. However, there were some essays in this theme which were novel and which I enjoyed, for instance, "The Laffer Curve" which talks about various half-truths that lie behind supply-side economics and the role of tax-cuts to rejuvenate the economy, which is a topic that seems to be one of the central dividing lines between left and right in politics.
In "WAP, SAP, PAP and FAP" Gardner takes on the proponents of the anthropic principle, and also "The Curious Mind of Allan Bloom" and "The Strange Case of Robert Maynard Hutchins" where the motivations behind a couple of tirades against the neglect of the `Great Books' of the western world in modern universities are explored with great dispatch.
Other enjoyable essays about literature and language were "The Irrelevance of Conan Doyle", "Lewis Carroll and his Alice books" , "H.G. Wells in Russia", "Coleridge and The Ancient Mariner", "Puzzles in Ulysses" (about James Joyce's penchant for wordplay) and "The Royal Historian of Oz" (Gardner wrote one of the first biographical essays about L. Frank Baum). These essays were very informative, although Gardner lacks Borges' flair for talking about literature. Also it is interesting that Gardner treats Lewis Carroll with the same cultural relativism that he argues against elsewhere (in "Beyond Cultural Relativism").
One common theme which I found particularly annoying was Gardner's belief which he is not shy to state as a fact, that there are some mysteries such as the nature of time ("Can Time Stop? The Past Change?"), or the nature of consciousness and `free will' ("The Mystery of Free Will", and "Computers near the Threshold?") which according to Gardner can never be discovered by puny human brains. Note that Gardner supports Penrose, even though what Penrose is saying is not that a theory of consciousness can never be discovered but that all will be explained when Penrose will discover how quantum mechanics relates to this issue. The Mysterians can only stand up, nay-say for a while, and then sit down and let the people working on these `unsolvable' problems continue with their work.
The back of this book has quotes from Noam Chomsky, Carl Sagan, Stephen Jay Gould, Raymond Smullyan, Douglas Hofstader and Arthur C. Clarke praising Martin Gardner and this book. High praise indeed.
%T The Night is Large %T :Collected Essays, 1938-1995 %A Martin Gardner %I St. Martins Press %D 1996 %G ISBN: 031214380X %P 586 %K science, philosophy
Review written: 1999/08/01