Ernst Specker is emeritus professor of mathematics at the Swiss Federal Institute of Technology (Eidgenössische Technische Hochschule, or ETH) in Zurich, Switzerland, and just recently celebrated his 82nd birthday. Nevertheless, this sprightly, if somewhat stooped, man is just as quick-witted and lucid as he was in the late 1960s when the author of this book was attending his lectures on linear algebra. The term “emeritus” does not sit comfortably with Specker, however. With healthy common sense he remarks that it only embellishes the true nature of affairs. In fact, he adds with a twinkle in his eye, he rather considers himself as demoted, because whenever he wants to give a lecture or organize a mathematics seminar, he now needs to ask for permission from the university. This remark shows, however, that the demotion some 15 years ago in no way diminished the man’s enthusiasm for work. Hardly a week has gone by since his retirement without one of his famous seminars on logic. But one day at the end of the academic year of 2001–2002, it really was over for good: The seminar series, first offered in Zurich some 60 years earlier, was not going to be on the academic calendar any longer. The decision had been made higher up.

Specker is a man of utmost kindness, friendliness, and good humor, a characterization to which a number of former students all over the world can attest from when they had to take their oral examinations at the ETH. A poor examinee, stuck with an incorrect answer and at a loss as to how to proceed, was lucky in such moments to have Specker in the examiner’s seat. The Herr Professor provided sufficient hints and tips until even the most nervous candidate could not help but stumble over the correct answer.

As a mathematician Specker is one of those enlightened men who was always ready to explore new, even outlandish, ideas. One such case that this reporter can attest to relates to his lectures on linear algebra. The professor would stand in front of the lecture hall and, chalk in hand, expound on systems of linear equations and matrices while busily covering the old-fashioned blackboard with equations and formulas. Before long he would run out of space, erase everything from the board, and repeat the process. So it went, week after week: write, erase, write, erase. Eventually Specker got fed up and sought a different way of presenting the subject to the class. He hit on what he thought was an ingenious idea: After covering the blackboard with white chalk, he resorted to a yellow chalk. Without wiping anything off the board, he simply proceeded to write all over the white equations with the yellow chalk, reminding his students to ignore the white writing and only pay attention to the yellow. As could be expected, within a very brief span of time an unbelievable chaos covered the blackboard. Of course, Specker, being the true mathematician that he is, soon realized that this innovation was unusable. He announced that he was discarding the idea, and an audible sigh of relief could be heard in the lecture hall.

As a youngster Specker suffered from tuberculosis and was forced to spend some of his childhood years in Davos, an Alpine resort in Switzerland known for its dry and clean air. There he studied at a private school before moving to Zurich to attend high school. There was no question in Specker’s mind but that he would follow in his father’s footsteps and pursue a career in law. Very soon, however, he realized that legal studies did not fulfill him. He was not taken by the manner in which lawyers searched for the truth. The way in which, by contrast, mathematicians sought and provided proof fascinated him. Thus, in 1940, he began his studies at the ETH. In 1949, at the age of 29, Specker was invited to spend a year at the Institute for Advanced Study in Princeton, New Jersey. At this legendary institution Specker got acquainted with such

luminaries as Kurt Gödel, Albert Einstein, and John von Neumann.

In the fall of 1950 Specker returned to Switzerland and was immediately offered a position as a lecturer at the ETH; after five years he was appointed full professor. It was then that he made a quite revolutionary discovery: The so-called Axiom of Choice does not hold in the formalized set theory of the Harvard philosopher Willard Van Orman Quine. Not surprisingly, this work caused quite a stir, and Specker promptly received an offer of a professorship from Cornell University in Ithaca, New York.

How does a mathematician get the inspiration for the proof of a problem that has been investigated for years and years? Specker’s answer is that one can never predict that. An idea may pop up in one’s head, just like that, in the middle of a bath or while shaving. It is important, though—and he lifts a finger to emphasize the point—to be completely relaxed. Tension, he maintains, is counter-productive. And another thing: One should never get frustrated by false starts. False starts, Specker emphasizes, very often contribute to further research or, indeed, even form the basis of it.

Specker declined the Cornell offer since his family preferred to remain in Switzerland, but the fact that he was invited to a prestigious university in America had repercussions back home. Specker’s hometown institution, the ETH, realized that they possessed in him an extremely valuable asset which needed looking after. The administration relieved him of the rather tedious introductory lectures that he, like all math professors, had to give to the engineering students. Instead, he was allowed to pursue his own specialties. Over the next 50 years Ernst Specker made revolutionary contributions to the fields of topology, algebra, combinatorial theory, logic, the foundations of mathematics, and the theory of algorithms.

One day the famous Hungarian mathematician Paul Erdös paid a visit to Zurich, and Specker coauthored a short paper with him. This publication earned him the coveted “Erdös number 1.” Approximately 500 mathema-

ticians around the world have been accorded this honor. The Erdös number evolved because the Hungarian mathematician collaborated with an unprecedented number of colleagues. To have an Erdös number 1, a mathematician must have published a paper with Erdös. To receive the number 2, he or she must have published with someone who has published with Erdös and so on. Specker, ever the mathematician, immediately translated this into a mathematical formula: Every author publishing a paper with an author possessing the Erdös number n automatically receives the Erdös number n + 1. As soon as Specker had become a member of the elite club of mathematicians with Erdös number 1, he immediately found himself the target of a whole slew of mathematicians asking him to coauthor papers. This would, of course, automatically entitle them to the honor of a coveted Erdös number 2. (There are about 4,500 mathematicians with Erdös number 2.)

Specker is frequently asked what logic can offer anybody in day-to-day life. Of course, it can help decide if and when an answer is correct, he says. But there are also other applications, such as linguistics or computer science, which have become rigorous branches of science only after first having been subjected to logical formalization.

A case in point would be the question of whether a computer program exists that can test other programs, as well as itself, as to whether they are correct. Thanks to logic the answer is obvious: No such programs exist. Then there are questions that deal with the complexity of a problem: Is it useful to know, for example, that one is able to decide a given question but that it would take an infinitely long time (or at least a few billion years) to actually calculate the answer? Finally, even issues arising in physics can be tackled by resorting to logical argumentation. For example, together with Simon Kochen from Princeton University, Specker proved, purely by logical argument, that so-called hidden variables cannot exist in quantum mechanics. Thus they cannot explain, as Einstein had hoped, certain quantum mechanical phenomena.

Specker continues to give lectures and to participate at colloquia all over the world, but his family comes first. He loves spending time with his eight grandchildren and fondly recalls a recent lunch he had with one of his granddaughters. He spent a most enjoyable few hours chatting with her about mathematics, and this, he recounts with a happy smile, was “a really wonderful experience.”