A Life of Logic and the Illogic of LifeMarch 1, 2005
Tarski as a student at the University of Warsaw.
Philip J. Davis
Alfred Tarski: Life and Logic. By Anita Burdman Feferman and Solomon Fef-erman, Cambridge University Press, Cambridge, UK, 2004, 432 pages, many snapshots, $34.99.
The Broadway playwright and director Moss Hart, a story goes, went out in the first flush of success and bought a boat, and a captain's hat to go with it. One day, he paraded before his mother, Lily Hart, in his new hat. Lily, less impressed, said to him:
"By you, you're a captain. By me, also, you're a captain. But are you a captain by the captains?"
Well, the captains of logic have spoken. Solomon Feferman, a professor of mathematics and philosophy at Stanford, and his wife, Anita Burdman Feferman, an independent scholar and biographer, open their book with the judgment that "Alfred Tarski was one of the greatest logicians of all times."
The genial Fefermans have produced a warm and adulatory biography. This is no surprise, as Sol was one of Tarski's PhD students and both he and Anita were professional and social acquaintances of Tarski for three decades. Reading Tarski's journals and publications, mining many archives, interviewing dozens and dozens of people, and traveling to Poland to visit original sites, the Fefermans have put together a story that is detailed and personal. They have painted a splendid portrait, Cromwellian warts and all, of an extraordinary individual.
Other captains of logic have strengthened the Fefermans' judgment. I've heard from more than one logician the opinion that the four most famous logicians of all times were Aristotle, Frege, Gödel, and Tarski. The library of the University of Warsaw now houses a life-sized statue of Tarski, surrounded by statues of other Polish philosopher-logicians.
I am ready to accept the judgment of those in the logic and set business, although I confess that I don't understand a good deal of what they have produced and am suspicious of a fraction of what I think I understand. In this regard, logic is hardly an exception---I don't understand most of what goes on in contemporary mathematics. Mathematics is such a patchwork quilt of specialties, each with its problems, goals, concepts, methodologies, terminologies, notations, and its coteries, that even with the overlap among the pieces, one can seriously question the vaunted "unity of mathematics," or indeed the "unity of science."
Speaking of the "unity of science," a meeting on this topic once saved Tarski's life. It came about in the following way. In Vienna in the 1930s, a group known as the Wienerkreis---the Vienna Circle---would gather to discuss the philosophy of mathematics and science. The participants were generally politically liberal and philosophically slanted toward logical positivism, a doctrine that consigned epistemological and metaphysical statements to the trash bin of meaninglessness. Though living and working in Poland, Tarski participated from time to time in the group's discussions. Otto Neurath (1882-1945), one of the leaders of the group, promoted the idea of the unity of science through meetings and publications.
The group scheduled a meeting for Cambridge, Massachusetts. Tarski pondered: Should he go or not? He went, leaving his wife and two children behind in Poland. The meeting took place September 3-9, 1939. Among those who attended, names I recognize include Otto Neurath, Haskell Curry, Stephen Kleene, J. Barkley Rosser (president of SIAM, 1965-66), and Willard V.O. Quine. Alfred North Whitehead (1861-1947), a key figure in the history of mathematical logic, was then in Cambridge, but I haven't found any indication that he attended the meeting. Twice retired by that time, Whitehead had in any case left logic for philosophy many years earlier.
Alfred Tarski (1971) at the University of Berkeley, where he taught from 1942 until the end of his life.
On September 2, 1939, the German army marched into Poland. With high probability, Tarski, a Jew, would have lost his life, although he had converted to the Catholic faith many years before. Miraculously, Tarski's wife and his son and daughter survived this terrible time. His parents, brother, and other relations were wiped out in 1944 by the Nazi savagery. As the Fefermans write:
"Stan Ulam recalled, 'I spent much of my time with the other Poles who had found their way to Cambridge---Tarski, Stefan Bergman and Alexander Wundheiler. They were all terribly unhappy. . . . We would sit in front of my little radio which I left on all day long and listen to the war news.'"
One wonders how these people, dislodged from country and family under such circumstances, were able to maintain their sanity. In later years, Tarski hardly ever mentioned the Holocaust.
In the third week of September 1939, I matriculated at Harvard. Tarski was there off and on between 1939 and 1941. In the fall of 1941, I took Quine's course in mathematical logic. I never met Tarski, either at that time or later, but I recall one of my classmates pointing to a figure walking in the distance: "There goes Tarski." I realized that this man was famous in circles that interested me.
Here is an abridged academic CV: Alfred Tarski, né Tajtelbaum (1902-1983). Born in Warsaw, Poland. Doctorate, University of Warsaw, 1924. Arrived in the USA, August 1939. Research associate at Harvard, 1939-41.Visiting professor at CCNY, 1941. Institute for Advanced Study, 1942. Professor at UC Berkeley, from 1942 to the end of his life. Major contributions (described at some length in a number of "Interludes" in the book under review): the Banach-Tarski paradox, the completeness and decidability of algebra and geometry, truth and definability, model theory, and algebras of logic. I comment on a couple of them in the second part of this review.
Stories of the man, his life, his family, his comings and goings, his work with students, his love affairs, his animosities, his multiple honors, his sufferings, are all told in somewhat tedious and occasionally embarrassing detail. What was the man like? A bon vivant, he was short in stature, Napoleonic, energetic, sensual, precise, self-confident, judgmental, forcefully approving or disapproving, aggressive, demanding, ambitious, competitive, talkative, theatrical, exasperating, suspicious, stingy, and, of course, tremendously brilliant. These are some of the adjectives applied by those who knew him. Such ingredients do not sound like a recipe for amiability, but there were many who loved him. He was surrounded by hero worshipers, one of whom summed him up as a "towering figure."
Given the rich trove of Tarskiana presented by the Fefermans, and in view of the increasing numbers of recent plays and movies that draw on the lives of mathematicians, I wonder whether Tarski's life would be an appropriate vehicle for such a production. What would be the agon, the life's struggle, in the sense of Aristotle's Poetics? Was it the struggle in which Tarski, Michelangelo-like, was able to extricate or release proper formulations and conclusions from the refractory material of logic? This would hardly appeal to the popcorn-munching crowds. What I infer from the Fefermans' biography is that what underlay Tarski's life was the constant struggle, conscious and unconscious, to suppress the bitter wounds he suffered as a young man from religious intolerance.
I suppose that your average mathematician recognizes the name Tarski as the second half of the Banach-Tarski paradox (1924). This theorem says that a solid ball can be divided into a finite number of pieces and reassembled by rigid motions to form two balls, each of the same size as the original. The pieces are not your ordinary segments, like those in, say, an orange, but rather are non-measurable sets (i.e., they have no definable volume) and have been called into being in an existential fashion via the Axiom of Choice (AC).
Thus, logic and its partner, set theory, provide us with a statement that is monstrously counter to common sense. A piece of mathematical humor, perhaps. It's an old story, known since the work of Gödel and Paul J. Cohen, that the Axiom of Choice is both compatible with and independent of other rather sensible axioms of set theory (e.g., Zermelo-Fraenkel: ZF). Moreover, as the Fefermans point out, "Solovay's theorem showed that the AC is indeed essential in the proof of the B-T paradox." So if we believe in ZF, we can accept or reject the AC as we wish, without stirring up our outrage at dealing with incompatibles. Hence, we can accept the Banach-Tarski paradox or we can reject it. I know full well what it would mean to reject it, and although I have read of the multifold consequences of the AC, I can't get a proper focus on what it would mean to accept it. I sometimes feel that formal logic can lead to a profound disregard of human responses, even though I admit that, from time to time, ignoring common sense turns out to be precisely what is required for mathematical or scientific advances.
Once you start reasoning about "completed" infinities, a lot of stuff is not common-sensical. For instance, Galileo noticed that there are "as many" even integers as there are integers (in that for every integer we can tick off its double). To the multitudes unbrainwashed by exposure to the language of set theory, all this might seem to be akin to the question of how many angels can dance on the head of a pin. In Cantor's "Paradise" (Hilbert's term), completed infinities hobnob with the dancing angels.
Set theory contains strange stuff, and when I come to things like the inaccessible cardinals, my head reels with ontological (i.e., existential) angst. Are we talking about things that have "real meaning," or, as the formalists would have us believe, are we merely moving symbols around according to certain agreed upon protocols?
Here's how Tarski positioned his philosophy:
"People have asked me 'How can you, a nominalist, do work in set theory and logic, which are theories about things you do not believe in?' . . . I believe there is value in fairy tales and in the study of fairy tales."
I turn now to the second of the Fefermans' Interludes: the completeness and decidability of algebra and geometry. The two questions are:
"Are the axioms complete, i.e., do they serve to determine the truth or falsity of every statement in the language of geometry?"
"Is geometry decidable---that is, do we have a systematic step-by-step method to determine the truth or falsity of any such statement?"
In 1930, Tarski answered these questions affirmatively by converting the problem to one of "generalizing Sturm's algorithm to determine for any finite system of polynomial equations and inequalities, how many real numbers satisfy them all." Of course, this result depends on what you allow as a statement in the language of geometry, and so there has to be at the creative stage a back-and-forth between formulating criteria of allowability and the desire for a positive result.
In later life, Tarski (along with many others) considered the decision procedure for algebra and geometry to be one of the two most important research contributions in his entire career, the other being his theory of truth.
As to the theory of truth, sketched out in Interlude III, I can comment only by reminding readers of Francis Bacon's quip: " 'What is truth?' said jesting Pilate, and would not stay for an answer." Tarski stayed and gave an answer in the restricted area of formalized languages.
A century has passed since Tarski's birth, and we may well ask what is our heritage from his work and, indeed, from the enterprise of logic---a subject that as now pursued lifted off with Boole and de Morgan in the mid-1800s. A decent evaluation would require a whole volume, and I have only a couple of opinions to contribute. Over the years, logic has taken its lumps. It was dislodged as a foundation for mathematics. Quine was not in the Department of Mathematics, but in the Department of Philosophy. Mathematics departments don't require logic as part of the core curriculum for majors. Yuri I. Manin, in the preface to his Mathematics and Physics, reports that a lecturer once began his sophomore course on logic as follows:
"Logic is the science of laws of thought. Now I must tell you what science is, what law is, and what thought is. But I will not explain what 'of' means."
I take this either as a parody or as something profound. Recall the question of what is "is" that came up during President Clinton's dies irae.
Thinking about Tarski, I wonder whether the principal heritage of his work, taken in a general sense, now lies in the field of computer science. I checked out this question with a number of friends in computer science departments and obtained mixed answers. My poll did not include people who worked in theorem proving or decision problems, where the relevance may be more obvious.
"In Comp Sci, the name of Tarski was mentioned much more 30 years ago than now."
"In AI, and in natural language processing, his influence is on the wane, being replaced by statistics and probability theory."
"I believe that Tarski, building, of course, on Frege, Whitehead and Russell, and Gödel, is largely responsible for standardizing the theory and notation of first-order logic, which I use extensively in my work. It was Tarski who, so to speak, clearly separated out first-order logic as a thing in itself, and wrapped it up neatly with a ribbon. The definition of the 'Tarskian semantics' for first-order logic has a clarity and simplicity that make it the gold standard for all formal semantic definitions. This kind of 'cultural' contribution is more influential than his technical results [i.e., the ones expounded in the Interludes] though harder to pin down because it may be 'too large to be seen.'"
"My belief is that foundations need to come to grips with probability. As for logic, I see some very interesting problems in logics that are adapted to "real" life, i.e., incorporating time, laws, incorrect beliefs as well as simple uncertainty."
Metamorphosis is a fact of life, and logic is by no means a static subject. It has developed non-Tarskian venues, such as polyvalued logics, fuzzy logics, modal logics, non-monotonic logics, paraconsistent logics. The late Gian-Carlo Rota dreamed of a logic that would incorporate intentionality. The future will tell whether these theories will infiltrate our daily lives or whether they will remain in the world of fairy tales.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at email@example.com.