# de Finetti, Bruno (1906–1985)

# DE FINETTI, BRUNO

*(1906–1985)*

Bruno de Finetti, an Italian mathematician, was born in Innsbruck, Austria. On the death of his father, the six-year-old de Finetti and his mother moved to Trento (then in Austrian possession). At thirteen he suffered severe osteomyelitis in the left leg; surgery left him permanently lame. In 1923 he entered the Politecnico di Milano to study engineering, his father's and grandfather's profession. In his third year he transferred to the new University of Milan, from which he graduated in 1927 with a degree in applied mathematics. While still an undergraduate he published the first of a series of articles on Mendelian population genetics, developing the first mathematical model with overlapping generations.

From graduation until 1931 de Finetti worked at Rome's Istituto Centrale di Statistica. This was a period of intense and productive research, resulting in publication of a series of mathematical and foundational works on probability. The mathematical works made his name internationally known. The foundational works set out the subjectivist interpretation of probability that he was to advocate all his life. Two stand out: "Sul significato soggetiva della probabilità" (1931) and the remarkable "Probabilismo" (1931), remarkable not least, but certainly not only, for its fascist peroration.

Between 1931 and 1946 de Finetti worked in the actuarial office of the Assicurazioni Generali insurance company in Trieste. At the same time he taught at the Universities of Padua and Trieste. In this period de Finetti's range widened to include actuarial and financial mathematics, economics, the automation of actuarial procedures (an interest reflected in the postwar years in his advocacy of computing and the use of simulation methods in statistics), and mathematics education. From the early 1950s his works became better known in the English-speaking world, thanks to the advocacy of the American statistician Leonard Savage. In 1947 de Finetti was appointed to the chair of financial mathematics in Trieste. In 1954 he moved to the Faculty of Economics at the University of Rome "La Sapienza"; in 1961 he transferred to the Faculty of Sciences in which he was a professor of the theory of probability until his retirement in 1976. De Finetti died in 1985.

In the 1970s de Finetti was active in Italian politics, standing as a parliamentary candidate for the Radical Party; for a while he edited the party's *Notizie Radicali*. On one occasion a judge ordered his arrest for antimilitarist campaigning.

What de Finetti's life exhibits is a concern for the tying of ideas to applications. The cornerstone of the radical subjectivist interpretation of probability, summed up in de Finetti's claim (in the preface to the English translation of his *Teoria delle probabilità* [1974]), "PROBABILITY DOES NOT EXIST" is that only concepts that can be given an operational, practical significance are meaningful. The radical subjectivist denies the meaningfulness of talk of objective, unknown probabilities. Probability is degree of belief/credence/conviction. De Finetti, as Frank Plumpton Ramsey before him (in work unknown to de Finetti), gave a Dutch book argument to show that a rational person's degrees of belief satisfy the axioms of the probability calculus: degrees of belief are revealed in the betting odds the person considers fair; a rational person does not bet so as to lose money with certainty; fair betting quotients avoid certain loss just if they satisfy the axioms of the probability calculus. Conditional probabilities are handled by conditional bets, bets that are canceled if a given event does not occur. (This led de Finetti to a logic of conditional events: *B* |*A* is true if *A* and *B* are both true, false if *A* is true and *B* is false, and neither if *A* is false, corresponding to the cases when the bet on *B* conditional on *A* is won, lost, and canceled. The idea has resurfaced from time to time in work on the indicative conditional of natural language and on production rules in computer science.)

One axiom is the subject of dispute. In Andrei Nikolajevich Kolmogorov's (1903–1987) *Foundations of the Theory of Probability* (1933) the axiom that probabilities add across a countably infinite partition is adopted as mathematically expedient. De Finetti urged its rejection. Much is known of the consequences of giving up this axiom, but de Finetti's line has not won general acceptance.

Not a philosopher by training, de Finetti found parallels to his thought in the Italian pragmatists Mario Calderoni and Giovanni Vailati (a mathematician), and the man-of-letters Giovanni Papini. Later he saw Humean connections in his influential work on exchangeable and partially exchangeable sequences of events and random variables. A sequence of events of *N* types is partially exchangeable if the probability that *n _{1}* events of the first type,

*n*events of the second type, …, and

_{2}*n*events of the

_{N}*N*th type all occur depends only on the numbers

*n*, …,

_{1}, n_{2}*n*. For exchangeability,

_{N}*N*= 1. De Finetti sees this notion as the subjective analogue of (and correction to) talk of independent trials with unknown probability and as making mathematically precise David Hume's account of induction and causation. This comes about through representation theorems. Take the case of an infinite sequence of exchangeable events. From the probability, for various

*n*, that

*n*events all yield favorable outcomes, one can infer the probabilities of

*r*favorable outcomes in

*n*trials, 0 ≤

*r*≤

*n*. The distributions of these relative frequencies for different

*n*tend, as

*n*increases, to a limit distribution that functions exactly as a distribution over an unknown probability, so that the probability of any definable event is the expectation with respect to this distribution of the probability it would have were one dealing with a sequence of independent events of constant probability. Exchangeability is preserved as one conditionalizes on the outcomes of any finite number of trials, so, provided the initial limit distribution assigns a nonzero probability to an interval containing it, one obtains a sequence of limit distributions increasingly weighted toward the observed relative frequency as the number of observed instances increases. This encapsulates de Finetti's account of learning from experience and inductive inference, his "translation into logic-mathematical terms of Hume's ideas" (1938, p. 194).

With the acceptance by today's philosophers of science of semantic realism and, increasingly, pluralism in the philosophy of probability, de Finetti's eliminativist reading of what is now called the de Finetti representation theorem is little in favor. But there has been a huge increase in the application both to scientific reasoning generally and to statistics in particular of the subjectivist interpretation of probability, usually under the name Bayesianism.

** See also ** Bayes, Bayes' Theorem, Bayesian Approach to Philosophy of Science; Calderoni, Mario; Hume, David; Mathematics, Foundations of; Papini, Giovanni; Probability and Chance; Ramsey, Frank Plumpton; Savage, Leonard; Vailati, Giovanni.

## Bibliography

### works by de finetti

"Probabilismo: Saggio critico sulla teoria della probabilità e sul valore della scienza." *Logos* (Naples) 14 (1931): 163–219. Translated by R. C. Jeffrey, M. C. DiMaio, and M. C. Galavotti as "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science." *Erkenntnis* 31 (2–3) (1989): 169–223.

"Sul significato soggetiva della probabilità." *Fundamenta Mathematicae* 17 (1931): 298–329. Translated as "On the Subjective Meaning of Probability." In *Probabilità e induzione/Induction and Probability*, edited by Paolo Monari and Daniela Cocchi, 291–321 (Bologna, Italy: Editrice Clueb, 1993).

"La logique de la probabilité." In *Actes du congrès international de philosophie scientifique*. Vol. 4, 31–39. Paris: Hermann, 1936. Translated by R. B. Angell as "The Logic of Probability." *Philosophical Studies* 77 (1995): 181–190.

"La prévision: ses lois logiques, ses sources subjectives." *Annales de l'Institut Henri Poincaré* 7 (1937): 1–68. Translated by Henry E. Kyburg Jr. as "Foresight: Its Logical Laws, Its Subjective Sources." In *Studies in Subjective Probability*. 2nd ed., edited by Henry E. Kyburg Jr. and Howard E. Smokler, 57–118. Huntington, NY: Robert E. Krieger, 1980.

"Sur la condition d'équivalence partielle." *Actualités scientifiques et industrieles No. 739* (*Coloque Genève d'Octobre 1937 sur la Théorie des Probabilités, 6ième partie* ), 5–18. Paris: Hermann, 1938. Translated by P. Benacerraf and R. C. Jeffrey as "On the Condition of Partial Exchangeability." In *Studies in Inductive Logic and Probability*, Vol. 2., edited by Richard C. Jeffrey, 193–205. Berkeley: University of California Press, 1980.

*Teoria delle probabilità: Sintesi introdutiva con appendice critica*. 2 vols. Turin, Italy: Einaudi, 1970. Translated by Antonio Machí and Adrian Smith as *Theory of Probability: A Critical Introductory Treatment*. 2 vols. New York: Wiley, 1974–1975.

*Probability, Induction, and Statistics: The Art of Guessing*. New York: Wiley, 1972.

"Probability and My Life." In *The Making of Statisticians*, edited by J. Gani, 3–12. New York: Springer-Verlag, 1982.

*Probabilità e induzione/Induction and Probability*, edited by P. Monari and D. Cocchi. Bologna, Italy: Editrice CLUEB, 1993.

*Filosofia dela probabilità*, edited by A. Mura. Milan, Italy: Il Saggiatore, 1995.

### works about de finetti

Cifarelli, Donato Michele, and Eugenio Regazzini. "De Finetti's Contribution to Probability and Statistics." *Statistical Science* 11 (4) (1996): 253–282.

Diaconis, Persi, and David Freedman. "De Finetti's Generalizations of Exchangeability." In *Studies in Inductive Logic and Probability*. Vol. 2, edited by Richard C. Jeffrey, 233–249. Berkeley: University of California Press, 1980.

Hintikka, Jaakko. "Unknown Probabilities, Bayesianism, and de Finetti's Representation Theorem." In PSA 1970 *In Memory of Rudolf Carnap*, edited by Roger C. Buck and Robert S. Cohen, 325–341. Dordrecht, Netherlands: D. Reidel, 1971.

Howson, Colin, and Peter Urbach. *Scientific Reasoning: The Bayesian Approach*. 2nd ed. Chicago: Open Court, 1993.

Jeffrey, Richard. *Subjective Probability: The Real Thing*. New York: Cambridge University Press, 2004.

Jeffrey, Richard C., and Maria Carla Galavotti, eds. "Bruno de Finetti's Philosophy of Probability." *Erkenntnis* 31 (2–3) (1989).

Milne, Peter. "Bruno de Finetti and the Logic of Conditional Events." *British Journal for the Philosophy of Science* 48 (1997): 195–232.

*Peter Milne (2005)*

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**de Finetti, Bruno (1906–1985)**