Francisco Antonio Doria
Encyclopedia
Francisco Antônio de Moraes Accioli Dória (born 1945, Rio de Janeiro, Brazil) is a Brazil
ian mathematician
, philosopher, and genealogist. Francisco Antônio Dória received his B.S. in Chemical Engineering
from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1968 and then got his doctorate from the Brazilian Center for Research in Physics (CBPF), advised by Leopoldo Nachbin in 1977. Dória worked for a while at the Physics Institute of UFRJ, and then left to become a Professor of the Foundations of Communications at the School of Communications, also at UFRJ. Dória held visiting positions at the University of Rochester
(NY), Stanford University
(CA) (here as a Senior Fulbright Scholar), and the University of São Paulo
(USP). His most prolific period spawned from his collaboration with Newton da Costa
, a Brazilian logician and one of the founders of paraconsistent logic
, which began in 1985. He is currently Professor of Communications, Emeritus, at UFRJ and a member of the Brazilian Academy of Philosophy.
His main achievement (with Brazilian logician and philosopher Newton da Costa
) is the proof that chaos theory
is undecidable (published in 1991), and when properly axiomatized within classical set theory, is incomplete in the sense of Gödel
. The decision problem for chaotic dynamical systems had been formulated by mathematician Morris Hirsch
.
More recently da Costa and Dória introduced a formalization for the P = NP hypothesis which they called the “exotic formalization,” and showed in a series of papers that axiomatic set theory together with exotic P = NP is consistent if set theory is consistent. They then prove:
If exotic P = NP together with axiomatic set theory is omega-consistent, then axiomatic set theory + P = NP is consistent.
(So far nobody has advanced a proof of the omega-consistency of set theory + exotic P = NP.) They also showed that the equivalence between exotic P = NP and the usual formalization for P = NP, is independent of set theory and holds of the standard integers. If set theory plus that equivalence condition has the same provably total recursive functions as plain set theory, follows the consistency of P = NP with set theory.
Dória is also interested in the theories of hypercomputation
and in the foundations of economic theory.
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
ian mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, philosopher, and genealogist. Francisco Antônio Dória received his B.S. in Chemical Engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...
from the Federal University of Rio de Janeiro (UFRJ), Brazil, in 1968 and then got his doctorate from the Brazilian Center for Research in Physics (CBPF), advised by Leopoldo Nachbin in 1977. Dória worked for a while at the Physics Institute of UFRJ, and then left to become a Professor of the Foundations of Communications at the School of Communications, also at UFRJ. Dória held visiting positions at the University of Rochester
University of Rochester
The University of Rochester is a private, nonsectarian, research university in Rochester, New York, United States. The university grants undergraduate and graduate degrees, including doctoral and professional degrees. The university has six schools and various interdisciplinary programs.The...
(NY), Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
(CA) (here as a Senior Fulbright Scholar), and the University of São Paulo
University of São Paulo
Universidade de São Paulo is a public university in the Brazilian state of São Paulo. It is the largest Brazilian university and one of the country's most prestigious...
(USP). His most prolific period spawned from his collaboration with Newton da Costa
Newton da Costa
Newton Carneiro Affonso da Costa is a Brazilian mathematician, logician, and philosopher. He studied engineering and mathematics at the Federal University of Paraná in Curitiba and the title of his 1961 Ph.D...
, a Brazilian logician and one of the founders of paraconsistent logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
, which began in 1985. He is currently Professor of Communications, Emeritus, at UFRJ and a member of the Brazilian Academy of Philosophy.
His main achievement (with Brazilian logician and philosopher Newton da Costa
Newton da Costa
Newton Carneiro Affonso da Costa is a Brazilian mathematician, logician, and philosopher. He studied engineering and mathematics at the Federal University of Paraná in Curitiba and the title of his 1961 Ph.D...
) is the proof that chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
is undecidable (published in 1991), and when properly axiomatized within classical set theory, is incomplete in the sense of Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
. The decision problem for chaotic dynamical systems had been formulated by mathematician Morris Hirsch
Morris Hirsch
Morris William Hirsch is an American mathematician, formerly at the University of California, Berkeley.A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...
.
More recently da Costa and Dória introduced a formalization for the P = NP hypothesis which they called the “exotic formalization,” and showed in a series of papers that axiomatic set theory together with exotic P = NP is consistent if set theory is consistent. They then prove:
If exotic P = NP together with axiomatic set theory is omega-consistent, then axiomatic set theory + P = NP is consistent.
(So far nobody has advanced a proof of the omega-consistency of set theory + exotic P = NP.) They also showed that the equivalence between exotic P = NP and the usual formalization for P = NP, is independent of set theory and holds of the standard integers. If set theory plus that equivalence condition has the same provably total recursive functions as plain set theory, follows the consistency of P = NP with set theory.
Dória is also interested in the theories of hypercomputation
Hypercomputation
Hypercomputation or super-Turing computation refers to models of computation that are more powerful than, or are incomparable with, Turing computability. This includes various hypothetical methods for the computation of non-Turing-computable functions, following super-recursive algorithms...
and in the foundations of economic theory.