Biographical Memoirs Volume 63 (1994) / Chapter Skim
Currently Skimming:
18. Julia Bowman Robinson
18. Julia Bowman Robinson
Pages 452-479
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 453... ...
AS A MATHEMATICIAN, Julia Bowman Robinson will long be remembered for her many important contributions to questions of algorithmic solvability and unsolvability of mathematical problems, in particular for her part in the negative solution of Hilbert's "Tenth Problem." Ancl, clespite her expressed wish, she will be remembered as the first woman to be electec! to the mathematical section of the National Academy of Sciences, as well as the first woman to be president of the American Mathematical Society.
|
From page 454... ...
Her mother died when she was two years old, and her father retired not long after, having lost interest in his machine tool and equipment business. Ralph Bowman remarried a few years later to Edenia Kridelbaugh, and the family moved first to Arizona and then to San Diego; a third daughter, Billie, joined Constance and Julia a few years later.
|
From page 455... ...
Julia received her A.B. degree in 1940 and, at the urging of Raphael Robinson, continued on with graduate studies at Berkeley.
|
From page 456... ...
A doctor there was the first to recognize that there was substantial scar tissue in the mitral valve of the heart, a residue of her early bout with rheumatic fever, and he strongly acivisecl her against becoming pregnant. He also told her mother in private that Julia would be lucky to live to age forty.
|
From page 457... ...
algebra, and algebraic theory. During a period of thirty-odd years following the war, he directed the doctoral work of a series of outstanding students, of whom Julia Robinson was one of the first.
|
From page 458... ...
Ion elements of S to S is said to be effectively computable if there is an algorithm which, given any element a eS as argument, leacis to the value (or "output")
|
From page 459... ...
By the remark above, once the general notion of effectively computable function is macle precise, the same applies to that of effectively clecidable predicate or relation. Several inclepenclent proposals for a general, mathematically precise definition of effectively computable function on the natural numbers N (or Nay were macle in the 1930s, and before long all were shown to be equivalent.
|
From page 460... ...
His definition also extends directly to that of effective computability on any set of expressions generated from a finite alphabet. The systematic development of the theory of effectively computable functions, assuming Church's Thesis, was initially clue to Kleene; in particular, working with the Herbranct(~ocleT definition he shower!
|
From page 461... ...
. ." It became convenient to work with Meene's schemata as a means for verifying various properties of the effectively computable functions.
|
From page 462... ...
For an explanation of Julia Robinson's dissertation work with Tarski ant! her later work on the Diophantine problem, we consider two further notions from recursive function theory.
|
From page 463... ...
as equality) each closed formula A of S has a definite truth value in M
|
From page 464... ...
This is where things stood when Julia Robinson took up, at Tarski's suggestion (via Raphael Robinson) , the question of clefinability of the set Nin Th(Q,+,.~.
|
From page 465... ...
The background to this problem is as follows. In the thirc7 century A.D., the Greek mathematician Diophantus worked on solving equations with arbitrary integer coefficients, for integer values.
|
From page 466... ...
= . Faced with this situation at the turn of the 20th century, David Hilbert proposed the following decision problem for arbitrary Diophantine equations: To devise a process accor~ling to which it can be determined by a finite number of operations whether the equation is solvable in integers.)
|
From page 467... ...
= 0] with P an integer polynomial was in line with his proclaimed optimism about the eventual solvability of all mathematical problems, but flew in the face of the fact that Diophantine equations were known to be exceptionally resistant to general methods of solution.
|
From page 468... ...
one can construct a recursively enumerable set A which is not recursive and hence not effectively decidable. Thus, elimination of the bounded universal quantifier in the form (2)
|
From page 469... ...
Around 1960, Julia received the ciraft of a paper by Martin Davis and Hilary Putnam in which they showed that if the famous hypothesis that there exist arbitrarily Tong arithmetic progressions containing only prime numbers were true, then every recursively enumerable (r.e.)
|
From page 470... ...
Raphael Robinson sometimes complained that "while other men's wives buy fur coats and diamond bracelets, E my] wife buys bicycles." (Autobiography, p.
|
From page 471... ...
to 9. There is still a wide gap between the positive information on algorithmically solvable Diophantine equations in two unknowns mentioned above and the negative results for 9 and more unknowns.
|
From page 472... ...
Robinson was chosen as the colloquium lecturer at the 84th Summer Meeting of the American Mathematical Society in 1980; her lectures covered the main areas of her interests between logic and number theory, and the notes for these
|
From page 473... ...
Moclest to the end, she let her character and achievements speak for themselves. ~ AM INDEBTED to Constance Reid, as well as to John Addison, Leon Henkin, and Raphael Robinson of the University of California at Berkeley for materials on the life, work, and career of Julia Robinson.
|
From page 474... ...
REFERENCES Alan Baker, "Contributions to the Theory of Diophantine Equations: I On the Representation of Integers by Binary Forms, II.
|
From page 475... ...
: 1 82-89. Carl Ludwig Siegel, "Zur Theorie der quadratischen Formen," Nach.
|
From page 476... ...
The decision problem for exponential Diophantine equations.
|
From page 477... ...
Two three-quantifier universal representations of recursively enumerable sets. (in Russian)
|
From page 478... ...
Diophantine equations: Positive aspects of a negative solution. Mathematical Developments Arising from Hilbert Problems, Proc.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.