Following
Newcomb
Greenleaf, 1966 the American mathematician Errett Bishop (1928-1983) proposed a
small revolution in the way that mathematicians think. Bishop called his
approach to mathematics **''constructive''** because of his belief that the
objects of mathematics are *constructed*, not received from on high.
Bishop supported the revolution with his great book, **Foundations of
Constructive Analysis (FCA)**, which appeared in 1967. His constructive
approach was welcomed by many computer scientists. Stanford’s computer science
pioneer Donald Knuth (*Algorithmic Thinking and Mathematical Thinking) *proposed
that computer science be known as ''algorithmics''. He shows in great detail how
a constructive mathematician can understand the triad (*Assumption
Proof Conclusion*) in terms of (*Input data Algorithm
Output data*). Knuth was
suggesting constructive math could serve as a bridge between classical
mathematics and computer science, an appealing intellectual unification.''
During
the last 50 five years one notices the flourishing of constructive approaches to mathematics
and the growth of a variety of research groups working on constructive
mathematics. During the
1920s L. E. J. Brouwer (1881-1966), the
renowned Dutch mathematician and philosopher, had urged mathematicians to
modify the logic used in mathematical arguments founding an entire school of mathematical
logic, known as intuitionism. It is (was) considered that Brouwer and
other constructivists were more successful in their criticism of classical
mathematics than in replacing it with a better alternative. In the words of Laura
Crosila ''The spark that started the present abundance of constructive
mathematics was the publication in 1967 of Errett Bishop's F*oundations of
Constructive Analysis*, **FCA**.''
The historical
and philosophical picture of constructive mathematics is
complex; a number of varieties of mathematics have been developed over
time including classical, Brouwerian and Russian constructive
mathematics – ''each
of the latter three forms of mathematics may be developed on the basis
of some
suitable extension of Bishop's mathematics by characteristic
principles.''
Following Arend Heyting, classical mathematics can be seen as a guide that helps the constructive
mathematician develop new mathematics.
References:
Crosila,
L. Bishop’s mathematics: A philosophical perspective, In *Handbook of Bishop's Mathematics*. Eds.
D. Bridges, H. Ishihara, M. Rathjen, and H.Schwichenberg, Cambridge University
Press, forthcoming.
Greenleaf,
N. (2020) Agony and Failure in Bishop’s Constructive Revolution
Heyting, A. (1956). *Intuitionism,
an Introduction*, North – Holland.
Websites
https://www.dsbridges.com/errett-bishop
https://www.britannica.com/biography/Errett-Bishop
https://mathshistory.st-andrews.ac.uk/Biographies/Bishop/ |