Constructivism (philosophy of mathematics)
In the
There are many forms of constructivism.
Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.[3] Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.
Constructive mathematics
Much constructive mathematics uses
For instance, in
In fact,
But there is no known proof that all of them are so, nor any known proof that not all of them are so; nor is it even known whether either a proof or a disproof of Goldbach's conjecture must exist (the conjecture may be undecidable in traditional ZF set theory). Thus to Brouwer, we are not justified in asserting "either Goldbach's conjecture is true, or it is not." And while the conjecture may one day be solved, the argument applies to similar unsolved problems. To Brouwer, the law of the excluded middle is tantamount to assuming that every mathematical problem has a solution.
With the omission of the law of the excluded middle as an axiom, the remaining
Example from real analysis
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.
In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
so that as n increases, the values of ƒ(n) get closer and closer together. We can use ƒ and g together to compute as close a rational approximation as we like to the real number they represent.
Under this definition, a simple representation of the real number
This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance,
is precisely the existence of the function computing the modulus of convergence. Thus the difference between the two definitions of real numbers can be thought of as the difference in the interpretation of the statement "for all... there exists..."
This then opens the question as to what sort of
Cardinality
To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show that the computable numbers are classically countable. And yet Cantor's diagonal argument here shows that real numbers have uncountable cardinality. To identify the real numbers with the computable numbers would then be a contradiction. Furthermore, the diagonal argument seems perfectly constructive.
Indeed Cantor's diagonal argument can be presented constructively, in the sense that given a
Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be
Axiom of choice
The status of the
In intuitionistic theories of
However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle (in the presence of other axioms), as shown by the
Measure theory
Classical
The place of constructivism in mathematics
Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".[5]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis,[2] worked to dispel these fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds. For example, constructive proofs in analysis may ensure
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic logic'" (page 31). "In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time" (page 28).
Mathematicians who have made major contributions to constructivism
- Leopold Kronecker (old constructivism, semi-intuitionism)
- L. E. J. Brouwer(founder of intuitionism)
- A. A. Markov(forefather of Russian school of constructivism)
- Arend Heyting (formalized intuitionistic logic and theories)
- Per Martin-Löf (founder of constructive type theories)
- Errett Bishop (promoted a version of constructivism claimed to be consistent with classical mathematics)
- Paul Lorenzen (developed constructive analysis)
- Martin Hyland (discovered the effective topos in realizability)
Branches
- Constructive logic
- Constructive type theory
- Constructive analysis
- Constructive non-standard analysis
See also
- Computability theory – Study of computable functions and Turing degrees
- Constructive proof – Method of proof in mathematics
- Finitism – Philosophy of mathematics that accepts the existence only of finite mathematical objects
- Game semantics – approach to formal semantics
- Inhabited set – Property of sets used in constructive mathematics
- Intuitionism – Approach in philosophy of mathematics and logic
- Intuitionistic type theory – Alternative foundation of mathematics
Notes
References
- ISBN 9783540121732.
- ISBN 4-87187-714-0.
- .
- ISBN 0-387-21978-1.
- Feferman, Solomon (1997). Relationships between Constructive, Predicative and Classical Systems of Analysis (PDF).
- Pradic, Pierre; Brown, Chad E. (2019-04-19). "Cantor-Bernstein implies Excluded Middle". arXiv:1904.09193 [math.LO].
- .
- ISBN 0-19-853163-X.
- ISBN 9780444702661.
- ISBN 9780444703583.
- Troelstra, Anne Sjerp (1991). History of constructivism in the 20th century (PDF). University of Amsterdam, ITLI Prepublication Series ML-91-05. Archived from the original on 2006-02-09. Retrieved 2019-07-09.
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