Constructivism (philosophy of mathematics)

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In the

existential quantifier
, which is at odds with its classical interpretation.

There are many forms of constructivism.

topos theory
.

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

law of non-contradiction
(which states that contradictory statements cannot both be true at the same time) is still valid.

For instance, in

quantifiers
, is a theorem (where x, y, z ... are the
bivalence does not extend to propositions that refer to infinite
collections.

In fact,

justification. For instance, Goldbach's conjecture is the assertion that every even number greater than 2 is the sum of two prime numbers
. It is possible to test for any particular even number whether it is the sum of two primes (for instance by exhaustive search), so any one of them is either the sum of two primes or it is not. And so far, every one thus tested has in fact been the sum of two primes.

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

existence property
that classical logic does not have: whenever is proven constructively, then in fact is proven constructively for (at least) one particular , often called a witness. Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.

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

e
is:

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,

standard constructive interpretation
of the mathematical statement

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

free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the computable functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers
.

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

recursively enumerable
.

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

law of the excluded middle, hence there can be no constructive proof of the theorem.[4]

Axiom of choice

The status of the

ZF set theory
without the axiom of choice." However, proponents of more limited forms of constructive mathematics would assert that ZF itself is not a constructive system.

In intuitionistic theories of

law of the excluded middle
.

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

Diaconescu-Goodman-Myhill theorem. Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice
in Myhill's set theory.

Measure theory

Classical

measure theory is fundamentally non-constructive, since the classical definition of Lebesgue measure does not describe any way how to compute the measure of a set or the integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real number and outputs a real number" then there cannot be any algorithm to compute the integral of a function, since any algorithm would only be able to call finitely many values of the function at a time, and finitely many values are not enough to compute the integral to any nontrivial accuracy. The solution to this conundrum, carried out first in Bishop (1967), is to consider only functions that are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence. An advantage of constructivizing measure theory is that if one can prove that a set is constructively of full measure, then there is an algorithm for finding a point in that set (again see Bishop (1967)). For example, this approach can be used to construct a real number that is normal to every base.[citation needed
]

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

internal language that is a constructive theory; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their homomorphisms
.

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

Branches

See also

Notes

References

  • .
  • Pradic, Pierre; Brown, Chad E. (2019-04-19). "Cantor-Bernstein implies Excluded Middle". ].

External links