Constructivism (philosophy of mathematics): Difference between revisions

=== Cardinality ===

=== Cardinality ===

To take the algorithmic interpretation above would seem at odds with classical notions of [[cardinal number|cardinality]]. By enumerating algorithms, we can show classically that the [[computable number]]s are countable. And yet [[Cantor’s diagonal argument]] shows that real numbers have uncountable cardinality. Furthermore, the diagonal argument seems perfectly constructive. To identify the real numbers with the computable numbers would then be a contradiction.

To take the algorithmic interpretation above would seem at odds with classical notions of [[cardinal number|cardinality]]. By enumerating algorithms, we can show that the [[computable number]]s are countable. And yet [[Cantor’s diagonal argument]] shows that real numbers have uncountable cardinality. To identify the real numbers with the computable numbers would then be a contradiction.

And in fact, Cantor’s diagonal argument ”is” constructive, in the sense that given a [[bijection]] between the real numbers and natural numbers, one constructs a real number that doesn’t fit, and thereby proves a contradiction. We can indeed enumerate algorithms to construct a function ”T”, about which we initially assume that it is a function from the natural numbers [[onto]] the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (”T” is a [[partial function]]), so this fails to produce the required bijection. In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor’s result as showing that the real numbers (collectively) are not [[recursively enumerable]].

Cantor’s diagonal argument , in the sense that given a [[bijection]] between the numbers and numbers, one constructs a real number , and thereby a contradiction. can enumerate algorithms to construct a function ”T”, about which we initially assume that it is a function from the natural numbers [[onto]] the reals. But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (”T” is a [[partial function]]), so this fails to produce the required bijection. In short, one who takes the view that real numbers are (individually) effectively computable interprets Cantor’s result as showing that the real numbers (collectively) are not [[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 [[Trivial (mathematics)|trivially]] represented as a real number, therefore the real numbers are ”no less than” countable. They are, therefore ”exactly” countable. However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the [[Cantor–Bernstein–Schroeder theorem]], is non-constructive. It has recently been shown that the [[Cantor–Bernstein–Schroeder theorem]] implies the [[law of the excluded middle]], hence there can be no constructive proof of the theorem.{{sfn|Pradic|Brown|2019}}

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 [[Trivial (mathematics)|trivially]] represented as a real number, therefore the real numbers are ”no less than” countable. They are, therefore ”exactly” countable. However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the [[Cantor–Bernstein–Schroeder theorem]], is non-constructive. It has recently been shown that the [[Cantor–Bernstein–Schroeder theorem]] implies the [[law of the excluded middle]], hence there can be no constructive proof of the theorem.{{sfn|Pradic|Brown|2019}}

Mathematical viewpoint that existence proofs must be constructive

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or “construct”) a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without “finding” that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called…