Essays On Constructive Mathematics

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle.

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With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property that classical logic does not have: whenever , often called a witness.

Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.

For instance, in Heyting arithmetic, one can prove that for any proposition p that does not contain quantifiers, is a theorem (where x, y, z ... In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence does not extend to propositions that refer to infinite collections. It is possible to test for any particular even number whether or not 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.

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 was tantamount to assuming that every mathematical problem has a solution.

From this argument mathematical induction is nearly useless in constructive setting ?

Normally we prove nearly 3/4 of stuff using this technique.

Lets say we are inductively defining Natural Numbers.

In extensional equality you have to prove anything about it, so u'll need principle of mathematical induction while in case of Intentional equality u can't use it (Or there is no such principle, ).


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