Yogyakarta State University
Wilder R.L.(1952) illustrates that as a complete rejection of Platonism, constructivism is not a product of the situation created by the paradoxes but rather a spirit which is practically present in the whole history of mathematics. The philo-sophical ideas taken go back at least to Aris¬totle's analysis of the notion of infinity. Kant's philosophy of mathematics can be interpreted in a constructivist manner. While constructivist 1 ideas
were presented in the nineteenth century-notably by Leopold Kronecker, who was an important forerunner of intuition¬ism-in opposition to the tendency in mathematics toward set-theoretic ideas, long before the paradoxes of set theory were discovered. Constructivist mathematics 2 proceed as if the last arbiter of mathematical existence and mathe¬matical truth were the possibilities of construction.
Mathematical constructions 3 are men¬tal and derive from our percep¬tion of external objects both mental and physical. However, the passage 4 from actuality to possibility and the view of possibility as of much wider scope perhaps have their basis in intentions of the mind-first, in the abstraction from concrete qualities and existence and in the abstraction from the limitations on generating sequences. In any case, in constructive mathematics, the rules by which infinite sequences are generated not merely a tool in our knowledge but part of the reality that mathe¬matics is about. Constructivism 5 is implied by the postulate that no mathematical proposition is true unless we can, in a non-miraculous way, know it true. For mathematical constructions, a proposition of all natural numbers can be true only if it is determined true by the law according to which the sequence of natural numbers is generated.
Mathematical constructions 6 is something of which the construction of the natural numbers. It is called an idealization. However, the construction will lose its sense if we abstract further from the fact that this is a process in time which is never com¬pleted. The infinite, in constructivism, must be potential rather than actual. Each individual natural number can be constructed, but there is no construction that contains within itself the whole series of natural numbers. A proof in mathematics 7 is said to be constructive if wherever it involves the men¬tion of the existence of something and provides a method of finding or constructing that object. Wilder R.L. maintains that the constructivist standpoint implies that a mathematical object exists only if it can be constructed. To say that there exists a natural number x such that Fx is that sooner or later in the generation of the sequence an x will turn up such that Fx.
Constructive mathematics is based on the idea that the logical connectives and the existential quantifier are interpreted as instructions on how to construct a proof of the statement involving these logical expressions. Specifically, the interpretation proceeds as follows:
1. To prove p or q (`p q'), we must have either a proof of p or a proof of q.
2. To prove p and q (`p & q'), we must have both a proof of p and a proof of q.
3. A proof of p implies q (`p q') is an algorithm that converts a proof of p into a proof of q.
4. To prove it is not the case that p (` p'), we must show that p implies a contradiction.
5. To prove there exists something with property P (` xP(x)'), we must construct an object x and prove that P(x) holds. 8
6. A proof of everything has property P (` xP(x)') is an algorithm that, applied to any object x, proves that P(x) holds.
Careful analysis of the logical principles actually used in constructive proofs led Heyting to set up the axioms for intuitionistic logic. Here, the proposition n P(n) n P(n) need not hold even when P(n) is a decidable property of natural numbers n. So, in turn, the Law of Excluded Middle (LEM): p p 9
1 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204
3 Brouwer in Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204
4 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204
5 Brouwer in Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204
7 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204 Bridges, D., 1997, “Constructive Mathematics”, Stanford Encyclopedia of Philosophy. Retrieved 2004