Oct 10, 2012

"Elegi Menggapai "Constructivism as the Epistemological Foundation of Mathematics"




By Marsigit
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

References:
1 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.204
2Ibid.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
6 Ibid.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
8Ibid.
9.Ibid

3 comments:

  1. Fabri Hidayatullah
    18709251028
    S2 Pendidikan Matematika B 2018

    Salah satu aliran yang menjadi fondamen bagi matematika ialah aliran konstruktivisme. Aliran ini sangat menolak platonisme. Konstruktivisme bukanlah produk dari suatu situasi yang diciptakan oleh paradoks melainkan semangat yang secara praktis disajikan di dalam sejarah matematika. Matematika konstruktivis berlaku seperti penengah dari keberadaan matematika dan kebenaran matematika, yaitu adanya kemungkinan pembentukan. Pembentukan matematika ialah secara mental dan diperoleh dari persepsi kita terhadap objek eksternal, baik secara mental maupun fisik. Pembentukan matematika juga merupakan sesuatu yang dibentuk dari hakekat bilangan, yang disebut sebagai idealisasi.

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  2. Amalia Nur Rachman
    18709251042
    S2 Pendidikan Matematika B UNY 2018

    Secara konstruktivis matematika meliputi beberapa hal sebagai berikut, seperti contoh terkait ide mengenai teori himpunan, kebenaran matematika sebagai dasar untuk membangun sebuah konstruktivisme. Objek-objek eksternal baik mental ataupun fisik juga berperan dalam pembangunan konstruktivisme. Aktualitas dalam abstraksi juga diperlukan dalam realitas matematika. Proporsi semua bilangan benar apabila ditentukan oleh hukum menurut ururtan bilangan yang dihasilkan. Idealisasi dari sebuah abstraksi yang jauh dari kenyataan akan menghasilkan runtuhnya pembangunan konstruktivisme

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  3. Janu Arlinwibowo
    18701261012
    PEP 2018

    Konstruktivis merupakan suatu aliran yang diadopsi menjadi suatu pendekatan dalam proses pendekatan. Konstruktivis dikeman menjadi suatu strategi pembelajaran yang baik dengan mengarahkan siswa untuk dapat membangun pengetahuannya sendiri sehingga konstruksinya lebih kokoh. Saat ini, metode ini dipandang sebagai metode yang dapat menyelesaikan masalah atau meningkatkan kualitas pendidikan karena berbasis pada pengalaman. Dimana pengalaman siswa dijadikan sebagai landasan untuk mengaji berbagai hal.

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