Oct 13, 2012

The Un-stability Foundation of Mathematics: Litlangs’,Bold’s, Thomson’s and Posy’s Arguments

By Marsigit
Yogyakarta State University

Litlangs, 2004, confronted that Aristotle disagreed with Plato; according to Aristotle, forms were not entities remote from appearance but something which entered into objects of the world. Aristotle claimed that when we can abstract oneness or circularity, it does not mean that these abstractions represent something remote and eternal.

For Aristotle1 , mathematics was simply reasoning about idealizations; and he looked closely at the structure of mathematics, distinguishing logic, principles used to demonstrate theorems, definitions and hypotheses. Plato2 also reflected on infinity, perceiving the difference between a potential infinity e.g. adding one to a number ad infinitum and a complete infinity e.g. number of points into which a line is divisible. Bold, T., 2004, claimed that both the intuitionist and the formalist assured that mathematics are just inventions and do not inform us with anything about the world; both take this approach to explain the absolute certainty of mathematics and reject the use of infinity. Bold3 noted that intuitionists admit this major similarity the formalist and note the difference as a disagreement on where mathematical exactness exist; the intuitionist says in the human intellect and the formalist says on paper. According to Arend Heyting 4, mathematics is a production of the human mind; he claimed that intuitionism claims mathematical propositions inherit their certainty from human knowledge that is based on empirical experience. Bold 5 maintained that since, infinity can not be experienced, the intuitionist refuses to push application of mathematics beyond finite; Heyting declared that faith in transcendental existence, unsupported by concepts, must be rejected as a means of mathematical proof. Similarly, Bold 6 found that Hilbert wrote that for logical inferences to be reliable it must be possible to survey these objects completely in their parts; since there is no such survey for infinity a reliable inference can only be based on a finite system. According to the formalists, the whole of mathematics consists of only arbitrary rules like those of chess.

Further, Bold, T., 2004, indicated that, on the other hand, the logicists came close to proving that mathematics was a branch of logic. According to Bold 7, the logisticians want to define mathematical concepts in terms of logical concepts and deduct mathematical propositions from logical axioms; as the basic elements of logic are sets and their properties, the logicists use sets to define mathematical concepts. Bold elaborated the following:

For example, the meaning of 0 is the class of all null sets, 1 is the class of all sets with 1 members, 2 is the class of all sets with a pair of members and etc. The problem of this definition is that it does not explain the concept of number 2. We need not a definition, but an explanation. “Number 2 is the class of all pairs” presupposes the concept of 2. Such definition is a mere construction of a formal system that is consistent with mathematics. It might be necessary or sufficient for the work on mathematical propositions, but it does not give us any insight into mathematical concepts. The most exciting parts of the logicist account hides in what Russell and Whitehead’s Principia Mathematica has to say about mathematical propositions and what Ramsey and Wittgenstein said about its flaws. However, the present paper is primarily on mathematical concepts only. “What is number?”, “What is infinity?” and “Why is the absolute certainty?” are the questions that are in need of philosophical interpretation. I am not holding a directly opposing position to these theories; at some points I agree with them. However, I will simply attempt to pick up where the above schools are unsatisfactory and offer a better account on the ensuing issues. 8

On the other hand, Posy, C., 1992, found that Hilbert actually put a structure on the intuitive part of mathematics, essentially that of finitary thought and formal systems; with Gödel's work. Thompson, P.,1993, argued that the Gödelian brand of Platonism, in particular, takes its lead from the actual experience of doing mathematics, and Gödel accounts for the obviousness of the elementary set-theoretical axioms by positing a faculty of mathematical intuition, analogous to sense-perception in physics, so that, presumably, the axioms 'force themselves upon us' much as the assumption of 'medium-sized physical objects' forces itself upon us as an explanation of our physical experiences. However, Thompson 9 stated that counterintuitive has acquired an ambiguous role in our language use that is when applied to a strange but true principle; counterintuitive can now mean anything on a continuum from intuitively false to not intuitively true, depending on the strength of the conjecture we would have been predisposed to make against it, had we not seen, and been won over by, the proof; and indeed, to our surprise, we often find out, in times of paradox, how weak and defeatable our ordinary intuitions are.

Thompson 10 claimed that the very idea that our intuitions should be both decisive and failsafe, derives historically from the maelstrom of senses which the term 'intuition' has acquired in a series of primitive epistemic theories in which some of these senses have been inherited from the large role introspection played in the indubitable bedrock of Cartesian-style philosophy, and some simply from the pervasiveness of out-moded theological convictions which seek to make certain modes of justification unassailable. On the other hand, Posy, C., 1992, insisted that Hilbert's formal system fits the theory of recursive functions. Posy 11 insisted that Brouwer was very much opposed to these ideas, especially that of formalizing systems; he even opposed the formalization of logic; Brouwer had a very radical view of mathematics and language's relationship. According to Brouwer 12, in language, we can communicate the output of mathematical construction, thus helping others recreate the mathematical experience; however, the proof itself is a pre-linguistic, purely conscious activity which is much more flexible than language. Brouwer thought formal systems could never be adequate to cover all the flexible options available to the creative mathematician; and thought that formalism was absurd. Posy 14 noted that, in particular, Brouwer 13 thought that it was crazy to think that codified logic could capture the rules for correct mathematical thought. Brouwer 15 showed particular rules of logic are inadequate with the most famous of the law of the excluded middle.

Thompson, P.,1993, noted that Brouwer that the common un-circumspect belief in the applicability of traditional logic to mathematics was caused historically; he next stated that by the fact that, firstly, classical logic was abstracted from the mathematics of subsets of a definite finite set, that, secondly, an a priori existence independent of mathematics was ascribed to this logic, and finally, on the basis of this suppositious apriority, it was unjustifiably applied to the mathematics of infinite sets. Furthermore, Posy, C., 1992, insisted that Brouwer hypothesized about the reason why philosophers and mathematicians included the law of the excluded middle; according to Brouwer, logic was codified when the scientific community was concerned only with finite objects. Brouwer 16 said that, considering only finite objects, the law of the excluded middle holds; however, a mistake was made when mathematics moved into the infinitary in which the rigid rules of logic were maintained without question. Brouwer 17 suggested that no rigid codification should come before the development of mathematics. Posy found that a second major distinction between Brouwer and Hilbert was that they disagreed on the position of logic in which Hilbert thought logic was an autonomous, finished science that could be freely applied to other mathematics, Brouwer argued that logic should only come after the mathematics is developed.

Litlangs, 2004, in his overview, insisted that profound questions of how varied of intellect faces difficulties in explaining mathematics internally i.e. their gaps, contradictions and ambiguities that lie beneath the most certain of procedures, leads to rough conclusion that mathematics may be no more logical than poetry; it is just free creations of the human mind that unaccountably give order to ourselves and the natural world. Litlangs 18 perceived that though mathematics might seem the clearest and most certain kind of knowledge we possess, there are problems just as serious as those in any other branch of philosophy about the nature of mathematics and the meaning of its propositions. Litlangs 19 found that Plato believed in forms or ideas that were eternal, capable of precise definition and independent of perception; among such entities Plato included numbers and the objects of geometry such as lines, points, circles, which were therefore apprehended not with the senses but with reason; he deals with the objects of mathematics with specific instances of ideal forms. According to Plato, as it was noted by Litlangs, since the true propositions of mathematics were true of the unchangeable relations between unchangeable objects, they were inevitably true that is mathematics discovered pre-existing truths "out there" rather than created something from our mental predispositions; and as for the objects perceived by our senses, they are only poor and evanescent copies of the forms.

Meanwhile, Litlangs, 2004, insisted that Leibniz brought together logic and mathematics; however, whereas Aristotle used propositions of the subject i.e. predicate form, Leibniz argued that the subject contains the predicate that is a view that brought in infinity and God. According to Leibniz 20, mathematical propositions are not true because they deal in eternal or idealized entities, but because their denial is logically impossible; they are true not only of this world, or the world of eternal forms, but of all possible worlds. Litlangs 21 insisted that unlike Plato, for whom constructions were adventitious aids, Leibniz saw the importance of notation, a symbolism of calculation, and so began what became very important in the twentieth century that is a method of forming and arranging characters and signs to represent the relationships between mathematical thoughts.

Litlangs, 2004, further stipulated that Immanuel Kant perceived mathematical entities as a-priori synthetic propositions, which of course provide the necessary conditions for objective experience; time and space were matrices, the containers holding the changing material of perception. According to Kant 22, mathematics was the description of space and time; if restricted to thought, mathematical concepts required only self-consistency, but the construction of such concepts involves space having a certain structure, which in Kant's day was described by Euclidean geometry. Litlangs 23 noted that for Kant, the distinction between the abstract "two" and "two pears" is about construction plus empirical matter; in his analysis of infinity, Kant accepted Aristotle's distinction between potential and complete infinity, but did not think the latter was logically impossible. Kant 24 perceived that complete infinity was an idea of reason, internally consistent, though of course never encountered in our world of sense perceptions. Litlangs further insisted that 25 Frege and Russell and their followers developed Leibniz's idea that mathematics was something logically undeniable; Frege used general laws of logic plus definitions, formulating a symbolic notation for the reasoning required. However, through the long chains of reasoning, these symbols became less intuitively obvious, the transition being mediated by definitions. Litlangs 26 noted that Russell saw them as notational conveniences, mere steps in the argument; while Frege saw them as implying something worthy of careful thought, often presenting key mathematical concepts from new angles. Litlangs 27 found that while in Russell's case the definitions had no objective existence, in Frege's case the matter was not so clear that is the definitions were logical objects which claim an existence equal to other mathematical entities. Litlangs 28 concluded that, nonetheless, Russell carried on, resolving and side-stepping many logical paradoxes, to create with Whitehead the monumental system of description and notation of the Principia Mathematica.

Meanwhile, Thompson, P.,1993, exposed the critical movement of Cauchy and Weierstrass to have been a caution or reserve over the mathematical use of the infinite, except as a façon de parler in summing series or taking limits, where it really behaved as a convenient metaphor, or mode of abbreviation, for clumsier expressions only involving finite numbers. Thompson 29 claimed that when Cantor came on the scene, the German mathematician Leopold Kronecker, who had already 'constructively' re-written the theory of algebraic number fields, objected violently to Cantor's belief that, so long as logic was respected, statements about the completed infinite were perfectly significant. According to Thompson 30, Cantor had further urged that we should be fully prepared to use familiar words in altogether new contexts, or with reference to situations not previously envisaged; Kronecker, however, felt that Cantor was blindly cashing finite schemas in infinite domains, both by attributing a cardinal to any aggregate whatsoever, finite or infinite, and worse still, in his subsequent elaboration of transfinite arithmetic. Thompson 31 insisted that although the interim strain on the intuition, at the time, was crucial to Euler's heuristic approach, this particular infinite detour had been analyzed out of his subsequent proofs of the result, which appeared almost 10 years after its discovery.

Thompson, P.,1993, clarified that Gödel's feeling is that our intuition can be suitably extended to a familiarity with very strongly axiomatic domains, such as extensions of ZF, or calculus on smooth space-time manifolds, thereby providing us with backgrounds for either accepting or rejecting hypotheses independently of our pre-theoretic prejudices or preconceptions about them. Thompson 32 indicated that the general reccursiveness, as in the Gödel and Herbrand sense, with regard to their claims to be collectively demarcating the limits of intuitive computability, is a feature of this particular problem that it is susceptible to a diversity of equally restrictive intuitive re-characterizations, whose unexpected confluence gives each of them a strong intuitive recommendation and this confluence turns out to be a surprisingly valuable asset in appraising our rather more recondite extensions of our intuitive concepts. Thompson 33 concluded that Gödel,34 with his basic trust in transcendental logic, likes to think that our logical optics is only slightly out of focus, and hopes that after some minor correction of it, we shall see sharp, and then everyone will agree that we are right; however, he who does not share such a trust will be disturbed by the high degree of arbitrariness in a system like Zermelo's, or even in Hilbert's system. Thompson 35 suggested that Hilbert will not be able to assure us of consistency forever; therefore we must be content if a simple axiomatic system of mathematics has met the test of our mathematical experiences so far.


1Litlangs, 2004, Math Theory, Poetry Magic: editor@poetrymagic.co.uk
3Bold, T., 2004, Concepts on Mathematical Concepts, http://www.usfca.edu/philosophy/ discourse/8/bold.doc
11Posy, C., 1992, Philosophy of Mathematics, http://www.cs.washington.edu/ homes/ gjb.doc/philmath.htm
18Litlangs, 2004, Math Theory, Poetry Magic: editor@poetrymagic.co.uk
29Thompson, P.,1993, The Nature And Role Of Intuition In Mathematical Epistemology, University College, Oxford University, U.K


    S2 P.MAT A 2016

    Thompson, P., 1993, menyatakan bahwa para filsuf matematika memiliki, selama ribuan tahun, berulang kali keterlibatan dalam perdebatan tentang paradoks dan kesulitan mereka dalam melihat fenomena yang muncul dari tengah-tengah keyakinan mereka yang kuat dan intuitif. Dari munculnya Geometri non-Euclidean, analisis teori kontinum, dan penemuan Cantor tentang bilangan transfinite, sistem Frege, matematikawan kemudian menyuarakan keprihatinan mereka bagaimana kita secara serampangan telah memikirkan sesuatu yang asing, dan dengan liar memperpanjang persoalan matematika kita dengan intuisi, atau kalau tidak kita telah menjadi rentan terhadap perangkap yang tak terduga dan sampai sekarang, dengan apa yang disebut kontradiksi.

  2. Rhomiy Handidcan
    PPs Pendidikan Matematika B 2016

    Mengembangkan matematika kita harus menggunakan intuisi yang kita miliki, intuisi yang kita punya dan pengalaman tidak bisa menjangkau bilangan infinit, sehingga kaum intuisionisme menolak bilangan infinit, hanya mengembangkan bilangan finit. Menurut Heyting sebagai penerus Brouwer menolak kenyataan transenden sebagai alat bukti matematika. Menurutnya bilangan infinit merupakan salah satu kenyataan transenden

  3. Bismillah
    Ratih Kartika
    PPS PEP B 2016

    Mari kita membahas satu persatu fondasi fondasi ketidakstabilan matematika dari beberapa ahli seperti Litlangs, Bold, Thomson dan Posy. Menurut Litlangs, ia tidak setuju dengan pernyataan Aristotel yang menolak pernyataan Plato bahwa menurut Aristotel bentuk itu entitas tidak terpisah dari penampakannya tetapi sesuatu yang termasuk dalam objek di dunia ini. Aristoteles juga menyatakan bahwa matematka adalah alasan sederhana tentang sebuah proses idealisasi. Sedangkan Bold mengindikasikan bahwa matematika adalah salah satu cabang dari logika. Bisa dilihat dari aksioma sebagai unsur dasar logika. Posy berargumen bahwa intuisi matematika adalah bagian penting dari pikiran. Thompson menyatakan bahwa intuisi kita harus bersifat menentukan karena sifat dasar intuisi itu sendiri.


  4. Erlinda Rahma Dewi
    S2 PPs Pendidikan Matematika A 2016

    Litlang menunjukkan bahwa meskipun matematika mungkin tampak paling jelas dari pengetahuan yang kita miliki, ada masalah serius seperti di cabang ilmu lain dari filsafat. Hal ini tidak mudah untuk menguraikan sifat matematika. Plato percaya, bahwa ada yang kekal, definisi yang tepat dan independen dari persepsi. Antara entitas tersebut, angka dan benda-benda geometri seperti garis, poin atau lingkaran yang ditangkap tidak dengan indra tetapi dengan alasan. Menurut Plato, mereka pasti benar, yang berarti bahwa lebih mudah menemukan matematika kebenaran yang sudah ada di luar sana daripada menciptakan sesuatu dari kecenderungan mental kita; karenanya, matematika berurusan dengan kebenaran dan realitas.

  5. Achmad Rasyidinnur
    PEP S2 B

    Perbedaan pandangan antara orang satu dengan orang lainnya dapat saja berbeda. Objek satu dengan objek lainnya tidak sama. Pada keadaan pada tempat yang samapun tetap kontradiksi kepercayaan dan keyakinan dapat berbeda. Karena perbedaan pikiran adalah akibat pengalaman indrawi maupun analitiknya.

  6. Achmad Rasyidinnur
    PEP S2 B

    Pemikiran Aristoteles tentang matematika itu hanya penalaran tentang idealisasi; dan ia melihat dekat pada struktur matematika, logika pembeda, prinsip-prinsip yang digunakan untuk menunjukkan teorema, definisi dan hipotesis.

  7. Aprisal
    PPs S2 Pendidikan Matematika Kelas A 2016

    Assalamu Alaikum Wr.Wb

    Litlangs 2004, menyitir ketidaksetujuan Aristoteles terhadap Plato; menurut Aristoteles, bentuk fisik tidaklah jauh berbeda dengan penampilannya tetapi sesuatu yang konkrit sajalah yang menjadi benda-benda dunia. Aristoteles menyatakan bahwa ketika kita mendapatkan sesuatu yang abstrak, bukan berarti bahwa abstraksi merupakan sesuatu yang jauh dan abadi. Bagi Aristoteles, matematika adalah hanya penalaran tentang idealisasi, dan ia melihat dekat pada struktur matematika, membedakan logika, prinsip yang digunakan untuk menunjukkan teorema, definisi dan hipotesis. Plato juga tercermin pada tak terhingga, memahami perbedaan antara potensi tak terbatas misalnya menambahkan satu ke bilangan infinit misalnya tak terbatas. Selain itu, Bold, T., 2004, menyatakan bahwa kedua intuisionis dan formalis meyakinkan bahwa matematika hanyalah penemuan dan mereka melakukannya dengan tidak menginformasikan kepada kami dengan apa-apa tentang dunia; keduanya mengambil pendekatan ini untuk menjelaskan kepastian mutlak matematika dan menolak penggunaan bilangan infinit. Bold mencatat bahwa intuitionists mengakui hal ini kesamaannya dengan formalis dan menganggap perbedaan yang ada sebagai perbedaan pendapat di mana ketepatan matematis memang ada; intuisionis mengatakannya sebagai kecerdasan manusia dan formalis mengatakannya sebagai hanya coretan di atas kertas

    Waalaikum salam wr.wb

  8. Syahrial
    S2 PEP kelas B 2016
    pada dassarnya untuk memahami pendapat dari para ahli Litlangs’,Bold’s, Thomson’s and Posy’s, menunjukkan bahwa setiap hal itu termasuk fondasi matematika itu selalu mengalami kontradiksi. setiap ahli memiliki pemikiran sendiri sesuai dengan kemampuan dan bidang yang ia miliki. namun pada dasarnya dari semua itu maka harus ada dari segi filsafat, karena filsafat adalah refleksi dari itu semua, sehingga kita memahami bahwa intuisi, logika, pengalaman dsb merupakan komponen yang perlu dalam membangun fondasi matematika.

  9. Siska Nur Rahmawati
    PEP-B 2016

    Aristoteles menyampaikan bahwa matematika itu adalah penalaran terhadap sebuah idealisasi. Logika digunakan sebagai pembeda untuk menunjukkan definisi dan hipotesis. Sebenarnya, dalam mempelajari matematika, kita perlu menggunakan intuisi, pengetahuan,pengalaman, dan logika.

  10. Nira Arsoetar
    PPS UNY Pendidikan Matematika
    Kelas A

    Dalam elegi ini, Thompson menjelaskan bahwa penryataan Gödel merupakan intuisi yang mencakup domain aksiomatik, seperti ekstensi dari ZF, atau kalkulus pada manifold ruang-waktu, sehingga memberikan latar belakang baik untuk menerima atau menolak hipotesis secara independen dari prasangka pra-teori Litlangs, Bolds, dan Posy atau prasangka tentang mereka.

  11. Dita Nur Syarafina
    NIM. 16709251003
    PPs Pendidikan Matematika Kelas A 2016

    Litlangs mengemukakan pandangan Plato dan Aristotle terhadap matematika. Menurut Aristotle, matematika hanya penalaran tentang idealisasi. Lebih rinci, Plato melihat pada struktur matematika, logika pembeda, dan prinsip-prinsip dalam matematika yang digunakan untuk menunjukkan teorema. Plato pada kepercayaan infinitinya memahami perbedaan infitnity. Di sisi lain, Bold T mengemukakan bahwa matematika hanyalah penemuan yangtidak menggambarkan dunia seharusnya. Itu hanyalah teorema dan ketetapan yang dibuat oleh para formalist.

    PPS2016PEP B
    The idea is this: there are a number of basic mathematical truths, called axioms or postulates, from which other true statements may be derived in a finite number of steps. It may take considerable ingenuity to discover a proof; but it is now held that it must be possible to check mechanically, step by step, whether a purported proof is indeed correct, and nowadays a computer should be able to do this. The mathematical statements that can be proved are called theorems, and it follows that, in principle, a mechanical device, such as a modern computer, can generate all theorems.

  13. Muh. Faathir Husain M.
    PPs PEP B 2016

    Pandangan Litlang, Bold, Thomson, and Posy mengenai matematika saling berbenturan satu sama lain. Mereka memiliki pandangan sendiri akan struktur yang terdapat dalam matematika. Itulah matematika yang memiliki sifat yang tidak tunggal dan tidak tertutup.

  14. 16701251016
    PEP B S2

    Memahami berbagai konsep matematika sebenarnya adalah bukan hanya memandang segi kontekstual apa yang tertuang didalamnya. Namun pemikiran jauh lebih luas emandang bahwa untuk sebuah pemahaman diperlukan analitik pemikiran yang logis serta empiris berdasarkan pembuktian kebenaran setiap konsepnya

  15. Asri Fauzi
    Pend. Matematika S2 Kelas A 2016
    Pada artikel tersebut menceritakan tentang Litlangs yang dihadapkan dengan ketidak setujuan Aristoteles terhadap Plato. Aristoteles menyatakan bahwa ketika kita mendapatkan sesuatu yang abstrak, bukan berarti bahwa abstraksi merupakan sesuatu yang jauh dan abadi. Bagi Aristoteles, matematika adalah hanya penalaran tentang idealisasi, dan ia melihat dekat pada struktur matematika, membedakan logika, prinsip yang digunakan untuk menunjukkan teorema, definisi dan hipotesis. Litlangs menyatakan bahwa meskipun matematika mungkin tampak sebagai jenis pengetahuan yang paling jelas dan tertentu dari pengetahuan yang kita miliki, ada masalah cukup serius yang terdapat di setiap cabang lain dari filsafat tentang hakekat matematika dan makna proposisi tersebut.

  16. Rospala Hanisah Yukti Sari
    S2 Pendidikan Matematika Kelas A Tahun 2016

    Assalamu’alaikum warohmatullahi wabarokatuh.

    Intuisi diperlukan untuk mengembangkan konsep matematika. Intuisi yang kita miliki dapat digunakan dalam kondisi-kondisi tertentu. Dengan intuisi pula, kita memiliki kemampuan untuk merespon cepat suatu konsep. Namun, intuisi juga dikembangkan berdasarkan kebenaran dan keakuratan konsep tersebut.

    Selain itu, matematika juga dikembangkan dalam pemikiran yang logis serta berdasarkan analitik suatu konsep, sehingga dalam butir-butir konsep dapat dipertanggungjawabkan kebenarannya.

    Wassalamu’alaikum warohmatullahi wabarokatuh.

  17. Taofan Ali Achmadi
    PPs PEP B 2016

    Dari pemaparan diatas membuktikan bagaimana landasan matematika itu bisa dikatakan tidak stabil terlihat jelas bahwa masing-masing pemikiran para tokoh masih bisa dibantahkan oleh pemikiran tokoh lain dan setiap dari para tokoh tersebut memiliki landasan masing-masing terkait matematika itu sendiri.

  18. Niswah Qurrota A'yuni
    NIM. 16709251023
    PPs S2 Pendidikan Matematika Kelas B 2016

    Assalamu'alaikum Wr.Wb.,

    Krisis landasan dalam matematika selalu diawali dengan munculnya paradoks atau antinomi dalam matematika. Krisis landasan matematika, terutama yang berlandaskan teori himpunan dan logika formal, memaksa para matematikawan mencari landasan filsafat yang ingin mengonstruksi seluruh massa matematika yang besar, sehingga dapat diperoleh landasan yang kokoh. Mereka terpecah ke dalam tiga aliran besar filsafat matematika: logistis, intuisionis, dan formalis.

    Wassalamu'alaikum Wr.Wb.

    S2 Pendidikan Matematika 2016 Kelas B

    Assalamualaikum Wr.Wb.

    Litlangs menyatakan bahwa meskipun matematika mungkin tampak sebagai jenis pengetahuan yang paling jelas dan tertentu dari pengetahuan yang kita miliki, ada masalah cukup serius yang terdapat di setiap cabang lain dari filsafat tentang hakekat matematika dan makna proposisi tersebut. Bold, T., 2004, mengklaim bahwa baik intuisionis dan formalis meyakinkan bahwa matematika hanya penemuan dan tidak memberitahu kami dengan apa-apa tentang dunia. Thompson menyatakan bahwa gagasan tentang intuisi kita yang harus baik, tegas dan benar. Posy menemukan bahwa perbedaan utama antara Brouwer dan Hilbert adalah bahwa mereka tidak setuju pada posisi logika di mana Hilbert pikir logika adalah ilmu pengetahuan, jadi yang otonom dapat secara bebas diterapkan pada matematika lain, sedangkan Brouwer berpendapat tidak demikian.

    Wassalamualaikum Wr.Wb.

  20. Azwar Anwar
    Pendidikan Matematika S2 Kelas B 2016

    Menurut Leibniz proposisi matematika adalah tidak benar karena mereka berurusan dengan entitas kekal atau ideal, tetapi karena penolakan mereka secara logika. Leibniz melihat pentingnya notasi, simbolisme perhitungan, dan mulai apa yang menjadi sangat penting dalam abad kedua puluh yang adalah metode membentuk dan mengatur karakter dan tanda-tanda untuk mewakili hubungan antara pikiran matematika. Menurut pandangan Leibniz ini bahwa matematika selalu di identikan dengan notasi, simbol dan lain-lain karena kita dapat langsung merasakan objek yang ada dalam berbagai cabang ilmu matematika.

  21. Konstantinus Denny Pareira Meke
    NIM. 16709251020
    PPs S2 Pendidikan Matematika Kelas A 2016

    Litlangs menyatakan bahwa meskipun matematika mungkin tampak sebagai jenis pengetahuan yang paling jelas dan tertentu dari pengetahuan yang kita miliki, ada masalah cukup serius yang terdapat di setiap cabang lain dari filsafat tentang hakekat matematika dan makna proposisi tersebut. Bold, T., 2004, menyatakan bahwa komponen penting dari matematika mencakup konsep angka integer, pecahan, penambahan, perpecahan dan persamaan; di mana penambahan dan pembagian terhubung dengan studi proposisi matematika dan konsep bilangan bulat dan pecahan adalah elemen dari konsep-konsep matematika. Posy, dalam hal pertanyaan ontologis, bertanya-tanya seberapa akurat gagasan bahwa himpunan adalah objek dasar matematika, sedangkan teori yang dihimpun terlalu kaya dan ada cara yang berbeda terlalu banyak untuk membangun matematika. Posy berpendapat bahwa elemen dasar tidak boleh sembarang dipilih, namun tidak menentukan pilihannya, dan menunjukkan bahwa, dalam pandangan modern tentang strukturalisme, unit dasar adalah struktur, yang bukan benar-benar objek.

  22. Nanang Ade Putra Yaman
    PPs PM B 2016

    saya kira matematika sebagai cabang ilmu, kebenarannya terbatas karna matematika hanya hasil pikiran manusia dari pengamatan terhadap objek-objek yang ada di alam. Bold, T., 2004, sebagaimana tulisan diatas mengklaim bahwa baik intuisionis dan formalis meyakinkan bahwa matematika hanya penemuan dan tidak memberitahu kami dengan apa-apa tentang dunia; baik mengambil pendekatan ini untuk menjelaskan kepastian yang mutlak matematika dan menolak penggunaan tak terbatas. Bold mencatat bahwa intuitionists mengakui kesamaan besar ini formalis dan perhatikan perbedaan sebagai ketidaksepakatan di mana ketepatan matematika ada; intuisionis mengatakan dalam akal manusia dan formalis yang tertulis di kertas. Menurut Arend Heyting 4, matematika adalah produksi dari pikiran manusia; ia mengklaim bahwa intuitionism mengklaim proposisi matematika mewarisi kepastian mereka dari pengetahuan manusia yang didasarkan pada pengalaman empiris.


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