Nov 25, 2012

Theory of Knowledge and The Foundation of Mathematics

THE ROLE OF KANT’S THEORY OF KNOWLEDGE IN SETTING UP THE EPISTEMOLOGICAL FOUNDATION OF MATHEMATICS

By Marsigit, Yogyakarta State University, Indonesia

Email: marsigitina@yahoo.com
Mathematics , by its nature, always has an inclination toward the right, and, for this reason, has long withstood the spirit of the time that has ruled since the Renaissance; i.e., the empiricist theory of mathematics, such as the one set forth by Mill, did not find much support. Indeed, mathematics has evolved into ever higher abstractions, away from matter and to ever greater clarity in its foundations e.g. by giving an exact foundation of the infinitesimal calculus and the complex numbers - thus, away from scepticism. However, around the turn of the century, it is the antinomies of set theory, contradictions that allegedly appeared within mathematics, whose significance is exaggerated by sceptics and empiricists and which are employed as a pretext for the leftward upheaval.

Epistemological foundationalism  seeks for a solid ground of mathematical cognition to shield it from arbitrariness, non-conclusiveness, and vulnerability to historical circumstances as well as to guarantee certainty and truth of mathematical knowledge. It is noted that in any version of epistemological foundationalism there is an element of absolutism viz.epistemological standards, such as truth, certainty, universality, objectivity, rationality, etc. According to an empiricist version of foundationalism , the basic elements of knowledge are the  truths  which  are  certain  because  of  their  causes 1rather than because of the arguments given for them.

The rising of Kant’s theory of knowledge, as the epistemological foundations in seeking the solid ground on knowledge , was influenced by at least two distinct epistemological foundations i.e. from its root of Empiricist foundationalism and  Rationalist foundationalism. According to an empiricist version of foundationalism  there are basic elements of knowledge that the truths are certain because of their causes rather than because of the arguments given for them; they believe in the existence of such truths because they presuppose that the object which the proposition is about imposes the proposition's truth.  It  is an empiricist version of foundationalism that can be characterized as consisting of two assumptions: (a) there are true claims which, if known, would allow us to derive all knowledge of the architecture of existence; and (b) such claims are given.

To discover which concepts and judgments are foundational for our knowledge,  rationalist foundationalism finds a source of cognitive act, that is, an act in which we discover  fundamental ideas and truths . This is of course an intellectual act but not an act of reasoning since it itself requires premises. It is either an act of intellectual intuition or an act of self-reflection, self-consciousness, i.e., Cartesian Cogito; such an act not only reveals the ultimate foundations of knowledge but also gives us certainty referring to their epistemological value: we know for sure that they are indubitable, necessary, and - consequently - true. The foundational ideas and judgments are indubitable since they are involved in acts of the clear and distinct knowing-that; they are necessary as ultimate premises from which other judgments are deducible; and they are true since they are involved in acts of correct and accurate knowing-about.   

If rationalists, such as Plato, Descartes, Leibniz, or Spinoza, believe that all knowledge is already present in human mind before any cognitive activity begins. They cannot construct their foundationalist program as the search for privileged, fundamental,  and true propositions. On the other hand, the search for foundations of cognition and knowledge does not mean for Kant to establish basic ideas and truths from which the rest of knowledge can be inferred. It means to solve the question of how is cognition as a relation between a subject and an object possible, or - in other words - how synthetic representations and their objects can establish connection, obtain necessary relation to one another, and, as it were, meet one another.

Relating to those problems, in his theory of knowledge, Kant proposes to epistemologically set mathematics upon the secure path of science. Kant  claims that the true method rests in the realization of mathematics can only have certain knowledge that is necessarily presupposed a priori by reason itself. In particular, the objective validity of mathematical knowledge, according to Kant, rests on the fact that it is based on the a priori forms of our sensibility which condition the possibility of experience. However, mathematical developments in the past two hundred years have challenged Kant's theory of mathematical knowledge in fundamental respects.

This research, therefore, investigates the role of Kant’s theory of knowledge in setting up the epistemological foundations of mathematics. The material object is the epistemological foundation of mathematics and the formal object of this research are Kant’s notions in his theory of knowledge. The research problem consists of: what and how is the epistemological foundation of mathematics?, what and how is the Kant’s Theory of Knowledge?; and, how and to what extent Kant’s theory of knowledge has its roles in setting up the epistemological foundation of mathematics.Having reviewed related references, the researcher found two other researches on similar issues. Those are of Thomas J. McFarlane, 1995, i.e. “Kant and Mathematical Knowledge”, and of Lisa Shabel, 2003, i.e. “Mathematics in Kant’s Critical Philosophy”. The first, McFarlanne, in his research, has critically examined the challenges non-Euclidean geometry poses to Kant's theory and then, in light of this analysis, briefly speculates on how we might understand the foundations of mathematical knowledge; by previously examined Kant's view of mathematical knowledge in some of its details. While in the second, Shabel has strived to clarify the conception of synthetic a priori cognition central to the critical philosophy, by shedding light on Kant’s account of the construction of mathematical concepts; and  carefully examined the role of the production of diagrams in the mathematics of the early moderns, both in their practice and in their own understanding of their practice. While, in this research, the researcher strives to investigate the role of Kant’s philosophy of mathematics in setting up the epistemological foundation of mathematics.

The philosophy of mathematics has it aims to clarify and answer the questions about the status and the foundation of mathematical objects and methods that is ontologically clarify whether there exist mathematical objects, and epistemologically clarify whether all meaningful mathematical statements have objective and determine the truth. Perceiving that the laws of nature and the laws of mathematics have a similar status, the very real world of mathematical objects forms the mathematical foundation; however, it is still a big question how we access them. Although some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense; some philosophers focused on human cognition as the origin of the reliability of mathematics and proposed to find mathematical foundations only in human thought, as Kant did, not in any objective outside mathematics.

While the empiricist and rationalist versions of foundationalism strive to establish the foundation of mathematics as justifying the epistemologically valid, i.e., certain, necessary, true, Kant established forms and categories to find the ultimate conditions of the possibility of the objectivity of cognition, i.e., of the fact that we cognize mathematics. The historical development of the reflection on knowledge and cognition of mathematics shows that the most notorious exemplars of foundationalism have overgrown themselves and gave birth to criticisms undermining their postulates and conclusions.  The ideal of justificationism has been abandoned in consequence of the demonstration that neither inductive confirmation nor deductive falsification can be conclusive. The foundationalist program of Descartes is also radically modified or even rejected.

Kant’s theory of knowledge  insists that instead of assuming that our ideas, to be true, must conform to an external reality independent of our knowing, objective reality should be known only in so far as it conforms to the essential structure of the knowing mind. Kant  maintains that objects of experience i.e. phenomena, may be known, but that things lying beyond the realm of possible experience i.e. noumena or things-in-themselves are unknowable, although their existence is a necessary presupposition. Phenomena, in which it can be perceived in the pure forms of sensibility, space, and time must possess the characteristics that constitute our categories of understanding. Those categories, which include causality and substance, are the source of the structure of phenomenal experience.

Kant  sums up that all three disciplines—logic, arithmetic and geometry—are synthetic as disciplines independent from one another. In the Critique of Pure Reason and the Prolegomena to Any Future Metaphysics, Kant  argues that the truths of geometry are synthetic a priori truths, and not analytic, as most today would probably assume. Truths of logic and truths that are merely true by definition are "analytic" because they depend on an "analysis", or breakdown, of a concept into its components, without any need to bring in external information. Analytic truths are thus necessarily a priori. Synthetic truths, on the other hand, require that a concept to be "synthesized" or combined with some other information, perhaps another concept or some sensory data, to produce something truly new.

Modern philosophy after Kant presents some important criteria that distinguish the foundations of mathematics i.e. pursuing the foundation in the logical sense, in the philosophy of mathematics, in the philosophy of language, or in the sense of epistemology. The dread of the story of the role of Kant’s theory of knowledge to the foundation of mathematics can be highlighted e.g. from the application of Kant’s doctrine to algebra and his conclusion that geometry is the science of space and since time and space are pure sensuous forms of intuition, therefore the rest of mathematics must belong to time and that algebra is the science of pure time.

This research employs literal study on Kant’s critical philosophy and on the philosophy of mathematics. This research employs various approaches to describe the works of Immanuel Kant, specifically epistemology, and to investigate its contribution to the philosophy of mathematics. Qualitative approach was employed to identify and describe coherently some related concepts, in a narrative form; in a certain occasion, the researcher needs to develop in depth of understanding and description provided by textual sources. The related references have several features to take into consideration in such away that understanding of Kant’s epistemology and its contribution to the philosophy of mathematics will be gained through a holistic perspective. The researcher, in striving to understand the concepts, at any occasion, strived to compare and contrast the notions from different philosophical threads.

The researcher strives to reflect and interpret the related texts of critical philosophy that is most of them are the works of Immanuel Kant as the primary data. Argumentations and expositions were in analytical and logical form in such away that the researcher formed the body of the essay fit together. On the other hand, the researcher need to be flexible and open because the purpose was to learn how past intentions and events were related due to their meaning and value. Lack of understanding of Kant’s theory of knowledge was over came by phenomenological approach in which the researcher began with the acknowledgement that there was a gap between author’s understanding and the clarification or illumination of the features.

Kant's most significant contribution to modern philosophy, by his own direct assertion, is the recognition that mathematical knowledge ,  holds the key to defeating the Humean skeptical work, as the synthetic a priori judgments are possible. The vicious dichotomy in the foundation of mathematics maintains that there are two absolutely incommensurable types of knowledge, the a priori analytic and a posteriori synthetic, each deriving from two radically distinct sources, reason and experience. This division makes the product of reason, logic, empty and relegates knowledge of nature as merely particular and therefore blind. Kant's insight is to recognize that mathematical knowledge seems to bridge this dichotomy in a way that defeats the skeptical claim. Mathematical thinking is both a priori in the universality and necessity of its results and synthetic in the expansively ampliative promise of its inquiry.

Kant’s proof consists of mathematical knowledge, in which under the action of cognition, leads to mathematical knowledge of the conclusion even though the truth-conditions of the former may not cover those of the latter. Kant’s epistemological mathematic is the principle that an inference is gapless when one grasps a mathematical architecture in which a mathematical justification of the conclusion is seen as a development of a mathematical justification of the premises. Prior to modern empiricism, analysis was the "taking apart" of a mathematical problem while synthesis referred to the process of reconstructing that same problem in a logically deductive form. Euclidean Geometry is erroneously taken as synonymous with both ancient geometry and this same synthetic form of presentation.

Kant's use of mathematics is not the same as that made by Frege; his basic claim was that Kant used a geometrical model to subsume arithmetic within the real of the synthetic a priori. Kant's model of mathematical reasoning only actually puts the smaller part of geometry in the synthetic a priori. By recognizing that the metaphysical gap can be represented as within the mathematical, rather than as between the mathematical and the dynamical uses of reason, we can realize, as did Plato and Descartes, that the productive synthesis of the transcendental schemata is exactly the mathematical or "figurative" representation of the metaphysical. The figurative schemata of mathematics can exemplify the pure concepts of reason as adeptly as they capture the possible objects of experience. Kant clearly had some part of this model in mind when he defined the mathematical as the constructio of concepts.

Kant believes the judgments of mathematics and Newtonian physics to be instances of genuine knowledge. Mathematics and science are objective and universally valid, because all human beings know in the same way. Mathematician after Kant, Frege, of course, agreed with Kant on geometry, and considered it to be synthetic a priori; but  different logicist might claim that their reduction laid out the `true nature' of geometrical objects as strictly logical. Taking a particularly logicist definition of rigour for granted; more than that of mathematician pre-Kant. Kant goes on to identify this illegitimate use of logic as organon as one of the primary sources of paradox in the contemporary philosophy of mathematics, and indeed one of the fundamental points of Kant’s theory of knowledge is that pure reason, including logic, cannot go beyond a close connection with objects as given in experience. This is precisely why the three alternatives to logicism can be considered as Kantian; logicists insist on attention to the connection between knowledge of mathematics and the content of that knowledge.

Kant’s theory of knowledge recognizes broad notions of mathematics and intuition, so that both algebra and geometry are accommodated, but it can easily do this while maintaining a distinction between the mathematical and the universal. So, it should now be obvious why the challenge that, e.g. the Poincarean epistemologist presents to logicism is much more daunting than a mere question of adequacy; rejecting the use of logic as an organon of knowledge means logicism is incoherent from the start. Even questions like the consistency of some proposed neo-Fregean system that attempts to demonstrate that geometry is analytic are meaningless on this view. And, even though logic is accepted as an organon among most contemporary philosophers of mathematics, this is no reason to consider the Kantian views vanquished.

Ultimately, it can be said that Kant’s theory of knowledge has inherently philosophical interest, contemporary relevance, and defensibility argument to remain essentially intact to the foundations of mathematics no matter what one may ultimately think about controversial of his metaphysics and epistemology of transcendental idealism. In term of the current tendency of the philosophy of mathematics, Kant’s theory of knowledge is in line with the perception that understanding of mathematics can be supported by the nature of perceptual faculties. Accordingly, mathematics should be intuitive. What is needed in the foundation of mathematics is then a more sophisticated theory of sensible intuition. Kronecker in Smith N.K., for example, advocated a return to an intuitive basis of mathematics due to the fact that abstract non-intuitive mathematics was internally inconsistent; and this is the other crisis in the epistemological foundation of mathematic.

The important results of the research can specifically be exposed as follows:

1.Kant’s Theory of Knowledge Synthesizes the Foundation of Mathematics

In the sphere of Kant’s ‘dogmatic’ notions, his theory of knowledge, in turn, can be said to lead to un-dogmatization and de-mythologization of mathematical foundations as well as rages the institutionalization of the research of mathematical foundations in which it encourages the mutual interactions among them. Kant  insists that a dogmatist is one who assumes that human reason can comprehend ultimate reality, and who proceeds upon this assumption; it expresses itself through three factors viz. rationalism, realism, and transcendence.

The role of Kant’s theory of knowledge, in the sense of de-mythologization of mathematical foundations, refers to history of the mathematical myth from that of Euclid’s to that of contemporary philosophy of mathematics. The myth of Euclid: "Euclid's Elements contains truths about the universe which are clear and indubitable", however, today advanced student of geometry to learn Euclid's proofs are incomplete and unintelligible.    Kant’s theory of knowledge implies to the critical examinations of those myths. In fact, being a myth doesn't entail its truth or falsity. Myths validate and support institutions in which their truth may not be determinable. Those latent mathematical myths are almost universally accepted, but they are not sef-evident or self-¬proving. From a different perspective, it is  possible to question, doubt, or reject them and some people do reject them.

Smith, N. K. concerns with Kant’s conclusion that there is no dwelling-place for permanent settlement obtained only through perfect certainly in our mathematical knowledge, alike of its objects themselves and of the limits which all our knowledge of object is enclosed. In other word , Kant’s theory of knowledge implies to un-dogmatization and de-mythologization of mathematical foundations as well as to rage the institutionalization of the research of mathematical foundations. In term of these perspectives, Kant considers himself as contributing to the further advance of the eighteenth century Enlightenment and in the future prospect of mathematics philosophy.

2.Kant’s Theory of Knowledge Contributes to Epistemological Foundation of Mathematics

Kant (1783) in “Prolegomena to Any Future Metaphysics”, claims that the conclusions of mathematicians proceed according to the law of contradiction, as is demanded by all apodictic certainty. Kant  says that it is a great mistake for men persuaded themselves that the fundamental principles were known from the same law. Further, Kant  argues that the reason that for a synthetical proposition can indeed be comprehended according to the law of contradiction but only by presupposing another synthetical proposition from which it follows.  Further, Kant  argues that all principles of geometry are no less analytical; and that the proposition “a straight line is the shortest path between two points” is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality.

Meanwhile, Shabel L. believes that Kant explores an epistemological explanation whether pure geometry ultimately provides a structural description of certain features of empirical objects. According to Shabel L. , Kant requires his first articulation that space is a pure form of sensible intuition and argues that, in order to explain the pure geometry without paradox, one must take the concept of space to be subjective, such that it has its source in our cognitive constitution. Kant  perceives that epistemological foundation of geometry is only possible under the presupposition of a given way of explaining our pure intuition of space as the form of our outer sense.

For Kant  and his contemporaries, the epistemological foundations of mathematics consists amount of a view to which our a priori mental representation of space-temporal intuition provides us with the original cognitive object for our mathematical investigations, which ultimately produce a mathematical theory of the empirical world. However , Kant’s account of mathematical cognition serves still remains unresolved issues. Shabel L.  concludes that the great attraction of  Kant’s theory of knowledge comes from the fact that other views seem unable to do any better.

3.Kant’s Theory of Sensible Intuition Contributes to  Constructive and Structural Mathematics

For Kant , to set up the foundation of mathematics we need to start from the very initially step analysis of pure intuition. Kant means by a “pure intuition” as an intuition purified from particulars of experience and conceptual interpretation. i.e., we start with experience and abstract away from concepts and from particular sensations. The impressions made by outward thing which is regarded as pre-established forms of sensibility i.e. time and space. Time  is no empirical conception which can be deduced from experience and a necessary representation which lies at the foundation of all intuitions.  It is given a priori and, in it alone, is any reality of phenomena possible; it disappears, but it cannot be annihilated. Space is an intuition, met with in us a priori, antecedent to any perception of objects, a pure, not an empirical intuition. These two forms of sensibility, inherent and invariable to all experiences, are subject and prime facts of consciousness in the foundation of mathematics. 

According to Kant , mathematics depends on those of space and time that means that the abstract ex¬tension of the mathematical forms embodied in our experi¬ence parallels an extension of the objective world beyond what we actually perceive. Wilder R.L.  points out the arguments for the claim that intuition plays an es¬sential role in mathematics are inevitably subjectivist to a degree, in that they pass from a direct consid¬eration of the mathematical statements and of what is required for their truth verifying them. The dependence of mathematics  on sensible intuition gives some plausibility to the view that the possibility of mathematical representation rests on the form of our sensible intuition. This conception  could be extended to the intuitive verification of elementary propositions of the arithmetic of small numbers. If these propositions really are evident in their full generality, and hence are necessary, then this conception gives some insight into the nature of this evidence.

Other writer, Johnstone H.W. in Sellar W. ascribes that Kant’s sensible intuition account the role in foundation of mathematics by the productive imagination in perceptual geometrical shapes. Phenomenological reflection  on the structure of perceptual geometrical shapes, therefore, should reveal the categories, to which these objects belong, as well as the manner in which objects perceived and perceiving subjects come together in the perceptual act. To dwell it we need to consider Kant's distinction between (a) the concept of an object, (b) the schema of the concept, and (c) an image of the object, as well as his explication of the distinction between a geometrical shape as object and the successive manifold in the apprehension of a geometrical shape. Johnstone H.W.  highlights Kant’s notion that the synthesis in connection with perception has two things in mind (1) the construction of mathematical model as an image, (2) the intuitive formation of mathematical representations as a complex demonstratives. Since mathematical intuitions have categorical form, we can find this categorical form in them and arrive at categorical concepts of mathematics by abstracting from experience.

4.The Relevance of Kant’s Theory of Knowledge to Contemporary Foundation of Mathematics

The relevance of Kant’s theory of knowledge to the contemporary foundation of mathematics can be traced from the notions of contemporary writers. Jørgensen, K.F.(2006) admits that a philosophy of mathematics must square with contemporary mathematics as it is carried out by actual mathematicians. This  leads him to define a very general notion of constructability of mathematics on the basis of a generalized understanding of Kant's theory of schema. Jørgensen, K.F. further states that Kant’s  theory of schematism  should be taken seriously in order to understand his Critique. It was science which Kant wanted to provide a foundation for. He says that one should take schematism to be a very central feature of Kant's theory of knowledge.

Meanwhile, Hanna, R. insists that Kant offers an account of human rationality which is essentially oriented towards judgment. According to her, Kant also offers an account of the nature of judgment, the nature of logic, and the nature of the various irreducibly different kinds of judgments, that are essentially oriented towards the anthropocentric empirical referential meaningfulness and truth of the proposition. Further, Hanna, R.  indicates that the rest of Kant's theory of judgment is then thoroughly cognitive and non-reductive.

In Kant , propositions are systematically built up out of directly referential terms (intuitions) and attributive or descriptive terms (concepts), by means of unifying acts of our innate spontaneous cognitive faculties. This unification is based on pure logical constraints and under a higher-order unity imposed by our faculty for rational self-consciousness. Furthermore  all of this is consistently combined by Kant with non-conceptualism about intuition, which entails that judgmental rationality has a pre-rational or proto-rational cognitive grounding in more basic non-conceptual cognitive capacities that we share with various non-human animals.  In these ways, Hanna, R.  concludes that Kant’s theory of knowledge is the inherent philosophical interest, contemporary relevance, and defensibility remain essentially intact no matter what one may ultimately think about his controversial metaphysics of transcendental idealism.

In the sense of contemporary foundation of mathematics, Hers R.   notifies  that  in  providing  truth  and certainty in mathematics Hilbert implicitly referred Kant. He.  pointed out that, like Hilbert, Brouwer was sure that mathematics had to be established on a sound and firm foundation in which mathematics must start from the intuitively given. The name intuitionism  displays its descent from Kant’s intuitionist theory of mathematical knowledge. Brouwer follows Kant in saying that mathematics is founded on intuitive truths. As it was learned that Kant though geometry is based on space intuition, and arithmetic on time intuition, that made both geometry and arithmetic “synthetic a priori”.   About geometry, Frege  agrees with Kant that it is synthetic intuition.  

The conclusion of this dissertation can be taken as follows:

1.The philosophy of mathematics has aims to clarify and answer the questions about the status and the foundation of mathematical objects and methods, that is, ontologically clarify whether there mathematical objects exist, and epistemologically clarify whether all meaningful mathematical statements have objective and determine the truth. A simplistic view of the philosophy of mathematics indicates that there are four main schools i.e. Platonism, Logicism, Formalism, and Intuitionism.

2.Pre-Kant’s philosophy of mathematics is organized as a debate between Rationalists and Empiricists. Kant begins the philosophy of mathematics with a focus on mathematics knowers and their epistemic relationship to theorems and proofs viz. epistemology of mathematics. The role of Kant’s theory of knowledge in setting up the epistemological foundation of mathematics emerges from Kant’s efforts to set forth epistemological foundation of mathematics based on the synthetical a priori principles in which he believes that the judgments of mathematics is an instances of genuine knowledge.

3.Kant's most significant contribution to modern philosophy is the recognition that mathematical knowledge is possible. The privileging of mathematical thought after Kant seems to derive from Kant's earlier distinguishing of the models of intuition and thought. Kant’s epistemological mathematic is the principle that an inference is gapless when one grasps a mathematical architecture in which a mathematical justification of the conclusion is seen as a development of a mathematical justification of the premises.

4.Kant believes that the judgments of mathematics and Newtonian physics to be instances of genuine knowledge. Mathematics and science are objective and universally valid, because all human beings know in the same way. Kant’s theory of knowledge recognizes broad notions of mathematics and intuition, so that both algebra and geometry are accommodated. Kant’s theory of knowledge has inherently philosophical interest, contemporary relevance, and defensibility argument to remain essentially intact to the foundations of mathematics no matter what one may ultimately think about controversial of his metaphysics and epistemology of transcendental idealism.

5.In term of the current tendency of the philosophy of mathematics, Kant’s theory of knowledge is in line with the perception that understanding of mathematics can be supported by the nature of perceptual faculties. There are at least two philosophical lines in which they have different position of epistemological problems in the epistemological foundation of mathematics. The first line perceives that mathematics should be limited by the nature of perceptual faculties. The second line perceives that problems in mathematics are not consistent with perceptual abilities, but do not limit mathematics to what is able to intuit.

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28 comments:

  1. Nama : Irna K.S.Blegur
    Nim : 16709251064
    kelas : PM D 2016(PPS)
    Kant mengemukakan bahwa ilmu matematika merupakan contoh yang paling cemerlang tentang bagaimana akal murni berhasil bisa memperoleh kesuksesannya dengan bantuan pengalaman.
    Kant bisa mengatakan tanpa berlebihan bahwa banyak logika mengikuti jalur tunggal sejak awal, dan bahwa sejak Aristoteles itu tidak harus menelusuri kembali satu langkah. Kant mengatakan bahwa logika silogisme adalah untuk semua tampilan lengkap dan sempurna.
    Di dalam Teori Pengetahuannya, Immanuel Kant berusaha meletakkan dasar epistemologis bagi matematika untuk menjamin bahwa matematika memang benar dapat dipandang sebagai ilmu. Kant menyatakan bahwa metode yang benar untuk memperoleh kebenaran matematika adalah memperlakukan matematika sebagai pengetahuan a priori. Menurut Kant, secara spesifik, validitas obyektif dari pengetahuan matematika diperoleh melalui bentuk a priori dari sensibilitas kita yang memungkinkan diperolehnya pengalaman inderawi.

    ReplyDelete
  2. Sumandri
    16709251072
    S2 Pendidikan Matematika D 2016

    Dalam filsafat matematika, adanya pertentangan antara kaum rasionalis dan kaum empiris menimbulkan pengakuan mendalam akan sintesis Immanuel Kant bahwa matematika adalah ilmu yang bersifat sintetik a priori. Pengetahuan matematika di satu sisi bersifat “subserve” yaitu hasil dari sistesis pengalaman inderawi; di sisi yang lain matematika bersifat “superserve” yaitu pengetahuan a priori sebagai hasil dari konsep matematika yang bersifat immanen dikarenakan didalam pikiran kita sudah terdapat kategori-kategori yang memungkinkan kita dapat memahami matematika tersebut.

    ReplyDelete
  3. Saepul Watan
    16709251057
    S2 P.Mat Kelas C 2016

    Bismilahir rahmaanir rahiim..
    Assalamualaikum wr..wb...

    Dalam artikel ini dijelaskan bagaimana peran teori pengetahuan Kant sebagai fondasionalisme dalam epistemologi Matematika. Teori pengetahuan Kant menjelaskan bahwa, sebagai dasar epistemologis dalam mencari landasan pengetahuan, dipengaruhi oleh dua dasar epistemologis yang berbeda yaitu dari fondasionalisme empiris dan fondasionalisme rasionalis. Hal ini berarti bahwa matematika merupakan pengetahuan yang berdasarkan pengalaman dan juga berdasarkan pemikiran rasionalis. Kebenaran dalam Matematika bukan karena argumen yang diberikan melainkan dari setiap pembuktian yang ada dan telah dibuktikan nilai kebenarannya. Peran teori pengetahuan Kant dalam mendirikan dasar epistemologis matematika muncul dari upaya Kant untuk menetapkan landasan epistemologis matematika berdasarkan sebuah prinsip apriori di mana ia percaya bahwa penilaian matematika merupakan contoh pengetahuan sejati. Mohon maaf Pak apabila ada kesalahan. Terima Kasih.

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  4. Wahyu Lestari
    16709251024
    PPs P.Matematika Kelas D

    Dari artike ini Meningkatnya teori pengetahuan Kant, karena fondasi epistemologis dalam mencari landasan pengetahuan yang kuat, paling tidak dipengaruhi oleh dua fondasi epistemologis yang berbeda, yaitu dari akar fundamentalis empiris dan dasar pemikiran rasionalis. Menurut versi empiris tentang foundationalism ada unsur dasar pengetahuan bahwa kebenaran dipastikan karena penyebabnya dan bukan karena argumen yang diberikan untuk mereka; Mereka percaya akan adanya kebenaran semacam itu karena mereka mengandaikan bahwa objek yang proposisinya adalah memaksakan kebenaran proposisi. Ini adalah versi empiris dari foundationalism yang dapat dicirikan sebagai terdiri dari dua asumsi: (a) ada klaim sejati yang, jika diketahui, akan memungkinkan kita untuk mendapatkan semua pengetahuan tentang arsitektur eksistensi; Dan (b) klaim tersebut diberikanUntuk menemukan konsep dan penilaian mana yang mendasar bagi pengetahuan kita, foundationalism rasionalis menemukan sumber tindakan kognitif, yaitu tindakan di mana kita menemukan gagasan dan kebenaran mendasar. Ini tentu saja merupakan tindakan intelektual tapi bukan tindakan penalaran karena hal itu sendiri memerlukan premis.

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  5. Wahyu Lestari
    16709251024
    PPs P.Matematika Kelas D

    dalam teorinya tentang pengetahuan, Kant mengusulkan untuk secara epistemologis menetapkan matematika di jalur sains yang aman. Kant mengklaim bahwa metode yang benar terletak pada realisasi matematika hanya bisa memiliki pengetahuan tertentu yang harus diprakirakan secara apriori oleh akal itu sendiri. Secara khusus, validitas obyektif pengetahuan matematika, menurut Kant, terletak pada kenyataan bahwa hal itu didasarkan pada bentuk apriori dari sensibilitas kita yang mengkondisikan kemungkinan pengalaman. Namun, perkembangan matematika dalam dua ratus tahun terakhir telah menantang teori pengetahuan matematika Kant dalam hal-hal mendasar. Sementara versi empiris dan rasionalis dari fondasionalisme berusaha untuk menetapkan dasar matematika sebagai pembenaran yang secara epistemologis valid, yaitu, pasti, perlu, benar, Kant membentuk bentuk dan kategori untuk menemukan kondisi akhir dari kemungkinan objektivitas kognisi, yaitu, Dari kenyataan bahwa kita mengenal matematika.

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  6. Cendekia Ad Dien
    16709251044
    PPs Pendidikan Matematika Kelas C 2016

    Filsafat modern setelah masa Immanuel Kant memberikan kriteria penting bagi pondasi matematika misalnya pondasi matematika harus bersifat logis, pondasi matematika harus berdasarkan kepada filsafat matematika, filsafat bahasa atau epistemologi matematika. Peranan Teori Pengetahuan Kant dapat disoroti dari implementasi doktrin Immanuel Kant terhadap aljabar dan geometri. Kesimpulannya bahwa aljabar adalah ilmu tentang waktu dan geometri adalah ilmu tentang ruang, karena waktu dan ruang berbentuk intuisi formal maka semua pengetahuan matematika lainnya harus dipelajari dalam ruang dan waktu.


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  7. Dessy Rasihen
    16709251063
    S2 P.MAT D

    Menurut Kant terdapat tiga disiplin yaitu logika, aritmatika dan geometri yang merupakan sintetik sebagai disiplin bebas antara satu sama lain. Dalam Critique of Pure Reason dan Proglegomena to Any Future Metapghysics, Kant memandang kebenaran geometri bahwa kebenaran yang sintetik apriori. Kant mengklaim bahwa metode yang benar berdasarkan pada realisasi matematika hanya dapat memiliki pengetahuan yang mensyaratkan apriori pada pengambilan alasannya. Selanjutnya, konstruksi matematika atas dasar pemahaman umum dari teori skema Kant menyatakan bahwa teori Kant schematism harus diambil serius untuk memahami kritik. Inilah landasan ilmu yang ingin diberikan Kant.

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  8. Nurwanti Adi Rahayu
    16709251067
    S2 Pendidikan Matematika Kelas D 2016

    Epistemologi matematika, yang menelaah matematika berdasarkan berbagai segi pengetahuan seperti kemungkinan, asal-mula, sifat alami, batas, asumsi dan landasan.
    Kemungkinan yang muncul,, asal-muasal matematika itu sendiri, batas yang diasas dalam matematika dan asumsi yang digunakan.
    Menurut Kant matematika sebagai ilmu adalah mungkin jika kita mampu menemukan intuisi murni sebagai landasannya; dan matematika yang telah dikonstruksinya bersifat sintetik a priori.

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  9. Loviga Denny Pratama
    16709251075
    S2 P.Mat D

    Dari artikel ini saya memahami bahwa dalam belajar matematika, kita tahu bahwa semakin tinggi level kelas suatu anak, maka juga akan mengalami level abstraksi yang semakin tinggi pula. Matematika telah berkembang menjadi kegiatan abstraksi yang lebih tinggi di atas kejelasan pondasinya seperti yang terjadi pada Kalkulus Infinitas dan Bilangan Kompleks yang telah mengambil jarak dari pandangan kaum skeptik. Tetapi pada abad yang lalu, dengan ditemukannya kontradiksi pada Teori Himpunan, kaum skeptik dan empiric mulai menggaungkan lagi pandangan-pandangan tentang pondasi matematika. Semakin berkembangnya zaman, muncullah Teori Pengetahuan dari Immanuel Kant, sebagai landasan epistemologis dari pengetahuan, dipengaruhi paling tidak oleh pengaruh dua aliran epistemologi yang masing-masing berakar pada pondasi empiris dan pondasi rasionalis. Menurut kaum pondasionalis empiris, terdapat unsur dasar pengetahuan dalam mana nilai kebenarannya lebih dihasilkan oleh hukum sebab-akibat dari pada dihasilkan oleh argumen-argumennya; mereka percaya bahwa keberadaan dari kebenaran tersebut disebabkan oleh asumsi bahwa obyek dari pernyataannyalah yang membawa nilai kebenaran itu.

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  10. Supriadi / 16709251048
    Kelas C 2016 Pendidikan matematika – S2

    Kant, menyimpulkan bahwa Kant menyimpulkan bahwa tiga disiplin yaitu logika, aritmatika dan geometri adalah sintetik sebagai disiplin bebas dari satu sama lain. Dalam Critique of Pure Reason dan Proglegomena to Any Future Metapghysics, Kant berpendapat bahwa kebenaran geometri adalah kebenaran yang sintetik apriori, dan tidak analitik,sebagaimana kebanyakan asumsi. Kebenaran logika dan kebenaran yang hanya benar secara definisi adalah analitik karena mereka bergantung pada analisis, atau kerusakan, dari konsep menjadi komponen-komponennya, tanpa perlu membawa informasi eksternal. kebenaran analitik dengan demikian tentu bersifat apriori. kebenaran sintetik, di sisi lain, mengharuskan konsep yang harus disintesis atau dikombinasikan dengan beberapa informasi lain, mungkin konsep lain atau beberapa data sensorik, untuk menghasilkan sesuatu yang benar-benar baru.


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  11. Primaningtyas Nur Arifah
    16709251042
    Pend. Matematika S2 kelas C 2016
    Assalamu’alaikum. Landasan teori epistemologis mencari landasan matematika kognitif yang kuat untuk melindunginya dari kesewenang-wenangan, ketidaktegasan, dan kerentanan terhadap keadaan historis serta untuk menjamin kepastian dan kebenaran pengetahuan matematika. Sebagai fondasi epistemologis dalam mencari dasar pengetahuan yang kokoh, dipengaruhi oleh setidaknya dua fondasi epistemologis yang berbeda, yaitu dari akar fondasionalisme Empiris dan fondasionalisme Rasionalis. Filsafat matematika memilikinya bertujuan untuk mengklarifikasi dan menjawab pertanyaan tentang status dan dasar objek dan metode matematika yang secara ontologis memperjelas apakah ada benda matematis, dan secara epistemologis menjelaskan apakah semua pernyataan matematika yang bermakna memiliki tujuan dan menentukan kebenaran.

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  12. Anwar Rifa’i
    PMAT C 2016 PPS
    16709251061

    Pada pembelajaran matematika, dibutuhkan suatu strukur yang tepat. Hal ini mengikuti sifat matematika yang tersusun begitu rapi dan terstruktur. Kemdian, Ide dasar dan penilaian yang bisa diragukan karena mereka terlibat dalam tindakan yang jelas dan berbeda; mereka diperlukan sebagai tempat utama dari mana penilaian lain deducible; dan mereka benar karena mereka terlibat dalam tindakan yang benar dan akurat. Akibatnya perencanaan pembelajaran matematika perlu dilakukan secara bersungguh-sungguh dan diperkirakan mampu untuk meningkatkan kemampuan siswa.

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  13. Lana Sugiarti
    16709251062
    PPs Pendidikan Matematika D 2016

    Dari paparan tersebut menjelaskan tentang filosofi matematika yang bertujuan untuk mengklarifikasi dan menjawab pertanyaan tentang status dan dasar objek dan metode matematika yang secara ontologis memperjelas apakah ada benda matematis, dan secara epistemologis menjelaskan apakah semua pernyataan matematika yang bermakna memiliki tujuan dan menentukan kebenaran. Beberapa teori modern dalam filsafat matematika menyangkal adanya fondasi dalam pengertian aslinya. Beberapa filsuf berfokus pada kognisi manusia sebagai asal mula reliabilitas matematika dan mengusulkan untuk menemukan dasar matematika hanya dalam pemikiran manusia, seperti yang Kant lakukan, tidak dalam tujuan di luar matematika.

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  14. Yosepha Patricia Wua Laja
    16709251080
    S2 Pendidikan Matematika D 2016

    Kant berpendapat bahwa penalaran matematis tidak dapat digunakan di luar ranah matematika yang tepat untuk penalaran semacam itu, karena ia memahaminya, harus diarahkan pada objek yang "ditentukan secara pasti dalam intuisi murni secara apriori dan tanpa data empiris" . Karena hanya objek matematis formal (yaitu besaran spasial dan temporal) dapat diberikan begitu saja, penalaran matematis tidak ada gunanya sehubungan dengan materi yang diberikan konten (walaupun kebenaran yang dihasilkan dari penalaran matematis tentang objek matematika formal dapat diterapkan dengan baik pada kandungan materi tersebut, yaitu Untuk mengatakan bahwa matematika benar-benar apriori dari penampilan.) Akibatnya, "landasan menyeluruh" yang ditemukan matematika dalam definisi, aksioma, dan demonstrasi tidak dapat "dicapai atau ditiru" oleh filsafat atau ilmu fisika.


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  15. Ahmad Wafa Nizami
    16709251065
    S2 Pendidikan Matematika D

    Jika matematika dianggap sebagai ilmu pengetahuan, maka filsafat matematika dapat dianggap sebagai cabang filsafat sains, disamping disiplin ilmu seperti filsafat fisika dan filsafat biologi. Namun, karena pokok bahasannya, filosofi matematika menempati tempat khusus dalam filsafat sains. Sedangkan ilmu alam menyelidiki entitas yang berada di ruang dalam waktu, sama sekali tidak jelas bahwa ini juga kasus objek yang dipelajari dalam matematika. Selain itu, metode investigasi matematika sangat berbeda dengan metode investigasi dalam ilmu pengetahuan alam. Sedangkan yang terakhir memperoleh pengetahuan umum dengan menggunakan metode induktif, pengetahuan matematika tampaknya diperoleh dengan cara yang berbeda: dengan deduksi dari prinsip dasar. Status pengetahuan matematika juga nampak berbeda dari status pengetahuan dalam ilmu alam. Teori-teori ilmu alam nampaknya kurang pasti dan lebih terbuka untuk revisi daripada teori matematika. Untuk alasan ini matematika menimbulkan masalah jenis yang cukup khas untuk filsafat. Oleh karena itu para filsuf telah memberikan perhatian khusus pada pertanyaan ontologis dan epistemologis mengenai matematika.( sumber: https://plato.stanford.edu/entries/philosophy-mathematics/ )

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  16. Annisa Hasanah
    16709251051
    PPs Pendidikan Matematika C 2016

    Teoripengetahuan Kant menegaskan bahwa dibanding asumsi bahwa ide-ide kita harus sesuai dengan realitas eksternal yang independen untuk menjadi kenyataandaripada mengetahui realitas kita, tujuan harus diketahui hanya sejauh itu sesuai dengan struktur penting dari pikiran mengetahui. Kant menyatakan bahwa objek pengalamana adalah fenomena, dapat diketahui, tapi itu hal berada di luar bidang pengalaman yang mungkin yaitu nomena atau hal-in-sendiri diketahui, meskipun keberadaan mereka adalah presuposisi diperlukan.

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  17. Nurwanti Adi Rahayu
    16709251067
    S2 Pendidikan Matematika Kelas D 2016

    Menurut Kant, intuisi, dengan macam dan jenisnya memegang peranan yang sangat penting untuk mengkonstruksi matematika
    Selain itu intusisi menyelidiki dan menjelaskan bagaimana matematika dipahami dalam bentuk geometri atau arithmetika.
    Sehingga intuisi berperan dalam pengetahuan lan landasan dalam matematika.

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  18. Ardeniyansah
    16709251053
    S2 Pend. Matematika Kelas C_2016

    Assalamualaikum wr. . wb.
    Landasan matematika merupakan bidang pengetahuan yang berkaitan dengan konsep dasar atau asas fundamental yang lebih prinsipil yang dipergunakan dalam matematika. Kemudian dengan prinsip dasar pada landasan matematika proses pengkajiannya akan sampai pada sifat alami matematika dan sampai juga pada metode matematika. Sedangkan meta-matematika yang dimaksudkan sebagai sebuah teori pembuktian untuk menetapkan ada atau tidaknya konsistensi dalam matematika dan menjawab permasalahan lainnya.

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  19. Desy Dwi Frimadani
    16709251050
    PPs Pendidikan Matematika Kelas C 2016

    Kant adalah sosok orang yang berpengaruh besar dalam filsafat modern. engakuan bahwa pengetahuan matematika adalah mungkin. Keistimewaan pemikiran matematika setelah Kant nampaknya berasal dari pembedaan model intuisi dan pemikiran Kant sebelumnya. Matematika epistemologis Kant adalah prinsip bahwa kesimpulan tidak ada celah ketika seseorang memahami sebuah arsitektur matematis di mana pembenaran matematis dari kesimpulan dipandang sebagai pengembangan pembenaran matematis dari premis.Untuk era saat ini diperlukan intuisi siswa yang tinggi untuk mencapai tujuan pembelajaran matematika. Oleh karena itu tugas gurulah untuk membangun intuisi dan mengasah intuisi yang ada pada diri siswa.

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  20. Syaifulloh Bakhri
    16709251049
    S2 Pendidikan Matematika C 2016

    Assalamu’alaikum wr.wb.
    Sintesis dari artikel tersebut sebagai berikut:
    1. Filsafat matematika bertujuan untuk mengklarifikasi dan menjawab pertanyaan tentang status dan dasar objek dan metode matematika baik secara ontologis maupun epistemologis.
    2. Filsafat Kant tentang matematika disusun sebagai perdebatan antara Rasionalis dan Empiris.
    3.Kant kontribusi paling signifikan terhadap filsafat modern adalah pengakuan bahwa pengetahuan matematika adalah mungkin.
    4.Kant percaya bahwa penilaian matematika dan fisika Newton menjadi contoh pengetahuan asli.
    5.Dalam istilah kecenderungan filosofi matematika saat ini, teori pengetahuan Kant sejalan dengan persepsi bahwa pemahaman matematika dapat didukung oleh sifat fakultas persepsi.

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  21. Windi Agustiar Basuki
    16709251055
    S2 Pend. Mat Kelas C – 2016

    Dalam Teori Pengetahuannya, Immanuel Kant berusaha meletakkan dasar epistemologis bagi matematika untuk menjamin bahwa matematika memang benar dapat dipandang sebagai ilmu.Kant menyatakan bahwa metode yang benar untuk memperoleh kebenaran matematika adalah memperlakukan matematika sebagai pengetahuan a priori. Menurut Kant, secara spesifik, validitas obyektif dari pengetahuan matematika diperoleh melalui bentuk a priori dari sensibilitas kita yang memungkinkan diperolehnya pengalaman inderawi.

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  22. Resvita Febrima
    16709251076
    P-Mat D 2016
    Matematika telah berkembang menjadi kegiatan abstraksi yang lebih tinggi di atas kejelasan pondasinya seperti yang terjadi pada Kalkulus Infinitas dan Bilangan Kompleks yang telah mengambil jarak dari pandangan kaum skeptik. Tetapi pada abad yang lalu, dengan ditemukannya kontradiksi pada Teori Himpunan, kaum skeptik dan empiric mulai menggaungkan lagi pandangan-pandangan tentang pondasi matematika. Munculnya Teori Pengetahuan dari Immanuel Kant, sebagai landasan epistemologis dari pengetahuan, dipengaruhi paling tidak oleh pengaruh dua aliran epistemologi yang masing-masing berakar pada pondasi empiris dan pondasi rasionalis. Menurut kaum pondasionalis empiris, terdapat unsur dasar pengetahuan dalam mana nilai kebenarannya lebih dihasilkan oleh hukum sebab-akibat dari pada dihasilkan oleh argumen-argumennya; mereka percaya bahwa keberadaan dari kebenaran tersebut disebabkan oleh asumsi bahwa obyek dari pernyataannyalah yang membawa nilai kebenaran itu. 

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  23. Kunny Kunhertanti
    16709251060
    PPs Pendidikan Matematika kelas C 2016

    Immanuel Kant memiliki peran yang sangat besar dalam filsafat modern. Dasar matematika adalah studi tentang dasar logis dan filosofis matematika , atau, dalam arti luas, penyelidikan matematika yang mendasari teori filosofis tentang sifat matematika. Perbedaan antara dasar matematika dan filsafat matematika ternyata cukup jelas. Semua hal tersebut didasarkan intuisi manusia dalam berolah pikir.

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  24. Heni Lilia Dewi
    16709251054
    PPs Pendidikan Matematika Kelas C 2016

    Salah satu filsuf yang memiliki andil yang besar dalam filsafat pendidikan matematika adalah Immanuel Kant. Dari artikel ini diperoleh pemikiran tentang teori pengetahuan dan pondasi matematika (secara epistemologi). Yang pertama bahwa ada beberapa pandangan dalam filsafat pendidikan matematika yaitu Platonism, Logicism, Formalism, and Intuitionism. Yang kedua adalah pentingnya intuisi dalam mencapai pengetahuan. Dan yang berikutnya adalah pandangan Kant bahwa pengetahuan itu diperoleh melalui synthetical a priori (sintetik a priori).

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  25. Muh Ferry Irwansyah
    15709251062
    Pendidikan Matematika PPS UNY
    Kelas D
    Fondasionalisme epistemologis berusaha sebagai dasar kognisi matematika untuk melindungi dari kesewenang-wenangan, non-conclusiveness, dan kerentanan terhadap situasi sejarah serta menjamin kepastian dan kebenaran pengetahuan matematika. Teori pengetahuan Kant sebagai dasar epistemologis dalam mencari dasar pengetahuan, dipengaruhi oleh setidaknya dua bagian epistemologis yang berbeda yaitu dari fondasionalisme empiris dan fondasionalisme Rasionalis. Versi empiris fondasionalisme terdiri dari dua asumsi yaitu adanya klaim yang benar yang, jika diketahui, akan memungkinkan kita untuk mendapatkan semua pengetahuan tentang arsitektur keberadaan dan yang kedua yaitu klaim tersebut diberikan. Untuk menemukan konsep dan penilaian yang merupakan dasar untuk pengetahuan kita, fondasionalisme rasionalis menemukan sumber tindakan kognitif, yaitu suatu tindakan di mana kita menemukan ide-ide dan kebenaran mendasar.

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  26. Lihar Raudina Izzati
    16709251046
    P. Mat C 2016 PPs UNY

    Kant memulai filosofi matematika dengan fokus pada pengetahuan matematika dan hubungan epistemik mereka untuk teorema dan bukti yaitu epistemologi matematika. Peran teori Kant dalam menyiapkan pengetahuan dasar epistemologis matematika muncul dari upaya Kant untuk mengatur epistemologis dasar matematika yang didasarkan pada prinsip-prinsip sintetis apriori dimana ia percaya bahwa penilaian matematika adalah contoh asli pengetahuan. Kontribusi Kant yang paling signifikan untuk filsafat modern adalah pengakuan bahwa pengetahuan matematika adalah mungkin. Munculnya teori pengetahuan dari Kant, sebagai landasan epistemologis dari pengetahuan, dipengaruhi paling tidak oleh dua aliran epistemologi yang masing-masing berakar pada pondasi empiris dan pondasi rasionalis. Menurut kaum pondasi empiris, terdapat unsur dasar pengetahuan dimana nilai kebenarannya lebih dihasilkan oleh hukum sebab-akibat dari pada dihasilkan oleh argumen-argumennya. Sedangkan kaum pondasionalis rasionalis berusaha mencari sumber dari kegiatan berpikir, yaitu kegiatan dimana kita dapat menemukan ide dasar dan kebenaran. Kant membangun teori pengetahuan baru untuk menjembatani perbedaan pandangan antara rasionalisme dan empirisme yang dikenal dengan sintetik apriori. Relevansi teori pengetahuan Kant dengan pondasi matematika saat ini adalah dalam mempelajari matematika kita haruslah bisa menggabungkan antara aktivitas rasio dan empiris sehingga kebenaran akan lebih valid.

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  27. Ahmad Bahauddin
    16709251058
    PPs P.Mat C 2016

    Assalamualaikum warohmatullahi wabarokatuh.
    Epistemologi, studi tentang teori pengetahuan, adalah salah satu bidang filsafat yang paling penting. "Intuitionists" percaya bahwa matematika hanyalah ciptaan pikiran manusia. Dalam hal ini Anda bisa berpendapat bahwa matematika ditemukan oleh manusia. Setiap objek matematika hanya ada dalam pikiran kita dan tidak memiliki eksistensi. "Platonis", di sisi lain, berpendapat bahwa ada benda matematis dan kita hanya bisa "melihat" mereka melalui pikiran kita. Oleh karena itu, dalam beberapa hal, para platonis akan memilih bahwa matematika telah ditemukan.

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  28. Wahyu Berti Rahmantiwi
    PPs Pendidikan Matematika Kelas C 2016
    16709251045

    Objek materi merupakan landasan epistemologi matematika sedangkan objek formal ialah gagasan pengetahuan. Filsafat matematika bertujuan untuk mengetahui status dan dasar dari objek dan metode matematika secara ontologis untuk menyatakan apakah pernyataan matematis mempunyai tujuan dan kebenaran. Objek dari pengalaman menurut Kant adalah fenomena sedangkan yang berada di luar pengalaman adalah noumena. Fenomena merupakan segala sesuatu yang dapat dirasakan oleh manusia berdasarkan indrawi manusia, ruang dan waktu sesuai dengan pemahaman sendiri-sendiri sedangkan noumena merupakan benda-benda yang tidak diketahui keberadaannya tetapi merupakan prasyarat diperlukan untuk membentuk pengalaman.

    ReplyDelete