Oleh: Marsigit
Orang Tua Berambut Putih Duduk Sendiri:
Aku duduk di sini. Maka aku yang telah mengaku sebagai orang tua berambut putih serta merta aku menyadari bahwa ketika aku duduk di sini, aku menjumpai ada aku yang duduk di sini, aku merasakan ada duduk dan aku mengerti ada di sini. Tetapi serta merta pula jika yang demikian aku renungkan maka aku menyadari ada diri yang bukan aku, ada diri yang sedang tidak duduk dan ada diri yang di sana. Kalau aku perjelas refleksiku itu maka aku menemukan bahwa aku yang duduk di sini menyebabkan ada yang bukan aku yang tidak duduk di sini maupun yang duduk di sana. Aku terkejut karena aku menemukan bahwa ada aku yang tidak duduk di sini, dan ada diri yang bukan aku duduk di sini. Padahal aku telah mengaku bahwa aku duduk di sisini. Maka aku bertanya siapakah aku yang tidak duduk di sini, dan siapakah diri bukan aku yang duduk di sini. Pertanyaan itu ternyata dapat saya lanjutkan, siapakah aku yang duduk di sini dan siapakah aku yang tidak duduk di sini? Saya juga bisa bertanya siapakah aku yang duduk di sini dan siapakah bukan aku yang duduk di sini? Ternyata pertanyaan itu menyebabkan pertanyaanku yang berikutnya. Kalau begitu apakah ada aku yang bukan aku, duduk yang tidak duduk, di sini tetapi tidak di sini?. Aku menjadi teringat dengan pepatah jawa “ngono ning ojo ngono”. Apakah ada ngono yang bukan ngono, dan ojo ning bukan ojo? Kalau orang jawa saja sejak nenek moyang sudah mempunyai ngono ning ojo ngono, maka enehkah jika sekarang aku bertanya aku tetapi bukan aku, duduk tetapi bukan duduk, dan di sini tetapi bukan di sini? Kalau ini masih dianggap aneh maka sesungguhnyalah kita telah kehilangan dan tidak mampu memahamiwarisan leluhur.
Orang Tua Berambut Putih (pertama) Bertemu Dengan Orang Tua Berambut Putih Yang Lain (kedua):
Orang Tua Berambut Putih kedua:
Salam. Sesungguhnyalah aku mengikuti segala yang engkau pikirkan. Akulah mungkin aku yang bukan dirimu. Kemudian engkau mungkin bertanya apakah aku itulah yang tidak duduk di sini, atau apakah akulah yang duduk di sana, atau apakah akulah yang tidak duduk di sana? Tetapi setidaknya engkau telah berkata bahwa aku duduk di sini. Maka aku pun kemudian bertanya siapakah yang engkau maksud sebagai aku yang duduk di sini? Siapakah yang engkau maksud sebagai bukan aku yang duduk di sini? Siapakah yang engkau maksud aku yang tidak duduk di sini? Siapakah yang engkau maksud aku yang duduk di sana? Siapakah yang engkau maksud aku yang tidak duduk di sana? Siapakah yang engkau maksud bukan aku yang tidak duduk di sana?
Orang Tua berambut Putih Pertama:
Salam kembali. Sesungguhnya pula aku juga menyadari bahwa engkau yang bukan aku telah mengetahui pikiranku sedari awal. Namun ketahuilah bahwa sebenar-benar yang terjadi bahwa aku juga mengerti tentang pikiranmu sedari awal seawal-awalnya. Maka aku juga ingin bertanya mengapa engkau bertanya tentang aku yang duduk di sini? Padahal engkau tahu bahwa aku telah duduk di sini dan engkau yang bukan aku juga duduk di sini. Jadi sebetulnya siapakah engkau yang duduk di sini? Apakah engkau yang tidak duduk di sini? Apakah engkau juga duduk tidak di sini? Apakah engkau juga tidak duduk tidak di sini?
Orang Tua Berambut Putih Kedua:
Sesungguh-sungguhnya aku telah mengerti bahwa engkau akan mengajukan pertanyaan seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti jawabanmu.
Orang Tua Berambut Putih Pertama:
Sesungguh-sungguhnya aku juga telah mengerti bahwa engkau akan memberi komentar seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti komentarmu.
Orang Tua Berambut Putih Kedua:
Sesungguh-sungguhnya aku juga telah mengerti bahwa engkau akan memberi komentar seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti komentarmu.
Orang Tua Berambut Putih Pertama:
Mengapa engkau selalu menirukanku?
Orang Tua Berambut Putih Kedua:
Mengapa engka selalu menirukanku?
Orang Tua Berambut Putih Pertama:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Kedua:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Ketiga Menjumpai Ada Dua Orang Berambut Putih Berkelahi:
Wahai para orang tua. Mengapa sesama orang tua seperti engkau berdua saling berhantam? Apa kau pikir yang ada di sini cuma engkau berdua? Sebenar-benar yang terjadi adalah bahwa aku telah mengerti pikiranmu berdua sedari awal. Apakah engkau berdua tidak mengerti bahwa sedari awal aku telah duduk di sini bersamamu? Maka apalah gunanya mengapa engkau saling bertengkar? Bukankah saling bertukar pikiran itu lebih baik dari pada berkelahi.
Orang Tua Berambut Putih Pertama dan Kedua Secara bersama-sama menjawab:
Sesungguhnya pula aku juga menyadari bahwa engkau yang bukan aku telah mengetahui pikiranku berdua sedari awal. Namun ketahuilah bahwa sebenar-benar yang terjadi bahwa aku berdua juga mengerti tentang pikiranmu sedari awal seawal-awalnya. Maka aku berdua juga ingin bertanya mengapa engkau bertanya tentang aku berdua yang duduk di sini? Padahal engkau tahu bahwa aku berdua telah duduk di sini dan engkau yang bukan aku berdua juga duduk di sini. Jadi sebetulnya siapakah engkau yang duduk di sini? Apakah engkau juga tidak duduk di sini? Apakah engkau juga duduk tidak di sini? Apakah engkau juga tidak duduk tidak di sini?
Orang Tua Berambut Putih Ketiga:
Sesungguh-sungguhnya aku telah mengerti bahwa engkau berdua akan mengajukan pertanyaan seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti jawabanmu berdua.
Orang Tua Berambut Putih Pertama dan Kedua Secara bersama-sama menjawab:
Sesungguh-sungguhnya aku berdua juga telah mengerti bahwa engkau akan memberi komentar seperti itu, tetapi aku berdua ragu apakah aku berdua mengerti atau tidak mengerti komentarmu.
Orang Tua Berambut Putih Ketiga:
Sesungguh-sungguhnya aku juga telah mengerti bahwa engkau berdua akan memberi komentar seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti komentarmu berdua.
Orang Tua Berambut Putih Pertama dan Kedua Secara bersama-sama menjawab:
Mengapa engkau selalu menirukanku?
Orang Tua Berambut Putih Ketiga:
Mengapa engkau selalu menirukanku?
Orang Tua Berambut Putih Pertama dan Kedua Secara bersama-sama menjawab:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Ketiga:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Keempat Menjumpai Ada Tiga Orang Berambut Putih Berkelai:
Wahai para orang tua. Mengapa sesama orang tua seperti engkau bertiga saling berhantam? Apa kau pikir yang ada di sini cuma engkau bertiga? Sebenar-benar yang terjadi adalah bahwa aku telah mengerti pikiranmu bertiga sedari awal. Apakah engkau bertiga tidak mengerti bahwa sedari awal aku telah duduk di sini bersamamu? Maka apalah gunanya mengapa engkau saling bertengkar? Bukankah saling bertukar pikiran itu lebih baik dari pada berkelahi.
Orang Tua Berambut Putih Pertama, Kedua dan Ketiga Secara bersama-sama menjawab:
Sesungguhnya pula aku juga menyadari bahwa engkau yang bukan aku telah mengetahui pikiranku berdua sedari awal. Namun ketahuilah bahwa sebenar-benar yang terjadi bahwa aku bertiga juga mengerti tentang pikiranmu sedari awal seawal-awalnya. Maka aku bertiga juga ingin bertanya mengapa engkau bertanya tentang aku bertiga yang duduk di sini? Padahal engkau tahu bahwa aku bertiga telah duduk di sini dan engkau yang bukan aku bertiga juga duduk di sini. Jadi sebetulnya siapakah engkau yang duduk di sini? Apakah engkau juga tidak duduk di sini? Apakah engkau juga duduk tidak di sini? Apakah engkau juga tidak duduk tidak di sini?
Orang Tua Berambut Putih Keempat:
Sesungguh-sungguhnya aku telah mengerti bahwa engkau bertiga akan mengajukan pertanyaan seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti jawabanmu bertiga.
Orang Tua Berambut Putih Pertama, Kedua dan Ketiga Secara bersama-sama menjawab:
Sesungguh-sungguhnya aku bertiga juga telah mengerti bahwa engkau akan memberi komentar seperti itu, tetapi aku bertiga ragu apakah aku bertiga mengerti atau tidak mengerti komentarmu.
Orang Tua Berambut Putih Keempat:
Sesungguh-sungguhnya aku juga telah mengerti bahwa engkau bertiga akan memberi komentar seperti itu, tetapi aku ragu apakah aku mengerti atau tidak mengerti komentarmu bertiga.
Orang Tua Berambut Putih Pertama, Kedua dan Ketiga Secara bersama-sama menjawab:
Mengapa engkau selalu menirukanku?
Orang Tua Berambut Putih Keempat:
Mengapa engkau selalu menirukanku?
Orang Tua Berambut Putih Pertama, Kedua dan Ketiga Secara bersama-sama menjawab:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Keempat:
Kalau begitu apa maumu?
Orang Tua Berambut Putih Kelima Menjumpai Ada Empat Orang Berambut Putih Berkelahi:
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Orang Tua Berambut Putih Keseribu Menjumpai Ada Sembilan Ratus Sembilan Puluh Sembilan Orang Tua Berambut Putih Berkelai.
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Orang Tua Berambut Putih Ke-n Menjumpai Ada n-1 Orang Berambut Putih Berkelahi:
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Ada diri yang bukan mereka menjumpai mereka berkelahi:
Aku sendiri tidak tahu siapakah diriku. Apakah diriku itu diriku atau bukan diriku. Aku juga tidak tahu apakah diriku adalah satu dari mereka? Aku juga tidak tahu apakah aku tahu pikiran mereka sedari awal? Aku juga tidak tahu apakah aku duduk? Aku juga tidak tahu apakah aku di sini atau tidak di sini? Aku tidak tahu apakah mereka mengetahui pikiranku sedari awal. Aku tidak tahu apakah mereka tahu atau tidak tahu aku di sini atau tidak di sini, aku duduk atau aku tidak duduk? Aku tidak tahu apakah yang akan mereka tanyakan? Aku tidak tahu apakah yang mereka akan komentari. Aku tidak tahu apakah mereka akan selalu menirukanku? Aku tidak tahu apakah aku akan selalu menirukannya. Aku tidak tahu apakah aku juga akan berkelahi atau tidak akan berkelahi dengan mereka? Aku tidak tahu apakah jika aku berkelahi dengan mereka maka akan ada diri yang bukan diriku dan bukan diri mereka yang juga mengetahui pikiranku dan pikiran mereka sedari awal? Itulah sebenar-benar bahwa aku tidaklah mengatahui siapakah diriku itu. Tetapi aku mengetahui bahwa setidaknya aku bisa mengucapkan kalimatku yang terakhir. Maka akupun tidak tahu apakah itu batas ku yang ingin ku gapai. Artinya aku tidak bisa menjawab apakah aku bisa menggapai batasku.
Saturday, January 31, 2009
Friday, January 2, 2009
The Metaphysic of Science
To be reviewed from many sources by Marsigit
Steven Kreis , 2001 in his Lectures on Modern European Intellectual History elaborated Giambattista Vico position in his “The New Science (1725)” some notions of the ontology and or metaphysic of science. Accordingly, Vico, G (1725) stated that Science or metaphysic, studying the common nature of nations in the light of divine providence, discovers the origins of divine and human institutions among the gentile nations, and thereby establishes a system of he natural law of the gentes, which proceeds with the greatest equality and constancy through the three ages which the Egyptians handed down to us as the three periods through which the world has passed up to their time; these are (1) The age of the gods, in which the gentiles believed they lived under divine governments, and everything was commanded them by auspices and oracles, which are the oldest institutions in profane history. (2) The age of the heroes, in which they reigned everywhere in aristocratic commonwealths, on account of a certain superiority of nature which they held themselves to have over the plebs. (3) The age of men, in which all men recognized themselves as equal in human nature, and therefore there were established first the popular commonwealths and then the monarchies, both of which are forms of human government.
Next, Vico (1725), as cited by Kreis (2001) indicated that peoples who have reached the point of premeditated malice, when they receive this last remedy of providence and are thereby stunned and brutalized, are sensible no longer of comforts, delicacies, pleasures, and pomp, but only of the sheer necessities of life. And the few survivors in the midst of an abundance of the things necessary for life naturally become sociable and, returning to the primitive simplicity of the first world of peoples, are again religious, truthful, and faithful; thus providence brings back among them the piety, faith, and truth which are the natural foundations of justice as well as the graces and beauties of the eternal order of God.
Meanwhile, Thales (1999) in SAINTS argued that there is no reason why modern religion shouldn't incorporate the latest discoveries of Psychical Research or Metaphysics; from the time of Aristotle (300 BCE) to the time of Galileo (1600 CE), nearly 2000 years, the worldview, the background of all thought, was that of Aristotle and Ptolemy. It made a large distinction between the heavens (i.e. stars, the moon, the sun, and the planets) and earth. Earth was made of four elements, earth, air, fire, and water, and was mutable and perishable. The heavenly bodies were made of a fifth element (quintessence) which was immutable, imperishable and eternal. Thus, the correct translation of this metaphor is "realm of the imperishable," or "realm of the quintessence."
Thales said that it is not unusual for religions to begin with the mystical teachings of the founder to a small circle of disciples; as the religion develops it is not unusual for it to absorb elements from other religions over the centuries (syncretism) and to incorporate fantastic fairy tales, which may incorporate some symbolic truth (mythology). According to him, the religions who gave up worldly concerns and went off into the desert as seekers of the illumination of fire often succeeded, and when they returned to the world (or when the world came to them), they were not only holy and wise, but they also had "miraculous" powers, such as healing, or walking on water. The miracles of one age are the science of the next. The age of faith passes, and the age of spiritual science begins.
Bryan Appleyard, 1992, in “Understanding the Present: Science and the Soul of Modern Man” clarified that Western science is not simply a neutral method of acquiring knowledge but that it is ‘a metaphysic like any other.’; the foundations of this metaphysic were laid by Galileo, for his discovery was that one of the most effective ways of understanding the world ‘is to pretend that we do not exist.’ He, further indicated that it is the history of science in which he traces the development of physics from Plato and Aristotle through Thomas Aquinas to Galileo, Descartes and Newton and their modern descendants; modern science gradually emerges not as the embodiment of reason but as a form of worldly mysticism whose zeal for accumulating knowledge about the inanimate and the non-human, and whose ‘rational’ commitment to technological power and material wealth has almost completely obscured its radical anti-humanism.
However, Appleyard, B., (1992), pointed out that the contradictions between science and religion are absolutely and irresolvable conflict; he, then stated that the most obvious problem here is that Islam developed directly out of the Judaeo-Christian tradition and shares much of its world-view with Judaism – whose prophets Muslims revere. On the other hand, according to him, at the same time modern science was the almost exclusive creation of zealous Christians who were seeking not to escape their faith but to confirm and magnify it. Descartes, Newton and Robert Boyle, to name but three representative figures, all believed they had triumphantly succeeded through their science in bearing witness to the majesty and rationality of God.
Appleyard, B., (1992), explained that one reaction to the failure to escape is for us all to throw up our hands and loudly proclaim our belief in the reality and complexity of the human soul in the hope that by doing so we can triumph over science; while, the other reaction is to think more carefully, more sensitively and more systematically about the very aspects of human reality which science has traditionally neglected. He, then concluded that only if we do this is it possible that our intellectual culture may yet triumph over its own history, and over the spiritual extremism which shaped modern rationalism and bequeathed to us a contempt for the ‘human element’ whose religious origins we too readily forget.
REFERENCE
Appleyard, B., 1992, “Understanding The Present: Science And The Soul Of Modern Man”: Picador
Giambattista Vico in Kreis, S., 2001, “The New Science: Lectures On Modern European Intellectual History”, The History Guide
Iranzo, V., 1995, “ Epistemic Values In Science” : Sorites
Katz, M, 2004, “Value Science Can Change The World (And Be Changed By It)” : Cristina Lafont
Meer, J.M.V.D., 1995, “The Struggle Between Christian Theism,
Metaphysical Naturalism And Relativism: How To Proceed In Science?”, Ontarion: Pascal Centre, Redeemer College
Wikipedia, The Free Encyclopedia.,
Wilson, F.L., 1999, “Plato, Science And Human Values”, Rochester Institute Of Technology: Physics Teacher.Org
Steven Kreis , 2001 in his Lectures on Modern European Intellectual History elaborated Giambattista Vico position in his “The New Science (1725)” some notions of the ontology and or metaphysic of science. Accordingly, Vico, G (1725) stated that Science or metaphysic, studying the common nature of nations in the light of divine providence, discovers the origins of divine and human institutions among the gentile nations, and thereby establishes a system of he natural law of the gentes, which proceeds with the greatest equality and constancy through the three ages which the Egyptians handed down to us as the three periods through which the world has passed up to their time; these are (1) The age of the gods, in which the gentiles believed they lived under divine governments, and everything was commanded them by auspices and oracles, which are the oldest institutions in profane history. (2) The age of the heroes, in which they reigned everywhere in aristocratic commonwealths, on account of a certain superiority of nature which they held themselves to have over the plebs. (3) The age of men, in which all men recognized themselves as equal in human nature, and therefore there were established first the popular commonwealths and then the monarchies, both of which are forms of human government.
Next, Vico (1725), as cited by Kreis (2001) indicated that peoples who have reached the point of premeditated malice, when they receive this last remedy of providence and are thereby stunned and brutalized, are sensible no longer of comforts, delicacies, pleasures, and pomp, but only of the sheer necessities of life. And the few survivors in the midst of an abundance of the things necessary for life naturally become sociable and, returning to the primitive simplicity of the first world of peoples, are again religious, truthful, and faithful; thus providence brings back among them the piety, faith, and truth which are the natural foundations of justice as well as the graces and beauties of the eternal order of God.
Meanwhile, Thales (1999) in SAINTS argued that there is no reason why modern religion shouldn't incorporate the latest discoveries of Psychical Research or Metaphysics; from the time of Aristotle (300 BCE) to the time of Galileo (1600 CE), nearly 2000 years, the worldview, the background of all thought, was that of Aristotle and Ptolemy. It made a large distinction between the heavens (i.e. stars, the moon, the sun, and the planets) and earth. Earth was made of four elements, earth, air, fire, and water, and was mutable and perishable. The heavenly bodies were made of a fifth element (quintessence) which was immutable, imperishable and eternal. Thus, the correct translation of this metaphor is "realm of the imperishable," or "realm of the quintessence."
Thales said that it is not unusual for religions to begin with the mystical teachings of the founder to a small circle of disciples; as the religion develops it is not unusual for it to absorb elements from other religions over the centuries (syncretism) and to incorporate fantastic fairy tales, which may incorporate some symbolic truth (mythology). According to him, the religions who gave up worldly concerns and went off into the desert as seekers of the illumination of fire often succeeded, and when they returned to the world (or when the world came to them), they were not only holy and wise, but they also had "miraculous" powers, such as healing, or walking on water. The miracles of one age are the science of the next. The age of faith passes, and the age of spiritual science begins.
Bryan Appleyard, 1992, in “Understanding the Present: Science and the Soul of Modern Man” clarified that Western science is not simply a neutral method of acquiring knowledge but that it is ‘a metaphysic like any other.’; the foundations of this metaphysic were laid by Galileo, for his discovery was that one of the most effective ways of understanding the world ‘is to pretend that we do not exist.’ He, further indicated that it is the history of science in which he traces the development of physics from Plato and Aristotle through Thomas Aquinas to Galileo, Descartes and Newton and their modern descendants; modern science gradually emerges not as the embodiment of reason but as a form of worldly mysticism whose zeal for accumulating knowledge about the inanimate and the non-human, and whose ‘rational’ commitment to technological power and material wealth has almost completely obscured its radical anti-humanism.
However, Appleyard, B., (1992), pointed out that the contradictions between science and religion are absolutely and irresolvable conflict; he, then stated that the most obvious problem here is that Islam developed directly out of the Judaeo-Christian tradition and shares much of its world-view with Judaism – whose prophets Muslims revere. On the other hand, according to him, at the same time modern science was the almost exclusive creation of zealous Christians who were seeking not to escape their faith but to confirm and magnify it. Descartes, Newton and Robert Boyle, to name but three representative figures, all believed they had triumphantly succeeded through their science in bearing witness to the majesty and rationality of God.
Appleyard, B., (1992), explained that one reaction to the failure to escape is for us all to throw up our hands and loudly proclaim our belief in the reality and complexity of the human soul in the hope that by doing so we can triumph over science; while, the other reaction is to think more carefully, more sensitively and more systematically about the very aspects of human reality which science has traditionally neglected. He, then concluded that only if we do this is it possible that our intellectual culture may yet triumph over its own history, and over the spiritual extremism which shaped modern rationalism and bequeathed to us a contempt for the ‘human element’ whose religious origins we too readily forget.
REFERENCE
Appleyard, B., 1992, “Understanding The Present: Science And The Soul Of Modern Man”: Picador
Giambattista Vico in Kreis, S., 2001, “The New Science: Lectures On Modern European Intellectual History”, The History Guide
Iranzo, V., 1995, “ Epistemic Values In Science” : Sorites
Katz, M, 2004, “Value Science Can Change The World (And Be Changed By It)” : Cristina Lafont
Meer, J.M.V.D., 1995, “The Struggle Between Christian Theism,
Metaphysical Naturalism And Relativism: How To Proceed In Science?”, Ontarion: Pascal Centre, Redeemer College
Wikipedia, The Free Encyclopedia.,
Wilson, F.L., 1999, “Plato, Science And Human Values”, Rochester Institute Of Technology: Physics Teacher.Org
Ritual Mathematics
By Marsigit
In Javanese society all aspect of daily life or activities can be viewed as underpine from religious perspectives. It is not surprising that, for example, in a certain village, the people in one week get invitation to come to ritual activities for more than seven. Some of the important activities are the celebration of the 'fifth' and the 'thirty fifth' baby birthday, the ritual feast to mark some one's death in the 'seventh day', 'forthyth day', 'a hundreth day' and 'a thousanth day'. The problem is the people, specifically the man who are responsible of carrying out the ritual meal, should exactly decide that the death has long been whether seven days, forthy days, one hundreth days, or one thousand days. They do it well and they learn it for generations. The people, who are concerned about it, sometimes involves in the dialog informally to justify whether the counting the numbers of tha days is right or wrong. Most of them are relatively correct. They just use simple formula which is ussualy spoken and not ever written, e.g. 'nomosarmo', 'norosarmo', nonemsarmo', etc.
They use three kinds of numbers system at the same time : 10 based numbers system, 7 based numbers system and 5 based numbers system, in the frame work of position system. Position system for numbers was found by Indian and Javanese people knew it before the Europen because it directly was brought by Indian to Indonesia. They use Numbers System Basis 10 when they should decide the duration of time; 35 days, 40 days, 100 days and 1000 days. They use Numbers System Basis 7 (Week system) when they use the name orderly of the days : Sunday, Monday, Tuesday, Wednesday, Thuersday, Friday and Saturday. They use Numbers System Basis 5 (Dino Pasaran system) when Javanese people have been using system 'Dino Pasaran' for along time before Western system of callendar came to Indonesia. In this system there are only five days orderly in a cycle of period of time called 'Pasar'. Those are 'Legi', 'Pahing', 'Pon', 'Wage', 'Kliwon'. One day in this system is equals to one day in Week system, that is 24 hours. Thus one Dino Pasaran has five days, two Dino Pasaran has 10 days, three Dino Pasaran has 15 days, etc.
Mathematically, for all of the system numbers, the notation of numbers 'm' can be written as polynomial from 'b' such as follows : m = a0 bn-1 + a1 bn-2 + a2 bn-3 + ... + an-1 b + an
where, b is any numbers greater than 1, and a is the basis. According to this formula, we can write any number at any system using the same pattern i.e. for basis 10, basis 7 and basis 5. In Basis 10 we have the numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In Basis 7 we have the numbers : 0, 1, 2, 3, 4, 5, 6
We can match the name of the day with those numbers orderly : 0 = Sunday 1 = Monday; 2 = Tuesday; 3 = Wednesday; 4 = Thuersday; 5 = Friday and 6 = Saturday. In Basis 5 we have the numbers : 0, 1, 2, 3, 4. We can match the name of the day in Dino Pasaran with those numbers orderly : 0 = Legi, 1 = Pahing, 2 = Pon, 3 = Wage, 4 = Kliwon. How can we write 35 of Basis 10 into Basis 7 ? The following is the formula 3510 = 5 x 71 + 0 x 70 = 507. How can we write 100 of Basis 10 into Basis 7 ? The formula is 10010 = 2 x 72 + 0 x 71 + 2 x 70 = 2027. How can we write 1000 of Basis 10 into Basis 7 ? The formula is 100010 = 2 x 73 + 6 x 72 + 2 x 71 + 6 x 70 = 26267
How can we write 35 of Basis 10 into Basis 5? The following is the formula 3510 = 7 x 51 + 0 x 50 = 705. How can we write 100 of Basis 10 into Basis 5? Thai is 10010 = 4 x 52 + 0 x 51 + 0 x 50 = 4005 How can we write 1000 of Basis 10 into Basis 5 ? That is 100010 = 1 x 54 + 3 x 53 + 0 x 52 + 0 x 51 + 0 x 50 = 130005
Javanese people always have two unseparated name for the day. For example : Sunda Legi, Sunday Kliwon, Monday Wage, Friday Pahing, Saturday Kliwon, Saturday Pon, etc. It is clear that the numbers of combinations is 35 names. If this day is Monday Pahing then the duration up to on the next Monday Pahing is 7 x 5 = 35 days. Thus when somebody wish to celebrate the 35 th of his son's birthday he just waiting for the next day with the same name. It is easy and he need not to do with mathematics at all in his mind. Forthyth days ritual feast to mark some one's death by transforming 4010 = 557 and 4010 = 805. Numbers 55 is ended by 5; it mean that 'the fourty days duration of time' will begin on the day i and ended on the day 5 th of Week System. Thus, if now is Sunday then forty days to come will be the fifth day from Sunday that is Thuersday. Javanese people called 'five' as 'limo' or briefly 'mo'. Number 80 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then it will be 40 days on the next Thuersday Kliwon.
If this day is Friday Legi then it will be 40 days on the next Tuesday Legi. If this day is Wednesday Pahing then it will be 40 day on the next Sunday Pahing, etc. Javanese people just called 'no mo; sar mo' that mean : no = dino = day; mo = limo = five; and sar = Pasar = Basis 5. Thus, no mo means 'the fifth day of Week System' sar mo means 'the fifth day of Pasar System'
For one hundred days ritual feast to mark some one's death as I described that :10010 = 2027 and 10010 = 4005. Number 202 is ended by 2; its mean that 'the one hundred days duration of time' will begin on the day i and ended on the day 2 th of Week System . Thus, if now is Sunday then forty days to come will be the 2 nd day from Sunday that is Monday. Javanese people called 'two' as 'loro' or briefly 'ro'. Number 400 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then the next 100 days will be on the next Monday Kliwon. If this day is Friday Legi then the next 100 days will be on Saturday Legi. If this day is Wednesday Pahing then the next 100 day will be on Sunday Pahing, etc. Javanese people just called 'no ro; sar mo' that mean : no = dino = day; ro = loro = dua; and sar = Pasar = Basis 5. Thus, no ro means 'the second day of Week System' sar mo means 'the fifth day of Pasar System'. One thousand days ritual feast to mark some one's death is calculated by the same way they use the formula 'no nem; sar mo' that mean 'the sixth day of Week Syatem and the fifth day of Pasar System'
In Javanese society all aspect of daily life or activities can be viewed as underpine from religious perspectives. It is not surprising that, for example, in a certain village, the people in one week get invitation to come to ritual activities for more than seven. Some of the important activities are the celebration of the 'fifth' and the 'thirty fifth' baby birthday, the ritual feast to mark some one's death in the 'seventh day', 'forthyth day', 'a hundreth day' and 'a thousanth day'. The problem is the people, specifically the man who are responsible of carrying out the ritual meal, should exactly decide that the death has long been whether seven days, forthy days, one hundreth days, or one thousand days. They do it well and they learn it for generations. The people, who are concerned about it, sometimes involves in the dialog informally to justify whether the counting the numbers of tha days is right or wrong. Most of them are relatively correct. They just use simple formula which is ussualy spoken and not ever written, e.g. 'nomosarmo', 'norosarmo', nonemsarmo', etc.
They use three kinds of numbers system at the same time : 10 based numbers system, 7 based numbers system and 5 based numbers system, in the frame work of position system. Position system for numbers was found by Indian and Javanese people knew it before the Europen because it directly was brought by Indian to Indonesia. They use Numbers System Basis 10 when they should decide the duration of time; 35 days, 40 days, 100 days and 1000 days. They use Numbers System Basis 7 (Week system) when they use the name orderly of the days : Sunday, Monday, Tuesday, Wednesday, Thuersday, Friday and Saturday. They use Numbers System Basis 5 (Dino Pasaran system) when Javanese people have been using system 'Dino Pasaran' for along time before Western system of callendar came to Indonesia. In this system there are only five days orderly in a cycle of period of time called 'Pasar'. Those are 'Legi', 'Pahing', 'Pon', 'Wage', 'Kliwon'. One day in this system is equals to one day in Week system, that is 24 hours. Thus one Dino Pasaran has five days, two Dino Pasaran has 10 days, three Dino Pasaran has 15 days, etc.
Mathematically, for all of the system numbers, the notation of numbers 'm' can be written as polynomial from 'b' such as follows : m = a0 bn-1 + a1 bn-2 + a2 bn-3 + ... + an-1 b + an
where, b is any numbers greater than 1, and a is the basis. According to this formula, we can write any number at any system using the same pattern i.e. for basis 10, basis 7 and basis 5. In Basis 10 we have the numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In Basis 7 we have the numbers : 0, 1, 2, 3, 4, 5, 6
We can match the name of the day with those numbers orderly : 0 = Sunday 1 = Monday; 2 = Tuesday; 3 = Wednesday; 4 = Thuersday; 5 = Friday and 6 = Saturday. In Basis 5 we have the numbers : 0, 1, 2, 3, 4. We can match the name of the day in Dino Pasaran with those numbers orderly : 0 = Legi, 1 = Pahing, 2 = Pon, 3 = Wage, 4 = Kliwon. How can we write 35 of Basis 10 into Basis 7 ? The following is the formula 3510 = 5 x 71 + 0 x 70 = 507. How can we write 100 of Basis 10 into Basis 7 ? The formula is 10010 = 2 x 72 + 0 x 71 + 2 x 70 = 2027. How can we write 1000 of Basis 10 into Basis 7 ? The formula is 100010 = 2 x 73 + 6 x 72 + 2 x 71 + 6 x 70 = 26267
How can we write 35 of Basis 10 into Basis 5? The following is the formula 3510 = 7 x 51 + 0 x 50 = 705. How can we write 100 of Basis 10 into Basis 5? Thai is 10010 = 4 x 52 + 0 x 51 + 0 x 50 = 4005 How can we write 1000 of Basis 10 into Basis 5 ? That is 100010 = 1 x 54 + 3 x 53 + 0 x 52 + 0 x 51 + 0 x 50 = 130005
Javanese people always have two unseparated name for the day. For example : Sunda Legi, Sunday Kliwon, Monday Wage, Friday Pahing, Saturday Kliwon, Saturday Pon, etc. It is clear that the numbers of combinations is 35 names. If this day is Monday Pahing then the duration up to on the next Monday Pahing is 7 x 5 = 35 days. Thus when somebody wish to celebrate the 35 th of his son's birthday he just waiting for the next day with the same name. It is easy and he need not to do with mathematics at all in his mind. Forthyth days ritual feast to mark some one's death by transforming 4010 = 557 and 4010 = 805. Numbers 55 is ended by 5; it mean that 'the fourty days duration of time' will begin on the day i and ended on the day 5 th of Week System. Thus, if now is Sunday then forty days to come will be the fifth day from Sunday that is Thuersday. Javanese people called 'five' as 'limo' or briefly 'mo'. Number 80 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then it will be 40 days on the next Thuersday Kliwon.
If this day is Friday Legi then it will be 40 days on the next Tuesday Legi. If this day is Wednesday Pahing then it will be 40 day on the next Sunday Pahing, etc. Javanese people just called 'no mo; sar mo' that mean : no = dino = day; mo = limo = five; and sar = Pasar = Basis 5. Thus, no mo means 'the fifth day of Week System' sar mo means 'the fifth day of Pasar System'
For one hundred days ritual feast to mark some one's death as I described that :10010 = 2027 and 10010 = 4005. Number 202 is ended by 2; its mean that 'the one hundred days duration of time' will begin on the day i and ended on the day 2 th of Week System . Thus, if now is Sunday then forty days to come will be the 2 nd day from Sunday that is Monday. Javanese people called 'two' as 'loro' or briefly 'ro'. Number 400 is ended by 0; it mean that 'the fourty days duration of time' will begin on the day of Pasar j and ended on the day j + 0 or ended on the same day. Thus If this day is Sunday Kliwon then the next 100 days will be on the next Monday Kliwon. If this day is Friday Legi then the next 100 days will be on Saturday Legi. If this day is Wednesday Pahing then the next 100 day will be on Sunday Pahing, etc. Javanese people just called 'no ro; sar mo' that mean : no = dino = day; ro = loro = dua; and sar = Pasar = Basis 5. Thus, no ro means 'the second day of Week System' sar mo means 'the fifth day of Pasar System'. One thousand days ritual feast to mark some one's death is calculated by the same way they use the formula 'no nem; sar mo' that mean 'the sixth day of Week Syatem and the fifth day of Pasar System'
Tuesday, December 23, 2008
PHILOSOPHICAL EXPLANATION ON MATHEMATICAL EXPERIENCES OF THE FIFTH GRADE STUDENTS
By Marsigit
Introduction
This article strives to explain philosophically on the students’ experiences on decimal numeration which were emerged in the research on The Effect of Epistemic Fidelity On Teaching Decimal Numeration With Physical Materials by Kaye Stacey et al (2001). The use of linear arithmetic blocks (LAB) was associated with more active engagement by students and deeper discussion than that of multi-base arithmetic blocks (MAB). Epistemic fidelity is critical to facilitate teaching with the models, but Stacey, K, et al (p.199-221, 2001) attributed the enhanced environment to the greater accessibility of the LAB material. This research and its results exhibits the writer to employ Greimas’ Structural Analyses, Kant’s theory of double-affection and other notions of philosophical explanation in order to uncover concepts behind the aspects of the process as well as the results of the research. The in-depth explanations of the nature of mathematical experiences, specifically about the effect of epistemic fidelity on teaching decimal numeration with physical materials, will expose not a single truth of its nature due to the fact that they will be put in the area of philosophy.
The level of philosophical discussion have their characteristics such as the need to cross-check as well as to compare with several point of views independently, to construct general theory of subject related. Mackenzie, J.S, (1917), stated that philosophy has to take account of the general results of the investigations of all sciences to endeavour or to construct a general theory. To achieve the purpose the writer employ some philosophical approaches such as interpretation, internal coherences, idealisation, comparison, analogy and description. Based on those approaches, accordingly, the writer adapts Greimas’ Hermenetics Structural Analyses to show the inter-relationship among the components of decimal numeration teaching with physical materials as it was carried out as part of the research of Kaye Stacey et al. To achieve the objective i.e the general theory of the related subject, the writer strive to implement the theory of ‘double-affection’ to the scheme of Greimas’ Hermenitics Structural Analyses with the context of the process and the results of the research, conducted by Stacey, K, et al, (2001), on the effect of epistemic fidelity and accessibility on teaching with physical material.
Greimas’ Hermenitics Structural Analyses
In that scheme, the student was put into the centre of the mathematical teaching learning activities; the teacher has a role as the ‘the sender’ as well as the ‘supporter’ in such a way that their students learn physical material as an object of learning; the ‘transaction’ between the teacher and their students happened if there is a motivation of the students to learn the objects i.e. physical material; the ‘constraints’ need to be considered and to be anticipated as well as to be found its solutions in such away that the students are able to interact with their physical material; the ‘anti-subject’ arises if there is extremely constraints such as bullying, un-expected accident etc. in such a way that the students are not able to interact with their physical material mathematical objects; the ‘receivers’ are the people or the agents that takes the benefit of the students’ interaction with their objects, therefore, the student him/herself cam be perceived as ‘receiver’.
The Myth Of Double Affection
The theory of double affection is a classical attempt to rescue Kant’s account of perceptual awareness from what is alleged to be a glaring inconsistency (Gram, S.M, in Werkmeister, W.H, 1975). According to Kant, ‘to be affected by anything ‘ is to experience the effect of an object upon the faculty of representation (ibid, p.29). Kant provides two kinds of objects which affect the subject: there are ‘thing in themselves’ which affect the self; and there are ‘appearances in themselves’ which act on our sensibility and are independent of whatever characteristics attach to our sensory receptors (Werkmeister, W.H, 1975). Facing this Kant’s notion, Gram, S.M, in Werkmeister, W.H, (1975) delivered the following argument:
“ Suppose we say that what affects our sensibility is ‘a thing in itself’. This account of what affects us, however, prevents us from distinguishing between a case in which somebody perceives an object and the quite different case in which an object exert a merely causal influence on the body of the perceiver. This can be seen by consulting an elementary fact of perception. The fact is that to perceive anything is to perceive it under ‘a certain description’. If this were not the case, then we could not distinguish between the perceiving of one object rather than another. But if we must always perceive something under a description, to say that we are affected by ‘a thing in itself’ when we perceive anything would imply that we perceive that objects satisfy certain descriptions. And this would contradict the claim that we cannot be perceptually acquainted with ‘a thing in itself’.
The above propositions were delivered to argue Kant’s description that affection as the experience of the ‘effect’ of an object on our sensory apparatus; whilst, the dilemma facing Kant’s theory has nothing to do with the quite separate issue of whether what is related to sensibility is the effect of an object rather than the object itself; and, the issue concerns the nature of the object which is immediately present to perceptual awareness rather than the casual relation in which it might stand to some further object. The notion of affection does not, however, become fully clear unless we can specify the kind of object which can stand in such a relation to our sensibility (ibid, p.29). He then erected the next dilemma as shown the following:
“If ‘a thing in itself’ can act upon our sensory organs even though we cannot perceive it to satisfy any description at all, we would not be able to distinguish between ‘the situation ‘ in which an object casually affects our bodies in certain ways and we do not perceive the effects of that action from the quite different situation in which the object exerts such as influence and we do perceive it. If the first affection is to hold between ‘a thing in itself’ and ‘an act of perceptual awareness, we would have to be able to perceive ‘thing in themselves’ under descriptions appropriate to them or obliterate the distinction between causation and perceptual awareness”.
What we can learn is that there should be any other relation between ‘thing in themselves’ and affection. Kant asserted that ‘space’ and ‘time’ are forms of our sensibility; what affects our sensibility is an object that has ‘spatial’ or ‘temporal’ characteristics i.e. a phenomenal object. If the object which affects the forms of our sensibility cannot itself have ‘spatio-temporal’ characteristics, then what affects us must, on Kant’s theory, be a thing in itself . Empirical affection does not require that the objects in our sensory field lack spatio-temporal characteristics; while, transcendental affection countenances the existence of objects which affect ego in themselves. However, the distinction between these two kinds of perception is still a myth (ibid 32-33).
Research on The Effect Of Epistemic Fidelity On Tea-ching Decimal Numeration With Physical Materials by Kaye Stacey et al (2001)
The results of the research on the effect of epistemic fidelity and accessibility on teaching with physical material (Stacey, K, et al, 2001) comes to some conclusion that: 1) the are numbers of favor differences of different model of physical material (LAB and MAB), 2) the most striking difference between the two models was their ability to model number density, with LAB found to be the superior model in this respect, 3) teaching with physical materials is an area of great difficulty for many students, 4) students did not attend to the volume relationships embedded in MAB and struggled to remember the names, rather than immediately appreciating the sense behind them, 5) MAB students experienced difficulty generalizing to numbers beyond the model due to their difficulties with volume and apparent dimensional shifts in their perceptions of the components, 6) LAB appeared to promote richer engagement in the classroom than MAB due to its greater accessibility (detail results of the research, refer to Educational Studies in Mathematics 47: 199-221, 2001).
It was acknowledged by the researchers that some manipulative materials can be distracting and open misinterpretation; teachers could overestimate the value of physical materials because they are already familiar with the concepts being presented (Ball in Kaye, et al, 2001). It also stated that, Meira (1998), the mechanical devices became ‘visible’ as things that required explanation, rather than ‘invisible’ resources for making the mathematics more accessible. Having considered those notions of the constraints in employing physical materials in teaching mathematics and having learnt the document of the process and the results of the research, the writer perceives that the research consists a lot of important critical concepts that need to be developed as the notions in the implementation of mathematics teaching as well as the notions of theoretical and or philosophical discussions. In term of theoretical concept, those important critical concepts consist of: 1) epistemic fidelity, 2) the posing problems devices, 3) the link between the features of the device and the target knowledge, 4) something objective, 5) students’ engagement, and 6) accessibility. From the explanation, it can be inferred that the objective of this paper is to investigate general theory of the aspects of mathematics teaching learning processes with the context of the process and the results of the research conducted by Stacey, K, et al, (2001), on the effect of epistemic fidelity and accessibility on teaching with physical material.
Philosophical Explanation on Mathematical Experience
In their theoretical review of the stated research, Stacey, K, et al, (2001) indicated that epistemic fidelity of the material is one of the factors influences the transparency of instructional material. They also indicated that epistemic fidelity of the material depends on the materials themselves in which the mathematical domain being represented does not depend on their use by students. Explicitly, they defined that the epistemic fidelity of an instructional material is a measure of the quality of analogical mapping between the features of the material and the target knowledge domain. Further, they stated that epistemic fidelity of a model depends on the relationship of features intrinsic in the model to target mathematical structure, and is independent of user characteristics. On the other hand, Gram, S.M. (1975) provides a clear and comprehensive statement, of the case that likely as what Stacey, K., et al infer as epistemic fidelity, that he called ‘double affection’. He claimed that what affects our sensibility is ‘a phenomenal object’; it allowing anything which has spatial or temporal characteristics to count as such an object. Further he stated that, according to Kant, sensibility is the capacity (that the researcher claimed as ‘quality’) for receiving representations through the mode in which we are affected by objects.
From those two points of view we may learn that although there similarities of the claim of the relation between subject and object of learning, although the writer could not identify what did they mean by ‘a measure of the quality of analogical mapping between the features of the material and the target knowledge domain’, except that of its category consists of excellent, good, satisfactory and unsatisfactory. If the researchers meant that epistemic fidelity is the capacity for receiving representations through the mode in which we are affected by objects, the next problem is that we need to clarify them. Kant implied that affection is to be partially defined in terms of a relation in which an object stands to certain spatio-temporal forms; and this kind of relationship is specified in terms of a connection between an object and these forms, not in term of an object exhibiting these forms and sensibility. It is important here to conclude that, according to Kant, if the object which affects the forms of our sensibility cannot itself have spatio-temporal characteristics, then what affects us must be ‘a thing in itself’(in which the researchers indicated it as ‘material in themselves’). It seemed that the researchers did not specify the affect of the different characteristics of the object in term of ‘appearances in themselves’ and ‘things in themselves’.
Next, they also indicated that the ‘accessibility’ of the materials is a collection or psychological factors that arise in the use of the materials by students but which are not specific to particular students (ibid. p. 2001); further it was stated that accessibility of a model of physical material depends on characteristics of likely users interacting with features of the model; accessibility, stands above the detailed analyses of particular tasks in particular classrooms that Meira (1998) in Stacy (2001) has traced in his quest for ‘transparency’. Accordingly, there are at least two issues (both social and psychological) that may impact of LAB and MAB. In LAB the issues consists of: 1) students’ confusing the organizer rods with the value of the component and 2) students’ confusing about the left-right positioning of the place value columns. It is clear that what the researcher infer by ‘accessibility’ is something related to the subject that what inferred by Kant as ‘sensibility’.
Differences accessibility were actually found that students in MAB group experienced confusion with remembering the new names components. There was no such confusion in the LAB group. How numbers are represented? In MAB group, the students did not understand that the components relative value is based on their volume. In term of ability to generalise beyond the model, the students were confused by the apparent dimensional shift and appeared to be looking for a forth dimension. Were the different learning outcomes related to differences in epistemic fidelity or accessibility? The LAB model was more effective on decimal numeration; the LAB model was found to more transparent model for numeration; the LAB model was more effective model of number density; the LAB model should also be better model for rounding decimal number.
In term of the differences between the group, the LAB model was more favourable and the LAB model appeared to promote richer engagement in the classroom due to greater accessibility. The Year 5 students appeared reluctant to use the MAB; it was a constant struggle to get them to use it. There was more discussion and exchange the ideas in the LAB group and there more significantly episodes of talk referring to the LAB model than the MAB model. There was evidence that LAB students spontaneously exploring new ideas, which did not occur with students using MAB. When LAB was not available, students made connections with other physical representations, such as ruler lengths and MAB; One student pointed out “LAB is another type of MAB”; “These are the exact same thing”. The LAB group scored higher than the MAB group on every measure of attitude(Likert items). In term of the attitude, the LAB group is typified by one student’s comment: ”Learning what the numbers mean –how big they were-just from length, was the best”.
Conclusion
The research has given the researchers an insight into the different roles of epistemic fidelity and accessibility of physical instructional material. The researchers hypothesise that epistemic fidelity is necessary for securely grounded teaching of concept with a model, whereas accessibility promotes rich classroom engagement. Epistemic fidelity and accessibility have different roles in establishment transparency. From all of those findings, the writer strives to develop the method to uncover what are there behind the concepts.
Over all, we regard to the students’ status of mathematical knowledge resulted by manipulating with physical materials, in the schema of Greimas’ Hermenetics Structural Analyses. If the distinction between the two kinds of perception is still a myth, then we can still argue it on the status of mathematical knowledge. As it was acknowledged by the researchers that some manipulative materials can be distracting and open misinterpretation; it can be explain with the theory of double-affection due to the fact that the teachers are already familiar with the concepts being presented. The writer perceives that Kant’s notion of appearance in them selves and thing in themselves are useful to explain the issues of visibility and /or invisibility of the mechanical device.
The writer emphasizes that the different context, i.e. in term of time and space as it was notified by Kant, may influence students perception of the objects. Therefore, teachers need to employ those kind of factors as supporting one in teaching learning of mathematics. The link between the features of the device and the target knowledge was very intensively to be discussed by Kant in his Critical of Pure Reason. General theory of the aspects of mathematics teaching learning processes is to pursue in term of the relation of student as a subject and physical material as an object in the schema of Greimas’ Hermenetics Structural Analyses. The effort to pursue those relationships will determine the extent of the quality of philosophical point of view.
References:
Haryatmoko, 2004, Research Methodology, Unpublished document of his lecturing in the Post Graduate Program of Philosophy Science, Gadjah Mada University
Kant, I., 1998, Critique of Pure Reason (trans. Meiklejohn, J.M, )
Kant, I., 1998, Prolegomena to Any Future Metaphysics(trans.)
Smith, N.K., 2003, A Commentary to Kant’s Critique of Pure Reason, New York: Palgrave Macmillan.
Stacey K., 2001, The Effect Of Epistemic Fidelity On Teaching Decimal Numeration With Physical Materials by Kaye Stacey
Werkmeister, W.H., 1975, Reflections on Kant’ Philosophy,Florida: University Presses of Florida.
Introduction
This article strives to explain philosophically on the students’ experiences on decimal numeration which were emerged in the research on The Effect of Epistemic Fidelity On Teaching Decimal Numeration With Physical Materials by Kaye Stacey et al (2001). The use of linear arithmetic blocks (LAB) was associated with more active engagement by students and deeper discussion than that of multi-base arithmetic blocks (MAB). Epistemic fidelity is critical to facilitate teaching with the models, but Stacey, K, et al (p.199-221, 2001) attributed the enhanced environment to the greater accessibility of the LAB material. This research and its results exhibits the writer to employ Greimas’ Structural Analyses, Kant’s theory of double-affection and other notions of philosophical explanation in order to uncover concepts behind the aspects of the process as well as the results of the research. The in-depth explanations of the nature of mathematical experiences, specifically about the effect of epistemic fidelity on teaching decimal numeration with physical materials, will expose not a single truth of its nature due to the fact that they will be put in the area of philosophy.
The level of philosophical discussion have their characteristics such as the need to cross-check as well as to compare with several point of views independently, to construct general theory of subject related. Mackenzie, J.S, (1917), stated that philosophy has to take account of the general results of the investigations of all sciences to endeavour or to construct a general theory. To achieve the purpose the writer employ some philosophical approaches such as interpretation, internal coherences, idealisation, comparison, analogy and description. Based on those approaches, accordingly, the writer adapts Greimas’ Hermenetics Structural Analyses to show the inter-relationship among the components of decimal numeration teaching with physical materials as it was carried out as part of the research of Kaye Stacey et al. To achieve the objective i.e the general theory of the related subject, the writer strive to implement the theory of ‘double-affection’ to the scheme of Greimas’ Hermenitics Structural Analyses with the context of the process and the results of the research, conducted by Stacey, K, et al, (2001), on the effect of epistemic fidelity and accessibility on teaching with physical material.
Greimas’ Hermenitics Structural Analyses
In that scheme, the student was put into the centre of the mathematical teaching learning activities; the teacher has a role as the ‘the sender’ as well as the ‘supporter’ in such a way that their students learn physical material as an object of learning; the ‘transaction’ between the teacher and their students happened if there is a motivation of the students to learn the objects i.e. physical material; the ‘constraints’ need to be considered and to be anticipated as well as to be found its solutions in such away that the students are able to interact with their physical material; the ‘anti-subject’ arises if there is extremely constraints such as bullying, un-expected accident etc. in such a way that the students are not able to interact with their physical material mathematical objects; the ‘receivers’ are the people or the agents that takes the benefit of the students’ interaction with their objects, therefore, the student him/herself cam be perceived as ‘receiver’.
The Myth Of Double Affection
The theory of double affection is a classical attempt to rescue Kant’s account of perceptual awareness from what is alleged to be a glaring inconsistency (Gram, S.M, in Werkmeister, W.H, 1975). According to Kant, ‘to be affected by anything ‘ is to experience the effect of an object upon the faculty of representation (ibid, p.29). Kant provides two kinds of objects which affect the subject: there are ‘thing in themselves’ which affect the self; and there are ‘appearances in themselves’ which act on our sensibility and are independent of whatever characteristics attach to our sensory receptors (Werkmeister, W.H, 1975). Facing this Kant’s notion, Gram, S.M, in Werkmeister, W.H, (1975) delivered the following argument:
“ Suppose we say that what affects our sensibility is ‘a thing in itself’. This account of what affects us, however, prevents us from distinguishing between a case in which somebody perceives an object and the quite different case in which an object exert a merely causal influence on the body of the perceiver. This can be seen by consulting an elementary fact of perception. The fact is that to perceive anything is to perceive it under ‘a certain description’. If this were not the case, then we could not distinguish between the perceiving of one object rather than another. But if we must always perceive something under a description, to say that we are affected by ‘a thing in itself’ when we perceive anything would imply that we perceive that objects satisfy certain descriptions. And this would contradict the claim that we cannot be perceptually acquainted with ‘a thing in itself’.
The above propositions were delivered to argue Kant’s description that affection as the experience of the ‘effect’ of an object on our sensory apparatus; whilst, the dilemma facing Kant’s theory has nothing to do with the quite separate issue of whether what is related to sensibility is the effect of an object rather than the object itself; and, the issue concerns the nature of the object which is immediately present to perceptual awareness rather than the casual relation in which it might stand to some further object. The notion of affection does not, however, become fully clear unless we can specify the kind of object which can stand in such a relation to our sensibility (ibid, p.29). He then erected the next dilemma as shown the following:
“If ‘a thing in itself’ can act upon our sensory organs even though we cannot perceive it to satisfy any description at all, we would not be able to distinguish between ‘the situation ‘ in which an object casually affects our bodies in certain ways and we do not perceive the effects of that action from the quite different situation in which the object exerts such as influence and we do perceive it. If the first affection is to hold between ‘a thing in itself’ and ‘an act of perceptual awareness, we would have to be able to perceive ‘thing in themselves’ under descriptions appropriate to them or obliterate the distinction between causation and perceptual awareness”.
What we can learn is that there should be any other relation between ‘thing in themselves’ and affection. Kant asserted that ‘space’ and ‘time’ are forms of our sensibility; what affects our sensibility is an object that has ‘spatial’ or ‘temporal’ characteristics i.e. a phenomenal object. If the object which affects the forms of our sensibility cannot itself have ‘spatio-temporal’ characteristics, then what affects us must, on Kant’s theory, be a thing in itself . Empirical affection does not require that the objects in our sensory field lack spatio-temporal characteristics; while, transcendental affection countenances the existence of objects which affect ego in themselves. However, the distinction between these two kinds of perception is still a myth (ibid 32-33).
Research on The Effect Of Epistemic Fidelity On Tea-ching Decimal Numeration With Physical Materials by Kaye Stacey et al (2001)
The results of the research on the effect of epistemic fidelity and accessibility on teaching with physical material (Stacey, K, et al, 2001) comes to some conclusion that: 1) the are numbers of favor differences of different model of physical material (LAB and MAB), 2) the most striking difference between the two models was their ability to model number density, with LAB found to be the superior model in this respect, 3) teaching with physical materials is an area of great difficulty for many students, 4) students did not attend to the volume relationships embedded in MAB and struggled to remember the names, rather than immediately appreciating the sense behind them, 5) MAB students experienced difficulty generalizing to numbers beyond the model due to their difficulties with volume and apparent dimensional shifts in their perceptions of the components, 6) LAB appeared to promote richer engagement in the classroom than MAB due to its greater accessibility (detail results of the research, refer to Educational Studies in Mathematics 47: 199-221, 2001).
It was acknowledged by the researchers that some manipulative materials can be distracting and open misinterpretation; teachers could overestimate the value of physical materials because they are already familiar with the concepts being presented (Ball in Kaye, et al, 2001). It also stated that, Meira (1998), the mechanical devices became ‘visible’ as things that required explanation, rather than ‘invisible’ resources for making the mathematics more accessible. Having considered those notions of the constraints in employing physical materials in teaching mathematics and having learnt the document of the process and the results of the research, the writer perceives that the research consists a lot of important critical concepts that need to be developed as the notions in the implementation of mathematics teaching as well as the notions of theoretical and or philosophical discussions. In term of theoretical concept, those important critical concepts consist of: 1) epistemic fidelity, 2) the posing problems devices, 3) the link between the features of the device and the target knowledge, 4) something objective, 5) students’ engagement, and 6) accessibility. From the explanation, it can be inferred that the objective of this paper is to investigate general theory of the aspects of mathematics teaching learning processes with the context of the process and the results of the research conducted by Stacey, K, et al, (2001), on the effect of epistemic fidelity and accessibility on teaching with physical material.
Philosophical Explanation on Mathematical Experience
In their theoretical review of the stated research, Stacey, K, et al, (2001) indicated that epistemic fidelity of the material is one of the factors influences the transparency of instructional material. They also indicated that epistemic fidelity of the material depends on the materials themselves in which the mathematical domain being represented does not depend on their use by students. Explicitly, they defined that the epistemic fidelity of an instructional material is a measure of the quality of analogical mapping between the features of the material and the target knowledge domain. Further, they stated that epistemic fidelity of a model depends on the relationship of features intrinsic in the model to target mathematical structure, and is independent of user characteristics. On the other hand, Gram, S.M. (1975) provides a clear and comprehensive statement, of the case that likely as what Stacey, K., et al infer as epistemic fidelity, that he called ‘double affection’. He claimed that what affects our sensibility is ‘a phenomenal object’; it allowing anything which has spatial or temporal characteristics to count as such an object. Further he stated that, according to Kant, sensibility is the capacity (that the researcher claimed as ‘quality’) for receiving representations through the mode in which we are affected by objects.
From those two points of view we may learn that although there similarities of the claim of the relation between subject and object of learning, although the writer could not identify what did they mean by ‘a measure of the quality of analogical mapping between the features of the material and the target knowledge domain’, except that of its category consists of excellent, good, satisfactory and unsatisfactory. If the researchers meant that epistemic fidelity is the capacity for receiving representations through the mode in which we are affected by objects, the next problem is that we need to clarify them. Kant implied that affection is to be partially defined in terms of a relation in which an object stands to certain spatio-temporal forms; and this kind of relationship is specified in terms of a connection between an object and these forms, not in term of an object exhibiting these forms and sensibility. It is important here to conclude that, according to Kant, if the object which affects the forms of our sensibility cannot itself have spatio-temporal characteristics, then what affects us must be ‘a thing in itself’(in which the researchers indicated it as ‘material in themselves’). It seemed that the researchers did not specify the affect of the different characteristics of the object in term of ‘appearances in themselves’ and ‘things in themselves’.
Next, they also indicated that the ‘accessibility’ of the materials is a collection or psychological factors that arise in the use of the materials by students but which are not specific to particular students (ibid. p. 2001); further it was stated that accessibility of a model of physical material depends on characteristics of likely users interacting with features of the model; accessibility, stands above the detailed analyses of particular tasks in particular classrooms that Meira (1998) in Stacy (2001) has traced in his quest for ‘transparency’. Accordingly, there are at least two issues (both social and psychological) that may impact of LAB and MAB. In LAB the issues consists of: 1) students’ confusing the organizer rods with the value of the component and 2) students’ confusing about the left-right positioning of the place value columns. It is clear that what the researcher infer by ‘accessibility’ is something related to the subject that what inferred by Kant as ‘sensibility’.
Differences accessibility were actually found that students in MAB group experienced confusion with remembering the new names components. There was no such confusion in the LAB group. How numbers are represented? In MAB group, the students did not understand that the components relative value is based on their volume. In term of ability to generalise beyond the model, the students were confused by the apparent dimensional shift and appeared to be looking for a forth dimension. Were the different learning outcomes related to differences in epistemic fidelity or accessibility? The LAB model was more effective on decimal numeration; the LAB model was found to more transparent model for numeration; the LAB model was more effective model of number density; the LAB model should also be better model for rounding decimal number.
In term of the differences between the group, the LAB model was more favourable and the LAB model appeared to promote richer engagement in the classroom due to greater accessibility. The Year 5 students appeared reluctant to use the MAB; it was a constant struggle to get them to use it. There was more discussion and exchange the ideas in the LAB group and there more significantly episodes of talk referring to the LAB model than the MAB model. There was evidence that LAB students spontaneously exploring new ideas, which did not occur with students using MAB. When LAB was not available, students made connections with other physical representations, such as ruler lengths and MAB; One student pointed out “LAB is another type of MAB”; “These are the exact same thing”. The LAB group scored higher than the MAB group on every measure of attitude(Likert items). In term of the attitude, the LAB group is typified by one student’s comment: ”Learning what the numbers mean –how big they were-just from length, was the best”.
Conclusion
The research has given the researchers an insight into the different roles of epistemic fidelity and accessibility of physical instructional material. The researchers hypothesise that epistemic fidelity is necessary for securely grounded teaching of concept with a model, whereas accessibility promotes rich classroom engagement. Epistemic fidelity and accessibility have different roles in establishment transparency. From all of those findings, the writer strives to develop the method to uncover what are there behind the concepts.
Over all, we regard to the students’ status of mathematical knowledge resulted by manipulating with physical materials, in the schema of Greimas’ Hermenetics Structural Analyses. If the distinction between the two kinds of perception is still a myth, then we can still argue it on the status of mathematical knowledge. As it was acknowledged by the researchers that some manipulative materials can be distracting and open misinterpretation; it can be explain with the theory of double-affection due to the fact that the teachers are already familiar with the concepts being presented. The writer perceives that Kant’s notion of appearance in them selves and thing in themselves are useful to explain the issues of visibility and /or invisibility of the mechanical device.
The writer emphasizes that the different context, i.e. in term of time and space as it was notified by Kant, may influence students perception of the objects. Therefore, teachers need to employ those kind of factors as supporting one in teaching learning of mathematics. The link between the features of the device and the target knowledge was very intensively to be discussed by Kant in his Critical of Pure Reason. General theory of the aspects of mathematics teaching learning processes is to pursue in term of the relation of student as a subject and physical material as an object in the schema of Greimas’ Hermenetics Structural Analyses. The effort to pursue those relationships will determine the extent of the quality of philosophical point of view.
References:
Haryatmoko, 2004, Research Methodology, Unpublished document of his lecturing in the Post Graduate Program of Philosophy Science, Gadjah Mada University
Kant, I., 1998, Critique of Pure Reason (trans. Meiklejohn, J.M, )
Kant, I., 1998, Prolegomena to Any Future Metaphysics(trans.)
Smith, N.K., 2003, A Commentary to Kant’s Critique of Pure Reason, New York: Palgrave Macmillan.
Stacey K., 2001, The Effect Of Epistemic Fidelity On Teaching Decimal Numeration With Physical Materials by Kaye Stacey
Werkmeister, W.H., 1975, Reflections on Kant’ Philosophy,Florida: University Presses of Florida.
Monday, December 22, 2008
Pondasi Matematika: Dari Plato sampai Godel
Oleh: Marsigit
Jika kita berkehendak melakukan kajian atau penelitian matematika secara mendalam maka kita tidak bisa terhindar untuk melakukan sintesis-sintesis dari tesis-tesis yang ada dengan cara memproduksi antitesis-antitesisnya. Dalam sejarahnya, Aristoteles tidak sependapat dengan gurunya Plato dalam prinsip-prinsip filsafat; namun demikian jika ditilik lebih lanjut sebetulnya silang pendapat juga meliputi bidang matematika. Menurut Aristoteles, form bukanlah entitas yang terpisah dengan data empiris. Menurutnya jika kita memikirkan suatu benda tidaklah berarti bahwa konsep benda tadi akan terpisah dengan benda tadi. Menurut Aristoteles, matematika adalah idealisasi dari benda-benda; dengan melakukan idealisasi kita dapat membuat definisi, menemukan struktur matematika, menemukan logika, menemukan teorema, dan melakukan hipotesis. Jika matematika bersifat given yang sudah ada di dalam ide kita maka implikasi pandanga Plato adalah bahwa matematika bersifat aktual. Seperti kita ketahui bahwa bilangan infinit, di satu sisi dapat dipandang sebagai aktual tetapi juga dapat dipandang sebagai potensial. Pandangan yang terakhir ini kemudian ditolak oleh Brouwer sebagai kaum intuisionis untuk mengembangkan matematika intuisionisme. Menurut Brouwer di dalam mengembangkan matematika kita harus menggunakan intuisi kita, sayangnya intuisi dan pengalaman kita tidak dapat menjangkau bilangan infinit. Itulah sebabnya kaum intuisionis hanya mengembangkan matematika untuk bilangan-bilangan finit atau berhingga dan menolak bilangan infinit atau bilangan tak hingga. Selanjutnya Heyting sebagai penerus Brouwer menolak kenyataan transenden sebagai alat bukti matematika. Menurutnya, bilangan infinit merupakan salah satu kenyataan transenden.
Pandangan umum setuju bahwa kebenaran matematika merupakan kebenaran yang bersifat bersyarat “necessary truth”. Tetapi hal demikian tidak mudah kita wujudkan untuk menunjukkan kebenaran konsep-konsep bilangan tak hingga atau infinit? Bagaimana manusia yang bersifat serba terbatas mampu memikirkan hal-hal yang bersifat tak terbatas? Ada paling tidak dua pandangan bagaimana memperoleh kebanaran matematika, pertama kebenaran matematika diperoleh murni menggunakan akal pikiran, kedua kebenaran matematika diperoleh berdasarkan pengalaman. Sudah sejak lama kaum rasionalis yang dipelopori oleh Rene Descartes dan Leibniz berpendapat bahwa konsep matematika bersifat melekat “innate” pada pikiran kita; sementara John Locke dan David Hume berpendapat bahwa pengetahuan matematika diturunkan berdasarkan pengalaman inderawi. Pandangan John Locke dan David Hume diteruskan oleh John Stuart Mill sebagai seorang empiris yang berpandangan bahwa pemahaman matematika diperoleh dari pengalaman dan kebenaran matematika diperoleh dengan melakukan generalisasi kegiatan penemuan konsep-konsep empiris. Di sisi lain dengan ditemukannya Geometri non-Euclides telah membuka cakrawala para matematisi dan para filsuf untuk mengevauasi kembali konsep geometri Euclides; dalam mana telah diakui selama lebih dari 2000 tahun bahwa geometri Euclides dianggap sebagai representasi alam semesta. Dalam hal tertentu kebenaran dan pembenaran pada geometri Euclides selaras dengan apa yang di perjuangkan Mill; namun penemuan geometri non_Euclides telah menyebabkan konsep Mill dan empirisisme pun kembali dipertanyakan.
Di 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. Namun krisis pondasi matematika tidak berhenti sampai di sini. Pada akhir abad ke 19 Cantor menemukan dan mengembangkan teori himpunan. Di dalam pengembangan teori himpunan tersebut Cantor menghadapi persoalan paradoks matematika, yang menambah panjang deretan krisis di dalam pondasi matematika. Pada awal abad ke 20, karya besar telah dicapai oleh para filsuf dan matematisi dengan diletakkannya logika sebagai pondamen matematika. Sampai akhirnya ditemukan pula paradoks dari logika; sehingga hal yang demikian menggagalkan usaha Hilbert untuk membangun matematika sebagai suatu sistem di atas satu pondasi yang kokoh. Adalah muridnya sendiri Kurt Godel yang berhasil menyimpulkan bahwa jika sistem matematika bersifat lengkap maka dia pasti tidak konsisten; dan jika sistem matematika konsisten maka dia tidak akan bisa lengkap. Era filsafat kontemporer telah mendorong para filsuf dan matematisi untuk melihat kenyataan bahwa matematika bersifat multi-facet.
Jika kita berkehendak melakukan kajian atau penelitian matematika secara mendalam maka kita tidak bisa terhindar untuk melakukan sintesis-sintesis dari tesis-tesis yang ada dengan cara memproduksi antitesis-antitesisnya. Dalam sejarahnya, Aristoteles tidak sependapat dengan gurunya Plato dalam prinsip-prinsip filsafat; namun demikian jika ditilik lebih lanjut sebetulnya silang pendapat juga meliputi bidang matematika. Menurut Aristoteles, form bukanlah entitas yang terpisah dengan data empiris. Menurutnya jika kita memikirkan suatu benda tidaklah berarti bahwa konsep benda tadi akan terpisah dengan benda tadi. Menurut Aristoteles, matematika adalah idealisasi dari benda-benda; dengan melakukan idealisasi kita dapat membuat definisi, menemukan struktur matematika, menemukan logika, menemukan teorema, dan melakukan hipotesis. Jika matematika bersifat given yang sudah ada di dalam ide kita maka implikasi pandanga Plato adalah bahwa matematika bersifat aktual. Seperti kita ketahui bahwa bilangan infinit, di satu sisi dapat dipandang sebagai aktual tetapi juga dapat dipandang sebagai potensial. Pandangan yang terakhir ini kemudian ditolak oleh Brouwer sebagai kaum intuisionis untuk mengembangkan matematika intuisionisme. Menurut Brouwer di dalam mengembangkan matematika kita harus menggunakan intuisi kita, sayangnya intuisi dan pengalaman kita tidak dapat menjangkau bilangan infinit. Itulah sebabnya kaum intuisionis hanya mengembangkan matematika untuk bilangan-bilangan finit atau berhingga dan menolak bilangan infinit atau bilangan tak hingga. Selanjutnya Heyting sebagai penerus Brouwer menolak kenyataan transenden sebagai alat bukti matematika. Menurutnya, bilangan infinit merupakan salah satu kenyataan transenden.
Pandangan umum setuju bahwa kebenaran matematika merupakan kebenaran yang bersifat bersyarat “necessary truth”. Tetapi hal demikian tidak mudah kita wujudkan untuk menunjukkan kebenaran konsep-konsep bilangan tak hingga atau infinit? Bagaimana manusia yang bersifat serba terbatas mampu memikirkan hal-hal yang bersifat tak terbatas? Ada paling tidak dua pandangan bagaimana memperoleh kebanaran matematika, pertama kebenaran matematika diperoleh murni menggunakan akal pikiran, kedua kebenaran matematika diperoleh berdasarkan pengalaman. Sudah sejak lama kaum rasionalis yang dipelopori oleh Rene Descartes dan Leibniz berpendapat bahwa konsep matematika bersifat melekat “innate” pada pikiran kita; sementara John Locke dan David Hume berpendapat bahwa pengetahuan matematika diturunkan berdasarkan pengalaman inderawi. Pandangan John Locke dan David Hume diteruskan oleh John Stuart Mill sebagai seorang empiris yang berpandangan bahwa pemahaman matematika diperoleh dari pengalaman dan kebenaran matematika diperoleh dengan melakukan generalisasi kegiatan penemuan konsep-konsep empiris. Di sisi lain dengan ditemukannya Geometri non-Euclides telah membuka cakrawala para matematisi dan para filsuf untuk mengevauasi kembali konsep geometri Euclides; dalam mana telah diakui selama lebih dari 2000 tahun bahwa geometri Euclides dianggap sebagai representasi alam semesta. Dalam hal tertentu kebenaran dan pembenaran pada geometri Euclides selaras dengan apa yang di perjuangkan Mill; namun penemuan geometri non_Euclides telah menyebabkan konsep Mill dan empirisisme pun kembali dipertanyakan.
Di 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. Namun krisis pondasi matematika tidak berhenti sampai di sini. Pada akhir abad ke 19 Cantor menemukan dan mengembangkan teori himpunan. Di dalam pengembangan teori himpunan tersebut Cantor menghadapi persoalan paradoks matematika, yang menambah panjang deretan krisis di dalam pondasi matematika. Pada awal abad ke 20, karya besar telah dicapai oleh para filsuf dan matematisi dengan diletakkannya logika sebagai pondamen matematika. Sampai akhirnya ditemukan pula paradoks dari logika; sehingga hal yang demikian menggagalkan usaha Hilbert untuk membangun matematika sebagai suatu sistem di atas satu pondasi yang kokoh. Adalah muridnya sendiri Kurt Godel yang berhasil menyimpulkan bahwa jika sistem matematika bersifat lengkap maka dia pasti tidak konsisten; dan jika sistem matematika konsisten maka dia tidak akan bisa lengkap. Era filsafat kontemporer telah mendorong para filsuf dan matematisi untuk melihat kenyataan bahwa matematika bersifat multi-facet.
Sunday, December 21, 2008
MATEMATIKA DILIHAT DARI BERBAGAI SUDUT PANDANG
Resensi buku “The Philosophy of Mathematics Education”, karya Paul Ernest
Oleh: Marsigit
Para absolutis teguh pendiriannya dalam memandang secara objektif kenetralan matematika, walaupun matematika yang dipromosikan itu sendiri secara implisit mengandung nilai-nilai. Abstrak adalah suatu nilai terhadap konkrit, formal suatu nilai terhadap informal, objektif terhadap subjektif, pembenaran terhadap penemuan, rasionalitas terhadap intuisi, penalaran terhadap emosi, hal-hal umum terhadap hal-hal khusus, teori terhadap praktik, kerja dengan fikiran terhadap kerja dengan tangan, dan seterusnya. Setelah mendaftar macam-macam nilai di atas maka pertanyaannya adalah, bagaimana matematisi berpendapat bahwa matematika adalah netral dan bebas nilai ? Jawaban dari kaum absolutis adalah bahwa niai yang mereka maksud adalah nilai yang melekat pada diri mereka yang berupa kultur, jadi bukan nilai yang melekat secara implisist pada matematika. Diakui bahwa isi dan metode matematika, karena hakekatnya, membuat matematika menjadi abstrak, umum, formal, obyektif, rasional, dan teoritis. Ini adalah hakekat ilmu pengetahuan dan matematika. Tidak ada yang salah bagi yang kongkrit, informal, subyektif, khusus, atau penemuan; mereka hanya tidak termasuk dalam sains, dan tentunya tidak termasuk di dalam matematika (Popper, 1979 dalam Ernest, 1991: 132).
Yang ingin ditandaskan di sini adalah bahwa pandangan kaum absolutis, secara sadar maupun tak sadar, telah merasuk ke dalam matematika melalui definisi-definisi. Dengan perkataan lain, kaum absolutis berpendapat bahwa segala sesuatu yang sesuai dengan nilai-nilai di atas dapat diterima dan yang tidak sesuai tidak dapat diterima. Pernyataan-pernyataan matematika dan bukti-buktinya, yang merupakan hasil dari matematika formal, dipandang dapat melegitimasikan matematika. Sementara, penemuan-penemuan matematika, hasil kerja para matematisi dan proses yang bersifat informal dipandang tidak demikian. Dengan pendekatan ini kaum absolutis membangun matematika yang dianggapnya sebagai netral dan bebas nilai. Dengan pendekatan ini mereka menetapkan kriteria apa yang dapat diterima dan tidak diterima. Hal-hal yang terikat dengan implikasi sosial dan nilai-nilai yang menyertainya, secara eksplisit, dihilangkannya. Tetapi dalam kenyataannya, nilai-nilai yang terkandung dalam hal-hal tersebut di atas, membuat masalah-masalah yang tidak dapat dipecahkan. Hal ini disebabkan karena mendasarkan pada hal-hal yang bersifat formal saja hanya dapat menjangkau pada pembahasan bagian luar dari matematika itu sendiri.
Jika mereka berkehendak menerima kritik yang ada, sebetulnya pandangan mereka tentang matematika yang netral dan bebas nilai juga merupakan suatu nilai yang melekat pada diri mereka dan sulit untuk dilihatnya. Dengan demikian akan muncul pertanyaan berikutnya, siapa yang tertarik dengan pendapatnya ? Inggris dan negara-negara Barat pada umumnya, diperintah oleh kaum laki-laki berkulit putih dari kelas atas. Keadaan demikian mempengaruhi struktur sosial para matematisi di kampus-kampus suatu Universitas, yang kebanyakan didominasi oleh mereka. Nilai-nilai mereka secara sadar dan tak sadar terjabarkan dalam pengembangan matematika sebagai bagian dari usaha dominasi sosial. Oleh karena itu agak janggal kiranya bahwa matematika bersifat netral dan bebas nilai, sementara matematika telah menjadi alat suatu kelompok sosial. Mereka mengunggulkan pria di atas wanita, kulit putih di atas kulit hitam, masyarakat strata menengah di atas strata bawah, untuk kriteria keberhasilan penguasaan pencapaian akademik matematikanya.
Suatu kritik mengatakan, untuk suatu kelompok tertentu, misalnya kelompok kulit putih dari strata atas, mungkin dapat dianggap matematika sebagai netral dan bebas nilai. Namun kritik demikian menghadapi beberapa masalah. Pertama, terdapat premis bahwa matematika bersifat netral. Kedua, terdapat pandangan yang tersembunyi bahwa pengajaran matematika juga dianggap netral. Di muka telah ditunjukkan bahwa setiap pembelajaran adalah terikat dengan nilai-nilai. Ketiga, ada anggapan bahwa keterlibatan berbagai kelompok masyarakat beserta nilainya dalam matematika adalah konsekuensi logisnya. Dan yang terakhir, sejarah menunjukkan bahwa matematika pernah merupakan alat suatu kelompok masyarakat tertentu. Kaum ‘social constructivits’ memandang bahwa matematika merupakan karya cipta manusia melalui kurun waktu tertentu. Semua perbedaan pengetahuan yang dihasilkan merupakan kreativitas manusia yang saling terkait dengan hakekat dan sejarahnya. Akibatnya, matematika dipandang sebagai suatu ilmu pengetahuan yang terikat dengan budaya dan nilai penciptanya dalam konteks budayanya.Sejarah matematika adalah sejarah pembentukannya, tidak hanya yang berhubungan dengan pengungkapan kebenaran, tetapi meliputi permasalahan yang muncul, pengertian, pernyataan, bukti dan teori yang dicipta, yang terkomunikasikan dan mengalami reformulasi oleh individu-individu atau suatu kelompok dengan berbagai kepentingannya. Pandangan demikian memberi konsekuensi bahwa sejarah matematika perlu direvisi.
Kaum absolutis berpendapat bahwa suatu penemuan belumlah merupakan matematika dan matematika modern merupakan hasil yang tak terhindarkan. Ini perlu pembetulan. Bagi kaum ‘social constructivist’ matematika modern bukanlah suatu hasil yang tak terhindarkan, melainkan merupakan evolusi hasil budaya manusia. Joseph (1987) menunjukkan betapa banyaknya tradisi dan penelitian pengembangan matematika berangkat dari pusat peradaban dan kebudayaan manusia. Sejarah matematika perlu menunjuk matematika, filsafat, keadaan sosial dan politik yang bagaimana yang telah mendorong atau menghambat perkembangan matematika. Sebagai contoh, Henry (1971) dalam Ernest (1991: 34) mengakui bahwa calculus dicipta pada masa Descartes, tetapi dia tidak suka menyebutkannya karena ketidaksetujuannya terhadap pendekatan infinitas. Restivo (1985:40), MacKenzie (1981: 53) dan Richards (1980, 1989) dalam Ernest (1991 : 203) menunjukkan betapa kuatnya hubungan antara matematika dengan keadaan sosial; sejarah sosial matematika lebih tergantung kepada kedudukan sosial dan kepentingan pelaku dari pada kepada obyektivitas dan kriteria rasionalitasnya. Kaum ‘social constructivist’ berangkat dari premis bahwa semua pengetahuan merupakan karya cipta. Kelompok ini juga memandang bahwa semua pengetahuan mempunyai landasan yang sama yaitu ‘kesepakatan’. Baik dalam hal asal-usul maupun pembenaran landasannya, pengetahuan manusia mempunyai landasan yang merupakan kesatuan, dan oleh karena itu semua bidang ilmu pengetahuan manusia saling terikat satu dengan yang lain. Akibatnya, sesuai dengan pandangan kaum ‘social constructivist’, matematika tidak dapat dikembangkan jika tanpa terkait dengan pengetahuan lain, dan yang secara bersama-sama mempunyai akarnya, yang dengan sendirinya tidak terbebaskan dari nilai-nilai dari bidang pengetahuan yang diakuinya, karena masing-masing terhubung olehnya.
Karena matematika terkait dengan semua pengetahuan dari diri manusia, maka jelaslah bahwa matematika tidaklah bersifat netral dan bebas nilai. Dengan demikian matematika memerlukan landasan sosial bagi perkembangannya (Davis dan Hers, 1988: 70 dalam Ernest 1991 : 277-279). Shirley (1986: 34) menjelaskan bahwa matematika dapat digolongkan menjadi formal dan informal, terapan dan murni. Berdasarkan pembagian ini, kita dapat membagi kegiatan matematika menjadi 4 (empat) macam, di mana masing-masing mempunyai ciri yang berbeda-beda:
a. matematika formal-murni, termasuk matematika yang dikembangkan pada Universitas dan matematika yang diajarkan di sekolah;
b. matematika formal-terapan, yaitu yang dikembangkan dalam pendidikan maupun di luar, seperti seorang ahli statistik yang bekerja di industri.
c. matematika informal-murni, yaitu matematika yang dikembangkan di luar institusi kependidikan; mungkin melekat pada budaya matematika murni.
d. matematika informal-terapan, yaitu matematika yang digunakan dalam segala kehidupan sehari-hari, termasuk kerajinan, kerja kantor dan perdagangan.
Dowling dalam Ernest (1991: 93), berdasar rekomendasi dari Foucault dan Bernstein, mengembangkan berbagai macam konteks kegiatan matematika. Dia membagi satu dimensi model menjadi 4 (empat) macam yaitu : Production (kreativitas), Recontextualization (pandangan guru dan dasar-dasar kependidikan), Reproduction (kegiatan di kelas) dan Operationalization (penggunaan matematika). Dimensi kedua dari pengembangannya memuat 4 (empat) macam yaoitu: Academic (pada pendidikan tinggi), School (konteks sekolah), Work (kerja) dan Popular (konsumen dan masyarakat).
Dengan memasukkan berbagai macam konteks matematika, berarti kita telah mengakui tesis D’Ambrosio (1985: 25) dalam ‘ethnomathematics’ nya. Tesis tersebut menyatakan bahwa matematika terkait dengan aspek budaya; secara khusus disebutkan bahwa kegiatan-kegiatan seperti hitung-menghitung, mengukur, mendesain, bermain, berbelanja, dst. Merupakan akar dari pengembangan matematika. Dowling dalam Ernest (1991: 120) mengakui bahwa pandangan demikian memang agak kabur; kecuali jika didukung oleh pembenaran tradisi matematika.
DAFTAR PUSTAKA
Ernest, P., 1991, The Philosophy of Mathematics Education, London : The Falmer Press.
Oleh: Marsigit
Para absolutis teguh pendiriannya dalam memandang secara objektif kenetralan matematika, walaupun matematika yang dipromosikan itu sendiri secara implisit mengandung nilai-nilai. Abstrak adalah suatu nilai terhadap konkrit, formal suatu nilai terhadap informal, objektif terhadap subjektif, pembenaran terhadap penemuan, rasionalitas terhadap intuisi, penalaran terhadap emosi, hal-hal umum terhadap hal-hal khusus, teori terhadap praktik, kerja dengan fikiran terhadap kerja dengan tangan, dan seterusnya. Setelah mendaftar macam-macam nilai di atas maka pertanyaannya adalah, bagaimana matematisi berpendapat bahwa matematika adalah netral dan bebas nilai ? Jawaban dari kaum absolutis adalah bahwa niai yang mereka maksud adalah nilai yang melekat pada diri mereka yang berupa kultur, jadi bukan nilai yang melekat secara implisist pada matematika. Diakui bahwa isi dan metode matematika, karena hakekatnya, membuat matematika menjadi abstrak, umum, formal, obyektif, rasional, dan teoritis. Ini adalah hakekat ilmu pengetahuan dan matematika. Tidak ada yang salah bagi yang kongkrit, informal, subyektif, khusus, atau penemuan; mereka hanya tidak termasuk dalam sains, dan tentunya tidak termasuk di dalam matematika (Popper, 1979 dalam Ernest, 1991: 132).
Yang ingin ditandaskan di sini adalah bahwa pandangan kaum absolutis, secara sadar maupun tak sadar, telah merasuk ke dalam matematika melalui definisi-definisi. Dengan perkataan lain, kaum absolutis berpendapat bahwa segala sesuatu yang sesuai dengan nilai-nilai di atas dapat diterima dan yang tidak sesuai tidak dapat diterima. Pernyataan-pernyataan matematika dan bukti-buktinya, yang merupakan hasil dari matematika formal, dipandang dapat melegitimasikan matematika. Sementara, penemuan-penemuan matematika, hasil kerja para matematisi dan proses yang bersifat informal dipandang tidak demikian. Dengan pendekatan ini kaum absolutis membangun matematika yang dianggapnya sebagai netral dan bebas nilai. Dengan pendekatan ini mereka menetapkan kriteria apa yang dapat diterima dan tidak diterima. Hal-hal yang terikat dengan implikasi sosial dan nilai-nilai yang menyertainya, secara eksplisit, dihilangkannya. Tetapi dalam kenyataannya, nilai-nilai yang terkandung dalam hal-hal tersebut di atas, membuat masalah-masalah yang tidak dapat dipecahkan. Hal ini disebabkan karena mendasarkan pada hal-hal yang bersifat formal saja hanya dapat menjangkau pada pembahasan bagian luar dari matematika itu sendiri.
Jika mereka berkehendak menerima kritik yang ada, sebetulnya pandangan mereka tentang matematika yang netral dan bebas nilai juga merupakan suatu nilai yang melekat pada diri mereka dan sulit untuk dilihatnya. Dengan demikian akan muncul pertanyaan berikutnya, siapa yang tertarik dengan pendapatnya ? Inggris dan negara-negara Barat pada umumnya, diperintah oleh kaum laki-laki berkulit putih dari kelas atas. Keadaan demikian mempengaruhi struktur sosial para matematisi di kampus-kampus suatu Universitas, yang kebanyakan didominasi oleh mereka. Nilai-nilai mereka secara sadar dan tak sadar terjabarkan dalam pengembangan matematika sebagai bagian dari usaha dominasi sosial. Oleh karena itu agak janggal kiranya bahwa matematika bersifat netral dan bebas nilai, sementara matematika telah menjadi alat suatu kelompok sosial. Mereka mengunggulkan pria di atas wanita, kulit putih di atas kulit hitam, masyarakat strata menengah di atas strata bawah, untuk kriteria keberhasilan penguasaan pencapaian akademik matematikanya.
Suatu kritik mengatakan, untuk suatu kelompok tertentu, misalnya kelompok kulit putih dari strata atas, mungkin dapat dianggap matematika sebagai netral dan bebas nilai. Namun kritik demikian menghadapi beberapa masalah. Pertama, terdapat premis bahwa matematika bersifat netral. Kedua, terdapat pandangan yang tersembunyi bahwa pengajaran matematika juga dianggap netral. Di muka telah ditunjukkan bahwa setiap pembelajaran adalah terikat dengan nilai-nilai. Ketiga, ada anggapan bahwa keterlibatan berbagai kelompok masyarakat beserta nilainya dalam matematika adalah konsekuensi logisnya. Dan yang terakhir, sejarah menunjukkan bahwa matematika pernah merupakan alat suatu kelompok masyarakat tertentu. Kaum ‘social constructivits’ memandang bahwa matematika merupakan karya cipta manusia melalui kurun waktu tertentu. Semua perbedaan pengetahuan yang dihasilkan merupakan kreativitas manusia yang saling terkait dengan hakekat dan sejarahnya. Akibatnya, matematika dipandang sebagai suatu ilmu pengetahuan yang terikat dengan budaya dan nilai penciptanya dalam konteks budayanya.Sejarah matematika adalah sejarah pembentukannya, tidak hanya yang berhubungan dengan pengungkapan kebenaran, tetapi meliputi permasalahan yang muncul, pengertian, pernyataan, bukti dan teori yang dicipta, yang terkomunikasikan dan mengalami reformulasi oleh individu-individu atau suatu kelompok dengan berbagai kepentingannya. Pandangan demikian memberi konsekuensi bahwa sejarah matematika perlu direvisi.
Kaum absolutis berpendapat bahwa suatu penemuan belumlah merupakan matematika dan matematika modern merupakan hasil yang tak terhindarkan. Ini perlu pembetulan. Bagi kaum ‘social constructivist’ matematika modern bukanlah suatu hasil yang tak terhindarkan, melainkan merupakan evolusi hasil budaya manusia. Joseph (1987) menunjukkan betapa banyaknya tradisi dan penelitian pengembangan matematika berangkat dari pusat peradaban dan kebudayaan manusia. Sejarah matematika perlu menunjuk matematika, filsafat, keadaan sosial dan politik yang bagaimana yang telah mendorong atau menghambat perkembangan matematika. Sebagai contoh, Henry (1971) dalam Ernest (1991: 34) mengakui bahwa calculus dicipta pada masa Descartes, tetapi dia tidak suka menyebutkannya karena ketidaksetujuannya terhadap pendekatan infinitas. Restivo (1985:40), MacKenzie (1981: 53) dan Richards (1980, 1989) dalam Ernest (1991 : 203) menunjukkan betapa kuatnya hubungan antara matematika dengan keadaan sosial; sejarah sosial matematika lebih tergantung kepada kedudukan sosial dan kepentingan pelaku dari pada kepada obyektivitas dan kriteria rasionalitasnya. Kaum ‘social constructivist’ berangkat dari premis bahwa semua pengetahuan merupakan karya cipta. Kelompok ini juga memandang bahwa semua pengetahuan mempunyai landasan yang sama yaitu ‘kesepakatan’. Baik dalam hal asal-usul maupun pembenaran landasannya, pengetahuan manusia mempunyai landasan yang merupakan kesatuan, dan oleh karena itu semua bidang ilmu pengetahuan manusia saling terikat satu dengan yang lain. Akibatnya, sesuai dengan pandangan kaum ‘social constructivist’, matematika tidak dapat dikembangkan jika tanpa terkait dengan pengetahuan lain, dan yang secara bersama-sama mempunyai akarnya, yang dengan sendirinya tidak terbebaskan dari nilai-nilai dari bidang pengetahuan yang diakuinya, karena masing-masing terhubung olehnya.
Karena matematika terkait dengan semua pengetahuan dari diri manusia, maka jelaslah bahwa matematika tidaklah bersifat netral dan bebas nilai. Dengan demikian matematika memerlukan landasan sosial bagi perkembangannya (Davis dan Hers, 1988: 70 dalam Ernest 1991 : 277-279). Shirley (1986: 34) menjelaskan bahwa matematika dapat digolongkan menjadi formal dan informal, terapan dan murni. Berdasarkan pembagian ini, kita dapat membagi kegiatan matematika menjadi 4 (empat) macam, di mana masing-masing mempunyai ciri yang berbeda-beda:
a. matematika formal-murni, termasuk matematika yang dikembangkan pada Universitas dan matematika yang diajarkan di sekolah;
b. matematika formal-terapan, yaitu yang dikembangkan dalam pendidikan maupun di luar, seperti seorang ahli statistik yang bekerja di industri.
c. matematika informal-murni, yaitu matematika yang dikembangkan di luar institusi kependidikan; mungkin melekat pada budaya matematika murni.
d. matematika informal-terapan, yaitu matematika yang digunakan dalam segala kehidupan sehari-hari, termasuk kerajinan, kerja kantor dan perdagangan.
Dowling dalam Ernest (1991: 93), berdasar rekomendasi dari Foucault dan Bernstein, mengembangkan berbagai macam konteks kegiatan matematika. Dia membagi satu dimensi model menjadi 4 (empat) macam yaitu : Production (kreativitas), Recontextualization (pandangan guru dan dasar-dasar kependidikan), Reproduction (kegiatan di kelas) dan Operationalization (penggunaan matematika). Dimensi kedua dari pengembangannya memuat 4 (empat) macam yaoitu: Academic (pada pendidikan tinggi), School (konteks sekolah), Work (kerja) dan Popular (konsumen dan masyarakat).
Dengan memasukkan berbagai macam konteks matematika, berarti kita telah mengakui tesis D’Ambrosio (1985: 25) dalam ‘ethnomathematics’ nya. Tesis tersebut menyatakan bahwa matematika terkait dengan aspek budaya; secara khusus disebutkan bahwa kegiatan-kegiatan seperti hitung-menghitung, mengukur, mendesain, bermain, berbelanja, dst. Merupakan akar dari pengembangan matematika. Dowling dalam Ernest (1991: 120) mengakui bahwa pandangan demikian memang agak kabur; kecuali jika didukung oleh pembenaran tradisi matematika.
DAFTAR PUSTAKA
Ernest, P., 1991, The Philosophy of Mathematics Education, London : The Falmer Press.
Kant on The Objectivity and Subjectivity of Knowledge
Collected from many resources by Marsigit
The Objective Form
Kant, 1788, claimed that reason is concerned with the grounds of determination of the will, which is a faculty either to produce objects corresponding to ideas, or to determine ourselves to the effecting of such objects whether the physical power is sufficient or not; that is, to determine our causality; and reason can at least attain so far as to determine the will, and has always objective reality in so far as it is the volition only that is in question. Kant argued that Practical principles are propositions which contain a general determination of the will, having under it several practical rules. They are subjective, or maxims, when the condition is regarded by the subject as valid only for his own will, but are objective, or practical laws, when the condition is recognized as objective, that is, valid for the will of every rational being. Further he indicated that the practical rule is always a product of reason, because it prescribes action as a means to the effect; but in the case of a being with whom reason does not of itself determine the will, this rule is an imperative that expresses the objective necessitation of the action and signifies that, if reason completely determined the will, the action would inevitably take place according to this rule.
According to Kant, 1788, imperatives, therefore, are objectively valid, and are quite distinct from maxims, which are subjective principles. For Kant, the objectively valid either determine the conditions of the causality of the rational being as an efficient cause that is merely in reference to the effect and the means of attaining it; or they determine the will only, whether it is adequate to the effect or not and it would be hypothetical imperatives, and contain mere precepts of skill. Further Kant noted that reason may give laws it is necessary that it should only need to presuppose itself, because rules are objectively and universally valid only when they hold without any contingent subjective conditions, which distinguish one rational being from another. Kant, 1788, claimed that the principle of determination would still be only subjectively valid and merely empirical, and would not possess the necessity which is conceived in every law that is an objective necessity arising from a priori grounds; unless, indeed, we hold this necessity to be not at all practical, but merely physical in which our action is as inevitably determined by our inclination. Kant argued that it would be better to maintain that there are no practical laws at all, but only counsels for the service of our desires, than to raise merely subjective principles to the rank of practical laws, which have objective necessity, and not merely subjective, and which must be known by reason a priori, not by experience. Kant claimed that even the rules of corresponding phenomena are only called laws of nature when we either know them really a priori or suppose that they would be known a priori from objective grounds if our insight reached further.
Kant, 1781, claimed that the categories have the function of prescribing the general form that this detailed order must take and belong to the very framework of knowledge; however, although they are indispensable for objective knowledge, the sole knowledge that the categories can yield is of objects of possible experience; they yield valid and real knowledge only when they are ordering what is given through sense in space and time. In the "Transcendental Dialectic" Kant turned to consideration of a priori synthetic judgments in metaphysics and claimed that the situation is just the reverse from what it was in mathematics and physics. Kant argued that metaphysics cuts itself off from sense experience in attempting to go beyond it and, for this very reason, fails to attain a single true a priori synthetic judgment. To justify this claim, Kant analyzed the use that metaphysics makes of the concept of the unconditioned. Reason, according to Kant, seeks for the unconditioned or absolute in three distinct spheres: first, in philosophical psychology it seeks for an absolute subject of knowledge; second, in the sphere of cosmology, it seeks for an absolute beginning of things in time, for an absolute limit to them in space, and for an absolute limit to their divisibility; and third, in the sphere of theology, it seeks for an absolute condition for all things. Kant, 1788, summed up that on the ground that we have no knowledge of any other rational beings besides man, we should have a right to suppose them to be of the same nature as we know ourselves to be that is we should really know them; then we omit to mention that universal assent does not prove the objective validity of a judgement that is its validity as a cognition and although this universal assent should accidentally happened, it could furnish no proof of agreement with the object; and on the contrary, it is the objective validity which alone constitutes the basis of a necessary universal consent.
Hegel, GWE., 1830, stated that Kant gives the name objective to what is thought, to the universal and necessary; thoughts, according to Kant, are only our thoughts that is separated by an impassable gulf from the thing, as it exists apart from our knowledge and the true objectivity of thinking means that the thoughts, far from being merely ours, must at the same time be the real essence of the things, and of whatever is an object to us. Hegel clarified that the specific ground of the categories is declared by the Critical system to lie in the primary identity of the ‘I’ in thought what Kant calls the transcendental unity of self-consciousness. Kant argued that the impressions from feeling and perception are a multiplicity or miscellany of elements and the multiplicity is equally conspicuous in their form; for sense is marked by a mutual exclusion of members and that under two aspects, namely space and time, which, being the forms, that is to say, the universal type of perception, are themselves a priori. Hegel noted Kant that this congeries, afforded by sensation and perception and must however be reduced to an identity or primary synthesis and to accomplish this the ‘I’ brings it in relation to itself and unites it there in one consciousness which Kant calls ‘pure apperception’ that is the specific modes in which the ego refers to itself the multiplicity of sense are the pure concepts of the understanding that is the Categories. Kant, 1788, designated that for it is every man's own special feeling of pleasure and pain that decides in what he is to place his happiness, and even in the same subject this will vary with the difference of his wants according as this feeling changes, and thus a law which is subjectively necessary is objectively a very contingent practical principle, which can and must be very different in different subjects and therefore can never furnish a law; since, in the desire for happiness it is not the form that is decisive, whether we are to expect pleasure in following the law, and how much. Principles of self-love may, indeed, contain universal precepts of skill, but in that case they are merely theoretical principles.
Hegel, GWE., 1830 elaborated that Kant therefore holds that the categories have their source in the ego and that the ego consequently supplies the characteristics of universality and necessity; if we observe what we have before us primarily, we may describe it as a congeries or diversity and in the categories we find the simple points or units, to which this congeries is made to converge. According to Kant, the world of sense is a scene of mutual exclusion: its being is outside itself that is the fundamental feature of the sensible; however, thought or ego occupies a position the very reverse of the sensible, with its mutual exclusions, and its being outside itself. Kant held that the ‘I’ is the primary identity at one with itself and all at home in itself and expresses the mere act of bringing that is to-bear-upon-self and whatever is placed in this unit or focus is affected by it and transformed into it. Kant claimed that the ‘I’ is as it were the crucible and the fire which consumes the loose plurality of sense and reduces it to unity and called this process as pure apperception in distinction from the common apperception, to which the plurality it receives is a plurality still; whereas pure apperception is rather an act by which the ‘I’ makes the materials that is mine.
Further, Hegel, GWE., 1830, maintained that Kant’s meaning of transcendental may be gathered by the way he distinguishes it from transcendent and the transcendent may be said to be what steps out beyond the categories of the understanding that is a sense in which the term is first employed in mathematics and thus in geometry we are told to conceive the circumference of a circle as formed of an infinite number of infinitely small straight lines. Hegel then specified that, according to Kant, characteristics which the understanding holds to be totally different, the straight line and the curve, are expressly invested with identity and another transcendent of the same kind is the self-consciousness which is identical with itself and infinite in itself, as distinguished from the ordinary consciousness which derives its form and tone from finite materials. He noted that Kant called that unity of self-consciousness as transcendental only; and Kant meant thereby that the unity was only in our minds and did not attach to the objects apart from our knowledge of them.
Hegel, GWE., 1830 indicated that Kant’s categories may be viewed in two aspects that are by sensing the perception their instrumentality to objectivity and experience and by uniting these notions to our consciousness merely in which they are consequently conditioned by the material given to them, and having nothing of their own they can be applied to use only within the range of experience; the categories originate in the unity of self-consciousness that any knowledge which is gained by their means has nothing objective in it, and that the very objectivity claimed for them is only subjective as well as that common type of idealism known as subjective idealism. According to Hegel, Kant
simply considered the abstract form of subjectivity and objectivity, and that even in such a partial way that the former aspect, that of subjectivity, is retained as a final and purely affirmative term of thought and in the second part, however, when Kant examines the application, as it is called, which reason makes of the categories in order to know its objects, the content of the categories, at least in some points of view, comes in for discussion: or, at any rate, an opportunity presented itself for a discussion of the question.
Hegel, GWE., 1830, also elaborated that in The Practical Reason, Kant defined a thinking Will as that that determines itself on universal principles in which its office is to give objective, imperative laws of freedom laws, that is, which state what ought to happen. According to Kant, the warrant for thus assuming thought to be an activity which makes itself felt objectively, that is, to be really a reason, is the alleged possibility of proving practical freedom by experience, that is, of showing it in the phenomenon of self-consciousness. According to Hegel, Kant perceived that this experience in consciousness is at once met by all that the necessitiest produces from contrary experience, particularly by the sceptical induction from the endless diversity of what men regard as right and duty that is from the diversity apparent in those professedly objective laws of freedom. Kant claimed that there must be no contradiction in the act of self- determination; however, the Practical Reason does not confine the universal principle of the Good to its own inward regulation; it first becomes practical, in the true sense of the word, when it insists on the Good being manifested in the world with an outward objectivity, and requires that the thought shall be objective throughout, and not merely subjective.
Kant, 1788, claimed that the reality of the concept of freedom is proved by an apodeictic law of practical reason, it is the keystone of the whole system of pure reason, even the speculative, and all other concepts which, as being mere ideas, remain in it unsupported, now attach themselves to this concept, and by it obtain consistence and objective reality; that is to say, their possibility is proved by the fact that freedom actually exists, for this idea is revealed by the moral law. Kant insisted that as far as speculative reason is concerned, is a merely subjective principle of assent, which, however, is objectively valid for a reason equally pure but practical, and this principle, by means of the concept of freedom, assures objective reality and authority to the ideas of God and immortality. Kant further denied objective reality to the supersensible use of the categories in speculation and yet admited this reality with respect to the objects of pure practical reason. Kant specified that there is a contradiction to try to extract necessity from a principle of experience and to try by this to give a judgment true universality without which there is no rational inference, not even inference from analogy, which is at least a presumed universality and objective necessity. Kant then insisted that to substitute subjective necessity, that is custom for objective, which exists only in a priori judgments, is to deny to reason the power of judging about the object that is of knowing it, and what belongs to it. Kant concluded that as to attempting to remedy the want of objective and consequently universal validity by saying that we can see no ground for attributing any other sort of knowledge to other rational beings, if this reasoning were valid, our ignorance would do more for the enlargement of our knowledge than all our meditation.
Kant, 1788, claimed that the principle of determination would still be only subjectively valid and merely empirical, and would not possess the necessity which is conceived in every law that is an objective necessity arising from a priori grounds; unless, indeed, we hold this necessity to be not at all practical, but merely physical in which our action is as inevitably determined by our inclination. Kant argued that it would be better to maintain that there are no practical laws at all, but only counsels for the service of our desires, than to raise merely subjective principles to the rank of practical laws, which have objective necessity, and not merely subjective, and which must be known by reason a priori, not by experience. Kant claimed that even the rules of corresponding phenomena are only called laws of nature when we either know them really a priori or suppose that they would be known a priori from objective grounds if our insight reached further.
The Subjective Forms
Chignell, 2004, described that in the Critique of the Transcendent Method, Kant asserted that the subject is endowed with a priori form is of thought or categories; while Kant notified of not acknowledging forms of existence in the external world and when we examine them well, we realize that there are forms of existence that correspond to the forms of thought. Therefore, Chignell, concluded that the form of time and space is not only a subjective form, but an objective form as well. In term of aesthetics, Kant makes clear that these are the only four possible aesthetic judgments, as he relates them to the Table of Judgment from the Critic of Pure Reason; they are purely subjective judgments, based on inclination alone. According to Kant, the beautiful and the sublime occupy a space between the agreeable and the good, therefore they are as "subjective universal" judgments.
Kant, 1790, in The Critique Of Judgment, claimed that in a judgment of taste the universality of delight is only represented as subjective. Kant supported this argument by elaborated that the particular form of the universality of an aesthetic judgment is a significant feature for the transcendental philosopher. Kant said that the taste of reflection has often enough to put up with a rude dismissal of its claims to universal validity of its judgment and capable of demanding this agreement in its universality; such agreement it does in fact require from every one for each of its judgments of taste the persons who pass these judgments not quarreling over the possibility of such a claim, but only failing in particular cases to come to terms as to the correct application of this faculty. Kant specified that a universality which does not rest upon concepts of the object does not involve any objective quantity of the judgment, except that it is subjective. Kant noted that the universality that is the expression of general validity, denotes the validity of the reference of a representation, not to the cognitive faculties, but to the feeling of pleasure or displeasure for every subject.
Kant, 1790, claimed that a judgment that has objective universal validity has always got the subjective also, that the judgment is valid for everything which is contained under a given concept, it is valid also for all who represent an object by means of this concept. Kant maintained that we feel to be associated in the mind with the representation of the object is nothing else than its subjective finality for judgment; since judgment can only be directed to the subjective conditions of its employment in general, it follows that the accordance of a representation with these conditions of the judgment must admit of being assumed valid a priori for every one. Kant said that in order to be justified in claiming universal agreement an aesthetic judgment merely resting on subjective grounds, it is sufficient to assume: first, that the subjective conditions of this faculty of aesthetic judgement are identical with all men in what concerns the relation of the cognitive faculties; and second, that the judgment has paid regard merely to this relation.
Reference:
Chignell, A., 2004, The Problem of Particularity in Kant’s Aesthetic Theory, Aesthetics and Philosophy of the Arts, http://www.bu.edu/wcp/MainAest.htm
Hegel, GWE., 1830, THE CRITICAL PHILOSOPHY: Part One
Hoover, A.J., 2004, Arguments for the Existence ofGod http://www.ditex.com/index.html
Kant, I., 1781, Critic of Pure Reason, Translatedby J.M.D. Meiklejohn
Kant, I., 1788, The Critic of Practical Reason, http://www.google.search
Kant, I., 1790, The Critic of Judgment, translated by James Creed Meredith
Catatan:
Arikel ini hanya diperuntukan bagi mahasiswa S2 yang sedang kuliah Filsafat Ilmu dari P. Marsigit
The Objective Form
Kant, 1788, claimed that reason is concerned with the grounds of determination of the will, which is a faculty either to produce objects corresponding to ideas, or to determine ourselves to the effecting of such objects whether the physical power is sufficient or not; that is, to determine our causality; and reason can at least attain so far as to determine the will, and has always objective reality in so far as it is the volition only that is in question. Kant argued that Practical principles are propositions which contain a general determination of the will, having under it several practical rules. They are subjective, or maxims, when the condition is regarded by the subject as valid only for his own will, but are objective, or practical laws, when the condition is recognized as objective, that is, valid for the will of every rational being. Further he indicated that the practical rule is always a product of reason, because it prescribes action as a means to the effect; but in the case of a being with whom reason does not of itself determine the will, this rule is an imperative that expresses the objective necessitation of the action and signifies that, if reason completely determined the will, the action would inevitably take place according to this rule.
According to Kant, 1788, imperatives, therefore, are objectively valid, and are quite distinct from maxims, which are subjective principles. For Kant, the objectively valid either determine the conditions of the causality of the rational being as an efficient cause that is merely in reference to the effect and the means of attaining it; or they determine the will only, whether it is adequate to the effect or not and it would be hypothetical imperatives, and contain mere precepts of skill. Further Kant noted that reason may give laws it is necessary that it should only need to presuppose itself, because rules are objectively and universally valid only when they hold without any contingent subjective conditions, which distinguish one rational being from another. Kant, 1788, claimed that the principle of determination would still be only subjectively valid and merely empirical, and would not possess the necessity which is conceived in every law that is an objective necessity arising from a priori grounds; unless, indeed, we hold this necessity to be not at all practical, but merely physical in which our action is as inevitably determined by our inclination. Kant argued that it would be better to maintain that there are no practical laws at all, but only counsels for the service of our desires, than to raise merely subjective principles to the rank of practical laws, which have objective necessity, and not merely subjective, and which must be known by reason a priori, not by experience. Kant claimed that even the rules of corresponding phenomena are only called laws of nature when we either know them really a priori or suppose that they would be known a priori from objective grounds if our insight reached further.
Kant, 1781, claimed that the categories have the function of prescribing the general form that this detailed order must take and belong to the very framework of knowledge; however, although they are indispensable for objective knowledge, the sole knowledge that the categories can yield is of objects of possible experience; they yield valid and real knowledge only when they are ordering what is given through sense in space and time. In the "Transcendental Dialectic" Kant turned to consideration of a priori synthetic judgments in metaphysics and claimed that the situation is just the reverse from what it was in mathematics and physics. Kant argued that metaphysics cuts itself off from sense experience in attempting to go beyond it and, for this very reason, fails to attain a single true a priori synthetic judgment. To justify this claim, Kant analyzed the use that metaphysics makes of the concept of the unconditioned. Reason, according to Kant, seeks for the unconditioned or absolute in three distinct spheres: first, in philosophical psychology it seeks for an absolute subject of knowledge; second, in the sphere of cosmology, it seeks for an absolute beginning of things in time, for an absolute limit to them in space, and for an absolute limit to their divisibility; and third, in the sphere of theology, it seeks for an absolute condition for all things. Kant, 1788, summed up that on the ground that we have no knowledge of any other rational beings besides man, we should have a right to suppose them to be of the same nature as we know ourselves to be that is we should really know them; then we omit to mention that universal assent does not prove the objective validity of a judgement that is its validity as a cognition and although this universal assent should accidentally happened, it could furnish no proof of agreement with the object; and on the contrary, it is the objective validity which alone constitutes the basis of a necessary universal consent.
Hegel, GWE., 1830, stated that Kant gives the name objective to what is thought, to the universal and necessary; thoughts, according to Kant, are only our thoughts that is separated by an impassable gulf from the thing, as it exists apart from our knowledge and the true objectivity of thinking means that the thoughts, far from being merely ours, must at the same time be the real essence of the things, and of whatever is an object to us. Hegel clarified that the specific ground of the categories is declared by the Critical system to lie in the primary identity of the ‘I’ in thought what Kant calls the transcendental unity of self-consciousness. Kant argued that the impressions from feeling and perception are a multiplicity or miscellany of elements and the multiplicity is equally conspicuous in their form; for sense is marked by a mutual exclusion of members and that under two aspects, namely space and time, which, being the forms, that is to say, the universal type of perception, are themselves a priori. Hegel noted Kant that this congeries, afforded by sensation and perception and must however be reduced to an identity or primary synthesis and to accomplish this the ‘I’ brings it in relation to itself and unites it there in one consciousness which Kant calls ‘pure apperception’ that is the specific modes in which the ego refers to itself the multiplicity of sense are the pure concepts of the understanding that is the Categories. Kant, 1788, designated that for it is every man's own special feeling of pleasure and pain that decides in what he is to place his happiness, and even in the same subject this will vary with the difference of his wants according as this feeling changes, and thus a law which is subjectively necessary is objectively a very contingent practical principle, which can and must be very different in different subjects and therefore can never furnish a law; since, in the desire for happiness it is not the form that is decisive, whether we are to expect pleasure in following the law, and how much. Principles of self-love may, indeed, contain universal precepts of skill, but in that case they are merely theoretical principles.
Hegel, GWE., 1830 elaborated that Kant therefore holds that the categories have their source in the ego and that the ego consequently supplies the characteristics of universality and necessity; if we observe what we have before us primarily, we may describe it as a congeries or diversity and in the categories we find the simple points or units, to which this congeries is made to converge. According to Kant, the world of sense is a scene of mutual exclusion: its being is outside itself that is the fundamental feature of the sensible; however, thought or ego occupies a position the very reverse of the sensible, with its mutual exclusions, and its being outside itself. Kant held that the ‘I’ is the primary identity at one with itself and all at home in itself and expresses the mere act of bringing that is to-bear-upon-self and whatever is placed in this unit or focus is affected by it and transformed into it. Kant claimed that the ‘I’ is as it were the crucible and the fire which consumes the loose plurality of sense and reduces it to unity and called this process as pure apperception in distinction from the common apperception, to which the plurality it receives is a plurality still; whereas pure apperception is rather an act by which the ‘I’ makes the materials that is mine.
Further, Hegel, GWE., 1830, maintained that Kant’s meaning of transcendental may be gathered by the way he distinguishes it from transcendent and the transcendent may be said to be what steps out beyond the categories of the understanding that is a sense in which the term is first employed in mathematics and thus in geometry we are told to conceive the circumference of a circle as formed of an infinite number of infinitely small straight lines. Hegel then specified that, according to Kant, characteristics which the understanding holds to be totally different, the straight line and the curve, are expressly invested with identity and another transcendent of the same kind is the self-consciousness which is identical with itself and infinite in itself, as distinguished from the ordinary consciousness which derives its form and tone from finite materials. He noted that Kant called that unity of self-consciousness as transcendental only; and Kant meant thereby that the unity was only in our minds and did not attach to the objects apart from our knowledge of them.
Hegel, GWE., 1830 indicated that Kant’s categories may be viewed in two aspects that are by sensing the perception their instrumentality to objectivity and experience and by uniting these notions to our consciousness merely in which they are consequently conditioned by the material given to them, and having nothing of their own they can be applied to use only within the range of experience; the categories originate in the unity of self-consciousness that any knowledge which is gained by their means has nothing objective in it, and that the very objectivity claimed for them is only subjective as well as that common type of idealism known as subjective idealism. According to Hegel, Kant
simply considered the abstract form of subjectivity and objectivity, and that even in such a partial way that the former aspect, that of subjectivity, is retained as a final and purely affirmative term of thought and in the second part, however, when Kant examines the application, as it is called, which reason makes of the categories in order to know its objects, the content of the categories, at least in some points of view, comes in for discussion: or, at any rate, an opportunity presented itself for a discussion of the question.
Hegel, GWE., 1830, also elaborated that in The Practical Reason, Kant defined a thinking Will as that that determines itself on universal principles in which its office is to give objective, imperative laws of freedom laws, that is, which state what ought to happen. According to Kant, the warrant for thus assuming thought to be an activity which makes itself felt objectively, that is, to be really a reason, is the alleged possibility of proving practical freedom by experience, that is, of showing it in the phenomenon of self-consciousness. According to Hegel, Kant perceived that this experience in consciousness is at once met by all that the necessitiest produces from contrary experience, particularly by the sceptical induction from the endless diversity of what men regard as right and duty that is from the diversity apparent in those professedly objective laws of freedom. Kant claimed that there must be no contradiction in the act of self- determination; however, the Practical Reason does not confine the universal principle of the Good to its own inward regulation; it first becomes practical, in the true sense of the word, when it insists on the Good being manifested in the world with an outward objectivity, and requires that the thought shall be objective throughout, and not merely subjective.
Kant, 1788, claimed that the reality of the concept of freedom is proved by an apodeictic law of practical reason, it is the keystone of the whole system of pure reason, even the speculative, and all other concepts which, as being mere ideas, remain in it unsupported, now attach themselves to this concept, and by it obtain consistence and objective reality; that is to say, their possibility is proved by the fact that freedom actually exists, for this idea is revealed by the moral law. Kant insisted that as far as speculative reason is concerned, is a merely subjective principle of assent, which, however, is objectively valid for a reason equally pure but practical, and this principle, by means of the concept of freedom, assures objective reality and authority to the ideas of God and immortality. Kant further denied objective reality to the supersensible use of the categories in speculation and yet admited this reality with respect to the objects of pure practical reason. Kant specified that there is a contradiction to try to extract necessity from a principle of experience and to try by this to give a judgment true universality without which there is no rational inference, not even inference from analogy, which is at least a presumed universality and objective necessity. Kant then insisted that to substitute subjective necessity, that is custom for objective, which exists only in a priori judgments, is to deny to reason the power of judging about the object that is of knowing it, and what belongs to it. Kant concluded that as to attempting to remedy the want of objective and consequently universal validity by saying that we can see no ground for attributing any other sort of knowledge to other rational beings, if this reasoning were valid, our ignorance would do more for the enlargement of our knowledge than all our meditation.
Kant, 1788, claimed that the principle of determination would still be only subjectively valid and merely empirical, and would not possess the necessity which is conceived in every law that is an objective necessity arising from a priori grounds; unless, indeed, we hold this necessity to be not at all practical, but merely physical in which our action is as inevitably determined by our inclination. Kant argued that it would be better to maintain that there are no practical laws at all, but only counsels for the service of our desires, than to raise merely subjective principles to the rank of practical laws, which have objective necessity, and not merely subjective, and which must be known by reason a priori, not by experience. Kant claimed that even the rules of corresponding phenomena are only called laws of nature when we either know them really a priori or suppose that they would be known a priori from objective grounds if our insight reached further.
The Subjective Forms
Chignell, 2004, described that in the Critique of the Transcendent Method, Kant asserted that the subject is endowed with a priori form is of thought or categories; while Kant notified of not acknowledging forms of existence in the external world and when we examine them well, we realize that there are forms of existence that correspond to the forms of thought. Therefore, Chignell, concluded that the form of time and space is not only a subjective form, but an objective form as well. In term of aesthetics, Kant makes clear that these are the only four possible aesthetic judgments, as he relates them to the Table of Judgment from the Critic of Pure Reason; they are purely subjective judgments, based on inclination alone. According to Kant, the beautiful and the sublime occupy a space between the agreeable and the good, therefore they are as "subjective universal" judgments.
Kant, 1790, in The Critique Of Judgment, claimed that in a judgment of taste the universality of delight is only represented as subjective. Kant supported this argument by elaborated that the particular form of the universality of an aesthetic judgment is a significant feature for the transcendental philosopher. Kant said that the taste of reflection has often enough to put up with a rude dismissal of its claims to universal validity of its judgment and capable of demanding this agreement in its universality; such agreement it does in fact require from every one for each of its judgments of taste the persons who pass these judgments not quarreling over the possibility of such a claim, but only failing in particular cases to come to terms as to the correct application of this faculty. Kant specified that a universality which does not rest upon concepts of the object does not involve any objective quantity of the judgment, except that it is subjective. Kant noted that the universality that is the expression of general validity, denotes the validity of the reference of a representation, not to the cognitive faculties, but to the feeling of pleasure or displeasure for every subject.
Kant, 1790, claimed that a judgment that has objective universal validity has always got the subjective also, that the judgment is valid for everything which is contained under a given concept, it is valid also for all who represent an object by means of this concept. Kant maintained that we feel to be associated in the mind with the representation of the object is nothing else than its subjective finality for judgment; since judgment can only be directed to the subjective conditions of its employment in general, it follows that the accordance of a representation with these conditions of the judgment must admit of being assumed valid a priori for every one. Kant said that in order to be justified in claiming universal agreement an aesthetic judgment merely resting on subjective grounds, it is sufficient to assume: first, that the subjective conditions of this faculty of aesthetic judgement are identical with all men in what concerns the relation of the cognitive faculties; and second, that the judgment has paid regard merely to this relation.
Reference:
Chignell, A., 2004, The Problem of Particularity in Kant’s Aesthetic Theory, Aesthetics and Philosophy of the Arts, http://www.bu.edu/wcp/MainAest.htm
Hegel, GWE., 1830, THE CRITICAL PHILOSOPHY: Part One
Hoover, A.J., 2004, Arguments for the Existence ofGod http://www.ditex.com/index.html
Kant, I., 1781, Critic of Pure Reason, Translatedby J.M.D. Meiklejohn
Kant, I., 1788, The Critic of Practical Reason, http://www.google.search
Kant, I., 1790, The Critic of Judgment, translated by James Creed Meredith
Catatan:
Arikel ini hanya diperuntukan bagi mahasiswa S2 yang sedang kuliah Filsafat Ilmu dari P. Marsigit
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