aziz87
18-01-2018, 03:40 PM
مفارقة بناخ - تارسكي Banach -Tarski Paradox
المفارقة الأكثر إثارة للحيرة والدهشة في http://montada.echoroukonline.com/data:image/jpeg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wCEAAkGBxIQEBUSEBMQERASEhIQEREQFRAPFhIWFhEWFxUVFR UYHTQgGCYmGxUVITEiJSkrLjEuFx8zRDMtNyguLjcBCgoKDg0O GhAQGzImICUtLS0tLTctLTcyNy0tLy0tLS0rMC0vLi8tLTItLS 8tLS0vLS0tLS0tLS0tLTEtLS0tLf/AABEIAKsBJwMBIgACEQEDEQH/xAAbAAEAAwEBAQEAAAAAAAAAAAAABQYHBAMCAf/EAD8QAAEDAgQEAwMICQQDAAAAAAEAAgMEEQUSITEGE0FRImFxg ZGhBxQjMlKxssEVM0JDYnLC0eEkgqLwU2OS/8QAGgEBAAMBAQEAAAAAAAAAAAAAAAIDBAEFBv/EACYRAQACAgICAQQCAwAAAAAAAAABAgMRBDESIUETIlGBFDIFc fD/2gAMAwEAAhEDEQA/ANxREQEREBERAREQEREBERAREQEREBERAREQEREBERAREQEREB ERAREQEREBERAREQEREBERAREQEREBERAREQEREBERAREQEREB ERAREQEREBERAREQEREBERAREQEREBERAREQEREBERAREQEREB FCY5xNDTO5TQ+oqiPBSwDPIfN3Rg83WU0EH6iKLxXGBBNTRZC4 1Urog64GTLGXkkW12QSiLhpcTbJKYwydpbzPE+KRjDkeGmzyLG 5Nx3AJ6LuQEXzJIGgucQ1rQSXOIAAG5JOyjW4yDXGkDDcUwquZ cWsZMmW3xuglEUGOIgaqGn5MrDOKk3lswtEDrXy63Dtwb7EKcQ EXDS4m2SQxiOdpbzPG+KRjDkeGmzyLG5Nx3AulJibZJHRhk7S3 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من اليمين : ستيفن بناخ ومن اليسار : ألفريد تارسكي.
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هل حاولت من قبل أن تحضر كرة لها قطر معين وتقوم بتجزئتها إلى 5 قطع، ومن ثم تجميعها مرة أخرى من دون أي تغيير في شكل القطع (يُسمح فقط تحريك القطع وتدويرها)، فتصبح كرتان، لهما نفس شكل وحجم الكرة الأصلي؟!
أو قمت بتجربة أن تحضر قالباً من الشيكولاته وتقوم بتجزئته ومن ثم عمل تدوير وتغيير لمكان القطع، فتجد أن الحجم سيقل عن الحجم الأصلي؟! عليك بالتجربة الآن..
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الحقيقة إنه شيء مدهش، وهذا هو المعروف بـ مفارقة باناخ – تارسكى Banach – Tarski Paradox.
تعريف المفارقة
وفقا لتعريف الفيلسوف الإنجليزي مارك سينسبري، المفارقة تعني:
خاتمة قد تبدو غير مقبولة، مستمدة من فرضيات قد تبدو مقبولة من خلال منطق قد يبدو مقبولاً
والكلمة مستوحاة من الثقافة اليونانية القديمة وتفسيرها: para بجانب، doxa إيمان، رأي. المعنى الأصلي هو شيء يبدو للوهلة الأولى بدون قيمة أو غير ممكن، ولكن بعد تفكير عميق يتضح أنه صحيح!
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والمفارفة في المنطق الرياضي هي جملة خبرية تناقض المنطق، فلا يمكن أن نقول عنها أنها صحيحة، كما لا يمكن أن نقول أنها خاطئة، فإذا أخذنا الجملة الآتية : “أنا أكذب الآن” فإذا افترضنا أنني صادق يعني أن الجملة صحيحة الأمر الذي يناقض كوني صادقاً، أما إذا افترضنا أنني كاذب فالجملة خاطئة ونفيها صحيح، أي أنني صادق ويناقض كوني كاذباً.
مفارقة باناخ – تارسكي
مفارقة باناخ – تارسكي تعتبر من أكثر المفارقات المثيرة للدهشة والحيرة فى علم الرياضيات. لذلك تُعرف باسم المفارقة السحرية.
نص المفارقة: إذا قمت بتقسيم كرة ذات حجم أو قطر يساوي (أ) بطريقة معينة ثم قمت بتجميع هذه الأجزاء بطريقة معينة، فإنه يمكنك أن تكون كرتين من الحجم أو القطر (أ).
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وهنا تكمن المشكلة فى نص المفارقة من خلال أنه حينما تريد أن تجمع الكرة مرة أخرى بعد تجزئتها، فتجد أن هناك حجماً آخر مضافاً لا يُعلم مصدره (جزء سحري). باناخ وتارسكي برهنا صحة وإمكانية وجود هذه الظاهرة رياضياُ ونظرياً ولكن فقط وفقاً لمبدأ بديهية الاختيار،ولقد اعتبراها نقداً لصحة هذا المبدأ الذي طالما كان مثيراً للجدل.
حيث أن جميع القوانين المستخدمة في الإثبات تحفظ الحجم وبهذا يكون الخطأ راجعاً للمبدأ. فعند استخدام مبدأ غرابة الاختيار -المرشح حديثاً كبديل- مثلاً لا يمكن إثبات المفارقة. ولكن علماء الرياضيات البحتة لا زالوا يتمسكون بالمبدأ لاعتقادهم بوجود خطأ في مكان ما في مثل هذا الإثباتات غير المنطقية.
فقد قمنا بسؤال علم الرياضيات عن هذة المفارقة، قال: نعم، ولكن لماذا لا يمكن تطبيق هذه النظريّة على أرض الواقع؟
لكن، أي مادة (كما المعروف)، لا يمكن تقسيمها إلى مالانهاية، ولو كان هذا مسموحاً كان من السهل تطبيق هذة المفارقة. لكن مع وجود هذة المفارقة نرى اختفاء لتعريف الحجم، بمعنى: إذا حاولنا مع هذه النظرية تعريف الحجم فيصبح التعريف المنطقي بما يناسب النظرية (2=1)، وهذا مايخالف المنطق الرياضي المتبع.
ولهذة النظرية شكل آخر: إذا كان معك كرة بحجم (كرة تنس الطاولة) وقمت بتجزئتها أيضاً ومن ثم إعادة تجميعها، فيصبح معك كرة بحجم قرص الشمس. وهذا هو الغريب والمدهش والعجيب في هذه المفارقة، في أنها تبحر بخيالك وتخبرك بأشياء بعيدة عن المنطق. لذلك نجد في علم الرياضيات ما قد يرفضه الواقع لكن يسمح به الخيال، ولا شيء يكون مرفوضاً في عالم الخيال.
علم الرياضيات مليء بالألغاز البعيدة عن المنطق البشري، فهو كان السبب الرئيسي لوضع الخوارزميات التى نحن نعيش فيها الآن، التي هي سبب كل التقدم والتكنولوجيا والازدهار الذي ينمو فيه الانسان الآن، فهو العلم الذي كان السبب في ظهور وتقدم الحواسب الشخصية والهواتف المحمولة وكل وسائل الاتصال، وكان سبب اتصالنا بالفضاء الخارجي وغيرها من مظاهر التقدم والرقي والتكنولوجيا.
هذة المفارقة من المفارقات التي تكون بعيدة كل البعد عن المنطق الرياضي المتبع لنا، ولكنها تصيبك بدهشة وحيرة حين تقرأ عنها. لكن أقول لك إنها بعيدة كل البعد عما نحن نتبعه من قوانين وتحليلات في علم الرياضيات، لأن هناك احتمالين الأول هو اختفاء تعريف الحجم، والثاني هو ظهور جزء غريب عنا سحري مخفي لانعرف عنه شيئاً من قبل في الواقع لكنه ظهر مع المفارقة، لكن لا شيء يكون بعيداُ عن كوننا المثير للدهشة، فكل شيء حولنا غامض ومخيف، فلا تحرم نفسك من متعة المشاهدة والتأمل!
---- أراجيك ----
شرح المفارقة:
s86-Z-CbaHA&t=1308s
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من اليمين : ستيفن بناخ ومن اليسار : ألفريد تارسكي.
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https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Banach-Tarski_Paradox.svg/350px-Banach-Tarski_Paradox.svg.png
هل حاولت من قبل أن تحضر كرة لها قطر معين وتقوم بتجزئتها إلى 5 قطع، ومن ثم تجميعها مرة أخرى من دون أي تغيير في شكل القطع (يُسمح فقط تحريك القطع وتدويرها)، فتصبح كرتان، لهما نفس شكل وحجم الكرة الأصلي؟!
أو قمت بتجربة أن تحضر قالباً من الشيكولاته وتقوم بتجزئته ومن ثم عمل تدوير وتغيير لمكان القطع، فتجد أن الحجم سيقل عن الحجم الأصلي؟! عليك بالتجربة الآن..
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الحقيقة إنه شيء مدهش، وهذا هو المعروف بـ مفارقة باناخ – تارسكى Banach – Tarski Paradox.
تعريف المفارقة
وفقا لتعريف الفيلسوف الإنجليزي مارك سينسبري، المفارقة تعني:
خاتمة قد تبدو غير مقبولة، مستمدة من فرضيات قد تبدو مقبولة من خلال منطق قد يبدو مقبولاً
والكلمة مستوحاة من الثقافة اليونانية القديمة وتفسيرها: para بجانب، doxa إيمان، رأي. المعنى الأصلي هو شيء يبدو للوهلة الأولى بدون قيمة أو غير ممكن، ولكن بعد تفكير عميق يتضح أنه صحيح!
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والمفارفة في المنطق الرياضي هي جملة خبرية تناقض المنطق، فلا يمكن أن نقول عنها أنها صحيحة، كما لا يمكن أن نقول أنها خاطئة، فإذا أخذنا الجملة الآتية : “أنا أكذب الآن” فإذا افترضنا أنني صادق يعني أن الجملة صحيحة الأمر الذي يناقض كوني صادقاً، أما إذا افترضنا أنني كاذب فالجملة خاطئة ونفيها صحيح، أي أنني صادق ويناقض كوني كاذباً.
مفارقة باناخ – تارسكي
مفارقة باناخ – تارسكي تعتبر من أكثر المفارقات المثيرة للدهشة والحيرة فى علم الرياضيات. لذلك تُعرف باسم المفارقة السحرية.
نص المفارقة: إذا قمت بتقسيم كرة ذات حجم أو قطر يساوي (أ) بطريقة معينة ثم قمت بتجميع هذه الأجزاء بطريقة معينة، فإنه يمكنك أن تكون كرتين من الحجم أو القطر (أ).
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وهنا تكمن المشكلة فى نص المفارقة من خلال أنه حينما تريد أن تجمع الكرة مرة أخرى بعد تجزئتها، فتجد أن هناك حجماً آخر مضافاً لا يُعلم مصدره (جزء سحري). باناخ وتارسكي برهنا صحة وإمكانية وجود هذه الظاهرة رياضياُ ونظرياً ولكن فقط وفقاً لمبدأ بديهية الاختيار،ولقد اعتبراها نقداً لصحة هذا المبدأ الذي طالما كان مثيراً للجدل.
حيث أن جميع القوانين المستخدمة في الإثبات تحفظ الحجم وبهذا يكون الخطأ راجعاً للمبدأ. فعند استخدام مبدأ غرابة الاختيار -المرشح حديثاً كبديل- مثلاً لا يمكن إثبات المفارقة. ولكن علماء الرياضيات البحتة لا زالوا يتمسكون بالمبدأ لاعتقادهم بوجود خطأ في مكان ما في مثل هذا الإثباتات غير المنطقية.
فقد قمنا بسؤال علم الرياضيات عن هذة المفارقة، قال: نعم، ولكن لماذا لا يمكن تطبيق هذه النظريّة على أرض الواقع؟
لكن، أي مادة (كما المعروف)، لا يمكن تقسيمها إلى مالانهاية، ولو كان هذا مسموحاً كان من السهل تطبيق هذة المفارقة. لكن مع وجود هذة المفارقة نرى اختفاء لتعريف الحجم، بمعنى: إذا حاولنا مع هذه النظرية تعريف الحجم فيصبح التعريف المنطقي بما يناسب النظرية (2=1)، وهذا مايخالف المنطق الرياضي المتبع.
ولهذة النظرية شكل آخر: إذا كان معك كرة بحجم (كرة تنس الطاولة) وقمت بتجزئتها أيضاً ومن ثم إعادة تجميعها، فيصبح معك كرة بحجم قرص الشمس. وهذا هو الغريب والمدهش والعجيب في هذه المفارقة، في أنها تبحر بخيالك وتخبرك بأشياء بعيدة عن المنطق. لذلك نجد في علم الرياضيات ما قد يرفضه الواقع لكن يسمح به الخيال، ولا شيء يكون مرفوضاً في عالم الخيال.
علم الرياضيات مليء بالألغاز البعيدة عن المنطق البشري، فهو كان السبب الرئيسي لوضع الخوارزميات التى نحن نعيش فيها الآن، التي هي سبب كل التقدم والتكنولوجيا والازدهار الذي ينمو فيه الانسان الآن، فهو العلم الذي كان السبب في ظهور وتقدم الحواسب الشخصية والهواتف المحمولة وكل وسائل الاتصال، وكان سبب اتصالنا بالفضاء الخارجي وغيرها من مظاهر التقدم والرقي والتكنولوجيا.
هذة المفارقة من المفارقات التي تكون بعيدة كل البعد عن المنطق الرياضي المتبع لنا، ولكنها تصيبك بدهشة وحيرة حين تقرأ عنها. لكن أقول لك إنها بعيدة كل البعد عما نحن نتبعه من قوانين وتحليلات في علم الرياضيات، لأن هناك احتمالين الأول هو اختفاء تعريف الحجم، والثاني هو ظهور جزء غريب عنا سحري مخفي لانعرف عنه شيئاً من قبل في الواقع لكنه ظهر مع المفارقة، لكن لا شيء يكون بعيداُ عن كوننا المثير للدهشة، فكل شيء حولنا غامض ومخيف، فلا تحرم نفسك من متعة المشاهدة والتأمل!
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شرح المفارقة:
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