4.3 – 4.4 Investigate Desargues’ Theorem and Its Dual
I hear and I forget;
I see and I remember;
I do and I understand.
—Ancient Proverb
『六識』、『六塵』、『六根』之說古早也。究竟什麼是『覺』？放下屠刀不謂『行』乎？當人間逍遙遊遇上文字作遊戲！將如何談『知覺難題』耶？？
知覺難題是一個嘗試解釋為什麼我們會擁有感質或「現象性經驗」的命題——通俗的講，我們的感觀是如何獲取類如顏色或味道等物質特徵的[1]。這一命題由David Chalmers提出並與「簡單問題」作對比[2]。Chalmers將分辨、信息整合、心境描述、專注等現象的解釋歸為簡單問題，之所以稱為簡單問題是因為所有這些現象都可以對應到某一行為機制。也就是說，無論這一問題的解釋可能有多複雜或多不利於理解，只要可以完全收斂於自然現象的現代唯物主義概念，那麼就稱其為「簡單問題」。Chalmers將「體驗」的解釋問題排除在簡單問題之外，因為他認為「即使所有的相關功能的行為問題都解決了，也無法解釋體驗的存在」[3]。
關於是否存在「難題」這一概念本身，哲學界尚存爭論[4][5]。
Chalmers在直面知覺問題中提到：[3]
不可否認，我們的某些器官是體驗的主體，但是這些體驗是如何形成的是未知的。為什麼我們的認識系統在處理諸如視覺或聽覺信息時我們會有視覺及聽覺的 體驗如：深藍色的質感或對中音C的感知。我們該如何解釋為何我們會懷有精神意象（mental image）或體驗到情緒。眾所周知體驗終歸還是由物理基礎產生，但是究竟為何且如何會產生體驗，我們還沒有一個合理的解釋。為何物理的處理過程會引起任 何一點內心體驗？客觀上講這完全不合理，但這的確發生了。
實際上知覺難題就是體驗的問題。當我們思考或意識到信息的處理過程時，實際上也是主觀的意識。
Raamy Majeed的備註說，實際上這個難題是要解釋兩個問題：[6]
- [PQ] 物理的處理過程產生對現象特質的體驗。
- [Q] 我們的現象特質是一般的。
第一點專注於物理和現象的關係，第二點專注於最根本的現象本身 。大部分對這個知覺難題的解釋實際上是針對這兩點的。
自家『信仰』自己『思辨』，置諸『哲學』果有解嘛！！
故退之以『止觀觀止』、『觀止止觀』矣。
以《止觀》來《觀止》，自能了解『整體』與『部份』的『自洽性』。就像拼『拼圖』、玩『數獨』以及猜『燈謎』一樣，所求總在『整體』和『部份』之『契合』裡。這樣容易明白，一九七二年英國獨立的科學家、環保主義者和未來學家詹姆斯‧洛夫洛克 James Lovelock 提出的『蓋亞假說』 Gaia hypothesis ︰
地球整個表面，包括所有生命，構成一個自我調節的整體，這就是我所說的『蓋亞』。
簡單地說，蓋亞假說是指在生命與環境的相互作用之下，能使得『地球』適合『生命持續』的生存與發展。
據維基百科上講︰
該觀點於 1972 年首次提出，主流科學家主要以其不夠嚴密為由堅決拒絕接受。 1981 年，這一觀點首次得到支持。當時，洛夫洛克創造出計算機模擬的反射或吸收太陽輻射的白色或黑色雛菊世界。由於雛菊的數量隨著普遍的表面溫度變化而相對改變，因此雛菊群維持全球氣溫均衡。此後，更多生物多樣性的複合模型提高了該系統的穩定性。
當可以知道人們對『可計算』與『能度量』的堅持，有時忘卻了『不可計算性』和『測不準』的『科學』。怕只是『一時』以及『長遠』『○□難免』之爭的吧！
□︰ 『求解問題』有樂趣？『止觀觀止』能休閒嗎？？
○︰ 煩惱即菩提︰
樂休『求不得』！
閒趣『止不了』！！
─── 摘自《M♪o 之 TinyIoT ︰ 《破題》》
樂借Timothy Peil 教授之所說︰
就『笛沙格定理』而言，『平面』證明，難於『立體』理解。
4.4 Desargues’ Theorem Printout
If Desargues, the daring pioneer of the seventeenth century, could have foreseen what his ingenious method of projection was to lead to, he might well have been astonished. He knew that he had done something good, but he probably had no conception of just how good it was to prove.
—E. T. Bell (1883–1960)
The French mathematician Gérard Desargues (1593–1662) was one of the earliest contributors to the study of synthetic projective geometry. Desargues was an engineer and architect, who had served in the French army. The importance of the theorem, that bears his name, is due to the relating of two aspects of projective geometry: perspectivity from a point and perspectivity from a line. Because of his many contributions to the field of projective geometry, the theorem was named after him even though his major work, Brouillon projet, was lost for nearly two centuries before another French geometer Michel Chasles (1793–1880) discovered a copy in 1845.
Though we are only studying plane projective geometry, we motivate Desargues’ Theorem with a triangular pyramid in three dimensions. The diagram on the left is a triangular pyramid with vertices A, B, C, and P. The triangles ABC and A’B’C’ are perspective from the point P. From Euclidean geometry, two nonparallel planes intersect in a line. Therefore, the two planes a and a’ determined by the triangles ABC and A’B’C’ intersect in a line l. Since AB and A’B’ are in planes a and a’, respectively, the point Q = AB · A’B’ must be on line l. Similarly, R = AC · A’C’ and S = BC · B’C’ must be on line l. Hence, the triangles ABC and A’B’C’ are perspective from the line l = QR.
Axiom 5. (Desargues’ Theorem) If two triangles are perspective from a point, then they are perspective from a line.
Click here for a dynamic illustration of Desargues’ Theorem GeoGebra or JavaSketchpad.
In plane projective geometry, Desargues’ Theorem cannot be proven from the other axioms; therefore, it is taken as an axiom. The proof of the theorem requires two triangles that are not in the same plane, as illustrated in the motivation example above. That is, Desargues’ Theorem can be proven from the other axioms only in a projective geometry of more than two dimensions. Since we have not listed the axioms for a projective geometry in 3-space, we will not discuss the proof of the theorem here, but the proof is similar to the argument made in the illustration above. Many books on projective geometry discuss the topic. (Reference Projective Geometry by Veblen and Young, 1938)
Dual of Desargues’ Theorem. If two triangles are perspective from a line, then they are perspective from a point.
順循 Christopher Cooper 先生的詮釋吧︰
