來源:蔻享學術
作者 | Frank Wilczek
(溫馨提示:文章內含中文版本、英文版本,滿足更多讀者需求喲!)弗蘭克·維爾切克是麻省理工學院物理學教授、量子色動力學的奠基人之一。因發現了量子色動力學的漸近自由現象,他在2004年獲得了諾貝爾物理學獎。
古希臘哲學家芝諾曾提出 3 個著名悖論,直到一千多年後,這些數學、物理領域的難題才被解開。
面對同一個問題,我們用不同的方式進行思考,卻得到了矛盾的結論,於是悖論就產生了。然而大自然是不可能自相矛盾的。所以,物理悖論的出現意味著我們的思考出現了錯誤,這也推動了我們不斷進步。
在物理學史上,悖論扮演著光輝的角色。16歲的少年愛因斯坦想像著如果他可以追上光,那麼他看到的現象會違背當時的物理學定律。經過10年的思考,他提出了狹義相對論,其中證明了光是不可能被追上的。
公元前5世紀,一位早於蘇格拉底的哲學家——埃利亞的芝諾(Zeno of Elea)——提出了幾個關於變化與運動本質的著名悖論。約2500年後,英國哲學家伯特蘭 · 羅素(Bertrand Russell)寫道 :「在某種形式上,芝諾的論辯奠定了從他那個時代至今的所有關於時間、空間與無限的理論基礎。」
芝諾的悖論可以歸結為3個方面 :無限悖論、零化度悖論和靜止悖論。
芝諾講的阿喀琉斯和烏龜賽跑的故事很好地演示了無限悖論。阿喀琉斯從A點開始跑,同時烏龜從A前面的B點出發。當阿喀琉斯跑到B的時候,烏龜已經到了C。阿喀琉斯追到C的時候,烏龜又往前跑了一小段距離。這樣的連續劇情反覆上演,每集的時長是有限的,而集數卻是無窮的。那麼阿喀琉斯是怎麼在有限的時間內追上烏龜的?
零化度悖論講的是 :線是由點組成的,而點的長度為零,無論多少個零加在一起,仍然是零。那麼線是怎麼獲得非零長度的?
在這兩個悖論中,「正確」的答案都是一個有限量,但根據芝諾的分析,前一個悖論中的時間是無窮的,後一個的線段長度為零。
對於芝諾悖論中的邏輯問題,正確的解答直到近代才最終出現。19世紀時,數學家學會了如何求解一個遞減的無窮級數。數學理論表明,如果級數是收斂的,那麼它就會趨向一個確定的有限值。
到了20世紀,數學家知道了如果將足夠多的零累加,確實可以得到非零值。解決這個問題需要用到高等數學中的測度論(measure theory)。該理論出名地晦澀難懂,需要大量技巧才能掌握。
「芝諾之箭」講述的是靜止悖論。在每個時刻,箭都在一個固定的位置,而箭尖則是一個零長度的點。無論把多少個時刻加在一起,我們都得不到一個有限的距離,所以箭沒有動。根據現代物理中的時空連續性,我們可以用與解決零化度悖論相同的數學技巧來解決靜止悖論。
而如果時空是離散的,那麼箭就能夠在每個時刻從一個點跳到另一個點。這是對靜止悖論的另一種解答。這個回答更加簡單,更契合我們數位化時代的精髓。就像一部電影,說到底,是由一幀幀離散的鏡頭構成的。
牛頓說 :「如果以較少的付出就能達到結果,大自然絕不會做更多的無用功 ;自然界喜歡簡單和高效,而不喜歡浮誇的排場。」可為什麼它沒有選擇數位化,從而避免芝諾悖論中所有的微妙疑惑呢?你可以說這又是個悖論——芝諾悖論依然存在。
The ancient philosopher's famous riddles highlightedproblems in math and physics that would takemillennia to solve.
Paradoxes arise when different ways of thinking about a situation lead to contradictory conclusions. Since nature cannot contain contradictions, physical paradoxes point to flaws in our thinking. They invite us to do better.
Paradoxes have played a glorious role in the history of physics. Young Albert Einstein, at the age of 16, saw that if he could catch up to a light beam he would see things that contradicted the existing laws of physics. Ten years of thinking later, his special theory of relativity showed why it was impossible to catch up with a light beam.
The pre-Socratic philosopher Zeno of Elea, who lived in the 5th century B.C., produced several famous paradoxes concerned with the nature of changeand motion. Almost 2,500 years later, Bertrand Russell wrote that "Zeno's arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own."
Zeno's paradoxes can be boiled down to three: the paradox of infinity, the paradox of nullity and the paradox of stasis.
Zeno's story about a race between Achilles and a tortoise nicely illustrates the paradox of infinity. Achilles starts atpoint A while the tortoise starts ahead, at point B. By the time Achilles gets to B the tortoise has moved on to point C. Rinse, lather, repeat. Each episode takes a finite amount of time, and there are an infinite number of episodes. How does Achilles manage to catch up (in a finite time)?
The paradox of nullity goes as follows. Lines are made from points. But the length of a point is zero, and no matter how many zeros you add together, you still get zero. So how do lines achieve nonzero length?
In both paradoxes the "right" answer is a finite quantity, but Zeno's analyses suggest infinite time in one case and zero space in the other.
Logically satisfactory answers to those paradoxes only emerged relatively recently. In the 19th century, mathematicians learned how to deal with infinite sums of decreasing terms. The theory shows that when the sum "converges," there is a well-defined, finite answer.
In the 20th century they learned to accommodate the fact that by adding zero enough times you actually can get past zero. This kind of question is treated in an advancedchapter of mathematics called measure theory, which is notoriously tricky and unintuitive.
Zeno's arrow introduces the paradox of stasis. At every instant an arrow is at one place, and the tip of the arrow, a point, takes up zero distance. No matter how many instants we add, we never achieve a finite distance, so thearrow cannot move. In present-day physics, based on a space-time continuum, the arrow paradox is solved using the same tricky mathematics that resolves the paradox of nullity.
If space and time were both discrete, then the arrow could just hop, at each tick of time, from one position to another. That simpler alternative seems more in tune with our digital age. A movie, after all, is really a series of snapshots.
According to Isaac Newton, "Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes." Why doesn't it go digital, thus avoiding Zeno's subtleties? You could say it’s a paradox. Zeno lives.
編輯:黃琦
編輯:夏至