科學光環與科學現實

Albert Einstein in 1921 (Source)

當年讀完廣義相對論（General Relativity）發現竟然沒有一個預測是可以嚴謹地從 Einstein field equations 中直接導出，頓時變得好失望。水星的近日點（perihelion）預測就是其中一個例子，除了需要用一個非常脫離現實的 Schwarzchild solution 去 approximate 之餘，竟然還要用 Newtonian mechanics，ad hoc 的程度非常嚴重［1］，理論物理學從此就失去了在我小時候從《時間簡史》中獲得的完美光環。這倒也不是理論物理的錯，只是我希望越大失望越大而已。不過，這一點就是我經常想跟外人解釋的一件事︰不要依賴科普知識去評價科學的成就，有時所謂預測準確至幾多個 significant figures，可能是 ad hoc approximation 的結果［2］，而不是理論原型一出來就這麼準確。和 classical physics 不同，modern physics 中理論與證據之間的邏輯關係，比很多人想像的要稀薄。所以，雖然科學仍是我們最好的知識獲得途徑，但不要把她想得太理想，要真的把握當中的細節和計算才能下一個恰當的評估。

我引一段 Paul Feyerabend 的講法︰

Ad hoc approximations abound in modern mathematical physics. They play a very important part in the quantum theory of fields and they are an essential ingredient of the correspondence principle [...] ad hoc approximations conceal, and even eliminate, qualitative difficulties. They create a false impression of the excellence of our science. It follows that a philosopher who wants to study the adequacy of science as a picture of the world, or who wants to build up a realistic scientific methodology, must look at modern science with special care. [3]

Feyerabend 的 proposal 很激進［4］，放在神棍特別多的廿一世紀並不合時宜，但他的分析確實有很多有見地的地方︰

Modern science has developed mathematical structures which exceed anything that has existed so far in coherence, generality and empirical success. But in order to achieve this miracle all the existing troubles had to be pushed into the relation between theory and fact, and had to be concealed, by ad hoc hypotheses, ad hoc approximations and other procedures.

Feyerabend 舉的例子是 von Neumann 的公理化量子力學，指出公理化量子力學雖然比之前的版本在邏輯上較嚴謹，但代價卻是在計算實際問題時需要動用更多 ad hoc procedure ，例如要任意地修改一些物理定律，以求得到想要的計算結果，甚至需要承認它無法解決一些非常普通的問題如 scattering problem 。

這一個批判也動搖了所謂「數學是宇宙的語言」的流行講法，現實似乎是我們強行迫令具體執行遷就我們的抽象框架，把邏輯上不太說得通的東西都收藏在細節上，使得世界好像完美地跟數學模型重疊。（有一段時間我都非常驚訝，為什麼 Hilbert Space 這種完全獨立地發明出來的數學框架，能完美地套用到量子力學上呢？）如果我們仔細地觀察，就能看出數學與數學應用之間的關係並不直接，然後所謂 "the unreasonable effectiveness of mathematics" 的迷思，也就沒有那麼神秘了。



後記︰

這篇文章其實不是想叫讀者不要看科普書，而是想講透過科普書我們很容易會獲得一種 idealised picture ，從而可能會產生一種過份美化的科學觀。科普書還是很值得讀的，只是要比較小心。（另外，科普書中的科學史也很有誤導性，因為很多時身為科學家的作者都沒有受過史學訓練，寫出來的科學史通常都是一種 Whig history ，這我也是要透過讀科學哲學才能認識到的錯誤）

我受純數的訓練較多，這也可能令我對物理學的 rigor 有較高（審美）要求的原因，我的一些純物理的同學對 ad hoc assumption 就沒有那麼敏感。（習慣哲學思維可能也有關）

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［1］這一點要 elaborate 一下，我直接用英文比較容易︰The so-called Schwarzchild solution of Einstein's equation is based on a simplified model in which the universe is centrally symmetrical, containing a singularity at the middle and nothing else. This is far from the case with our solar system. If we say the empirical evidences must correspond to the picture as described by GR for GR to be scientifically verified, problems arise as there is never a direct correspondence between the theory of GR and the evidences for it. Their relation is always mediated by various simplification, axillary assumption and ad hoc modification. (One can even say, these procedures constitute a cluster of secondary, ad hoc theories designed specifically for specific evidence to correspond to.) There is no such thing as an universal procedure which brings one from theory to facts or the other way round. Whether the theory per se entails the observable result, or the theory is forced to give the desired result, it is hard to say. Furthermore, Newtonian mechanics is supposed to be "reduced" from GR, but the fact is it is often necessary to presuppose Newtonian mechanics for many actual calculations to be carried out. The idea of increasing generality is thus (once again) shaken.

另外 Feyerabend 自己的解釋也可以貼出來以供參考︰ "[In the case of approximating the perihelion of Mercury,] making the required approximations would mean calculating the full n-body problem relativistically (including long-term resonances between different planetary orbits), omitting magnitudes smaller than the precision of observation reached, and showing that the theory thus curtailed coincides with classical celestial mechanics as corrected by Schwarzschild. This procedure has not been used by anyone simply because the relativistic n-body problem has as yet withstood solution. When the argument started, there were not even approximate solutions for important problems such as, for example, the problem of stability (one of the first great stumbling blocks for Newton's theory). The classical part of the explanations, therefore, did not occur just for convenience, it was absolutely necessary. And the approximations made were not a result of relativistic calculations, they were introduced in order to make relativity fit the case. One may properly call them ad hoc approximations."

當然，這是用一個 rationalism 的標準去看物理學的結果，而其實 Feyerabend 也不是要指責物理學家的邏輯令人失望，而是要說明 rationalist 理想中的 argumentation 並不是放諸四海皆有效的。

［2］關於 ad hoc hypothesis 可看 Wiki

［3］Feyerabend, Against Method, part 5

［4］我的看法是他的分析跟他的 proposal（revival of ancient myths, etc）可以分開考慮，理論上我是認同 methodological pluralism 的，但我認為他對研究資金這回事看得太樂觀，如果隨便什麼新舊理論都可以拿到資金研究，那大學早就破產了。而且科學測量的精確度要得以改進，也需要有 Kuhn 所講的常態科學（normal science）那個階段，這可以當作是權宜之道，但不能簡單視為「霸權」的。