Incommensurable Possibilities of Mathematics

I think that people engaged in research in mathematics today are doing so the same way it was done 200 years ago. This is partly because we don』t choose mathematics as our profession, but rather it chooses us. And it chooses a certain type of person, of which there are no more than several thousand in each generation, worldwide. And they all carry the stamp of those sorts of people mathematics has chosen.

Yuri Manin interviewed by Mikhail Gelfand [1]

Priori

最近在閱讀 Yuri Manin 的一本書 [2]，除了重新發現自己既不懂數學也不懂物理之外，還讓我重新審視自己的計算理論和知乎專欄。

畢竟我已經很久沒有更新專欄了，雖然在這段時間裡專欄的讀者終於超過了一千。不太了解我的人大概會以為我很忙，但事實並非如此。我有充分的時間來思考和計算，甚至閑得有時間去玩遙控航模 [2]。只不過，我今年做的分析和計算偏離了過去三年所確定的基調，以至於我一直在糾結是不是應該新開一個專欄來討論新的成果。

雖然，Yuri Manin 的散文集已經對我造成了智力上的碾壓，但是我還是成功地（假裝）從他的訪談中獲得了一些啟發，並戰勝（緩解）了自己的拖延症。

Divergence between Mathematics and Physics

根據我的心情，我會自稱是民科或者民數，並從來沒有把自己當作一個所謂的「Data Scientist」。大多數時候，我都能把自己當作一個喜歡從現代物理學中剽竊計算框架的程序員。但是很多時候，我分不清楚數學和理論物理的邊界，有時我看物理學家嘲笑數學家的迂腐覺得很有道理，有時我看數學家鄙視物理學家瞎算也覺得過癮。很顯然，這不能都歸咎於我的無知——這畢竟是個歷史遺留問題。就此，Yuri Manin 給出了一個最簡單明了的解釋 [1]——

Gelfand: What about the relationship between mathematics and theoretical physics? How is that structured?

Manin: This relationship has changed during my own lifetime.

It is important to note that in the time of Newton, Euler, Lagrange, Gauss, the relationship was so close that the same people did research in both mathematics and theoretical physics. They might have considered themselves more as mathematicians or more as physicists, but they were exactly the same people. This lasted until about the end of the nineteenth century. The twentieth century revealed significant differences. The story of the development of the general theory of relativity is a striking example. Not only did Einstein not know the mathematics he needed, but he didn』t even know that such mathematics existed when he started understanding the general theory of relativity in 1907 in his own brilliantly intuitive language. After several years dedicated to the study of quanta, he returned to gravitation and in 1912 wrote to his friend Marcel Grossmann: 「You』ve got to help me, or I will go out of my mind!」 Their first article was called 「A sketch of a theory of general relativity and a theory of gravity: I. Physics Part by Albert Einstein; II. Mathematics Part by Marcel Grossmann.」

This attempt was half successful. They found the right language but had not yet found the right equations. In 1915 the right equations were found by Einstein and David Hilbert. Hilbert derived them by finding the right Lagrangian density—the importance of this problem, it seems, for some time eluded Einstein as well. It was a great collaboration of two great minds that unfortunately prompted historians to start silly fights about priorities. The creators themselves have been grateful and generous in recognizing each other』s insights.

For me, this story marks the period in which mathematics and physics parted ways. This divergence continued until about the 1950s. The physicists dreamed up quantum mechanics, in which they found a need for Hilbert space, Schr?dinger』s equations, the quantum of action, the uncertainty principle, the delta function. This was a completely new type of physics and a completely new type of philosophy. Whatever pieces of mathematics were necessary—they developed them themselves.

Meanwhile, the mathematicians did analysis, geometry, started creating topology and functional analysis. The important thing at the beginning of the century was the pressure by philosophers and logicians, trying to clarify and 「purify」 the insights of Cantor, Zermelo, Whitehead, et al., about sets and infinity. Somewhat paradoxically, this line of thought generated both what came to be known as the 「crisis in foundations」 and, somewhat later, computer science. The paradox of a finite language that can give us information about infinite things—is this possible? Formal languages, models and truth, consistency, (in)completeness—very important things were developed, but quite disjoint from physicists』 preoccupations of that time.

And Alan Turing appeared, to tell us: 「The model of a mathematical deduction is a machine, not a text.」 A machine! Brilliant. In ten years, we had von Neumann machines and the principle of separation of programs (software) and hardware. Twenty years more—and everything was ready.

During the first third of the century, except for particular minds—von Neumann was undoubtedly both a physicist and a mathematician, and I know of no other person with a mind on that scale in the twentieth century—mathematics and physics developed in parallel and after a while stopped taking notice of each other. In the 1940s Feynman wrote about his wonderful path integral, a new means of quantifying things, and worked on it in a startlingly mathematical way—imagine something like the Eiffel Tower, hanging in the air with no foundation, from a mathematical point of view. So it exists and works just right, but standing on nothing we know of. This situation continues to this very day. Then, in the 1950s the quantum field theory of nuclear forces started to appear, and it turned out that mathematically the respective classical fields are connection forms. The classical equation of stationary action for them was known in differential geometry. The equation of Yang-Mills entered the scene, mathematicians began to look askance at the physicists, and the physicists at the mathematicians. It turned out, paradoxically—and for me pleasantly—that we began to learn more from the physicists than they learned from us. It turned out that with the help of quantum field theory and the apparatus of the Feynman integral they developed cognitive tools that allowed them to discover one mathematical fact after another. These weren』t proofs, just discoveries. Later the mathematicians sat themselves down, scratched their heads, and reshaped some of these discoveries in the form of theorems and began trying to prove them in our honest manner. This shows that what the physicists do is indeed mathematically meaningful. And the physicists say, 「We always knew that, but of course, thanks for your attention.」 But in general, as a result, we learned from the physicists what questions to ask, and what answers we might presuppose—as a rule, they turn out to be correct. The renowned physicist and mathematician Freeman Dyson in his Gibbs lectures 「Missed opportunities」(1972) has beautifully described many cases when 「mathematicians and physicists lost chances of making discoveries by neglecting to talk to each other.」 Especially striking for me was his revelation that he himself 「missed the opportunity of discovering a deeper connection between modular forms and Lie algebras, just because the number theorist Dyson and the physicist Dyson were not speaking to each other.」

Then Witten appeared, with his unique gift for the production of glorious mathematics from this very Eiffel Tower that hangs in the air. I looked in Wikipedia: before getting his Ph.D. in physics in 1976, when he was twenty-five, he was planning to engage in political journalism, then economics…until he finally heard the call of mathematics and physics.

He is the master of such astonishing mental equipment, which produces mathematics of unlikely strength and force, but arising from physical insights. And the starting point of his insights is not the physical world, as it is described by experimental physics, but the mental machinery developed for the explanation of this world by Feynman, Dyson, Schwinger, Tomonaga, and many other physicists—machinery that is entirely mathematical but that has very weak mathematical foundation. It is such an earthshaking heuristic principle, not at all some triviality, but, I must say again, an enormous structure without a foundation, at least of the kind we have gotten accustomed to.

限於篇幅，我拒絕（懶得）做全文翻譯。這裡面需要特別指出，Yuri Manin 明確指出在 1950 年之後，物理學家開始不斷地發展物理學所需要的數學工具，雖然這些數學工具的嚴格性達不到數學家的標準。順便提一句，同樣的事情也發生在計算領域，這一輪人工智慧的熱潮主要是由深度神經網路這類數學根基不牢固的計算工具所驅動的。

在某個階段，我對數學和理論物理的膜拜大概可以用 Cargo Cultism [4] 來描述。從我的專欄就能看出，我總是在門檻外思考統計力學和量子力學的普適性——雖然我推崇的 Jaynes 和 Prigogine 都是科學界中的異類。正因為如此，當我發展出一套沒有根基的全新計算工具的時候[5]，我茫然不知所措，只是按照指鹿為馬的老習慣給它套上一個杜撰的學名。

A New Kind of Mathematician

在另一次訪談中，Yuri Manin 提到了 Donald Knuth 的觀點 [6]。

I have once translated a talk by Donald Knuth into Russian. In Uzbekistan there was a meeting dedicated to Alkhorezmi. Knuth started his talk with a funny statement. In his opinion the primary importance of computers for the mathematical community is that those people finally took to mathematics who were interested in mathematics but had an algorithmic sort of mind. Now they were able to do what they wanted. Before that, this subculture didnt exist. And Knuth was describing himself as a person whose mind is specifically designed for writing software and how happy he was that, finally, he could what he wanted to. I take this argument quite seriously and I do believe that among the community of future potential mathematicians there is a sub-community whose minds are better for writing computer programs than proving theorems.

很顯然，我就是這麼一個更喜歡寫程序而不是證明定理的人。但是弔詭的是，我為什麼會從一個數學家那裡讀到 Donald Knuth 的觀點並且深以為然？畢竟，我從本科開始開始關注計算數學相關的研究，並且假裝是 Stephen Wolfram 和 Benoit Mandelbrot 的門徒。大概，我過去幾年入戲太深，真的以為自己是一個統計力學/量子力學的民間物理學家？不管怎麼說，這大概也是一種福音，我這種寫碼的也有機會沐猴而冠成為一種新品種的數學家了！

當然，Stephen Wolfram 對此持有另一種觀點，作為一個物理學家，他更喜歡把新的計算學派稱為一種科學 [7]。與 Yuri Manin 這樣的柏拉圖主義者[8]不同，Stephen Wolfram 認為數學是一種人造物，是文化的一部分[9]。

Ive been interested for a long time at questions about the essence of mathematics. I make a living by building this thing called Mathematica which attempts to cover in the broadest possible sense the kinds of things that mathematics might encompass. But so a question that Ive been interested in also from the point of view of basic science is "is the mathematics that we practice today the only possible mathematics" or "is it a mathematics that is sort of a great artifact of our civilization". The conclusion that Ive so resoundingly come to is that mathematics that we have today is in fact really a historical artifact. Now thats not historically in the tradition of mathematics itself thats not what people have tended to conclude. Theyve tended to think mathematics is sort of the most general possible formal abstract system. If you look at the history of mathematics thats certainly not how it originally started out.

在這一集 Closer to Truth 的訪談中，Stephen Wolfram 還斷言，存在無窮多可能的公理系統和數學，人類在有限的時間和資源限制下不可能窮盡這些可能性。如果 Yuri Manin 的觀點成立，現有的數學文化通過選擇新一代的數學家構成了一個正反饋的閉環，主流的數學發展不可避免地被其歷史所限制。

此外，Gregory Chaitin 也有類似的觀點，他用資訊理論給出了更形式化的解釋[10]：

Unlike G?dels approach, mine is based on measuring information and showing that some mathematical facts cannot be compressed into a theory because they are too complicated. This new approach suggests that what G?del discovered was just the tip of the iceberg: an infinite number of true mathematical theorems exist that cannot be proved from any finite system of axioms.

所有可能的數學系統和自洽的數學定理具有無限的複雜性，它們不可能由有限的定理來證明。這一觀點和 E. T. Jaynes 完全一致，由於專欄中已經介紹過[11]，這裡不再贅述。

順便提一句，Stephen Wolfram 的傲慢和商業背景惹惱了一些人，他的粉絲中也不乏民科和神棍，以至於我這麼一個腦殘粉都不自覺地跟他劃清了界限。別的不說，他折騰出來的 Mathematica 還是很好用的，我都忍不住做了免費的網路水軍[12]。我非常認同 Gregory Chaitin 的某些觀點，不過我對他的著述了解非常有限，因此不做更多的介紹和評價。

Mechanics under the clouds

E. T. Jaynes 在他的遺著第十章提到了一個有趣的假設[13]，如果地球像金星一樣覆蓋著厚厚的雲層，天文學家沒有辦法積累足夠的觀測數據，即便是有牛頓這樣的天才也無法發展出牛頓力學。這當然是 Jaynes 為了調侃頻率主義者編的段子，但是其基本結構和 Stephen Wolfram 類似。根據我模糊的記憶，Henri Poincaré 也在他的書中提到過類似的假設。

就我個人的觀點而言，物理學家無疑是幸運的，正如 Yuri Manin 所說他們甚至「dreamed up quantum mechanics」。但是對於計算學派而言，「Mechanics under the clouds」 並非一個隱喻，而是亟待解決的問題。尤其是，對於我來說，最讓我著迷的一個領域甚至已經成了知乎上被調侃的明日黃花[14]

3、然後這領域的早期，人們嘗嘗關注的是仨問題：（1）孤子；（2）分形；（3）混沌。其中孤子理論進展最大，現在已經有不少工程應用了。然後分形則基本仍是一種數學理論，物理上的應用不多，在實際工程中的應用更少，主要貢獻就是畫一些好看的圖。。。

我對此持有保留意見，僅僅想指出一點，好看的圖本身就能激發想像力。按照這個邏輯，數據可視化整個領域都是在畫一些好看的圖。

當然，僅僅畫一些好看的圖是遠遠不夠的。現代手機的多頻段天線就是基於分形的

當然，我並沒有任何奢望做到撥雲見日，僅僅想通過這個專欄做一點微小的貢獻。

此外，我確信在厚厚的雲層以下，仍然會有類似 E. T. Jaynes 的物理學家（而非數學家）發展出類似的（應用）概率論。

[1] http://www.ams.org/notices/200910/rtx091001268p.pdf

[2] Mathematics as Metaphor, 2007, Mathematics as Metaphor: Selected Essays of Yuri I. Manin

[3] 空門：職業選擇，飛行員or程序員？

[4] Cargo cult - Wikipedia

[5] Redistribution Group

[6] Mathematics as Metaphor, 2007, P210

[7] A New Kind of Science

[8] 在訪談中，Yuri Manin 自稱「emotional Platonist」

[9] https://www.youtube.com/watch?v=RlMMeqO7wOI

[10] The Limits of Reason

[11] Paradox

[12] 空門：Mathematica 到底有多厲害？

[13] Probability Theory, The Logic of Science, E. T. Jaynes, 2003, Part I, 10.8

[14] 放際：有哪些以前的科學研究熱點已經逐漸被人淡忘？