A36 · Language and Science 語言與科學
Are Numbers Invented or Discovered? 數學是發明還是發現？
Nobody has ever seen the number seven, and everything we build depends on it. Plato’s unchanging world, the geometries that broke Euclid’s monopoly, Russell’s paradox, the carpenter’s square of the Zhoubi, and a triangle whose angles refuse to add up to 180.
從來沒有人見過「七」這個數；而我們建造的每一樣東西都靠它。柏拉圖那個不變的世界、打破歐幾里得獨佔地位的幾種幾何、羅素悖論、《周髀》裡的那把矩尺，以及一個內角和怎麼樣都不肯等於 180 度的三角形。
This lesson closes the unit on language and science. The last two asked how science tests its claims and how it changes. Mathematics seems to do neither. Its results are proved once and stand forever, and no experiment is consulted. We should like to know what such truths are about, and why the physical world obeys them.
這一課是「語言與科學」單元的最後一課。前兩課問的是科學怎麼檢驗它的主張、又怎麼改變。數學似乎兩樣都不做。它的結果一旦被證明，就永遠成立；而且不必請教任何實驗。我們會想知道，這樣的真理是關於什麼的，以及物理世界為什麼服從它們。
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The essay · 課文
Are Numbers Invented or Discovered? 數學是發明還是發現？
01A strange kind of knowledge 一種奇怪的知識
That seven and five make twelve is as certain as anything we know. We did not learn it by counting a great many sevens and fives, and no future count could overturn it. If someone added seven apples to five and found thirteen, we would look for the extra apple. Bertrand Russell put the point in 1912: “All pure mathematics is a priori,” meaning that its truth “can be neither proved nor disproved by experience.” A child meets “two and two are four” in particular cases, a pair of couples here and another there, “until at last it becomes possible to see that it is true of any pair of couples.” What is seen at that moment is general and necessary, and the senses deliver only particular things that might have been otherwise.
七加五等於十二，跟我們知道的任何事情一樣確定。我們並不是數過很多很多的七與五才學到它的；而未來的任何一次點數，也推翻不了它。如果有人把七顆蘋果加上五顆，卻數出十三顆，我們會去找多出來的那顆蘋果。伯特蘭・羅素在 1912 年把這一點說了出來：「所有的純數學都是先驗的」；意思是它的真「既不能被經驗證明，也不能被經驗否證」。一個孩子是在個別的情況裡遇到「二加二等於四」的：這裡兩對，那裡又兩對；「直到最後，他才有可能看出，這對任何兩對都成立」。那一刻被看見的東西，是普遍而必然的；而感官所交付的，只有本來可以是別的樣子的個別事物。
Mathematicians, asked what they are doing, almost always say that they are discovering. A proof feels like finding out how things stand. And yet the things in question cannot be located. No one has bumped into the number seven. A circle drawn on paper is only roughly round, and the theorems are about the circle that no one has ever drawn.
問數學家他們在做什麼，他們幾乎總是說自己在「發現」。一個證明給人的感覺，像是查出了事情本來是怎麼樣。然而，那些「事情」卻找不到在哪裡。沒有人撞見過「七」這個數。畫在紙上的圓只是大致圓；而定理講的，是從來沒有人畫出來過的那個圓。
02Plato’s answer 柏拉圖的答案
Plato took the mathematicians at their word. Numbers and figures exist, he held, in a realm that is outside space and time and is grasped by the mind alone. They are more real than the things we touch, since they do not change or pass away. Russell, in his early years, described this world with affection. It is “unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life.”
柏拉圖把數學家的話當真了。他主張，數與形存在於一個在空間與時間之外的領域，只能由心智來掌握。它們比我們摸得到的東西更真實，因為它們不變、不滅。羅素早年曾經帶著感情描述這個世界。它是「不變的、嚴整的、精確的；數學家、邏輯學家、形上學體系的建造者，以及所有愛完美勝過愛生命的人，都為它著迷」。
The view explains why mathematics is certain and why it feels like discovery. Its difficulty is the one that Plato’s critics have pressed ever since. We come to know stars and bacteria because they affect our instruments. Objects outside space and time affect nothing. If numbers are like that, it is a mystery how creatures with brains made of cells ever came to know anything about them.
這個看法解釋了數學為什麼確定，以及它為什麼感覺像發現。它的困難，就是從那時起柏拉圖的批評者一直緊追的那一個。我們之所以認識星星與細菌，是因為它們影響了我們的儀器。在空間與時間之外的對象，什麼也影響不了。如果數是那樣的東西，那麼大腦由細胞構成的生物，到底是怎麼知道關於它們的任何事的，就成了一個謎。
03The triangle that broke the rule 打破規則的三角形
For two thousand years geometry was the model of certain knowledge. Euclid had derived hundreds of theorems from five simple starting points, and everyone took those starting points to be self-evident truths about space. One of them, about parallel lines, was less obvious than the rest, and generations of mathematicians tried to prove it from the other four. In the 1820s several of them, working apart, tried something else. They replaced it with its opposite and looked for a contradiction. None appeared. What they found were complete and consistent geometries in which the angles of a triangle add up to less than two right angles, or to more.
兩千年間，幾何學是確定知識的典範。歐幾里得從五個簡單的出發點，推導出了幾百條定理；而人人都把那些出發點當成關於空間的、不證自明的真理。其中有一條，是關於平行線的，沒有其他幾條那麼明顯；一代又一代的數學家，試著用另外四條來證明它。1820 年代，有幾位數學家各自獨立地試了別的辦法。他們把它換成它的反面，然後去找矛盾。矛盾沒有出現。他們找到的，是完整而一致的幾種幾何；在其中，三角形的內角和小於兩個直角，或者大於。
This changed the question. If there are several geometries, each as consistent as the others, then “the angles of a triangle add up to 180 degrees” is no longer a truth about space. It is what follows if you start from Euclid’s rules. A new view of mathematics grew from this, called formalism. Mathematics, on this view, is the study of what follows from what. The axioms are ours to choose, and nothing needs to exist for the theorems to be about. Russell caught the spirit of it in a joke: mathematics is “the subject in which we never know what we are talking about, nor whether what we are saying is true.”
這改變了問題。如果有好幾種幾何，每一種都跟別的一樣一致，那麼「三角形的內角和是 180 度」就不再是一條關於空間的真理了。它是「如果你從歐幾里得的規則出發」會推出來的東西。由此長出了一種對數學的新看法，叫做形式主義。照這個看法，數學研究的是「從什麼可以推出什麼」。公理由我們來選；定理並不需要有任何存在的東西來讓它「關於」。羅素用一個笑話抓住了它的精神：數學是「這樣的一門學科：我們在其中從來不知道自己在談的是什麼，也不知道自己說的是不是真的」。
04Russell’s paradox 羅素悖論
Russell himself hoped for more than a game. With the German logician Gottlob Frege he believed that all of mathematics could be derived from logic alone, so that its certainty would be the certainty of logic. In 1901, examining Frege’s system, he asked a simple question. Most collections are not members of themselves. The collection of teaspoons is not a teaspoon. Consider the collection of all collections that are not members of themselves. If it is a member of itself, it is not, and if it is not, it is. Frege’s logic allowed such a collection, and so Frege’s logic was contradictory. Russell wrote to him, and Frege replied that arithmetic was tottering.
羅素自己期望的不只是一場遊戲。他與德國邏輯學家戈特洛布・弗雷格一樣，相信整個數學都可以單單從邏輯推導出來；這樣一來，數學的確定性就是邏輯的確定性。1901 年，他在檢視弗雷格的系統時，問了一個簡單的問題。大多數的集合，並不是它自己的成員。「所有茶匙的集合」不是一根茶匙。現在考慮這個集合：所有「不是自己的成員」的集合所構成的集合。如果它是自己的成員，它就不是；如果它不是，它就是。弗雷格的邏輯容許這樣的集合；所以弗雷格的邏輯是矛盾的。羅素寫信告訴他；弗雷格回信說，算術正在動搖。
Russell and Alfred North Whitehead spent ten years rebuilding the foundations so that the paradox could not arise. Their Principia Mathematica needed several hundred pages to reach the proposition that one and one are two. In 1931 Kurt Gödel proved that the project could not be completed in the way they had hoped. In any consistent system of rules strong enough for arithmetic there are true statements that the system cannot prove. Mathematical truth, it turned out, is larger than any game of proof that we can write down. Platonists took this as a point in their favor.
羅素與阿弗烈・諾斯・懷海德花了十年重建基礎，使那個悖論無法產生。他們的《數學原理》用了好幾百頁，才到達「一加一等於二」這個命題。1931 年，庫爾特・哥德爾證明了：這個計畫無法照他們期望的方式完成。在任何一個一致的、強到足以表達算術的規則系統裡，都有一些真的陳述，是這個系統證明不了的。結果，數學的真理比我們寫得出來的任何一場證明遊戲都大。柏拉圖主義者把這當作對自己有利的一點。
05The carpenter’s square 那把矩尺
Chinese mathematics grew from a different root, and it suggests a third answer. The oldest text, the Zhoubi suanjing, opens with a question put by the Duke of Zhou: the sky cannot be climbed and the earth cannot be measured with a ruler, so where do numbers come from? The answer he is given starts from a tool. The methods of number come from the circle and the square, the square comes from the carpenter’s try square, and when the try square is folded, “the breadth is three, the height four, and the diagonal five.” The Nine Chapters on the Mathematical Art, the standard textbook for a thousand years, is a collection of 246 problems about fields, granaries, taxes, and canals, each with its answer and its method. It states no axioms. In the third century Liu Hui supplied proofs, mostly by cutting figures up and rearranging the pieces.
中國的數學長自不同的根；而它提示了第三個答案。最古老的典籍《周髀算經》，是從周公提出的一個問題開始的：「夫天不可階而升，地不可得尺寸而度，請問數安從出？」他得到的回答，是從一件工具講起的：「數之法，出于圓方，圓出于方，方出于矩……故折矩，以為句廣三，股修四，徑隅五。」一千年間的標準教科書《九章算術》，是 246 道題目的匯編，講的是田畝、倉廩、賦稅與溝渠；每一題都附有答案與算法（術）。它沒有陳述任何公理。到了三世紀，劉徽補上了證明；多半是把圖形切開，再把各塊重新拼合。
On this picture mathematics is a refinement of things people do: measuring, sharing out, building. It begins as a craft, and its objects are idealized versions of ropes, rulers, and tallies. This makes it less mysterious that mathematics applies to the world, since it was drawn from the world in the first place. It makes it harder to explain how the same craft produced the theory of infinite sets. The physicist Eugene Wigner called the fit between abstract mathematics and the laws of physics “a wonderful gift which we neither understand nor deserve.”
照這個圖像，數學是對人們所做的事情的精煉：測量、分配、建造。它起初是一門手藝；它的對象，是繩子、尺與籌碼的理想化版本。這使得「數學能用在世界上」沒那麼神祕了，因為它本來就是從世界裡提煉出來的。但它也使另一件事更難解釋：同一門手藝，怎麼會產生出無窮集合的理論？物理學家尤金・維格納，把抽象數學與物理定律之間的那份契合，稱為「一份奇妙的禮物；我們既不理解它，也不配得它」。
06Both, perhaps 也許兩者都是
Many mathematicians today would give a divided answer. We invent the concepts and choose the axioms, as the formalists said. Once they are chosen, what follows from them is no longer up to us, and finding it out is discovery in every sense that matters. No one decided that the primes should thin out as they do. Which mathematics describes the physical world is a third question, and it is settled by observation. Einstein found in 1915 that space near a heavy body is better described by one of the non-Euclidean geometries. As he put it, as far as the laws of mathematics are certain, they do not refer to reality, and as far as they refer to reality, they are not certain.
今天許多數學家會給出一個分成幾部分的答案。概念是我們發明的，公理是我們選的；這一點形式主義者說對了。一旦選定，從它們推得出什麼，就不再由我們作主了；而把它查出來，在每一個要緊的意義上都是發現。沒有人決定過質數該照它們現在的方式越來越稀疏。至於「哪一種數學描述了物理世界」，是第三個問題；而它由觀察來決定。愛因斯坦在 1915 年發現：重物附近的空間，用其中一種非歐幾何來描述比較好。照他的說法：數學的定律只要是確定的，就不指涉實在；只要它們指涉實在，就不是確定的。
Thought experiment · 思想實驗
Try it: add up the angles 親手做一次：把內角加起來
On the flat sheet, drag the corners of the triangle anywhere you like and watch the sum of its angles. Then move to the globe, where a triangle has one corner at the north pole and two others that you can slide apart and down toward the equator.
在平面上，把三角形的角隨便拖到哪裡，看看它的內角和。然後換到球面上；那裡的三角形有一個角在北極，另外兩個角可以讓你拉開，也可以往赤道的方向滑下去。
Standard form · 標準形式
The argument, laid out 論證攤開來
The argument for mathematical Platonism in its modern form, due to Quine and Putnam. Click a step to see what can be said against it.
現代形式的數學柏拉圖主義論證；出自蒯因與普特南。點任何一步，看看別人可以怎麼反對它。
Primary text · 原典
In their own words 原典怎麼說
Russell at his most Platonic, and the passage in which a Zhou dynasty official explains where numbers come from.
最「柏拉圖」的時候的羅素；以及一位周代的官員解釋「數從哪裡來」的那一段。
Bertrand Russell, The Problems of Philosophy (1912), chapters 7 and 9; “The Study of Mathematics” (1902); Zhoubi suanjing, opening 羅素《哲學問題》第七、九章；〈數學的研究〉；《周髀算經》卷上 · Russell in the original English (public domain); the Zhoubi translated for this series
The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm. Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
「有」的世界是不變的、嚴整的、精確的；數學家、邏輯學家、形上學體系的建造者，以及所有愛完美勝過愛生命的人，都為它著迷。「存在」的世界是流逝的、模糊的，沒有清晰的邊界，也沒有任何清楚的計畫或安排；但它包含了一切思想與感受、一切感官的材料，以及一切物理對象：每一樣能造成好處或傷害的東西。 數學，如果看得正確，不只擁有真，還擁有至高的美：一種冷峻而嚴肅的美，像雕塑的美；它不訴諸我們天性裡較軟弱的任何部分，沒有繪畫或音樂那種華麗的裝飾，卻純淨得崇高，並且能夠達到一種只有最偉大的藝術才顯示得出來的、嚴格的完美。
Russell later gave up the view that universals and numbers have being in a world of their own, under the influence of Wittgenstein, and said that he had come to see mathematics as a body of tautologies. He never lost the feeling described in the second passage.
羅素後來在維根斯坦的影響之下，放棄了「共相與數在它們自己的世界裡有其存有」的看法；並且說，他後來把數學看成一堆重言式。但第二段話裡描述的那種感受，他始終沒有失去。
Long ago the Duke of Zhou asked Shang Gao: “I have heard that you, sir, are skilled in numbers. May I ask how Fu Xi in ancient times laid out the degrees of the heavens? The sky cannot be climbed by steps, and the earth cannot be measured with a foot rule. May I ask where numbers come from?” Shang Gao said: “The methods of number come from the circle and the square. The circle comes from the square, the square comes from the try square, and the try square comes from nine nines making eighty-one. So fold the try square, and let the breadth be three, the height four, and the diagonal five.”
昔者周公問于商高曰：「竊聞乎大夫善數也。請問古者包犧立周天歷度，夫天不可階而升，地不可得尺寸而度，請問數安從出？」商高曰：「數之法，出于圓方，圓出于方，方出于矩，矩出于九九八十一。故折矩，以為句廣三，股修四，徑隅五。」
The question is Plato’s question, and the answer points to a carpenter’s tool and the multiplication table. The 3-4-5 triangle is the simplest case of the theorem that Europe names after Pythagoras, which is why the Chinese name for it is 勾股, “base and height.” The Zhoubi reached its present form about the first century BCE.
這個問題就是柏拉圖的問題；而回答指向的是一件木匠的工具，以及九九乘法表。三、四、五的三角形，是歐洲以畢達哥拉斯命名的那條定理最簡單的情形；這就是為什麼它的中文名字叫「勾股」。《周髀》大約在公元前一世紀有了現在的樣子。
Both sides · 正反兩面
Objections and replies 反駁與回應
Mathematics is just a language. Asking whether numbers exist is like asking whether the letter Q exists.
數學只是一種語言。問數存不存在，就像問字母 Q 存不存在。
A language can say false things, and its rules are ours to change. English could have used another letter. But we cannot make seven divisible by two by agreeing to. Something resists us in mathematics that does not resist us in spelling, and a theory of mathematics has to say what it is. The formalist says that it is logical consequence. The Platonist says that it is the objects.
一種語言可以說出假的東西；而它的規則是我們可以改的。英文本來可以用另一個字母。但我們沒辦法靠著約定，使七能被二整除。數學裡有某樣東西在抵抗我們，而拼字裡沒有；一個關於數學的理論，必須說出那是什麼。形式主義者說，那是邏輯上的推論關係。柏拉圖主義者說，那是那些對象。
If we had evolved differently, or lived in a different culture, we would have a different mathematics.
如果我們演化成不同的樣子，或者生活在不同的文化裡，我們就會有不同的數學。
We would have a different notation, different interests, and different starting points. The Maya counted in twenties and the Babylonians in sixties. Chinese mathematicians worked with counting rods and solved systems of equations by a method Europe reached eighteen centuries later. Yet where two traditions took up the same question they reached the same answer. The Nine Chapters and Euclid agree about right triangles. This agreement across cultures that had no contact is the strongest evidence that something in mathematics is found and not made up.
我們會有不同的記號、不同的興趣，以及不同的出發點。馬雅人用二十進位；巴比倫人用六十進位。中國的數學家用算籌，並且用一種歐洲晚了十八個世紀才達到的方法來解方程組。然而，凡是兩個傳統處理同一個問題的地方，它們得到的是同一個答案。《九章算術》與歐幾里得，對直角三角形的看法是一致的。這種在毫無接觸的文化之間的一致，是「數學裡有某樣東西是被找到的，而不是被編出來的」最強的證據。
Gödel showed that mathematics is uncertain and incomplete. So it is no better than opinion.
哥德爾證明了數學是不確定、不完備的。所以它並不比意見好到哪裡去。
Gödel’s theorem is itself a piece of mathematics, proved with complete rigor. It shows that no single system of axioms captures every arithmetical truth. It does not show that any proved theorem is doubtful. Seven and five are still twelve. The result limits a certain hope about method, that all of mathematics could be generated mechanically from a fixed list of rules, and leaves the certainty of what has been proved exactly where it was.
哥德爾的定理本身就是一件數學，是以完全的嚴格性證明的。它顯示的是：沒有任何單一的公理系統，能囊括每一條算術的真理。它並沒有顯示任何一條已被證明的定理是可疑的。七加五還是十二。這個結果限制的，是某一種關於方法的期望：以為整個數學都可以從一張固定的規則清單裡機械地產生出來。至於已經被證明的東西的確定性，它原封不動地留在原處。
Terms of art · 哲學術語
The vocabulary of the subject 這門學問的用語
L. ‘from what comes before’
Known independently of experience, as mathematics and logic are held to be.
不依賴經驗而被知道的；人們認為數學與邏輯就是這樣。
The view that mathematical objects exist independently of us, outside space and time.
一種看法：數學對象獨立於我們而存在，在空間與時間之外。
The view that mathematics is the derivation of consequences from chosen axioms by fixed rules.
一種看法：數學就是依照固定的規則，從選定的公理推導出結論。
The program of Frege and Russell to derive all mathematics from logic.
弗雷格與羅素的計畫：把整個數學從邏輯推導出來。
Gk. axiōma ‘what is thought worthy’
A statement taken as a starting point and not proved from anything else.
被當作出發點、而不從任何別的東西來證明的陳述。
A consistent geometry in which Euclid’s rule about parallels is replaced by another.
一種一致的幾何；其中歐幾里得關於平行線的那條規則被換成了另一條。
Gödel’s result that any consistent system strong enough for arithmetic leaves some truths unprovable.
哥德爾的結果：任何一個一致的、強到足以表達算術的系統，都有一些真理是它證明不了的。
Ch. 句股 gōugǔ
The two shorter sides of a right triangle, and the Chinese name for the theorem about them.
直角三角形較短的兩邊；也是關於它們的那條定理的中文名字。
Advanced English · 高級英文
Words and phrases worth keeping 值得帶走的字與片語
No future count could overturn it.
未來的任何一次點數，也推翻不了它。
Plato took the mathematicians at their word.
柏拉圖把數學家的話當真了。
No one has bumped into the number seven.
沒有人撞見過「七」這個數。
Everyone took the starting points to be self-evident.
人人都把那些出發點當成不證自明的。
Russell caught the spirit of it in a joke.
羅素用一個笑話抓住了它的精神。
Frege replied that arithmetic was tottering.
弗雷格回信說，算術正在動搖。
What follows from the axioms is no longer up to us.
從公理推得出什麼，就不再由我們作主了。
The primes thin out as the numbers grow.
數越大，質數越稀疏。
Mathematics was drawn from the world in the first place.
數學本來就是從世界裡提煉出來的。
You watched the sum come out.
你是看著那個內角和跑出來的。
Reading check · 理解測驗
Did it land? 讀懂了嗎
1. What does it mean to say that mathematics is a priori?
2. What is the main difficulty for Plato’s view of mathematics?
3. What did the non-Euclidean geometries show?
4. What is Russell’s paradox about?
5. From what does the Zhoubi say that the methods of number come?
Write · 你怎麼想？
When you learned that the angles of a triangle add up to 180°, were you shown it, told it, or did you see why? Describe how you would teach it now, having tried the globe. About 150 words.
當你學到「三角形內角和是 180°」的時候，是有人做給你看、有人告訴你，還是你自己看出了為什麼？試過了球面之後，描述你現在會怎麼教它。約 150 個字。
A student asks: “Did someone make up negative numbers, or were they always there?” Write the answer you would give.
一個學生問：「負數是有人編出來的，還是它們本來就在那裡？」寫下你會給的回答。
The Nine Chapters teaches through problems and methods, and Euclid through axioms and proofs. In 200 words, say what each approach is good for, in mathematics or in any subject you teach.
《九章算術》用題目與算法來教；歐幾里得用公理與證明來教。用 200 個字說明：在數學上，或者在你所教的任何一科上，這兩種取徑各自擅長什麼。
Dharma and the West · 佛法與西方哲學
A view from the Dharma 佛法怎麼看
Buddhist thought has no single position on mathematics, and its general account of concepts bears on the question. Number depends on division. A chariot is one thing or two wheels or a hundred parts according to how it is counted, and nothing in the wood settles which count is right. On this view “one” and “many” are designations applied by a mind for a purpose. They are neither found ready-made in things nor simply invented, since once a way of dividing is fixed, the count is fixed too. This is close to the divided answer that many mathematicians now give. We choose the concepts, and we do not choose what follows.
佛教思想對數學並沒有單一的立場；而它對「概念」的一般說明，與這個問題有關。數取決於怎麼劃分。一輛車是一樣東西，是兩個輪子，還是一百個零件，要看怎麼數；而木頭裡沒有任何東西能決定哪一種數法才對。照這個看法，「一」與「多」是心為了某個目的而加上去的假名。它們既不是現成地在事物裡被找到的，也不是純粹編出來的；因為一旦劃分的方式定了，數出來的結果也就定了。這很接近今天許多數學家給的那個分成幾部分的答案。概念是我們選的；而推出什麼，不是我們選的。
The tradition also took a practical interest in number. Indian Buddhist texts count in units far beyond any use, with names for ten to the power of fifty and more, and say of the merit of a single good thought that it cannot be reckoned. The purpose is to stretch the mind until its ordinary measures give out. In this the texts stand with Russell’s feeling, in the passage quoted above, that mathematics lifts us out of the scale of human concerns. They would add a caution that he came to accept in his own way. The exact and changeless world is a product of abstraction, and to prefer it to the world of thoughts and feelings, as those who “love perfection more than life,” is one more form of clinging (Dharma and Western Philosophy D3).
這個傳統對數也有實用上的興趣。印度的佛教經典，用遠遠超出任何用途的單位來計數，為十的五十次方以上的數都取了名字；並且說到一念善心的功德時，說它「不可稱計」。目的是把心撐開，直到它平常的量尺都不夠用。在這一點上，這些經典與羅素在上面那段引文裡的感受是站在一起的：數學把我們提到人間關切的尺度之外。它們會補上一句提醒，而羅素後來也以他自己的方式接受了它。那個精確而不變的世界，是抽象的產物；像那些「愛完美勝過愛生命」的人一樣，偏愛它勝過思想與感受的世界，是又一種形式的執取（佛法與西方哲學 D3）。
Go further · 延伸
People and reading 人物與延伸閱讀
- Bertrand Russell, The Problems of Philosophy (1912), chapters 7–10 ↗On a priori knowledge and the world of universals. Short and clear. 談先驗知識與共相的世界。短而清楚。
- Bertrand Russell, “The Study of Mathematics,” in Mysticism and Logic ↗The essay with the passage on mathematical beauty. 有「數學之美」那一段的文章。
- Stanford Encyclopedia of Philosophy, “Philosophy of Mathematics” ↗Platonism, formalism, logicism, and what came after Gödel. 柏拉圖主義、形式主義、邏輯主義，以及哥德爾之後的發展。
- 《周髀算經》卷上；《九章算術》卷九〈句股〉 ↗The dialogue of the Duke of Zhou and Shang Gao, and the right-triangle problems. 周公與商高的對話；以及直角三角形的那些題目。
- Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (1960) ↗A physicist’s statement of the puzzle. The link is to a summary. 一位物理學家對這個謎的陳述。連結是一篇摘要。
- Pythagoras, in this series ↗The discovery that the diagonal of a square is not a ratio of whole numbers. 正方形的對角線不是整數之比的那個發現。