Since ancient times, mankind has hotly debated

自從遠古時代，人類就開始激烈的辯論

whether mathematics was discovered or invented.

是我們發現了數學還是我們發明了數學。

Did we create mathematical concepts to help us understand the universe around us,

是我們創造了數學的概念 來幫助我們理解周圍的世界，

or is math the native language of the universe itself,

還是數學本來就是宇宙的一種語言，

existing whether we find its truths or not?

我們是否發現了它的真理？

Are numbers, polygons and equations truly real,

數字，形狀和等式真實存在，

or merely ethereal representations of some theoretical ideal?

還是它們僅僅是 理論上想法的縹緲的代表？

The independent reality of math has some ancient advocates.

‘數學獨立於現實’在古代有不少擁護者。

The Pythagoreans of 5th Century Greece believed numbers were both

5世紀希臘的畢達哥拉斯相信

living entities and universal principles.

數字既是存在的實體，也是宇宙的原理。

They called the number one, "the monad," the generator of all other numbers

他們把數字“一”叫做單個體。 它是其他所有數字的創造者，

and source of all creation.

也是所有創造的源泉。

Numbers were active agents in nature.

數字是自然界活躍的特工。

Plato argued mathematical concepts were concrete

比拉圖認為數學的概念應是具體的，

and as real as the universe itself, regardless of our knowledge of them.

就像宇宙本身那樣真實， 無論我們是否意識到。

Euclid, the father of geometry, believed nature itself

幾何之父－歐幾里得相信自然本身

was the physical manifestation of mathematical laws.

是數學定律的物理表現。

Others argue that while numbers may or may not exist physically,

其他人則認為不管數字是否存在實體，

mathematical statements definitely don't.

數學的命題完全不是真實存在的。

Their truth values are based on rules that humans created.

它們的真實價值急於人類所創立的原則。

Mathematics is thus an invented logic exercise,

因此數學是一種被發明的邏輯練習，

with no existence outside mankind's conscious thought,

在人類理性的想法之外並不存在，

a language of abstract relationships based on patterns discerned by brains,

它只是一種被大腦識別、 用特殊格式所書寫的抽象語言，

built to use those patterns to invent useful but artificial order from chaos.

用來避免發生混亂的。

One proponent of this sort of idea was Leopold Kronecker,

這種理論的支持者之一 是利奧波德·克羅內克，

a professor of mathematics in 19th century Germany.

他是十九世紀德國數學教授。

His belief is summed up in his famous statement:

他的信仰可以總結如下：

"God created the natural numbers, all else is the work of man."

“上帝創造了自然界的數字， 除此之外都是人類的工作。“

During mathematician David Hilbert's lifetime,

在數學家大衛·希爾伯特的一生中，

there was a push to establish mathematics as a logical construct.

他曾急著把數學作為邏輯來構建。

Hilbert attempted to axiomatize all of mathematics,

希爾伯特曾嘗試 把所有數學的概念都變成公理，

as Euclid had done with geometry.

就像歐幾里德在幾何上的成就一樣。

He and others who attempted this saw mathematics as a deeply philosophical game

他和其他嘗試這樣做的人 將數學視作一種深層次的哲學遊戲，

but a game nonetheless.

但仍然是一個遊戲。

Henri Poincaré, one of the father's of non-Euclidean geometry,

非歐幾里得幾何之父 亨利·龐加萊

believed that the existence of non-Euclidean geometry,

認為非歐幾里得幾何地存在

dealing with the non-flat surfaces of hyperbolic and elliptical curvatures,

處理非水平的雙曲線表面， 以及橢圓曲度，

proved that Euclidean geometry, the long standing geometry of flat surfaces,

證明歐幾里德幾何學，非水平表面的幾何學

was not a universal truth,

並不是一個普遍的事實，

but rather one outcome of using one particular set of game rules.

還不如用以一套特定遊戲規則所得出的結果。

But in 1960, Nobel Physics laureate Eugene Wigner

但在1960年，後來的諾貝爾物理獎 獲得者尤金·維格納

coined the phrase, "the unreasonable effectiveness of mathematics,"

套用了一句老話，“數學離譜得有效率，“

pushing strongly for the idea that mathematics is real

把“數學證實存在”的想法硬推出來，

and discovered by people.

並被人們所發現。

Wigner pointed out that many purely mathematical theories

維格納指出，許多僅僅是憑空想出的數學理論，

developed in a vacuum, often with no view towards describing any physical phenomena,

大多沒有任何觀點描述任何物理現象，

have proven decades or even centuries later,

並在幾十年，甚至幾世紀後被證明，

to be the framework necessary to explain

成為有必要解釋宇宙是如何

how the universe has been working all along.

獨立運作的結構。

For instance, the number theory of British mathematician Gottfried Hardy,

比如，英國數學家戈弗雷·哈代的數論。

who had boasted that none of his work would ever be found useful

戈弗雷自誇稱自己描述任何在真實世界的現象

in describing any phenomena in the real world,

都不會對建立

helped establish cryptography.

密碼學有幫助。

Another piece of his purely theoretical work

他的另一個理論性的成果，

became known as the Hardy-Weinberg law in genetics,

戈弗雷·哈代遺傳定律， 為大家所知，

and won a Nobel prize.

並獲得了諾貝爾獎。

And Fibonacci stumbled upon his famous sequence

斐波那契在看一組被理想化的兔子總數時，

while looking at the growth of an idealized rabbit population.

磕磕絆絆得出了他的著名的數列。

Mankind later found the sequence everywhere in nature,

人類後來發現那個數列砸大自然中到處都是，

from sunflower seeds and flower petal arrangements,

從向日葵的種子和花瓣排列規律，

to the structure of a pineapple,

到菠蘿的結構，

even the branching of bronchi in the lungs.

甚至是肺上的支氣管分支， 無處不在。

Or there's the non-Euclidean work of Bernhard Riemann in the 1850s,

還有十九世紀50年代的 波恩哈德·黎曼的非歐裡機得研究成果，

which Einstein used in the model for general relativity a century later.

愛因斯坦一世紀後才在研究遺傳關聯性的時候 才在模型中使用到它。

Here's an even bigger jump:

這兒甚至有一個更大的跳躍：

mathematical knot theory, first developed around 1771

紐結理論。 它在1771年左右形成，

to describe the geometry of position,

用來描述位置的幾何學，

was used in the late 20th century to explain how DNA unravels itself

在20世紀晚期被用來解釋 DNA在自我複製過程中

during the replication process.

是如何解開自己的。

It may even provide key explanations for string theory.

它甚至會為弦理論提供關鍵性的證明。

Some of the most influential mathematicians and scientists

人類歷史上 最有影響力的機位數學家和科學家

of all of human history have chimed in on the issue as well,

也已經就這個問題發表了自己的看法，

often in surprising ways.

而且經常還是以令人驚訝的方式。

So, is mathematics an invention or a discovery?

那麼，數學是一個發明還是一個發現呢？

Artificial construct or universal truth?

是人工構建物還是普遍的真理？

Human product or natural, possibly divine, creation?

是人類產物，還是自然 （或者是上帝）的創造物？

These questions are so deep the debate often becomes spiritual in nature.

這些問題十分深奧， 辯論常常會變成自然的靈歌。

The answer might depend on the specific concept being looked at,

答案也許隨著研究的特定概念的變化而變化，

but it can all feel like a distorted zen koan.

但它可以像是一個扭曲的禪宗公案。

If there's a number of trees in a forest, but no one's there to count them,

如果森林里有很多樹，但沒人去數，

does that number exist?

那麼數字到底存在嗎？