历史上伟大的数学家

by Elena Nisioti

由Elena Nisioti

数学历史上的10个尴尬时刻 (10 awkward moments in math history)

We have all experienced our awkward moments. Something unexpected happens, there is some social tense and a personal uneasiness and you would really like to get over it or forget that it ever happened. But what if you are a rigorous mathematician and you just got your world disproven?

我们都经历了尴尬的时刻。 出乎意料的事情发生了，有一些社交时态和个人不安，您真的想克服它或忘记它曾经发生过。 但是，如果您是一位严格的数学家，而您只是被世界证明了该怎么办？

Math has always been about the pursuit of understanding the world through logic and expressing it in a strictly defined, mathematical language. It is really indicative, educative, and fun, to observe mathematics when it stopped (momentarily) making sense.

数学一直是关于通过逻辑理解世界并以严格定义的数学语言表达的追求。 当数学停止(暂时)有意义时，观察数学确实是指示性的，教育性的和有趣的。

1.非理性数的发现 (1. The discovery of irrational numbers)

As the origins of mathematical rigor lie in ancient Greece, mathematical thought started close to religious beliefs, thus numbers were attributed divine characteristics.

由于严格的数学起源于古希腊，数学思想开始接近宗教信仰，因此数字被赋予神圣的特征。

The School of Pythagoras, an occult team of early mathematicians that pushed mathematical knowledge forward, like all cults, was based on some fundamentalistic beliefs. Astonished by the applicability of ratios to every practical problem, they believed that ratios (yes, simple divided numbers) are divine, as they can explain anything that is happening in the world.

毕达哥拉斯学校是一个由早期数学家组成的神秘团队，像所有邪教一样，它推动了数学知识的发展，它是基于一些原教旨主义的信念。 比率对每个实际问题的适用性感到惊讶，他们认为比率(是的，简单的除数)是神圣的，因为它们可以解释世界上正在发生的一切。

Accordingly, everything that is happening in the world should be able to be expressed as a ratio, right?

因此，世界上发生的一切都应该能够以比率表示，对吗？

Now, imagine their surprise when they discovered the number square root of 2, while applying the newly formulated Pythagorean Theorem. This irrational number (irrational meaning that it cannot be expressed as the ratio of two numbers) defied world order as expressed by divinity of ratios and questioned their whole philosophy.

现在，想象一下他们在应用新公式化的勾股定理时发现2的平方根时感到惊讶。 这个不合理的数字(不合理的含义是它不能表示为两个数字的比率)违背了由比率的神圣性所表达的世界秩序，并质疑了它们的整体哲学。

Terrified by the consequences of this revolutionary discovery, they decided not to tell anyone about it. It is also said, that they even drowned the man who did the discovery, Hippasus. Quiet scientific, don’t you think?

他们对这一革命性发现的后果感到震惊，决定不告诉任何人。 也有人说，他们甚至淹死了做发现的人希帕索斯。 安静的科学，你不觉得吗？

2.无限 (2. Infinity)

The discovery of irrational numbers, being already bad as it was, brought the Greeks in front of a more terrifying discovery: infinity. As irrational numbers are characterized by having an infinite number of decimal digits, Greeks had to come up with an explanation for how a never-ending series of numbers can be created. The notion of infinity is difficult to understand today, let alone an age when religion was connected with science and a mathematical belief should not challenge our understanding of God. So, what did the Greeks do? Philosophers, like Aristotle and Plato, rejected the notion of an absolut infinity and mathematicians came up with inventive ways to circumvent the need for infinity in geometry, like Eudoxus of Cnidus who developed the method of exhaustion to calculate the area of shapes. It was not until the late 17th century that Newton and Leibniz encouraged taking infinity into account through their use of infinitesimals and John Wallis introduced the well-known symbol of infinity in 1655.

非理性数字的发现本来就很糟糕，但使希腊人走到了一个更可怕的发现面前：无限。 由于非理性数字的特征是具有无限数量的十进制数字，因此希腊人不得不提出一种解释，说明如何创建无休止的数字系列。 今天，无限的概念很难理解，更不用说宗教与科学联系在一起的时代了，数学信仰不应挑战我们对上帝的理解。 那么，希腊人做了什么？ 像亚里斯多德和柏拉图这样的哲学家都拒绝绝对无穷大的概念，而数学家想出了一些创造性的方法来规避几何无穷大的需要，例如Cnidus的Eudoxus开发了用力法来计算形状的面积。 直到17世纪末，牛顿和莱布尼兹通过使用无穷小鼓励使用无穷大，约翰·沃利斯(John Wallis)于1655年引入了著名的无穷大符号。

3.芝诺悖论 (3. Zeno’s paradoxes)

The Greeks certainly went to extremes when it came to philosophical reasoning.

当谈到哲学推理时，希腊人当然走到了极端。

After his predecessor Heraclitus claimed that everything in the world is constantly changing, Parmenides claimed that nothing changes. As a result, motion is a mere illusion and therefore, using mathematics, the language of truth according to the Greeks, to describe it should be impossible.

在他的前任赫拉克利特宣称世界上的一切都在不断变化之后，帕门尼德斯声称一切都没有改变。 结果，运动仅仅是一种错觉，因此，用数学(希腊人所说的真理语言)来描述运动应该是不可能的。

Zeno, one of Parmenides’ students, devised a series of paradoxes aimed to prove the irrationality of motion. The most famous one, Achilles and his tortoise, goes like this: Achilles is racing against a tortoise, which being significantly slower is given the advantage of starting the race 100 meters ahead of him.

帕梅尼德斯大学的学生之一芝诺(Zeno)设计了一系列悖论，旨在证明运动的不合理性。 最有名的阿喀琉斯和他的乌龟是这样的：阿喀琉斯正在与一只乌龟赛跑，这明显要慢得多，这得益于他领先100米开始比赛的优势。

If we assume, for the shake of simplicity, that the speeds of the two contestants are constant and Achilles is 10 times faster than the tortoise then we can say that when Achilles reaches the starting point of the tortoise, this will have run 10 meters. So, Achilles will try to catch up and by the time he reaches this next point, the tortoise will have moved an extra one meter.

如果为简单起见，假设两个参赛者的速度恒定并且阿喀琉斯比乌龟快10倍，那么我们可以说当阿喀琉斯到达乌龟的起点时，它将跑10米。 因此，阿喀琉斯将努力追赶，直到他到达下一个点时，乌龟会再移动一米。

This high-school math problem, being as simple and clear as it is, leads us to the following paradoxical conclusion: Achilles will never reach the tortoise no matter how much faster he is. Congratulations Zeno, you made motion sound illogical.

这个简单而清晰的高中数学问题使我们得出以下矛盾的结论：阿基里斯永远不会到达乌龟，无论他有多快。 恭喜，Zeno，您使动作听起来不合逻辑。

Zeno’s paradoxes were believed to exist in the realm of metaphysics and troubled philosophers and mathematicians for ages, but today they can be explained with calculus, a mathematical tool the Greeks did not possess. Let’s “move” on then.

人们认为芝诺的悖论存在于形而上学领域，并且困扰着哲学家和数学家多年，但如今，可以用微积分(希腊人所不具备的数学工具)来解释它们。 让我们继续前进。

4.莫比乌斯 地带 (4. Möbius strip)

The funny looking Möbius strip, which was also independently discovered in 1858 by the unlucky Listing whose name left the history of mathematics untouched, is a surface with only one side and only one boundary, often used to puzzle young math students.

看上去很有趣的莫比乌斯地带，也是由不幸的Listing在1858年独立发现的，其名称使数学史保持不变。它的表面只有一侧，只有一个边界，通常用来迷惑年轻的数学学生。

You can easily create it by taking a strip of paper, twisting it and then joining the ends of the strip.

您可以轻松地创建它，方法是取一条纸，将其扭曲，然后将纸条的两端连接起来。

Being the first example of a surface without orientation it did not shake the grounds of mathematics as much as the other discoveries of this list did, yet it provided a lot of practical applications, such as a resistant belt, and inspired mathematicians to come up with unorientable surfaces, like the Klein bottle. (The name of this surface possibly comes from a double coincidence: Klein, its conceptor, originally named it Fläche, which means surface in German and sounds similar to Flasche, which means bottle. The fact that it also looked like a bottle seems to have sealed the renaming).

作为没有方向的表面的第一个例子，它没有像列表中的其他发现那样动摇数学基础，但是它提供了许多实际应用，例如阻力带，并激发了数学家的想法。无法定向的表面，例如Klein瓶子。 (此表面的名称可能来自双重巧合：其概念设计者克莱因(Klein)最初将其命名为Fläche，这意味着德语为表面，听起来与Flasche相似，即为瓶子。看起来也像瓶子的事实似乎具有密封重命名)。

5. Cantor的实数不可数 (5. Cantor’s uncountability of real numbers)

Dealing with infinity already being a drag, Cantor proved in 1874 that there are in fact different kinds of infinity. In particular, proving the uncountability of real numbers, Cantor proved that this set is bigger to the already infinite set of natural numbers.

应对无限性已经是一种阻力，康托尔在1874年证明实际上存在着各种不同的无限性。 尤其是证明了实数的不可数性，康托尔证明了该集合比已经无限的自然数更大。

In 1891, he also provided the diagonal argument, one proof so elegant that it was later adopted as a tool to prove through the use of a paradox. His remark gave birth to the theory of cardinal numbers, as well as paradoxes dealing with the question: how many infinities can you handle?

在1891年，他还提供了对角线论证，这一论证是如此优雅，以至于后来被采纳为通过使用悖论进行论证的工具。 他的话催生了基数理论，以及处理这个问题的悖论 ：您可以处理多少个无穷大？

6.罗素悖论 (6. Russell’s paradox)

In 1901 Russell discovered a weak spot in Cantor’s so-far well-established set theory, which led him to a contradiction the mathematical world could not oversee. According to this theory, any collection of things can be a set.

1901年，罗素(Russell)在康托尔迄今为止建立良好的集合论中发现了一个弱点，这使他陷入了数学界无法监督的矛盾。 根据这个理论，任何事物的集合都可以是一个集合。

Russell’s contradictory example, also called the Barber’s paradox, goes as follows: imagine a town that has a special rule; every man that is not shaved by himself must be shaved by the barber of the town. The awkward question, which you can try to answer yourselves, is: who shaves the barber?

拉塞尔矛盾的例子，也称为理发师悖论，如下：想象一个有特殊规则的小镇； 每个不自己剃光的人都必须被镇上的理发师剃光。 您可以尝试回答自己的尴尬问题是：谁剃了理发师？

This discovery led him to questioning the mere foundations of the previous set theory and creating a new one, which being more complicated than the later proposed Zermelo-Fraenkel set theory, did not catch up.

这一发现使他开始质疑先前的集合论的基础并创建了一个新的集合论，该理论要比后来提出的Zermelo-Fraenkel集合论复杂得多，但没有赶上。

7.哥德尔不完备定理 (7. Gödel’s incompleteness theorems)

If the previous events seemed to create slightly uncomfortable moments, wait for the following awkward turtle (and this is worse than the one of Achilles).

如果先前的事件似乎造成了些许不舒服的时刻，请等待下一个笨拙的乌龟(这比阿喀琉斯的那只更糟糕)。

We are talking about the 20th century. People did not just want to know. They wanted to know if it is possible to know, and prove it. Unluckily for them, and the human need for understanding the universe, Gödel published in 1931 two theorems, known as the incompleteness theorems.

我们正在谈论20世纪。 人们不只是想知道。 他们想知道是否有可能知道并证明这一点。 不幸的是，对于人类以及人类对于理解宇宙的需求，哥德尔在1931年发表了两个定理，即不完全性定理。

Explaining the technicalities of them is as difficult as coming in terms with their conclusions, as what Gödel proved was that, considering a consistent and complete system, such as the language of arithmetic, there are statements that are both true and cannot be proven. He illustrated the truth of his theorem with this simple statement, inspired by the liar’s paradox: “This statement cannot be proven”. If this is true, then this statement is true and cannot be proven. If this is false, then this statement can be proven, which contradicts to the original argument that it cannot be proven.

解释它们的技术性与得出结论一样困难，因为哥德尔证明的是，考虑到一致而完整的系统(例如算术语言)，有些陈述是真实的，无法证明。 在受骗子悖论的启发下，他用一个简单的陈述说明了他定理的真理：“这一陈述无法得到证明”。 如果是这样，则此陈述是正确的，无法得到证明。 如果这是错误的，则可以证明该陈述，这与无法证明它的原始论点相矛盾。

These were very bad news for mathematics, depriving them of their original glare of explaining the absolute truth. It was also a terrible comeback to Hilbert’s quest for knowledge, expressed in his statement “We must know, we will know”.

对于数学来说，这是非常不好的消息，使他们丧失了解释绝对真理的本能。 这也是希尔伯特对知识的追求的一次可怕的复出，希尔伯特在声明中表示“我们必须知道，我们将知道”。

8.塔斯基的不可定理 (8. Tarski’s undefinability theorem)

It seems that Tarski was inspired by the despair created by Gödel. In 1936 he provided proof for the undefinability problem.

塔斯基(Tarski)似乎受到了哥德尔(Gödel)的绝望的启发。 1936年，他为无法定义问题提供了证据。

Although the observations made by Tarski are also included in Gödel’s work, it is argued that Tarski’s work has a more profound philosophical impact. Tarski managed to reach the general conclusion that a language cannot define truth in itself. Although this is an important limitation, he suggests that using a more powerful meta-language is sufficient to define truth in the simpler language.

尽管塔斯基的观点也包含在哥德尔的著作中，但有人认为塔斯基的著作具有更深远的哲学影响。 塔尔斯基设法得出一个普遍的结论，即一种语言本身无法定义真理。 尽管这是一个重要的限制，但他建议使用更强大的元语言足以用更简单的语言定义真理。

Now, an ordinary person may think that this solves the problem, but for a mathematician looking for the “one language to rule them all” this is not that consoling.

现在，一个普通人可能认为这可以解决问题，但是对于寻找“一种语言来统治所有人”的数学家来说，这并不是那么令人安慰。

9.停顿问题 (9. The halting problem)

Alan Turing attempted to tackle the decision problem, which, in simple words, dealt with finding an algorithm that can answer whether a statement is true or not. In order to tackle this conceptually simple, but hard-to-solve problem, he rephrased it to the halting problem: is there a machine that can tell you whether a program will halt on a given problem?

艾伦·图灵(Alan Turing)试图解决决策问题，用简单的话来说，就是寻找一种可以回答陈述是否正确的算法。 为了解决这个概念上简单但难以解决的问题，他将其重新表述为停顿问题：是否有一台机器可以告诉您程序是否会在给定问题上停顿？

Halting means that it will not loop forever. But how do you prove the infeasibility of a machine you know so little about? This is where paradoxes come handy.

停止意味着它不会永远循环。 但是，如何证明您所知甚少的机器不可行呢？ 这就是悖论派上用场的地方。

Alan Turing began by assuming the existence of a machine which given an input program and a problem answers the question of whether it will halt or not. He then augmented this machine by looping its output back to itself if the answer was yes and halting if the answer was no.

艾伦·图灵(Alan Turing)首先假设存在一台提供输入程序的机器，然后一个问题回答了该机器是否会停止的问题。 然后，如果答案是肯定的，则将其输出循环回自身，然后如果答案是否定的，则停止输出，从而扩大了该机器。

So, will the augmented machine halt on the halting problem? Alan’s answer is: if yes then no, if no then yes. Sounds like bad news for logic.

那么，增强型机器是否会因停止问题而停止运行？ 艾伦的答案是：如果是，则否，如果不是，则是。 听起来对逻辑来说是个坏消息。

10.没有免费的午餐定理 (10. The No Free Lunch Theorem)

The passage to the 21st century signified a transfer from pure, almost philosophical mathematics, to applied areas, such as statistics and optimization.

进入21世纪，标志着从纯粹的，几乎是哲学的数学向应用领域的转变，例如统计和优化。

If you consider yourself being fond of optimization, don’t you think this will make you a perfectionist? And wouldn’t a perfectionist want to find the optimal way to optimize things?

如果您认为自己喜欢优化，那么您是否认为这会使您成为完美主义者？ 完美主义者会不会想找到优化事物的最佳方法？

It seems that David Wolpert and William Macready sensed this need and came up with an answer, which, of course, was not encouraging at all (otherwise it would not be in our list). According to their No free lunch theorem for Optimization, published in 1997, “any two optimization algorithms are equivalent when their performance is averaged across all possible problems.”

似乎David Wolpert和William Macready意识到了这一需求，并想出了一个答案，这当然一点也不令人鼓舞(否则就不会出现在我们的清单中)。 根据他们在1997年发布的“没有免费的午餐优化定理”，“当在所有可能的问题上平均其性能时，任何两个优化算法都是等效的。”

Heart-breaking this may be, it does not mean optimization is futile. We’ll just never find a generally optimal way to do it.

这可能令人心碎，但这并不意味着优化是徒劳的。 我们永远不会找到通常最佳的方法。

These moments made the world of mathematics feel awkward, which is a light term for the feelings of despair and chaos that scientists tend to experience when the universe stops making sense. But shock is the way to move science forward.

这些时刻使数学世界感到尴尬，这是对当宇宙变得无意义时科学家倾向于经历的绝望和混乱感的轻描淡写。 但是震惊是推动科学前进的方式。

Mathematical fields were created, we got the Turing Machine, fancy looking surfaces and, most importantly, the ability to re-examine our perceptions and adapt our tools accordingly.

创建了数学领域，获得了图灵机，精美的表面，最重要的是，还具有重新检查我们的看法并相应地调整工具的能力。

These questioning moments helped us evolve intellectually.

这些疑问时刻帮助我们在智力上发展。

Except for the incompleteness theorems. These were just devastating.

除了不完全性定理。 这些只是毁灭性的。

翻译自: https://www.freecodecamp.org/news/10-awkward-moments-in-math-history-d364706d902d/

历史上伟大的数学家