How Writing Changes Mathematical Thought
David E.
大卫·E。
Dunning explores how choices of notation — such as using Roman or Hindu-Arabic numerals — changes what you can do with math.
邓宁探讨了符号的选择（例如使用罗马数字或印度阿拉伯数字）如何改变你可以用数学来做的事情。
I t’s natural to think of math as being fundamentally abstract.
人们很自然地认为数学本质上是抽象的。
Whether it’s invented or discovered, its truths are so literally universal that even aliens would agree (so the thinking goes) that 2 and 2 make 4.
无论它是发明的还是发现的，它的真理是如此普遍，甚至连外星人也会同意（这样的想法）2加2等于4。
The actual work of mathematics, though, typically involves something utterly earthbound: “making marks on paper or blackboards,” said David E.
然而，数学的实际工作通常涉及一些完全接地气的事情：“在纸上或黑板上做标记，”大卫·E·
Dunning , a historian of mathematics and curator of the history of science at the Smithsonian’s National Museum of American History.
邓宁，史密森美国国家历史博物馆的数学历史学家和科学史馆长。
And that mark-making, or “notation” — which might include anything from notches on a stick to obscure typographical symbols on a screen — produces not only intellectual ripple effects, but practical, material, and social ones too.
这种标记或“符号”——可能包括从棍子上的凹口到屏幕上模糊的印刷符号的任何东西——不仅产生智力上的连锁反应，而且还产生实用的、物质的和社会的连锁反应。
“Mathematicians are doing something when they explore this realm that we experience as abstract,” Dunning said.
“当数学家探索这个我们认为抽象的领域时，他们正在做一些事情，”邓宁说。
“Ideas have developed in tandem with different ways of representing them in written form.
“想法与以书面形式表达它们的不同方式同时发展。
This is where I find focusing on notation so useful — I see [it] as the way of grounding what mathematicians do in lived activity, in the physical world.”
这就是我发现关注符号非常有用的地方——我认为[它]是数学家在生活活动和物理世界中所做工作的基础方式。”
Dunning, who studies the social effects of notation, found his calling gradually, beginning with a double major in math and English as an undergraduate.
研究符号社会影响的邓宁逐渐找到了自己的使命，从本科时主修数学和英语双学位开始。
“They felt related to me,” he said.
“他们感觉和我有联系，”他说。
“They were both about the worlds and structures that can be summoned by writing.” After pursuing both interests separately, he realized that graduate study in the history of science could unite them.
“它们都是关于可以通过写作召唤的世界和结构。”在分别追求这两种兴趣之后，他意识到科学史的研究生学习可以将他们结合起来。
In particular, he was inspired by the sociologist David Bloor’s treatise Knowledge and Social Imagery to learn more about “how the creation of mathematical knowledge is something that people do in interaction with each other, deeply embedded in their context.”
特别是，他受到社会学家大卫·布洛尔（David Bloor）的论文《知识与社会意象》（Knowledge and Social Imagery）的启发，更多地了解“数学知识的创造如何是人们在彼此互动中所做的事情，并深深嵌入到他们的背景中。”
In other words, you don’t have to believe that math itself is all relative — which, to be clear, Dunning doesn’t — in order to engage with the social contingencies of its various forms of notation.
换句话说，你不必相信数学本身都是相对的——明确地说，邓宁并不这么认为——才能应对各种形式的符号所带来的社会偶然性。
Those forms “have to be invented, we have to learn to use them, they have affordances and limitations,” he said.
他说，这些形式“必须被发明，我们必须学会使用它们，它们有可供性和局限性”。
“It is a technology in a lot of ways.”
“从很多方面来说，这都是一项技术。”
Quanta spoke with Dunning about how that technology has influenced mathematical thought — about why Roman numerals are hard to use, how symbols beat shapes in calculus, and why arguments about writing things down may have produced some of our most fundamental insights about logic and computation.
广达与邓宁讨论了这项技术如何影响数学思想——为什么罗马数字很难使用，符号如何在微积分中击败形状，以及为什么关于写下来的争论可能产生了我们对逻辑和计算的一些最基本的见解。
The conversation has been condensed and edited for clarity.
为了清晰起见，对话已被压缩和编辑。
Dunning studied both English and math as an undergraduate.
邓宁本科时学习英语和数学。
“They felt related to me,” he said.
“他们感觉和我有联系，”他说。
“They were both about the worlds and structures that can be summoned by writing.”
“它们都是关于可以通过写作召唤的世界和结构。”
A notation is always a set of practices.
符号始终是一组实践。
The most fundamental type of mathematical notation is numerals — being able to write down numbers.
最基本的数学符号是数字——能够写下数字。
On the level of what squiggles you make, there’s always something arbitrary about it.
就你所画的曲线而言，它总是有一些随意性。
But notation isn’t just the specific squiggles.
但符号不仅仅是特定的曲线。
It’s the rules, it’s what you do with it.
这就是规则，这就是你用它做什么。
And different numeral systems have different strengths and weaknesses.
不同的数字系统有不同的优点和缺点。
The one we use, the Hindu-Arabic numerals, developed in India, traveled to the Arab world, and then spread to Europe largely through communities of merchants.
我们使用的印度-阿拉伯数字，在印度发展起来，传播到阿拉伯世界，然后主要通过商人社区传播到欧洲。
In comparison with the Roman numerals that were previously widespread in Europe, the Hindu-Arabic numerals facilitate the reckoning that merchants need to do for commerce much more easily.
与以前在欧洲广泛使用的罗马数字相比，印度-阿拉伯数字使商人更容易进行商业计算。
It’s not impossible to conceptualize arithmetic in Roman numerals, but it doesn’t lend itself to arithmetical activity in the same way.
用罗马数字来概念化算术并非不可能，但它并不适合以同样的方式进行算术活动。
With Roman numerals, you need a new symbol every time you reach a new order of magnitude.
对于罗马数字，每次达到新的数量级时都需要一个新的符号。
If you’re writing years, that’s not a problem, but it has a limit.
如果你写了很多年，那不是问题，但它有一个限制。
Whereas if you understand the Hindu-Arabic numerals, then with only 10 symbols, you can represent potentially infinitely many numbers.
然而，如果您了解印度-阿拉伯数字，那么只需 10 个符号，您就可以表示潜在的无限多个数字。
You can, in theory, understand any natural number.
理论上，你可以理解任何自然数。
Physical models, such as these seen at the Smithsonian’s National Museum of American History, were long used to gain new insights into surfaces and other mathematical objects.
物理模型，例如在史密森美国国家历史博物馆看到的模型，长期以来一直被用来获得对表面和其他数学对象的新见解。
“They used to be in every math department,” Dunning said.
“他们曾经在每个数学系都有，”邓宁说。
“There was an attitude in which part of studying math was to cultivate this physical intuition for the forms that equations represented.”
“有一种态度认为，学习数学的一部分就是培养对方程所代表形式的物理直觉。”
More importantly, they’re not just a system for static representation.
更重要的是，它们不仅仅是一个静态表示的系统。
With them comes the algorithms we know for addition, multiplication, and so on — a way of calculating.
随之而来的是我们所知的加法、乘法等算法——一种计算方式。
We take that for granted: Schoolchildren learn how to carry digits and multiply large numbers together.
我们认为这是理所当然的：小学生学习如何进位和如何将大数相乘。
But I think that brings into focus how the Hindu-Arabic numerals are an amazing technology.
但我认为这让人们注意到印度-阿拉伯数字是一项令人惊叹的技术。
We are so used to a world where that technology has been widely available for so long, but it’s important to realize that historically, multiplying large numbers was a difficult thing to do — until you had a system that made it easy.
我们已经习惯了这种技术已经广泛应用了很长时间的世界，但重要的是要认识到，从历史上看，大数相乘是一件困难的事情——直到你有一个可以轻松实现这一点的系统。
That is the power of the notation.
这就是符号的力量。
When you go further back, there are also complex systems for representing numbers that are not writing per se.
当你再往前追溯时，你会发现还有一些复杂的系统来表示数字，但它们本身并不是书写的。
I’m thinking in particular of the Incan quipus , which are systems of knotted cords that can encode sophisticated numerical information.
我特别想到印加 quipus，这是一种可以编码复杂数字信息的打结绳索系统。
And there’s a Roman finger-counting system, which continued to be pretty widely used through the Middle Ages in Europe, that could go up to 9,999 on two hands.
还有一种罗马手指计数系统，在整个中世纪的欧洲继续得到广泛使用，两只手的计数可达 9,999。
An illustration — found in the Summa de Arithmetica , an Italian treatise on algebra published in 1494 — of a Roman finger-counting system that allowed people to represent numbers up to 9,999 using two hands.
罗马手指计数系统的插图（见于 1494 年出版的意大利代数专着《算术大全》），允许人们用两只手表示 9,999 以内的数字。
There is some relevant history we can think about to get at this idea — in the different ways that Newton and Leibniz approached calculus and its notation.
我们可以思考一些相关的历史来理解这个想法——牛顿和莱布尼茨研究微积分及其符号的不同方式。
That’s right.
这是正确的。
Newton understood his calculus to be much more rooted in geometry than Leibniz did.
牛顿认为他的微积分比莱布尼茨更加植根于几何。
Newton’s masterpiece, the Principia , doesn’t introduce symbolic calculus — like Euclid, it opens with definitions and axioms and then has a lot of diagrams.
牛顿的杰作《原理》没有引入符号微积分——就像欧几里得一样，它以定义和公理开头，然后有很多图表。
It’s important to realize that [at the time] Euclid was still the foundational text of mathematics, while algebra was often seen as sort of a hack to get the value you wanted for a problem that was, in its “truer” form, geometric.
重要的是要认识到，（当时）欧几里得仍然是数学的基础文本，而代数通常被视为一种黑客手段，可以从“更真实”的几何形式的问题中获得你想要的值。
“Ideas have developed in tandem with different ways of representing them in written form,” Dunning said.
“想法与以书面形式表达它们的不同方式同时发展，”邓宁说。
Leibniz wanted a much more algebraic, symbolic, notation-focused calculus.
莱布尼茨想要一种更加代数、符号、注重符号的微积分。
There’s this quote you see from him: “I dare say that this is the last effort of the human mind, and when this project shall have been carried out, all that men will have to do will be to be happy.” I mean, he sounds like an AI hype guy or something — this idea that our systems of writing can do the thinking for us.
你可以从他那里看到这样一句话：“我敢说，这是人类思想的最后努力，当这个项目完成时，人们所要做的就是幸福。”我的意思是，他听起来像是一个人工智能炒作者之类的——他认为我们的写作系统可以为我们思考。
That’s really how Leibniz understood notation.
这确实是莱布尼茨理解符号的方式。
Yes — most significantly the integral sign, which is the most recognizable imagery of calculus.
是的——最重要的是积分符号，这是微积分中最容易识别的图像。
It’s an elongated “S,” for “sum.” Leibniz’s calculus got used a lot more in continental Europe, and it just grew and was fertile in a way that Newton’s wasn’t.
这是一个拉长的“S”，代表“总和”。莱布尼茨的微积分在欧洲大陆得到了更多的应用，而且它的发展和丰富程度是牛顿微积分所没有的。
I do think to some degree that’s true.
我确实认为在某种程度上这是真的。
One thing people point to is that Leibniz’s dy / dx notation [for differentiation] invited “playing with” in ways that Newton’s doesn’t.
人们指出的一件事是，莱布尼茨的 dy / dx 符号（用于微分）以牛顿的方式没有邀请“玩弄”。
Notes that Gottfried Leibniz wrote about integral calculus, found in his archive in Hanover, Germany.
指出戈特弗里德·莱布尼茨 (Gottfried Leibniz) 撰写了有关积分的文章，可在他位于德国汉诺威的档案中找到。
His notation later gave rise to the notation we use in calculus today.
他的符号后来产生了我们今天在微积分中使用的符号。
Gottfried Wilhem Leibniz Bibliothek/Stephen Wolfram/Public Domain
Gottfried Wilhem Leibniz 图书馆/Stephen Wolfram/公共领域
But I don’t want to reduce it to that.
但我不想把它简化为那样。
Leibniz’s notation catches on because his circle of collaborators takes it and runs with it, and their successors — people like Euler and Lagrange and Laplace — develop analysis into a whole discipline in continental Europe over the next century.
莱布尼茨的符号之所以流行，是因为他的合作者圈子接受了它并遵循它，而他们的继任者——如欧拉、拉格朗日和拉普拉斯——在下个世纪将分析发展成为欧洲大陆的一门完整的学科。
In England, Newton’s mathematical physics gets positioned as important, but not as the foundation of a research culture.
在英国，牛顿的数学物理学被视为重要的，但不是研究文化的基础。
So you get this situation where the English tradition that so venerates Newton isn’t actually keeping up with what Newtonian mechanics is like in other places.
所以你会遇到这样的情况，英国如此崇敬牛顿的传统实际上并没有跟上其他地方的牛顿力学。
In the early 19th century, young mathematics students in England are frustrated that they’re still being taught in Newton’s notation when all of math is happening in Leibniz’s.
19世纪初，英国年轻的数学学生感到沮丧，因为当所有的数学都在莱布尼茨的符号中发生时，他们仍然用牛顿的符号进行教学。
This is also a time when the French Revolution and the Napoleonic Wars make cultural exchange between England and Paris especially difficult.
这也是法国大革命和拿破仑战争使英国和巴黎之间的文化交流变得尤其困难的时期。
So [the notation] is not something that they can just quickly switch, and they don’t have the power to.
所以[符号]不是他们可以快速切换的东西，他们也没有权力这样做。
It’s more of a gradual transition, as that generation works their way up to be in the position to set exam questions.
这更像是一个渐进的过渡，因为那一代人不断努力，直到能够设置考试问题。
And by the middle of the 19th century, English mathematics switches over to Leibniz’s notation, and the transformation is really complete.
到了19世纪中叶，英国数学转向了莱布尼茨的记数法，并且转变真正完成了。
Dunning works with the Smithsonian’s mathematics and computer collections.
邓宁研究史密森尼的数学和计算机藏品。
Not always, and not always so clearly.
并不总是如此，也不总是那么清楚。
Mathematical logic is an especially important example where you see a huge proliferation of notations — and how comfortable authors seem [to be] with it.
数理逻辑是一个特别重要的例子，你可以看到符号的大量扩散，以及作者似乎对它的适应程度。
Eventually they converge, more or less, but you also just see coexistence.
最终它们或多或少地融合在一起，但你也只是看到了共存。
One reason is that [logicians] have very different goals, especially early on.
原因之一是[逻辑学家]有非常不同的目标，尤其是在早期。
George Boole, who publishes his first book on mathematical logic in 1847, believes that logic is mathematical, and that you can do the same logic as Aristotle more efficiently by representing syllogisms as equations.
乔治·布尔 (George Boole) 于 1847 年出版了他的第一本关于数理逻辑的书，他认为逻辑是数学的，通过将三段论表示为方程，可以更有效地实现与亚里士多德相同的逻辑。
So for him, it’s very important to use existing algebraic notation: the system readers already know.
所以对他来说，使用现有的代数符号非常重要：系统读者已经知道了。
But in 1879, Gottlob Frege, the German mathematician, publishes his Begriffsschrift , which means “concept script.” For him, the goal is the opposite: It’s to show that mathematics really is logic.
但在 1879 年，德国数学家戈特洛布·弗雷格 (Gottlob Frege) 出版了他的 Begriffsschrift，意思是“概念脚本”。对他来说，目标恰恰相反：证明数学确实是逻辑。
To do that, his notation for logic needs to have no mathematical notation in it at all, because he’s going to eventually rebuild mathematics.
为此，他的逻辑符号中根本不需要包含任何数学符号，因为他最终将重建数学。
And so he invents a system that looks nothing like prior mathematical notation.
因此，他发明了一种与以前的数学符号完全不同的系统。
A page from Gottlob Frege’s Begriffsschrift , in which he develops a formal notation for logic.
Gottlob Frege 的 Begriffsschrift 中的一页，他在其中开发了逻辑的形式符号。
For a while, this multiplicity [of notations] just becomes how logic works.
有一段时间，这种[符号]的多样性就成为了逻辑的运作方式。
Part of it is because logic doesn’t have one very clear application or one unified purpose.
部分原因是逻辑没有一个非常明确的应用程序或一个统一的目的。
Different writers think it matters for different reasons, and that is reflected in what system of notation they think best serves their purposes.
不同的作者出于不同的原因认为它很重要，这反映在他们认为哪种符号系统最能满足他们的目的。
Keeping up with the literature meant constantly moving between these systems and thinking about what they could and couldn’t do.
跟上文献意味着不断地在这些系统之间移动并思考它们能做什么和不能做什么。
This proliferation of notations isn’t unique to logic, but it uniquely matters.
这种符号的激增并不是逻辑所独有的，但它却具有独特的重要性。
In the 1930s, you see this culminating period where you have Kurt Gödel and Alan Turing and Alonzo Church putting forward really important theorems [about incompleteness and computation ] — where what a system of writing can do is the subject matter, is what you would prove a theorem about.
在 20 世纪 30 年代，你会看到这个巅峰时期，库尔特·哥德尔 (Kurt Gödel)、艾伦·图灵 (Alan Turing) 和阿朗佐·丘奇 (Alonzo Church) 提出了非常重要的定理（关于不完备性和计算）——书写系统可以做的就是主题，就是你要证明的定理。
And these kinds of meta-questions, I think, started by having this field where nobody writes the same way.
我认为，这些类型的元问题始于这个领域，没有人以同样的方式写作。
It is not random that the context [they’re] working in is this tradition where there’s so many notations and you’re constantly paying attention to what they can do.
他们所工作的环境是这种传统，其中有如此多的符号，并且你不断地关注他们可以做什么，这并不是随机的。
I doubt that we are hitting the ceiling.
我怀疑我们是否已经达到了天花板。
Computers allow for all kinds of modeling, and I think we will probably see more and more areas of mathematics where the result is something dynamic —[models or simulations] of objects and processes that you just can’t print.
计算机允许进行各种建模，我认为我们可能会看到越来越多的数学领域，其结果是动态的——你无法打印的对象和过程的[模型或模拟]。
But this is not unprecedented in the history of math.
但这在数学史上并不是史无前例的。
We talked about paradigms other than writing in the more distant past, and in the late 19th century there was a real heyday of physical models.
在更遥远的过去，我们谈论的不是写作，而是范式，而在 19 世纪末，物理模型迎来了真正的鼎盛时期。
There were a lot of plaster geometric models, and we have a ton here at the [Smithsonian] museum.
有很多石膏几何模型，我们在[史密森尼]博物馆里有很多。
They used to be in every math department: You had your set of models for all sorts of different surfaces, and there was an attitude in which part of studying math was to cultivate this physical intuition for the forms that equations represented.
它们曾经存在于每个数学系：你有一套针对各种不同表面的模型，并且有一种态度，其中学习数学的一部分就是培养对方程所代表的形式的物理直觉。
You can see the model-making practice is actually a type of research exploration.
可以看到，模型制作实践实际上是一种研究探索。
Analogously to that, computers open up a ton of possibilities for representations that are not typographic and will enable new questions to be asked.
与此类似，计算机为非印刷的表示形式提供了大量的可能性，并且可以提出新的问题。
A qualification we should have: We’re talking about elite mathematical discourse, and it’s a useful shorthand to call that “math,” but that actually leaves a lot out.
我们应该具备的资格：我们正在谈论精英数学话语，将其称为“数学”是一个有用的简写，但这实际上遗漏了很多内容。
The knowledge somebody uses in a grocery store, dealing with the prices of things and their budget, that is also mathematical.
人们在杂货店使用的知识，处理物品的价格和预算，这也是数学的。
We have these notational technologies that allow us to not see this as significant math, but it is.
我们拥有这些符号技术，使我们无法将其视为重要的数学，但事实确实如此。
Another one that comes to mind is the use of x as a variable.
我想到的另一个问题是使用 x 作为变量。
It takes a long time for that practice to develop in mathematical history.
这种实践在数学史上需要很长时间才能发展起来。
And when children in grade school first arrive at the idea that we’re going to treat letters as if they’re numbers, it still hits [them] as something esoteric.
当小学的孩子们第一次意识到我们要把字母当作数字来对待时，他们仍然认为这是深奥的东西。
But [now] it is very widely learned, and all sorts of people who would not consider themselves mathematical are comfortable using x in a sentence to stand in for the unknown.
但[现在]它已被广泛学习，各种不认为自己是数学家的人都愿意在句子中使用 x 来代表未知数。
You can say, “Suppose I have x pounds of apples,” and someone who wants nothing to do with the actual manipulating of an equation is not put off by that way of speaking.
你可以说，“假设我有 x 磅苹果”，而那些不想参与实际操作方程的人不会因为这种说法而被推迟。
That’s what notation makes possible — the esoteric.
这就是符号使之成为可能的原因——深奥的。
And what counts as esoteric can change.
深奥的东西可能会改变。
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Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.
《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。
Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.
辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。
Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.
版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。