Would mathematics exist if people didn't? Since ancient times, mankind has hotly debated whether mathematics was discovered or invented. Did we create mathematical concepts to help us understand the universe around us, or is math the native language of the universe itself, existing whether we find its truths or not? Are numbers, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient advocates. The Pythagoreans of 5th Century Greece believed numbers were both living entities and universal principles. They called the number one, "the monad," the generator of all other numbers and source of all creation. Numbers were active agents in nature. Plato argued mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them. Euclid, the father of geometry, believed nature itself was the physical manifestation of mathematical laws. Others argue that while numbers may or may not exist physically, mathematical statements definitely don't. Their truth values are based on rules that humans created. Mathematics is thus an invented logic exercise, with no existence outside mankind's conscious thought, a language of abstract relationships based on patterns discerned by brains, built to use those patterns to invent useful but artificial order from chaos. One proponent of this sort of idea was Leopold Kronecker, a professor of mathematics in 19th century Germany. His belief is summed up in his famous statement: "God created the natural numbers, all else is the work of man." During mathematician David Hilbert's lifetime, there was a push to establish mathematics as a logical construct. Hilbert attempted to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a deeply philosophical game but a game nonetheless. Henri Poincaré, one of the father's of non-Euclidean geometry, believed that the existence of non-Euclidean geometry, dealing with the non-flat surfaces of hyperbolic and elliptical curvatures, proved that Euclidean geometry, the long standing geometry of flat surfaces, was not a universal truth, but rather one outcome of using one particular set of game rules. But in 1960, Nobel Physics laureate Eugene Wigner coined the phrase, "the unreasonable effectiveness of mathematics," pushing strongly for the idea that mathematics is real and discovered by people. Wigner pointed out that many purely mathematical theories developed in a vacuum, often with no view towards describing any physical phenomena, have proven decades or even centuries later, to be the framework necessary to explain how the universe has been working all along. For instance, the number theory of British mathematician Gottfried Hardy, who had boasted that none of his work would ever be found useful in describing any phenomena in the real world, helped establish cryptography. Another piece of his purely theoretical work became known as the Hardy-Weinberg law in genetics, and won a Nobel prize. And Fibonacci stumbled upon his famous sequence while looking at the growth of an idealized rabbit population. Mankind later found the sequence everywhere in nature, from sunflower seeds and flower petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of Bernhard Riemann in the 1850s, which Einstein used in the model for general relativity a century later. Here's an even bigger jump: mathematical knot theory, first developed around 1771 to describe the geometry of position, was used in the late 20th century to explain how DNA unravels itself during the replication process. It may even provide key explanations for string theory. Some of the most influential mathematicians and scientists of all of human history have chimed in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? Artificial construct or universal truth? Human product or natural, possibly divine, creation? These questions are so deep the debate often becomes spiritual in nature. The answer might depend on the specific concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a forest, but no one's there to count them, does that number exist?
如果没有人的存在还会有数学吗? 在古时侯,人类就为此热烈地争辩 数学到底是被发明的或是被发现的 人们创造数学概念是为了 更好得了解周围的世界呢 还是数学本就是宇宙的语言, 一直存在于这世界上 不管人类的介入或不 数字,多边形和对等是真的吗? 或只是些空洞的理论概念? 数学的独立存在有很多古代支持者 五世纪希腊的毕达哥拉斯相信数字既是 存活的实体又是宇宙的规则。 他们把数字1唤作 ”单子,“ 是所有其它数字的启动器 是所有东西的来源 数字是大自然的活性剂 柏拉图认为 数学概念是具体的 数学概念就像宇宙自身一样真实, 不管我们是否意识到它们的存在 欧几里德,几何之父, 相信自然本身 就是数学定律的物理表现。 而有些人却说因为数字并非一定有实体, 所以数学的命题绝对不会有 它们的真实价值是 基于人类所创立的规则 数学是一种 被发明的逻辑练习, 在人类的理性的思想之外, 并不会存在 它是一种能被大脑识别的 基于某种格式的抽象语言, 利用这些模式 在混乱中来发明有用的人为秩序 这种理论的支持者 是Leopold Kronecker 一位十九世纪德国的数学教授 他的信条可在他著名的宣言 中总结如下: “上帝创造了自然数, 除此而外都是人类的工作。“ 在数学家David Hilbert的一生中, 对将数学看做一种逻辑的建树 有很大的推动 Hibert曾尝试将所有的数学公理化 就像欧几里德在几何上所做的 他和其他尝试这件事的数学家把数学 看成是一场深奥的哲学游戏 但依旧只是一个游戏。 Henri Poincaré,,是 非欧几里德几何之父, 他相信非欧几里德几何的存在 用于处理非平面的 双曲线和椭圆曲率 从而证明欧几里德了的平面几何 这一长时间被认同的理论 并不是全部的宇宙真相, 只是遵从了游戏规则的一种的结果 但在1960年,诺贝尔物理学奖得主 Eugene Wigner 创造了名言,“无理的 数学效率,” 强烈得灌输了数学得真实存在 并且是由人们发现的 Wigner指出很多纯粹的数学理论 是在真空里发展出来的,常常 无视任何物理现象, 这些理论在几十年或几个世纪 后被证明 它们仅仅是空空的骨架, 需要进一步地阐述 整个宇宙是如何 一直维持运行的。 比如,英国数学家 Gottfried Hardy的数字理论, 他曾自嘲说, 他的作品 在描述实用现象上的价值 没有一件是有用的 但是他帮助建立密码学 这是他的另一个纯理论成果 也变成了著名的遗传学上的 Hardy-Weinberg定律 并且赢得了诺贝尔奖。 费伯纳齐突破至他最有名的序列是 在观察假设的兔群增长时 而人类后来发现自然中到处都存在序列, 从葵花籽到葵花花瓣的排列 以及菠萝的结构, 甚至肺中的支气管分支。 另外在1850年,伯奈德瑞曼的非欧几里德 成果 在一个世纪后, 爱因斯坦用此为模版创立了广义相对论。 这儿甚至有着更大的飞跃: 数学结的理论,开创的时候是1771年 用以描述几何形状的方位, 这在二十世纪的后期用来解释DNA, 在复制的过程中,如何解开它的螺旋结构 这甚至为弦理论提供了关键的解释 人类史上一些最有影响力的 数学和科学家们 都以令人吃惊的方式 倾向于这个说法。 所以,数学是一种发明 还是一种发现? 是人工的构建或是宇宙的真相? 是人类的产物或是自然或 神圣的创造? 这些问题让争辩更为深入, 而成为自然的精髓。 问题的答案也许在于审视数学时的一个具体概念, 但它让所有人都感到像 扭曲的禅宗公案。 如果在森林中有很多树, 但没有人去数它们, 那么数字会存在吗?