Still debating with Plato

Think too hard about it, and mathematics starts to seem like a mighty queer business. For example, are new mathematical truths discovered or invented? Seems like a simple enough question, but for millennia, it has provided fodder for arguments among mathematicians and philosophers.

Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?

On the other hand, if math is invented, then why can\'t a mathematician legitimately invent that 2 + 2 = 5?

Many mathematicians simply set nettlesome questions like these aside and get back to the more pleasant business of proving theorems. But still, the questions niggle and nag, and every so often, they rise to attention. Several mathematicians will ponder the question of whether math is invented or discovered in the June European Mathematical Society Newsletter.

Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. “The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

The Platonic perspective fits well with an aspect of the experience of doing mathematics, says Barry Mazur, a mathematician at Harvard University, though he doesn\'t go so far as to describe himself as a Platonist. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”

But where are those hunting grounds? If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, Mazur describes the Platonic view as “a full-fledged theistic position.” It doesn’t require a God in any traditional sense, but it does require “structures of pure idea and pure being,” he says. Defending such a position requires “abandoning the arsenal of rationality and relying on the resources of the prophets.”

Indeed, Brian Davies, a mathematician at King\'s College London, writes that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article “Let Platonism Die.”

If mathematics is the perception of this realm of pure ideas, then doing mathematics requires our brains to somehow reach beyond the physical world. Davies argues that brain-imaging studies are making this belief steadily less plausible. He points out that our brains integrate many different aspects of visual perception with memory and preconceptions to create a single image — not always correctly, as optical illusions make clear. He also says that brain-imaging studies are beginning to show the biological basis of our numeric sense.

But Reuben Hersh of the University of New Mexico isn’t convinced that studies like these logically destroy the Platonic notion of an intuitive faculty to perceive mathematics. Nevertheless, he rejects the Platonic view, arguing instead that mathematics is a product of human culture, not fundamentally different from other human creations like music or law or money.

The challenge, he admits, is to explain why it is that mathematical statements can be definitively true or false, not subject to taste or whim. With simple statements like “2 + 2 = 4,” this is because of the connection between mathematics and physics, he says. Such a statement describes, for example, the way that coins or buttons behave. For more abstract statements that are further removed from the physical world, he points to the structure of our brains and our penchant for logic.

But Mazur finds that explanation unsatisfying. “We should keep an eye on the stealth word ‘our,’” he writes. “Is the we meant to be each and every one of us, given our separate and perhaps differing and often faulty faculties?” In this case, mathematics itself has to vary as individuals do.

On the other hand, if “we” means a kind of abstraction of our individual capabilities — the common thing that binds us together without actually being any of us — he says that we are verging back toward the Platonic notion of a realm of abstract ideas.

But the notion of invention also captures something true about the experience of doing mathematics, in his view. “At times,” he says, “I seem to be engaged in an analysis of my thought processes or other people’s thought processes while doing mathematics.” All aspects of these experiences, he argues, need to be included in these discussions.

“One thing is — I believe — incontestable,” he writes. “If you engage in mathematics long enough, you bump into The Question, and it won’t just go away. If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had better pay attention to it.”

与柏拉图的争论仍在继续

数学住在哪里呢？

对于这个问题，人们百思不得其解，数学正在变成一件奇怪的物事。例如，新的数学真理是被发现还是被发明的呢？这个看似简单的问题，却成为数学家和哲学家千百年来争论不休的话题。

那些“发现说”的支持者们指出，数学陈述的对和错与个人的信仰无关，从而表现出某种客观现实性。但这却由此而引发出一些奇怪的想法。数学真理究竟栖身何处呢？数学真理真的在我们的想象之前就存在？

从另一方面说，如果数学是被发明的，为什么2+2不等于5呢？

许多数学家只是把这些恼人的问题简单地弃置一旁，而忙于类似定理证明这样更有趣的事务。但是，这个问题并不会就此消失，而是时时跳出来吸引人们的注意。在《欧洲数学学会通讯》6月版上，几位数学家仔细琢磨了数学发现和发明的问题。

柏拉图是“发现说”信徒们的旗手。柏拉图主义认为，数学所表现出的泰然自若之境界乃宇宙之根本。追随数学内在逻辑的脚步，数学家可以超越人类观察的局限，脱离客观存在的暂态，而发现永恒的真理。Ulf Persson说：“由于长期的亲密关系，数学家所处的抽象世界比其恰好所坐的椅子要更加真实。”Ulf Persson来自瑞典Chalmers技术大学，自我标榜为柏拉图主义者。

哈佛大学的数学家Barry Mazur认为，这种柏拉图主义的观点与解答数学的经历尤为符合，尽管他并没有把自己形容为柏拉图主义者。他说，专注于一个定理的感觉，就像“一个猎人，一个数学概念的收集者。”

但是狩猎场又在何处呢？如果数学思想在某处等待着被发现，那么即使人类从未构想，一个纯粹抽象的概念也必须存在。正因为如此，Mazur将柏拉图的观点置于“一个成熟的有神论的位置上。”他说，这个观点不需要一个传统意义上的神，但它确实需要“纯粹想法和纯粹存在的结构”，捍卫这样的立场，需要“放弃理性这个武器，而依赖先知的力量。”

事实上，伦敦国王学院的Brian Davies认为，“柏拉图主义不是现代科学，而与神秘宗教有更多共同之处。”在他名为“让柏拉图主义死亡”的文章中，他坚信，现代科学可以提供证据来证明柏拉图的观点是错误的。

如果数学是对纯理念王国的感知的话，那么研究数学就需要让自己的大脑以某种方式抵达现实世界之外的某地。Davies认为，脑成像研究正在让这个理念越来越站不住脚。他指出，我们的大脑将记忆，创造单个图像以及视觉的其他方面整合在一起。当视觉错误占据上风，则单个图像也并非正确。他还指出，脑成像研究正逐步揭示出我们数字感的生理学基础。

新墨西哥大学的Reuben Hersh不确信，类似的研究是否在逻辑上消灭了数学感知的直觉能力。然而，他反对柏拉图的观点，认为数学是人类文化的产物，与诸如音乐、法律、金钱等产物没有根本上的不同。

他承认，挑战在于如何解释数学陈述的真假与口味或者一时兴起无关。类似于“2+2=4”这样的简单陈述，是因为数学与物理之间的联系。这样的陈述描述了硬币或者按钮的行为。对于从真实世界引申出的更抽象的陈述，他指出这与我们大脑的结构，以及对于逻辑的追求有关。

但是Mazur对此解释并不满意。他说，“我们总是着眼于‘我们’这个词。鉴于我们是独立、不同甚至常常是错误的，‘我们’意味着我们每一个人吗？”在这种情况下，每个人心中都有一个“数学”。

另一方面，如果“我们”意味着我们个体能力的一种抽象，让我们除去身份走到一起的共同点，那么我们又回到了抽象理念的范畴——这种柏拉图式的概念。

在Mazur看来，“发明”这个概念与数学研究中的某些经历吻合。他说：“有时，在研究数学的过程中，我似乎是在分析我自己或是其他人的思想过程。”他认为，这些经历的各个方面都应该包括在这些讨论中。

他写道：“我认为，有一件事情是毋庸置疑的。如果你从事数学这么长时间，你遇到一个问题，这个问题不会就此走开。如果这个问题让数学思考如此美好，而我们又想向这种充满激情的经验表示敬意，那么我们最好留心这个问题。”