ABSTRACT In this article, I take sixteenth-century mathematics as a test case for considering at its most general level what doing theory entails. Ideally, doing theory aims to change rather than merely describe reality, and in order to do so, it represents that reality differently. Such transformations, I argue, need special techniques of representation. In early modern mathematics, two techniques in particular enable new theoretical content: symbolic notation, and the real continuum. Together they create a language that can redescribe mathematical objects, for example, so that a single general formula can solve for all specific instances, and simple counting numbers can be reidentified in the real continuum as limits. I argue for the formative contribution of the imagination in such developments, that is, for an ability to reassemble givens of a problem in such a way as to solve it; and to concoct new objects or possibilities that do not exist in reality. In the sixteenth century, imaginary numbers (roots of negative values) most explicitly demonstrate the importance of speculating beyond what is real. As we consider what doing theory in the twenty-first century entails, it is worth remembering, in light of the article’s analysis, how form or technique itself has agency, and the ability to transform theoretical content.

中文翻译：

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他们当时进行理论的方式

摘要在本文中，我将十六世纪的数学作为测试用例，以从最一般的角度考虑理论的含义。理想情况下，做理论旨在改变而不是仅仅描述现实，为了做到这一点，它以不同的方式代表了现实。我认为，这样的转变需要特殊的表示技术。在早期的现代数学中，两种技术尤其启用了新的理论内容：符号表示法和真实连续体。他们共同创造了一种语言，可以重新描述数学对象，例如，这样一个通用公式就可以解决所有特定情况，并且简单的计数数字可以在真实的连续体中重新标识为极限。我主张在这种发展中想象力的形成性贡献，即 具有以解决问题的方式重新组合问题的能力；并炮制现实中不存在的新物体或可能性。在十六世纪，虚数（负值的根）最清楚地表明了超越真实范围进行推测的重要性。当我们考虑二十一世纪的理论时，根据本文的分析，形式或技术本身如何发挥作用以及转化理论内容的能力是值得记住的。