This chapter lays out the ways in which Hans Christian Ørsted (1777–1851) influenced the development of the concept of thought experiment. Ernst Mach (1838–1916) is currently more often credited with laying the foundations of contemporary views, and he is sometimes thought to have been little (if at all) influenced by Ørsted. Against these standard accounts, I will show that Ørsted’s and Mach’s descriptions have key features in common. Both thinkers hold that thought experiments: (1) are a method of variation, (2) require the experimenter’s free activity, and (3) are useful in educational contexts for guiding students to arrive at certain conclusions on their own (i.e., to genuinely appropriate new concepts). The process of variation is guided by the search for invariants, some of which do not directly appear in experience. Since it is important that teachers and students be able to bring the same ideal objects to mind, thought experiments play a key role for both Ørsted and Mach in math education. While Ørsted’s emphasis on the role of thought experiments in math has been proposed as a reason why his descriptions are not relevant for contemporary use of thought experiments, I will show how their role in mathematical thinking – stemming from Kant’s descriptions of the method of construction in geometry – are part of a wider account of thought experiments that encompasses their role in the sciences and also philosophy.

The conclusion outlines key points in the book. On a Kantian-Kierkegaardian account, thought experiments lead to presentations that justify calling a thought experiment an experience, though not an empirical or sensory one. Without a sensible object, we can nevertheless have givenness, or presence, but not existence. Cognition, for that reason, is not necessarily truthful: although object-directed, cognition on its own does not justify belief in any particular object’s existence. My conclusion that thought experiments provide cognition rather than knowledge echoes Michael Stuart’s claim that thought experiments increase understanding. Cognition is a common basis for knowledge in Kant, but it is not the same thing as knowledge. The conclusion also draws implications for how we understand faith (religious belief) in Kierkegaard and how thought experiments make sense of the complexity of reality.

Kierkegaard and Ørsted were not just contemporaries but personally knew each other. In this chapter, I argue that Kierkegaard probably learned the term Tankeexperiment from Ørsted. This chapter contextualizes Kierkegaard’s use of “imaginary construction” (Experiment) in his work as a whole, including his well-known uses of paradoxes. I will show how the core elements of Ørsted’s account – thought experiment as a method of variation, the need for free and active constitution, and the use of thought experiments for facilitating genuine thought – are echoed in Kierkegaard’s discussions. Along the way, I will describe some decisions on how to translate Experiment and Tankeexperiment that are unfortunate in some ways and fortuitous in others, as I will explain. In these ways, Kierkegaard indirectly takes up Kant’s proposal that “construction” (i.e., Experiment in Danish) is a means of achieving cognition.

In philosophy of science, Mach’s account of thought experiments is more often described as relevant for contemporary usage than Ørsted’s. In this chapter, I survey recent Kantian accounts of thought experiment, arguing that the leading views inspired by Kant in philosophy of science remain broadly empiricist. This tendency may be due to their focus primarily on the role of thought experiments in the sciences. In later chapters, I will argue – against recent Kantian views – that Kant understood cognition more broadly to include not only sensory perception but also mathematical construction. Acknowledging that cognition does not always require empirical fulfillment opens new ways of understanding how thought experiments work in philosophy, which may rightly differ from their use in the sciences.