Search

This article discusses the important and influential views of John Burgess on the nature of mathematical rigour and John Norton on the nature of thought experiments. Their accounts turn out to be surprisingly similar in spite of different subject matters. Among other things both require a reconstruction of the initial proof or thought experiment in order to officially evaluate them, even though we almost never do this in practice. The views of each are plausible and seem to solve interesting problems. However, both have problems and would seem not able to do justice to some interesting examples. They fail in similar ways. More pluralistic accounts of proof and of thought experiment could embrace aspects of each, while rejecting their claims to universality. An ideal account (not provided here) would contribute to explanation and understanding as well as evidence. These are important topics for future work.

A thought experiment involving an omniscient being and quantum mechanics is used to justify non-deductive methods in mathematics. The twin prime conjecture is used to illustrate what can be achieved.

Mathematics and ethics are surprisingly similar. To some extent this is obvious, since neither looks to laboratory experiments nor sensory experience of any kind as a source of evidence. Both are based on reason and something commonly call “intuition.” This is not all. Interestingly, mathematics and ethics both possess similar distinctions between pure and applied. I explore some of the similarities and draw methodological lessons from them. We can use these lessons to explore how and why Freiling’s refutation of the continuum hypothesis might be justified.

It is widely agreed that pharmaceutical research has serious problems, but there is no consensus on what to do about them. The first of these is concerned with issues of political morality, while the second focuses on scientific methodology, yet they turn out to be surprisingly related. The general idea is that everyone's medical needs are covered free of cost to the individual. On utilitarian grounds, socialized medicine does the most good for the least cost. Once aware of the issues involved in medical research, especially pharmaceutical, any reasonable person will be horrified and want to make changes. In his contribution to this volume Reiss argues for two important points. The first is to banish patents entirely, on the grounds that they do not promote innovation but rather stifle it. The second major suggestion Reiss makes is that randomized controlled trials are overrated when it comes to evidence in medical research.

In this paper, I analyze the relation between visualization and explanatory understanding in thought experiments. I discuss two thought experiments from special relativity. Both cases lead to a paradox, which can, however, be explained (away) as non-contradictory. The resolution in one of the two cases sheds light on the nature of thought experiments more generally by indicating what sort of thing it is that we see when we are observing events in the mind’s own laboratory. And it will in turn shed light on the notions of explanation and understanding as they arise in thought experiments.

This chapter is a brief survey of some of the reasons, including a new one, for thinking. Space (or spacetime) is a thing in its own right. This means it is not reducible to or emergent from matter/energy. Space (spacetime) interacts with matter/energy, but it has a life of its own.