• Corpus ID: 39474807

The Crisis in the Foundations of Mathematics

@inproceedings{Ferreirs2007TheCI,
  title={The Crisis in the Foundations of Mathematics},
  author={Jos{\'e} Ferreir{\'o}s},
  year={2007}
}

• J. Ferreirós
• Published 2007
• Philosophy

The foundational crisis is a celebrated affair among mathematicians and it has also reached a large nonmathematical audience. A well-trained mathematician is supposed to know something about the three viewpoints called “logicism,” “formalism,” and “intuitionism” (to be explained below), and about what Gödels incompleteness results tell us about the status of mathematical knowledge. Professional mathematicians tend to be rather opinionated about such topics, either dismissing the foundational discussion as irrelevant—and thus siding with the winning party— or defending, in principle or as an intriguing option, some form of revisionist approach to mathematics. But the real outlines of the historical debate are not well-known and the subtler philosophical issues at stake are often ignored. Here we shall mainly discuss the former, in the hope that this will help bring the main conceptual issues into sharper focus. Usually, the foundational crisis is understood as a relatively localized event in the 1920s, a heated debate between the partisans of “classical” (meaning late-nineteenthcentury) mathematics, led by Hilbert, and their critics, led by Brouwer, who advocated strong revision of the received doctrines. There is, however, a second, and in my opinion very important, sense in which the “crisis” was a long and global process, indistinguishable from the rise of modern mathematics and the philosophical/methodological issues it created. This is the standpoint from which the present account has been written. Within this longer process one can still pick out some noteworthy intervals. Around 1870 there were many discussions about the acceptability of non-Euclidean geometries, and also about the proper foundations of complex analysis and even the real numbers. Early in the twentieth century there were debates about set theory, about the concept of the continuum, and about the role of logic and the axiomatic method versus the role of intuition. By about 1925 there was a crisis in the proper sense, during which the main opinions in these debates were developed and turned into detailed mathematical research projects. And in the 1930s Gödel proved his incompleteness results, which could not be assimilated without some cherished beliefs being abandoned. Let us analyse some of these events and issues in greater detail.