Many of this week’s circumstances are limiting the time I have to write but I would like to point to a few sources that contain very nice accounts of what is known as the foundational crisis in mathematics. One of them was written by Paul Bernays in 1935. Understanding the nature of some of the 19th and 20th century disputes among mathematicians reveals quite a lot about the character of mathematical objects and what motivates their exploration – things that could easily slip by even the conscientious math student.
The discussions that kept me reading were those centered around the following:
*understanding how there was once a debate among mathematicians about whether mathematical objects should ultimately be understood conceptually or computationally.
*a persistent Platonism or how, or to what extent, a mathematical object can be detached from the thinking subject
*what is it that makes modern mathematics modern
I would like to devote a post to each of these topics. Because of the limitations on my time today, I will only provide some links.
Jose Ferreiros’ essay The Crisis in the Foundations of Mathematics appears in Princeton’s Companion to Mathematics edited by Timothy Gowers. A pdf version is available here. In it he says:
The foundational debate has also contributed in a definitive way to clarifying the peculiar style and methodology of modern mathematics, especially the so-called “Platonism” or “existential character” of modern mathematics…
And he then refers to Bernays’ 1935 paper. In this paper, Bernays talks about
a restricted Platonism which does not claim to be more than, so to speak, an ideal projection of a domain of thought.
By this he means that the ideal resides in and is the product of thought rather than living in some other ideal world. But it is an ideal none-the-less and will not changed by the thoughts of an individual contemplating it.
Both the Ferreiros essay and a review of a book by Ferreiros (The Architecture of Modern Mathematics) from Andrew Arana talk about Riemann’s preference for conceptually defined mathematical objects over computationally defined ones. Arana says clearly about the 19th century struggle with how best to think about complex numbers,
This was (roughly) a struggle between two camps, one led by Riemann, the other led by Weierstrass. Weierstrass and his followers thought complex analysis should be `arithmetized’, meaning in particular that analytic functions should be defined as functions representable by power series. By contrast, Riemann and his followers favored representation-independent definitions, which in practice meant defining analytic functions as those satisfying the Cauchy-Riemann equations. The Riemannian approach then develops the theory of analytic functions without having to consider particular explicitly defined analytic functions. Riemann pioneered the term “geometric” for this kind of approach to analysis, but this wasn’t a stretch: he encouraged visualization in analysis, developing the notion of a Riemann surface to help.
Ferreiros’ essay makes the point that the defining characteristics of the modern methods of the 19th and 20th centuries are the following:
the notion of an ‘arbitrary’ function (proposed by Dirichlet), i.e. one whose relation is not written explicitly.
infinite sets and a higher infinite
conceptual definitions with an emphasis on structure
a reliance on purely existential methods of proof
It would be interesting to consider the intellectual shifts in modern mathematics alongside what has been said about the intellectual shifts in modern art. I will do some thinking about this.
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