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Can there be an Alternative Mathematics? Should there be?
Last Updated: 2026-06-03 00:07:47
Abstract
The course explores, with a focus on examples, the apparent contradiction between the proliferation of seemingly alternative mathematics (historical revolutions, nonstandard mathematics, mathematics based on nonclassical logics, proposals for a liberatory mathematics, and more), and the dominant 'uniqueness narrative': there is only one mathematics.
Objective
Exposure to a concrete diversity of mathematical practices; learning to engage with the questions raised by this diversity (in particular, how putative alternatives can arise, and how they can be defended, dismissed, or assimilated); gaining experience in approaching and contextualizing critiques of mathematics; learning theoretical tools to locate different practices in relation to a mainstream.
Content
In "Knowledge and social imagery", the sociologist David Bloor posed the question of whether there could be an alternative mathematics, an idea controversial enough to be called a “monstrous absurdity”. Yet there are many apparent candidates for alternativeness: historical revolutions, nonstandard mathematics, mathematics based on nonclassical logics, proposals for a liberatory mathematics, and so on. Some of the questions tackled in this course, through discussion of various examples from the literature, are: what does it mean for a mathematics to be “alternative”? what does the existence of alternative mathematics entail for mathematics and its role in society? can any alternative mathematics resist exclusion or assimilation under the mainstream paradigm? where does the resistance to recognizing alternatives come from? should an alternative mathematics be adopted? what is the relationship between alternatives and revolutions in mathematics? The course will set the stage by looking at different definitions of alternative mathematics in the literature, and discussing David Bloor’s influential treatment and historical examples. It will then move on to discuss a variety of contemporary examples: these may include nonstandard mathematics (e.g. nonstandard analysis and non-well-founded set theory), intuitistionistic/constructive mathematics, inconsistent mathematics, Ferguson&al’s ‘feminist arithmetic’, and Van Bendegem’s ‘really alternative’ mathematics. For each example, special attention will be given to the ways in which the example fails (or does not fail) to be alternative, and the theoretical stances for which the alternativeness of the example plays an evidential role (e.g. social constructivism, logical pluralism, feminist epistemology, the contingency of mathematics). The course is open to anyone with an interest in mathematics. No familiarity with advanced mathematics or the philosophy of mathematics is expected. The focus is going to be on the philosophical and social aspects of alternative mathematics, although students will be welcome to dive into the more technical aspects of specific examples if they so wish.
General Information
- Language
- English
- Levels
- DS , MSC
Examination
- Type
- graded semester performance
Registration & Places
- Max Places
- 30
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| seminar | Can there be an Alternative Mathematics? Should there be? | No time listed | 2 h weekly |
Offered In
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Science in Perspective (In “Science in Perspective”-courses students learn to reflect on ETH’s STEM subjects from the perspective of humanities, political and social sciences. Only the courses listed below will be recognized as "Science in Perspective" courses.)
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Type A: Enhancement of Reflection Competence (SiP courses are recommended for bachelor students after their first-year examination and for all master- or doctoral students. All SiP courses are listed in Type A. Courses listed under Type B are only recommendations for enrollment for specific departments.)
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