Axiomatics: Mathematical thought and high modernism
Axiomatics by Alma Steingart explores the evolution of mathematical thought from the late 19th century to 1980, emphasizing historical context, the influence of Modernism, and the shift from pure to applied mathematics.
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Axiomatics: Mathematical Thought and High Modernism, authored by Alma Steingart and published by the University of Chicago Press, explores the evolution of mathematical thought from the late 19th century to around 1980. The book emphasizes the historical context of mathematics rather than its technical aspects, focusing on the tension between pure and applied mathematics. Steingart highlights the influence of the Modernist movement in art on mathematicians, particularly in their shift towards abstraction. The work discusses the impact of David Hilbert's axiomatization of geometry, which redefined the understanding of axioms and led to a predominance of pure mathematics until World War II. The war catalyzed a rise in applied mathematics, creating a divide that persisted post-war, exacerbated by federal funding for scientific research. The book also addresses the increasing interest in the history of mathematics and the philosophical debates it sparked, referencing key figures like Thomas Kuhn and André Weil. Steingart's narrative is rich with anecdotes about mathematicians and artists, making it a valuable resource for both historians and mathematicians, particularly those trained during the mid-20th century when abstraction was paramount.
- The book examines the historical development of mathematical thought rather than mathematics itself.
- It highlights the influence of Modernism on the abstraction in mathematics.
- The work discusses the shift from pure to applied mathematics during and after World War II.
- It addresses the growing interest in the history of mathematics and related philosophical debates.
- The narrative includes stories of key figures in mathematics and their intellectual conflicts.
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5 commentsMy best example of the split is https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives Wikpedia notes that "The list of unsuccessful proposed proofs started with Euler's, published in 1740,[3] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification." The split between pure (Euler) and applied(Bernoulli) is already there.
The result is hard to prove because it isn't actually true. A simple proof will apply to a counter example, so cannot be correct. A correct proof will have to use the additional hypotheses needed to block the counter examples, so cannot be simple.
Since the human life span is 70 years, I face an urgent dilemma. Do I master the technique needed to understand the proof (fun) or do I crack on and build things (satisfaction)? Pure mathematicians are planning on constructing long and intricate chains of reasoning; a small error can get amplified into a error that matters. From a contradiction one can prove anything. Applied mathematics gets applied to engineering; build a prototype and discover problems with tolerances, material impurities, and annoying edge cases in the mathematical analysis. A error will likely show up in the prototype. Pure? Applied? It is really about the ticking of the clock.
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