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I Origins of Formal Structure. The Natural Numbers. Infinite Sets.
Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Saunders Mac Lane, who died on 14 April, was one of the few mathematicians to create a major new concept that has had a lasting effect on the subject. In , he and Sammy Eilenberg introduced the basic ideas of category theory, often described as a language of mathematics that allows the description of transformations from one area of the subject into another by distilling their common properties. Concentrating on common properties of these structures by stripping away the non-essential aspects of a problem clears the path to new results.
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Don't have an account? He studied algebra with Emmy Noether, and both mathematics and philosophy with Hermann Weyl. The essay relates all of this to current philosophical structuralism, especially concerning isomorphisms and automorphisms of structures. At the same time he absorbed a structural method from Emmy Noether. He had wanted to do that since he was undergraduate at Yale.
Mathematics, Form and Function is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane. Throughout his book, and especially in chapter I. The following table is adapted from one given on p. The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From. Also see the related diagrams appearing on the following pages of Mac Lane , , , , ,
The Form and Function of Duality Phenomena covered by the term duality occur throughout the history of mathematics in all of its branches, from the duality of polyhedra to Langlands duality. We approach these questions by means of a category theoretic understanding. We try to show by some examples that in the first case, dual objects tend to be more ideal epistemologically more remote than original ones, while this is not necessarily so in the second case. This fascination has only increased with the passage of time right up to the current intense investigation of Langlands duality. A broader perspective orients us towards general dualities between algebra and geometry, and between syntax and semantics, and teaches us much about the content of mathematics. Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies.
Front Matter. Pages i-xi. PDF · Introduction. Saunders Mac Lane · Origins of Formal Structure. Saunders Mac Lane · From Whole Numbers to Rational Numbers.
Mathematics, Form and Function is a survey of the whole of mathematics , including its origins and deep structure, by the American mathematician Saunders Mac Lane. Throughout his book, and especially in chapter I. The following table is adapted from one given on p.
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