File Name: saunders maclane mathematics form and function .zip
During his first 10 years as a mathematician, while he worked in many sub- jects, the unity was more abstractly philosophical and logical than mathematical. Yet he gained experience.
Then came his collaboration with Samuel Eilenberg on one specific problem. They worked to explain how some pure algebra by Mac Lane had arrived at the cohomology of the p-adic solenoid, an infinitely tangled compact metric space. To do this they created category theory.
They organized the functorial basis for homology and cohomology in topology. And they created group cohomology in its full functorial form. His s work in arithmetic algebraic geometry took on new importance in light of his homological algebra of the s. He also learned that Emmy Noether was pro- ducing great new mathematical results by thinking about the foundations of algebra although he did not yet learn much of her work.
As a graduate student at the Uni- versity of Chicago he was deeply impressed by the founder of the mathematics department, Eliakim Hastings Moore. Moore followed the Chicago tradition by taking a strong interest in education as well as research. Mac Lane would do the same. Moore had an ambitious project to draw the greatest benefits for advanced mathematics from the new logic and set theory.
This work led Moore to state his principle on analogies and general theories quoted above. Mac Lane explained the paper to a seminar. Afterwards: Moore took me aside and explained what the paper was really all about and what I should have said. That was an occasion on which I learned a great deal thanks to Professor Moore: I learned from him how to give a talk on mathematics, and I learned about sets as a foundation for mathematics.
He was an amazing professor. Mac Lane , p. Moore had studied in Germany. Hilbert was developing logic there as Noether was developing her algebra.
There he found two difficulties. Still he got his doctorate with a dissertation on how to simplify formal logical proofs and make them more usable in practice. He also found Noether a confusing lecturer because she was creating her ideas as she spoke. Schilling made factor sets basic to their search for a non-Abelian class field theory. The two also worked together on valuations and factorization in algebraic number theory and algebraic geometry.
Certainly he believed the deepest analogy between central parts of arithmetic and geometry lay in the theory of valuations. He wanted to find the general theory. None of this work is known today, for an obvious reason: It was all swept away by modern functorial cohomology and the Abelian category methods that Mac Lane introduced. For valuations, Lefschetz and Zariski see Mac Lane , pp. Mac Lane was passionate about organizing and building the knowledge of category theory. In its simplest form the Yoneda lemma says every representable functor hX on a category C has a universal element, namely the identity arrow 1X on the represent- ing object X.
But this is only the beginning. The dual calculation shows this is natural in X so that h is a full and faithful functor from C to the category of contravariant set-valued functors on C. Essentially one fact takes three forms. But there is a catch. You must ignore foundations or else find some foundation more sophisticated than standard set theory to apply the lemma to large categories C, that is categories with a proper class of objects such as all groups or all topological spaces.
Mac Lane was adept at ignoring foundations as well as finding sophisticated ones. To say the least, the Yoneda lemma expresses the wide scope of categorical thinking as Mac Lane saw it in the s. It applies to every category and lies at the confluence of universal elements, representable functors, and adjunctions.
It appears in all advanced technical applications: for Mac Lane in the s that meant primarily homological algebra and homotopy. But more, the lemma makes a quick link between advanced technical applications and the wider more fundamental uses of categories that Mac Lane promoted in the s especially after meeting William Lawvere.
The lemma is central to the diagrammatic thinking which became natural and even delightful to Mac Lane. The Yoneda lemma is the proof that the two intuitions agree. And the most powerful calculating devices of mathematics are clarified by using universals, as for example spectral sequences in his great textbook on homology Mac Lane , chapter XI. I 7, — Related Papers.
By Colin McLarty. Saunders Mac Lane, the Knight of Mathematics. By Semen Kutateladze. By Antti Veilahti. By Luis Loza. Download pdf. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link.
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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.
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