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I was reading the great Thurston's article on Mathematical Education and one thing he points out is:

People appreciate and catch on to a mathematical theory much better after they have first grappled for themselves with the questions the theory is designed to answer.

In this sense, what do you think are the questions Topology is designed to answer?

P.S.: My focus here is Topology, but feel welcomed to give examples from other areas as well.

rschwieb
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Douglas
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    By a “theory” I think Thurston has in mind something much narrower than “topology,” which is not really a theory so much as a very broad, vast umbrella under which a whole lot of more specific theories are arranged. Something like the theory of dimension, or homotopy theory (even there the subject is vast), is probably closer to what Thurston has in mind. But either way, you can often get a good sense about the questions a mathematical idea (including a topology) was designed to answer by studying the history of the subject. See, eg, the book History of Topology edited by I.M. James. – symplectomorphic Sep 26 '21 at 17:27
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    I don't really agree with Thurston, taken literally. Topology, for example, is not designed to answer any questions: it is designed to provide the right abstract framework for posing a large and interesting collection of questions. I think what he means, is that topology is best understood after grappling with interesting examples like Euclidean topology in $n$-dimensional space. – Rob Arthan Sep 26 '21 at 21:39
  • Topology is a wide-ranging area of mathematics. You might get a better answer if you ask a more focused question, for instance: "Could you give me examples of problems from other areas of mathematics (or even outside of mathematics) that are solved using tools of topology." However, such a question might be a duplicate, cf. https://math.stackexchange.com/questions/73690/real-life-applications-of-topology, https://math.stackexchange.com/questions/1121338/surprising-applications-of-topology?noredirect=1&lq=1 – Moishe Kohan Sep 27 '21 at 19:06

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The question might be:

How much of "geometry" (our intuition of "shapes") works without using numbers (i.e. when we are not allowed to use coordinates, measure distances, etc.)?

Torsten Schoeneberg
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