Most mathematical theories (Exceptions are unknown to me) were created by answering one or more open ended non-trivial questions. For example, The Brachiostome problem lead to the calculus of variations, Three pendulum problem leaded to chaos theory (somewhat physical) etc.
What are the questions that spawned most of the mathematics of 16~21st century mathematics ?
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One of the big questions in algebra (around 1800) was whether a general quintic polynomial had a solution in terms of radicals. This led Galois to investigate and create what would later be named after him (Galois Theory), and finally Abel's impossibility theorem.
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Newton developed calculus, the generalized binomial theorem, and general power series primarily to solve problems in mechanics, especially regarding orbits with an inverse-square gravity. It has been said that while others asked what keeps the moon up, Newton asked what keeps it down, that is, why it doesn't fly off at a tangent to its orbit.
Georg Cantor developed set theory after studying "exceptional sets" in the question of characterizing the set of points at which a function's Fourier series may fail to converge to the function.
Paul Cohen invented Forcing to prove the consistency (in ZFC) of the negation of the Continuum Hypothesis, and the consistency (in ZF) of the negation of the Axion of Choice. His method led to an explosion of new results. As Kunen wrote, "Infinitary Combinatorics is the branch of mathematics that was called Set Theory before there were independence proofs."
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The attempts to prove Euclid's fifth postulate led to the idea that alternatives to the fifth postulate could be consistent. Hence, the development of the non-Euclidean geometries.
Georg Cantor investigated the idea that there are different sizes of infinity. Hence, the development of set theory.
A consequence of Cantor's work was the attempt by Russell and Whitehead to axiomatize arithmetic (and thus all of mathematics). But Godel's Incompleteness Theorem crushed those attempts.
The Seven Bridges of Königsberg laid the foundations of graph theory, despite Euler originally perceiving the problem as nothing more than a puzzle.
A general solution to the degree five (quintic) equation led to Galois Theory and the Abel–Ruffini theorem.
The problem of area under a curve led to the creation of calculus. Isaac Newton and Gotfried Leibniz are usually credited with this feat, but early Greek thinkers such as Archimedes also though about area in a similar sense.