At the very foundation, chaining sentences together is what we call logic.

Chaining unrelated sentences is retarded. Chaining sentences like most people is common sense. Chaining sentences airtight is math.

You ask what a true mathematician does. He chains sentences like everyone else but with an effort to make them airtight.

Today, even understanding what new mathematics is being done in a particular zone of the mathematical universe appears to require a four-year graduate program of constant study just to be able to follow some mathematician’s original work - and that’s only going to give you a window into a rather narrow subsection of mathematics. The days when people of great talent like Euler and Gauss could contribute to many areas of mathematics are long gone.

But mere mortals can still derive great satisfaction from following along in the footsteps of past pioneers, possibly adapting their work to new problems in a minor way, or just creating educational visualizations and tools that help other people understand things like Galois theory, Poincare phase space or Markov chains, which can be applied to quantum mechanics, orbital dynamics, or protein sequence analysis. That’s valuable, even if no Fields Medals will be coming your way.

For the core discipline, though, I’d mostly worry about lack of opportunities for serious mathematicians to practice their craft in the USA due to the trends of academic budget cuts, anti-intellectual rhetoric, insistence on profit generation as the only rationale for doing anything, etc. Looks a bit 1930s Germany to me, at least here in the USA.

so good

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