I’m puzzled by the anchoring on understanding. I learned Stirling’s approximation for n! as a sophomore with no proof. I’m pretty sure I understood it. Then I learned lots of probability theory without understanding very analytical proofs which still strike me as uninteresting and ‘technical’ - like proofs created by a machine. Yet I could apply the results happily, showing I understood, I think. In set theory, we learned Cohen’s forcing methods and understood ‘how’ the proofs work to establish independence results like the continuum hypothesis. But nobody understood quite ‘why’ the method worked, it was opaque. Kind of like Feynman on quantum mechanics. Understanding proofs is just one kind of mathematical understanding. As Wittgenstein would say it’s a family resemblance term. Most literate people understand positional value without understanding just how it works. We are often machine like in our understanding of lots of math. Indian mathematics eschewed proofs for centuries. Then there is Ramanujan. What of his mysterious understanding? Part of the difficulty is the shared social status of mathematical knowledge, making it possible to take advantage of others’ understanding for our own purposes. Machine learning is like that too. I’m also puzzled why there’s no controversy over computer ‘learning’. Polya showed learning is the royal road to mathematical understanding. Understanding as a corollary to learning so to speak. Here the machine seems less human. But then there’s Stirling’s formula….
Вымысел — не есть обман.
Замысел — еще не точка.
Realization is neither a deception nor is it t a full stop.
I’m puzzled by the anchoring on understanding. I learned Stirling’s approximation for n! as a sophomore with no proof. I’m pretty sure I understood it. Then I learned lots of probability theory without understanding very analytical proofs which still strike me as uninteresting and ‘technical’ - like proofs created by a machine. Yet I could apply the results happily, showing I understood, I think. In set theory, we learned Cohen’s forcing methods and understood ‘how’ the proofs work to establish independence results like the continuum hypothesis. But nobody understood quite ‘why’ the method worked, it was opaque. Kind of like Feynman on quantum mechanics. Understanding proofs is just one kind of mathematical understanding. As Wittgenstein would say it’s a family resemblance term. Most literate people understand positional value without understanding just how it works. We are often machine like in our understanding of lots of math. Indian mathematics eschewed proofs for centuries. Then there is Ramanujan. What of his mysterious understanding? Part of the difficulty is the shared social status of mathematical knowledge, making it possible to take advantage of others’ understanding for our own purposes. Machine learning is like that too. I’m also puzzled why there’s no controversy over computer ‘learning’. Polya showed learning is the royal road to mathematical understanding. Understanding as a corollary to learning so to speak. Here the machine seems less human. But then there’s Stirling’s formula….