> In particular, one should abandon the dichotomy between
conjecture and theorem.
Wasn't that the status quo before the 20th century? It's strange to suggest that working with infinite (or rather, ideal) objects is stupid. The sheer amount of progress in 20th century mathematics provides incontrovertible evidence that ideal objects are a useful reasoning tool. This is even true in combinatorics: working with generating functions is working with an algebraic structure on infinite streams...
This essay is written to incite... there is so much in there that invites comment from everyone who has ever spent five minutes thinking about these things. At the same time it is lacking in examples for ways in which a mathematical world without rigor would be better than what we have today. So instead of getting worked up about the essay itself, does anybody here have concrete examples where a lack of rigor lead to faster progress?
Wasn't that the status quo before the 20th century? It's strange to suggest that working with infinite (or rather, ideal) objects is stupid. The sheer amount of progress in 20th century mathematics provides incontrovertible evidence that ideal objects are a useful reasoning tool. This is even true in combinatorics: working with generating functions is working with an algebraic structure on infinite streams...
This essay is written to incite... there is so much in there that invites comment from everyone who has ever spent five minutes thinking about these things. At the same time it is lacking in examples for ways in which a mathematical world without rigor would be better than what we have today. So instead of getting worked up about the essay itself, does anybody here have concrete examples where a lack of rigor lead to faster progress?