Should mathematics prove to be in need of a philosophical insight, it will not come from a philosopher.
The reason is simple. Most fields of study only find themselves in need of philosophical insights when they are internally driven to crisis. But when this happens, it is those who are part of said crisis that have the best insight as to the actual needs revealed, and it is from them that critical ideas come. Even if a philosopher happens to make a good point in advance, the philosopher will be ignored until the field of study has driven itself in crisis. After the crisis has passed, the philosopher's insight may be discovered to be relevant. But it is unlikely to have been noticed at the time.
This is not just an abstract theory. It is borne out by history. For example calculus as envisioned by Newton and Leibniz had foundations in handwaving and wishful thinking. Bishop George Berkeley was absolutely right to complain about "ghosts of departed quantities". But it didn't become a crisis until the following decades when Joseph Fourier added together a bunch of sin and cosine terms (all well-behaved functions) and came up with a square wave (emphatically NOT a function by then current standards).
While Fourier proved Berkeley right, Berkeley's complaint did nothing to help solve the crisis at hand. Instead we followed a decades long path through Cauchy defining infinitesmals in terms of sequences approaching zero, and then Weierstrass came up with limits when that followed. We dutifully put a passing mention to Berkeley in a lot of textbooks for being so prescient, but he was otherwise irrelevant.
And so it proved again at the end of the 1800s. As mathematics again had a foundational crisis we came up with the three basic schools of mathematical philosophy that are still discussed. Platonism is a description of what mathematicians believed by default, though those who officially hold to it tend to be religious like Kurt Gödel. Formalism was essentially created by David Hilbert, a mathematician. Its opponent, Intutionism (which falls under Constructivism), was put forth by L. E. J. Brouwer - another mathematician.
And so it has proven in other fields. When physics had a philosophical crisis with interpreting quantum mechanics, most of the important interpretations were produced by physicists. When evolution theory had foundational crises around how to incorporate genetics, most of the critical contributors to The Modern Synthesis were biologists. (Though one of the critical early contributors was R. A. Fisher, a statistician.) But philosophers were conspicuous by their absence.
All that said, the people who come up with those key philosophical insights are often personally interested in philosophy. Having a mind that is capable of such philosophizing does seem correlated with curiosity about philosophy. BUT engagement with subject expertise during the period of crisis seems to be the more important criterion for important philosophical contributions.
Berkeley wasn’t the only one to criticize the foundations calculus. And insofar as his criticism of calculus goes, it is IMO hardly more philosophical rather than mathematical compared to criticism from mathematicians like Rolle. Insofar as the “philosophical” side of his argument goes, the general shtick behind Berkeley’s criticism seems to be the desire to show that the foundations of religion were no worse than those of math (he was a bishop after all). So I don’t see how his philosophical insights could help math to develop.
First, you're right about Berkeley's motivation. Specifically he was rebutting Edmund Halley. But his criticism was correct. And he was also correct that through compounding errors mathematicians arrived at truth, if not science. Furthermore he demonstrated that he had a better grasp of calculus than most who criticized him at the time.
You're also right that what he said could not help math to develop. But that is not due to any shortcoming on his part. Until mathematicians realized that there was a crisis, nothing could have made them pay attention. And once a crisis was realized, what would help them find a path forward depended on the details of said crisis.
Now you pointed to Rolle. Yes, Rolle criticized infinitesmal calculus. But he did so before it was well understood, and all of his criticisms were rebutted. In fact Rolle wound up convinced about calculus, and is known for an important theorem. (That said, his original version was considerably less general than the current one is.)
He was not alone. Lagrange also criticized calculus. His alternative was to found everything on formal power series. But his replacement didn't hold up. In fact Fourier's examples were more of a problem for Lagrange's approach than infinitesmals! So from Lagrange's criticism we got the f'(x) notation, but nothing that could help later.
Lagrange's failure to be able to address the future crisis, even though he recognized the problems that lead to it, demonstrates that Bishop Berkeley never had a hope. Lagrange was one of the top mathematicians of the 1700s. If he could not anticipate what would be needed, how could a bishop be expected to do better?
Should mathematics prove to be in need of a philosophical insight, it will not come from a philosopher.
The reason is simple. Most fields of study only find themselves in need of philosophical insights when they are internally driven to crisis. But when this happens, it is those who are part of said crisis that have the best insight as to the actual needs revealed, and it is from them that critical ideas come. Even if a philosopher happens to make a good point in advance, the philosopher will be ignored until the field of study has driven itself in crisis. After the crisis has passed, the philosopher's insight may be discovered to be relevant. But it is unlikely to have been noticed at the time.
This is not just an abstract theory. It is borne out by history. For example calculus as envisioned by Newton and Leibniz had foundations in handwaving and wishful thinking. Bishop George Berkeley was absolutely right to complain about "ghosts of departed quantities". But it didn't become a crisis until the following decades when Joseph Fourier added together a bunch of sin and cosine terms (all well-behaved functions) and came up with a square wave (emphatically NOT a function by then current standards).
While Fourier proved Berkeley right, Berkeley's complaint did nothing to help solve the crisis at hand. Instead we followed a decades long path through Cauchy defining infinitesmals in terms of sequences approaching zero, and then Weierstrass came up with limits when that followed. We dutifully put a passing mention to Berkeley in a lot of textbooks for being so prescient, but he was otherwise irrelevant.
And so it proved again at the end of the 1800s. As mathematics again had a foundational crisis we came up with the three basic schools of mathematical philosophy that are still discussed. Platonism is a description of what mathematicians believed by default, though those who officially hold to it tend to be religious like Kurt Gödel. Formalism was essentially created by David Hilbert, a mathematician. Its opponent, Intutionism (which falls under Constructivism), was put forth by L. E. J. Brouwer - another mathematician.
And so it has proven in other fields. When physics had a philosophical crisis with interpreting quantum mechanics, most of the important interpretations were produced by physicists. When evolution theory had foundational crises around how to incorporate genetics, most of the critical contributors to The Modern Synthesis were biologists. (Though one of the critical early contributors was R. A. Fisher, a statistician.) But philosophers were conspicuous by their absence.
All that said, the people who come up with those key philosophical insights are often personally interested in philosophy. Having a mind that is capable of such philosophizing does seem correlated with curiosity about philosophy. BUT engagement with subject expertise during the period of crisis seems to be the more important criterion for important philosophical contributions.