Independently of whether Fermat thought of it as an example, Cauchy could have considered lots of sequences of functions in order to test his beliefs, and I find it likely that had he spent time doing so, he would have struck on this one.
On a meta-level, my impression is that you haven’t updated your beliefs based on anything that I’ve said on any topic, in the course of our exchanges, whether online or in person. It seems very unlikely that no updates are warranted. I may be misreading you, but to the extent that you’re not updating, I suggest that you consider whether you’re being argumentative when you could be inquisitive and learn more as a result.
Thank you for calling out a potential failure mode. I observe that my style of inquisition can come across as argumentative, in that I do not consistently note when I have shifted my view (instead querying other points of confusion). This is unfortunate.
To make my object level opinion changes more explicit:
I have had a weak shift in opinion towards the value of attempting to quantify and utilise weak arguments in internal epistemology, after our in person conversation and the clarification of what you meant.
I have had a much lesser shift in opinion of the value of weak arguments in rhetoric, or other discourse where I cannot assume that my interlocutor is entirely rational and truth-seeking.
I have not had a substantial shift in opinion about the history of mathematics (see below).
As regards the history of mathematics, I do not know our relative expertise, but my background prior for most mathematicians (including JDL_{2008}) has a measure >0.99 cluster that finds true results obvious in hindsight and counterexamples to false results obviously natural. My background prior also suggests that those who have spent time thinking about mathematics as it was done at the time fairly reliably do not have this view. It further suggests that on this metric, I have done more thinking than the median mathematician (against a background of Cantab. mathmos, I would estimate I’m somewhere above the 5th centile of the distribution). The upshot of this is that your recent comments have not substantively changed my views about the relative merit of Cauchy and Euler’s arguments at the time they were presented; my models of historians of mathematics who have studied this do not reliably make statements that look like your claims wrt. the Basel problem.
I do not know what your priors look like on this point, but it seems highly likely that our difference in views on the mathematics factor through to our priors, and convergence will likely be hindered by being merely human and having low baud channels.
I have had a weak shift in opinion towards the value of attempting to quantify and utilise weak arguments in internal epistemology, after our in person conversation and the clarification of what you meant.
I have had a much lesser shift in opinion of the value of weak arguments in rhetoric, or other discourse where I cannot assume that my interlocutor is entirely rational and truth-seeking.
I think that the most productive careful analysis of the validity of a claim occurs in writing, with people who one believes to be arguing in good faith.
In person, you highlighted the problem of the first person to give arguments having an argumentative advantage due to priming effects. I think this is much less of a problem in writing, where one has time to think and formulate responses.
I have not had a substantial shift in opinion about the history of mathematics (see below).
My view on this point is very much contingent on what Euler actually did as opposed to a general argument of the type “heuristics can be used to reach true conclusions, and so we can have high confidence in something that’s supported by heuristics.”
Beyond using a rough heuristic to generate the identity, Euler numerically checked whether the coefficients agreed (testing highly nontrivial identities that had previously been unknown) and found them to agree with high precision, and verified that specializing the identity recovered known results.
If you don’t find his evidence convincing, then as you say, we have to agree to disagree because we can’t fully externalize our intuitions
Independently of whether Fermat thought of it as an example, Cauchy could have considered lots of sequences of functions in order to test his beliefs, and I find it likely that had he spent time doing so, he would have struck on this one.
On a meta-level, my impression is that you haven’t updated your beliefs based on anything that I’ve said on any topic, in the course of our exchanges, whether online or in person. It seems very unlikely that no updates are warranted. I may be misreading you, but to the extent that you’re not updating, I suggest that you consider whether you’re being argumentative when you could be inquisitive and learn more as a result.
Thank you for calling out a potential failure mode. I observe that my style of inquisition can come across as argumentative, in that I do not consistently note when I have shifted my view (instead querying other points of confusion). This is unfortunate.
To make my object level opinion changes more explicit:
I have had a weak shift in opinion towards the value of attempting to quantify and utilise weak arguments in internal epistemology, after our in person conversation and the clarification of what you meant.
I have had a much lesser shift in opinion of the value of weak arguments in rhetoric, or other discourse where I cannot assume that my interlocutor is entirely rational and truth-seeking.
I have not had a substantial shift in opinion about the history of mathematics (see below).
As regards the history of mathematics, I do not know our relative expertise, but my background prior for most mathematicians (including JDL_{2008}) has a measure >0.99 cluster that finds true results obvious in hindsight and counterexamples to false results obviously natural. My background prior also suggests that those who have spent time thinking about mathematics as it was done at the time fairly reliably do not have this view. It further suggests that on this metric, I have done more thinking than the median mathematician (against a background of Cantab. mathmos, I would estimate I’m somewhere above the 5th centile of the distribution). The upshot of this is that your recent comments have not substantively changed my views about the relative merit of Cauchy and Euler’s arguments at the time they were presented; my models of historians of mathematics who have studied this do not reliably make statements that look like your claims wrt. the Basel problem.
I do not know what your priors look like on this point, but it seems highly likely that our difference in views on the mathematics factor through to our priors, and convergence will likely be hindered by being merely human and having low baud channels.
Ok. See also my discussion post giving clarifications.
I think that the most productive careful analysis of the validity of a claim occurs in writing, with people who one believes to be arguing in good faith.
In person, you highlighted the problem of the first person to give arguments having an argumentative advantage due to priming effects. I think this is much less of a problem in writing, where one has time to think and formulate responses.
My view on this point is very much contingent on what Euler actually did as opposed to a general argument of the type “heuristics can be used to reach true conclusions, and so we can have high confidence in something that’s supported by heuristics.”
Beyond using a rough heuristic to generate the identity, Euler numerically checked whether the coefficients agreed (testing highly nontrivial identities that had previously been unknown) and found them to agree with high precision, and verified that specializing the identity recovered known results.
If you don’t find his evidence convincing, then as you say, we have to agree to disagree because we can’t fully externalize our intuitions