Observationally, the vast majority of mathematical papers do not make claims that are non-rigorous but as well supported as the Basel problem. They split into rigorous proofs (potentially conditional on known additional hypotheses eg. Riemann), or they offer purely heuristic arguments with substantially less support.
It should also be noted that Euler was working at a time when it was widely known that the behaviour of infinite sums, products and infinitesimal analysis (following Newton or Leibnitz) was without any firm foundation. So analysis of these objects at that time was generally flanked with “sanity check” demonstrations that the precise objects being analysed did not trivially cause bad behaviour. Essentially everyone treated these kinds of demonstrations as highly suspect until the 1830′s and a firm foundation for analysis (cf. Weierstrass and Riemann). Today we grandfather these demonstrations in as proofs because we can show proper behaviour of these objects.
On the other hand, there were a great many statements made at that time which later turned out to be false, or require additional technical assumptions once we understood analysis, as distinct from an calculus of infinitesimals. The most salient to me would be Cauchy’s 1821 “proof” that the pointwise limit of continuous functions is continuous; counterexamples were not constructed until 1826 (by which time functions were better understood) and it took until 1853 for the actual conditions (uniform continuity) to be developed properly. This statement was at least as well supported in 1821 as Euler’s was in 1735.
As to confidence in modern results: Looking at the Web of Science data collated here for retractions in mathematical fields suggests that around 0.15% of current papers are retracted.
The most salient to me would be Cauchy’s 1821 “proof” that the pointwise limit of continuous functions is continuous; counterexamples were not constructed until 1826 (by which time functions were better understood) and it took until 1853 for the actual conditions (uniform continuity) to be developed properly. This statement was at least as well supported in 1821 as Euler’s was in 1735.
That it worked in every instance of continuous functions that had been considered up to that point, seemed natural, and extended many existing demonstrations that a specific sequence of continuous functions had a continuous limit.
A need for lemmas of the latter form are endemic, for a concrete class of examples, any argument via a Taylor series on an interval implicitly requires such a lemma, to transfer continuity, integrals and derivatives over. In just this class, you get numerical evidence came from the success of perturbative solutions to Newtonian mechanics, and theoretical evidence in the existence of well behaved Taylor series for most functions.
I guess we’ll have to agree to disagree here :-). I find Euler’s evidence for the product formula for sine to be far more convincing than what was available to Cauchy at the time.
Edit: I say more here, where I highlight how different the two situations are.
Not 100% sure, but pretty sure. The situation isn’t so much that I think that the evidence for the limit of a continuous function being continuous is weak, as much as that the evidence for the product formula for sine is very strong.
The result (and its analog) imply two formulas for pi that had been proved by other means, and predicts infinitely many previously unknown numerical identities, which can be checked to be true to many decimal places. What more could you ask for? :-)
Polya reports on Euler performing such checks. I don’t know how many he did – one would have to look at the original papers (which are in Latin), and even they probably omit some of the plausibility checks that Euler did.
That it worked in every instance of continuous functions that had been considered up to that point,
In ~1659, Fermat considered the sequence of functions f(n,x) = x^n for n = 0, 1, 2, 3, …. Each of these is a continuous function of x. If you restrict these functions to the interval between 0 and 1, and take the limit as n goes to infinity, you get a discontinuous function.
So there’s a very simple counterexample to Cauchy’s ostensible theorem from 1821, coming from a sequence of functions that had been studied over 150 years before. If Cauchy had actually looked at those examples of sequences of function that had been considered, he would have recognized his ostensible theorem to be false. By way of contrast, Euler did extensive empirical investigation to check the plausibility of his result. The two situations are very, very different.
Fermat considered the sequence of functions f(n,x) = x^n for n = 0, 1, 2, 3, ….
Only very kind of. Fermat didn’t have a notion of function in the sense meant later, and showed geometrically that the area under certain curves could be computed by something akin to Archimedes’ method of exhaustion, if you dropped the geometric rigour and worked algebraically. He wasn’t looking at a limit of functions in any sense; he showed that the integral could be computed in general.
The counterexample is only “very simple” in the context of knowing that the correct condition is uniform convergence, and knowing that the classical counterexamples look like x^n, n->\infty or bump functions. Counterexamples are not generally obvious upfront; put another way, it’s really easy to engage in Whig history in mathematics.
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
It should also be noted that Euler was working at a time when it was widely known that the behaviour of infinite sums, products and infinitesimal analysis (following Newton or Leibnitz) was without any firm foundation.
Could you give a source for this claim? “Foundation” sounds to me anachronistic for 1735.
It possible that “were known in general to lead to paradoxes” would be a more historically accurate phrasing than “without firm foundation”.
For east to cite examples, there’s “The Analyst” (1734, Berkeley). The basic issue was that infinitesimals needed to be 0 at some points in a calculation and non-0 at others. For a general overview, this seems reasonable. Grandi noticed in 1703 that infinite series did not need to give determinate answers; this was widely known in by the 1730′s. Reading the texts, it’s fairly clear that the mathematicians working in the field were aware of the issues; they would dress up the initial propositions of their calculi in lots of metaphysics, and then hurry to examples to prove their methods.
Observationally, the vast majority of mathematical papers do not make claims that are non-rigorous but as well supported as the Basel problem. They split into rigorous proofs (potentially conditional on known additional hypotheses eg. Riemann), or they offer purely heuristic arguments with substantially less support.
It should also be noted that Euler was working at a time when it was widely known that the behaviour of infinite sums, products and infinitesimal analysis (following Newton or Leibnitz) was without any firm foundation. So analysis of these objects at that time was generally flanked with “sanity check” demonstrations that the precise objects being analysed did not trivially cause bad behaviour. Essentially everyone treated these kinds of demonstrations as highly suspect until the 1830′s and a firm foundation for analysis (cf. Weierstrass and Riemann). Today we grandfather these demonstrations in as proofs because we can show proper behaviour of these objects.
On the other hand, there were a great many statements made at that time which later turned out to be false, or require additional technical assumptions once we understood analysis, as distinct from an calculus of infinitesimals. The most salient to me would be Cauchy’s 1821 “proof” that the pointwise limit of continuous functions is continuous; counterexamples were not constructed until 1826 (by which time functions were better understood) and it took until 1853 for the actual conditions (uniform continuity) to be developed properly. This statement was at least as well supported in 1821 as Euler’s was in 1735.
As to confidence in modern results: Looking at the Web of Science data collated here for retractions in mathematical fields suggests that around 0.15% of current papers are retracted.
What was the evidence?
That it worked in every instance of continuous functions that had been considered up to that point, seemed natural, and extended many existing demonstrations that a specific sequence of continuous functions had a continuous limit.
A need for lemmas of the latter form are endemic, for a concrete class of examples, any argument via a Taylor series on an interval implicitly requires such a lemma, to transfer continuity, integrals and derivatives over. In just this class, you get numerical evidence came from the success of perturbative solutions to Newtonian mechanics, and theoretical evidence in the existence of well behaved Taylor series for most functions.
I guess we’ll have to agree to disagree here :-). I find Euler’s evidence for the product formula for sine to be far more convincing than what was available to Cauchy at the time.
Edit: I say more here, where I highlight how different the two situations are.
Are you sure you aren’t suffering from hindsight bias?
Not 100% sure, but pretty sure. The situation isn’t so much that I think that the evidence for the limit of a continuous function being continuous is weak, as much as that the evidence for the product formula for sine is very strong.
The result (and its analog) imply two formulas for pi that had been proved by other means, and predicts infinitely many previously unknown numerical identities, which can be checked to be true to many decimal places. What more could you ask for? :-)
And did Euler check them?
Polya reports on Euler performing such checks. I don’t know how many he did – one would have to look at the original papers (which are in Latin), and even they probably omit some of the plausibility checks that Euler did.
In ~1659, Fermat considered the sequence of functions f(n,x) = x^n for n = 0, 1, 2, 3, …. Each of these is a continuous function of x. If you restrict these functions to the interval between 0 and 1, and take the limit as n goes to infinity, you get a discontinuous function.
So there’s a very simple counterexample to Cauchy’s ostensible theorem from 1821, coming from a sequence of functions that had been studied over 150 years before. If Cauchy had actually looked at those examples of sequences of function that had been considered, he would have recognized his ostensible theorem to be false. By way of contrast, Euler did extensive empirical investigation to check the plausibility of his result. The two situations are very, very different.
Only very kind of. Fermat didn’t have a notion of function in the sense meant later, and showed geometrically that the area under certain curves could be computed by something akin to Archimedes’ method of exhaustion, if you dropped the geometric rigour and worked algebraically. He wasn’t looking at a limit of functions in any sense; he showed that the integral could be computed in general.
The counterexample is only “very simple” in the context of knowing that the correct condition is uniform convergence, and knowing that the classical counterexamples look like x^n, n->\infty or bump functions. Counterexamples are not generally obvious upfront; put another way, it’s really easy to engage in Whig history in mathematics.
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
Could you give a source for this claim? “Foundation” sounds to me anachronistic for 1735.
It possible that “were known in general to lead to paradoxes” would be a more historically accurate phrasing than “without firm foundation”.
For east to cite examples, there’s “The Analyst” (1734, Berkeley). The basic issue was that infinitesimals needed to be 0 at some points in a calculation and non-0 at others. For a general overview, this seems reasonable. Grandi noticed in 1703 that infinite series did not need to give determinate answers; this was widely known in by the 1730′s. Reading the texts, it’s fairly clear that the mathematicians working in the field were aware of the issues; they would dress up the initial propositions of their calculi in lots of metaphysics, and then hurry to examples to prove their methods.