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.

What was the evidence?

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.

Do you really think that their pledge total is something other than the (undiscounted) sum of what those 291 people have pledged to give over the course of their lives? You think they're just making it up?

Hi Jonathan,

First off, thanks for putting so much time into writing this extensive list of questions and doubts you have about CEA. Unlike for-profit activities, we don't have immediate feedback effects telling us when we're doing well and when we're doing badly, so criticism is an important countermeasure to make sure we do things as well as possible. We therefore really welcome people taking a critical eye to our activities.

As the person who wrote the original CEA material here on LessWrong, and the person who you mention above, I feel I should be the one to collate a response to your questions. However, because of other commitments (managing; fundraising; writing my first piece for a magazine column), it will be a few days before I can get this to you in a form I'd feel happy with. I hope that's ok.

Before then I'll just mention a few things in order to make things a bit clearer to the audience.

• In what you wrote a couple of comments made it sound as if you'd had an in-depth conversation with me on these issues; whereas really the context of the only exchange we've had is my giving a short talk to a group of about 15 people, of very varied backgrounds. You asked a few questions and there was discussion afterwards, but this must have only taken up about 10-15 minutes of time. Though I would very much like to, I haven't ever spoken with you or Alexey one-on-one.

• Similarly, in your response to Luke you say that Adam works full-time at CEA. I think there's some disagreement between the two of you on the extent to which he had signed off on the content. But, at any rate, it's worth noting that Adam is an intern at CEA. This means he does contribute a full working week for CEA, but he is not an employee. He's therefore not the person to go when it comes to high-level evaluation of CEA.

• You mention an internal estimate of £1700 for the value of a new pledge. None of us are familiar with this figure, and we're confused about where it could have come from.

• You suggest that CEA has ~4000 people on its mailing lists. The correct figure is less than half that (unless you include TLYCS, which you might have been thinking of, which does have in excess of 4000 on its mailing list).

• You estimate GWWC's research capacity at 6 staff for last year. This is actually more than an order of magnitude higher than the true figure. In fact, the average number of paid employees (full-time equivalent) we have had working on all aspects of 80,000 Hours and Giving What We Can over the last six months is only 3.7.

As a more general point, I think we should also be careful to distinguish whether CEA has acted optimally in terms of utility-maximization (to which the answer is certainly not), and whether it gets a return on investment which is better than 1:1.

In my follow-up comment, I'll talk about some of the many concerns you've raised that we share, and the issues over which we might be making big mistakes. I'll also be able to give a bit more background about our activities, and I'll be able to answer your questions. Thanks again for taking the time to comment.

Best Wishes,

Will

Did you send this article to Will or somebody else at CEA before posting it? Holden Karnofsky let me comment on a copy of his critique of SI before he published it. That procedure is what I would call "common courtesy," and also it reduces the chance that you'll grossly mislead readers about an organization that you know far less about than the organization's principals do.

Talking about effective altruism is a constraint, as is talking about mathematics.

One constrains you to a subject with thousands of high-status practitioners and hundreds of students - the other restricts you to a subject with one high-status practitioner and no students.

The GWWC site, for example, claims that from 291 members there will be £72.68M pledged.

Qn: On what basis does GWWC assert that its near 300 members are credibly precommitted to donating £72.68M?

While I don't know what the official answer would be, I think there is basically a linguistic one. To have pledged does not require one to have credibly precommitted. I can pledge to resist torture for 40 days just by saying so, but this doesn't mean I've credibly precommitted to doing so.

Of course, this only clears GWWC of accusations of excessive optimism. The question of how many members will maintain the pledge remains, and I don't see how we can get a very good idea for a few years (untill we have enough data points to extrapolate the perameters of the distribution).

Of that, there are around 300 members and a few dozen are coming to each event. By comparison, enterprising college societies in Cambridge (TMS, TCSS) have well in excess of 1000 undergraduates on their mailing lists, and get 80-100 people to their talks. When TCSS advertised an event to 1/3 of Cambridge, upwards of 600 people attended.

I agree that it'd be good to get more people. However, the chapters are operating under a constraint that other socieities are not. Science societies can put on any talk they think will be interesting; GWWC has to put on talks about Effective Altruism that people will find interesting.

And while I don't know about TCSS, when OUSS holds talks, they'll get fewer than 100 people - there's at most a factor of 2 there. Again, I don't know about TCMS, but OUIS get an average of 40 people or so, not suggesting any capacity for improvement. True, they'll get around 150 for the very big names (think Sir Roger Penrose), but there just isn't anyone equivalently famous for GWWC to use, except Peter Singer - and I think when he was invited there was a very big crowd (though I forget how large).

disclaimer: I volunteer for GWWC.

...something like 20-25% of those involved in 80K have changed or substantially rethought their career choice. This implies immediately that 75-80% haven't, and in practise that number will be higher care of the self-reporting. This substantially reduces the likely impact of 80,000 hours as a program. Indeed, it seems to be a near fatal problem for GWWC, in that if the 80,000 hours population is representative of pledges, then most of the GWWC pledges are earning in line with typical post grads, which makes it much harder to raise the mean value of each pledge to £250K as is required.

This doesn't follow. If you were already going to be an Investment Banker, you wouldn't change your goals based on 80k's advice, but you'd still earn more thanthe average graduate.

Meditation:

It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.