I mean, it sort of deserves being made fun of. 18 years ago Google existed, surely you'd search for "area under a curve" before going through all the effort of writing a paper reinventing integrals?
Edit: actually the paper was written in 1994, not sure what the "18 years" was referring to. But still, peer review existed and so did maths books... Even if the author can be excused somewhat (and that's already a stretch), peer reviewers should definitely not let this fly.
Unfortunately quite common to see serious mathematical issues in the medical literature. I guess due to a combination of math being essential to interpreting medical data and trial results, but most practitioners not having much depth of math knowledge.
Just this week I came across the quote "Frequentist 95% CI: we can be 95% confident that the true estimate would lie within the interval." This is an incorrect interpretation of confidence intervals, but the amusing part is that it is from a tutorial paper about them, so the authors should have known better. And cited by 327!
https://pmc.ncbi.nlm.nih.gov/articles/PMC6630113/
Does it make you less of a peer to others who found it before ? At leas the author showed ability to think creative for himself , not paralyzed by the great stagnation like the rest of us.
And what makes you less of a peer is not knowing the basics. And being so unaware of apparently not knowing the basics, and/or uninterested, that you don't bother to check something that is highly checkable.
This is why peer review exists. One can not known everything themselves. It's fairly common for CS paper submissions to reinvent algorithms and then tone down the claims after reviewers suggest that variants already exist.
Calculus textbooks existed in 1994. It just took me 30 seconds to find “area under a curve” in the index of the one I own, and another 30 seconds to find the section on numerical integration, which describes the trapezoidal approximation.
So you already know the particular area of the larger topic of mathematics that you need to look for, you already have a textbook for that particular subject in your possession, meaning you don't need to go to the library and somehow figure out the right book to choose from the thousands the 510 section; you know what you are looking for exists, and then you aren't surprised you can find it?
I know how to find the area under the curve, but there's so much biology I don't know jack shit about. Back in 1994, It would have been hopeless for me to know the Michaelis-Menten model even existed if it had been relevant to my studies in computer science. That you can right click on those words in my comment in 2025 and get an explanation of what that is and can interrogate chatgpt to get a rigorous understanding of it shouldn't make it seem like finding someone in the math department to help you in 1994 was easier than just thinking logically and reinventing the wheel.
There is thing called "higher education". Ostensibly one of its chief purposes is to arm you with all that interconnected knowledge and facts that is useful in your chosen field of study. To the boot, you get all of that from several different human beings who you can converse with, to improve the scope and precision of the knowledge you're receiving. You know, "standing on the shoulders of the giants" and all of that stuff?
> So you already know the particular area of the larger topic of mathematics that you need to look for
So did the author of the paper. The paper’s title itself mentions the area under a curve. It would not have been difficult to find information about how to calculate an approximation of the area under a curve in the library.
I'd argue this is an argument against purely peer review, as her peers also weren't mathematicians.
Some of us when learning calculus wonder if we'd been alive before it was invented, if we'd be smart enough to invent it. Dr. Tai provably was. (the trapezoid rule, anyway) So I choose to say xkcd 1053 to her,
rather than bullying her for not knowing advanced math.
No, we have no proof of that. We just know that she published a paper explaining the trapezoidal rule.
(A) That approximation for 'nice' curves was known long before calculus. Calculus is about doing this in the limit (or with infinitesimals or whatever) and also wondering about mathematical niceties, and also some things about integration. (B) I'm fairly certain she would have had a bit of calculus at some point in her education, even if she remembered it badly enough to think she found something new.
I mean, it's possible she reinvented the wheel because what she really needed in her life is for the math department to laugh at her, but that seems far fetched to me.
> Bone structure is, as far as I can tell as a layperson, the major determinant of how people look. I found it quite surprising as I thought it would be the other way around.
How would it work the other way
around? You don't have a "look" before your bone structure exists right?
Right, but naively I would think that your bones are the foundation and your skin and muscles are the house on top. But really, the skin and muscles are more of the paint and trim, and the bones are the foundation, walls, and even part of the roof. Even your nose is largely determined by the angle and width of your facial bones, which is quite surprising to me, given that obviously there's no bone in it past the bridge.
For me the question is, why would it be static? A piece of fruit decomposing (on a tree or otherwise) clearly doesn't stay static, why would anything else?
The expectation that the majestiy of death respects the dignity of humans?
(Or too many bad movies and a society that tries to banish the fought of death, instead of accepting it as part of the natural cycle. Also, corpes are usually filled up to the top with formaldehyd.)
> Maybe the standard should be "less false than the average human produced work."
I don't think so. Lots of people blindly trust LLMs more than they trust the average human, probably for bad reasons (including laziness and over-reliance on technology).
Given that reality, it's irresponsible to make LLMs that don't reach a much better standard, since it encourages people to misinform themselves.
> Arguing that you can't use an LLM for Christian apologetics because it "might not be true" overemphasizes the definition of "truth" when it comes to scripture and those teaching Christian apologetics, which is entirely influenced by what doctrine you subscribe to.
But the author is pretty explicit about wanting a high standard (e.g. insisting on using the best sources possible), and doesn't think that using LLMs is compatible with that goal.
> Some cats are afraid of cucumbers, presumably because the shape and color resembles a snake. Here’s a funny compilation: https://youtu.be/oDpQ2uGLUKU
It's funny in a way, but if you think about it it's actually abusive.
Would you think it's funny if you were terrified of snakes and someone randomly put a fake snake next to you when you were just relaxing?
I'm kind of in the opposite camp. If Schanuel's conjecture is true, then e^iπ = 0 would be the only non-trivial relation between e, π, and i over the complex numbers. And the fact that we already found it seems unlikely.
Do you have a citation for the rationality of e^pi - pi? I couldn't find anything alluding to anything close to that after some cursory googling, and, indeed, the OEIS sequence of the value's decimal expansion[1] doesn't have notes or references to such a fact (which you'd perhaps expect for a rational number, as it would eventually be repeating).
Is the joke here that if you lie to people (on the Internet or otherwise), they’ll take it at face value for a little bit and then decide you’re either a moron or an asshole once they realize their mistake?
The number you are looking for is e^(sqrt(163) pi). According to Wikipedia:
In a 1975 April Fool article in Scientific American magazine,[8] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it – hence its name.
Actually `e^(sqrt(n) pi)` is very close to being an integer for a couple of different `n`s, including 67 and 163. For 163 it's much closer to an integer, but for 67 you get something you can easily check in double precision floats is close to an integer, so I thought it worked better as a joke answer :)
It's not really "of course", and I don't think we have such a theorem in general. But in this case, I believe the fact that it's not an integer follows from the same theorem that says it's very close to an integer. See eg https://math.stackexchange.com/questions/4544/why-is-e-pi-sq...
Basically e^(sqrt(163)*pi) is the leading term in a Laurent series for an integer, and the other (non-integer) terms are really small but not zero.
Charitably it was a joke, as was my quip about `e^(sqrt(67) pi)`. It is a funnier joke without a disclaimer at the end, but unlike GP I couldn't bring myself to leave one out and potentially mislead some people...
What I meant was that I didn't know that `e^pi - pi` is another transcendental expression that is very close an integer. You might think this is just an uninteresting coincidence but there's some interesting mathematics around such "almost integers". Wikipedia has a quick overview [1]. I didn't realize it before, but they have GP's example and also the awesome `e + pi + e pi + e^pi + pi^e ~= 60`.
I mean you just have to get to the point where all of the trailing decimal places (bits) form a repeating pattern with finite period. But since there are infinitely many such patterns it becomes extremely hard to rule out without some mechanism of proof.
Is the result of the addition or multiplication of an irrational number with any other real number not equal to it (and non-zero in the case of multiplication) always irrational? ex: pi + e, pi * e, but also sqrt(2) - 1 or sqrt(3) * 2.54 ?
No, sqrt(5)*sqrt(16*5)=20. More trivially, there's always a number y such that z = x*y for a given irrational x. You can give similar examples for all the other basic operations.
Definitely not; consider the formula for calculating the log of any base given only the natural logarithm. That can result e.g. in two irrational numbers, the ratio of which are integers.
There is an important distinction to be made here. Examples in this thread show cases of irrational numbers multiplied by or added to other irrational numbers producing real numbers, but in the special case of a rational number added to or multiplied by an irrational number, the result is always irrational.
Otherwise, supposing for instance that (n/m)x is rational for integers n, m, both non-zero, and irrational x, we can express (n/m)x as a ratio of two integers p, q, q non-zero: (n/m)x = p/q if and only if x = (mp)/(qn). Since integers are closed under multiplication, x is rational, against supposition; thus by contradiction (n/m)x is irrational for any rational r = (n/m), with integers n, m both non-zero. Similarly for the case of addition.
To put the first equation more formally, we know that ℚ is closed under addition¹, so given k∊ℝ\ℚ, l∊ℚ then if k+l=m∊ℚ, then m-l=m+(-l)∊ℚ, but m-l=k which is not in ℚ so k+l∉ℚ.
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1. For p,q∊ℚ, let p=a/b, q=c/d, a,b,c,d∊ℤ, then p+q=(ad+bc)/bd, but the products and sums of integers are integers, so p+q∊ℚ
That one's "just" a special case of how complex numbers happen to work. I think the really cool relationship between e and pi is the fact that the Gaussian integral acts as a fixpoint/attractor when sampling and summing data from any distribution (this is the central limit theorem):
∫(−∞ to ∞) e^(-x²) dx = √π
I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.
> because these two numbers are not supposed to be related
Says who? They’re not known to be related in that way, but it’s not like nature set out to prevent such a thing, or that large parts of mathematics would break down if it happened to be the case.
> If it's 11:55, you would usually mentally subtract and conclude: the meeting is in 5 minutes. That's how I always do it myself anyway! But the most probable estimate given the available information is actually 4'30"!
The way I like to think about it is "the meeting is in less than 5 minutes". Which is always correct since my reaction time to seeing the clock switching to 11:55 is greater than zero.
It could even be less than 4 minutes if it has already switched to 11:56 and I haven't had time to react to that change, but that's OK - my assessment that I have less than 5 minutes to get to the meeting is still correct.
Edit: actually the paper was written in 1994, not sure what the "18 years" was referring to. But still, peer review existed and so did maths books... Even if the author can be excused somewhat (and that's already a stretch), peer reviewers should definitely not let this fly.
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