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Comment by nyc111 | Hacker News Reader

I checked the links and I think it's amazing and it answers with Latex formatted notation.

But I was curious and I asked something very simple, Euclid's first postulate and I got this answer:

Euclid's Postulate 1: "Through any two points, there is exactly one straight line."

In fact Euclid's Postulate 1 is "To draw a straight line from any point to any point." http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.h...

I think AI answer is not correct, it may be some textbook interpretation but I was expecting Euclid's exact wording.

Edit: Google's Gemini gives the exact wording of the postulate and then comments that this means that you can draw one line bitween two points. I think this is better

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The original text is: Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν. Roughly: let it be required that from any point to any point it is possible to draw a straight line.

Both gpt4o and o1 roughly know the correct original text, so prompting, the model’s background memory, or random chance may influence your outcomes, though hopefully (in an improved model) you should never get you incorrect info.

https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

Edit: in case it isnt clear, I could not reproduce this error on my end with o1-mini

it's definitely wrong though, "exactly one" straight line between two points is a different postulate and a stronger one.

Euclid has been translated, restated, and re-presented in enough books and textbooks that I'd expect a big-enough LLM to have actually memorized this correctly tbh

Agreed. That is what the original poster said. I didnt manage to reproduce the error on my end, but I dont know the full context or maybe the memory on my end changes the output.

> I think AI answer is not correct, it may be some textbook interpretation but I was expecting Euclid's exact wording

It was written before English even existed. That said, the original never implied "exactly one", so I agree its a bad translation.

Regardless of the wording being exact or not, ChatGPT’s answer is incorrect in its contents. The statement “exactly one” requires the parallel postulate, since otherwise it’s not necessarily true. Specifically in spherical geometry, which is considered to be consistent with Euclid’s first four postulates (i.e. without the parallel postulate).

The bottom line is, you can’t take any single LLM statement at face value, even in seemingly easy to answer cases like this.

it's the point-line postulate, you can use it as part of a set of axioms equivalent to Euclid but it definitely not one of Euclid's

https://en.wikipedia.org/wiki/Point%E2%80%93line%E2%80%93pla...

Euclid wrote in ancient Greek, so the "exact wording" in English does not exist.

Euclid's Elements is less pervasive on the Internet then content produced for Liberal Arts math courses. As those courses tend to emphasize critical thinking and problem-solving math over pure theory and advanced concepts, they tend to be far more common and tend to win compared to more domain specific meanings.

Examples:

https://en.wiktionary.org/wiki/Euclidean_geometry

https://www.cerritos.edu/dford/SitePages/Math_70_F13/Postula...

Problems with polysemy across divergent, more advanced theories has been one of my biggest challenges in probing some of my areas of intrest.

Funny enough, one of my pet areas of obscure interest, riddled basins, is constantly muddied not by math, but LSAT questions, specifically non-math content directed at a reading comprehension test: "September 2006 LSAT Section 1 Question 26"

IMHO a lot of the prompt engineering you have to do with these highly domain specific problems is avoiding the most common responses in the corpus.

LLM responses will tend to reflect common usage, not academic terminology unless someone cares enough to change that for a specific case.

“Exact wording” would be Ancient Greek. Euclid did not even write in English. You’re checking whether the model matches a specific translation, which is not valuable. If you search around you’ll find many sources that choose a more intelligible phrasing.

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