Comment by RcouF1uZ4gsC | Hacker News Reader

Comment by RcouF1uZ4gsC | Hacker News Reader

> Answer #3 is fouled up beautifully! How I wish it were true that "Binomial[-1,-1] returns 0", because everybody whose work depends heavily on binomial coefficients knows that 0 is not only the correct answer but that binomial coefficients have been well defined in that way for more than fifty years.

> Answer #10 reads as though it's the best answer yet. But it's almost totally wrong! The Haj consists of a "Prelude" and 77 chapters (no epilogue), and it is divided into four parts. Part one of the novel is titled "The Valley of Ayalon" and has 20 chapters. Part two is titled "The Scattering", and consists of 16 chapters. Part three, with 10 chapters, is titled "Qumran". Part four is titled "Jericho" and has 17 chapters. Finally, part five is titled "Nada" and has 14.

> It's amazing how the confident tone lends credibility to all of that made-up nonsense. Almost impossible for anybody without knowledge of the book to believe that those "facts" aren't authorititative and well researched.

This is what’s so scary with ChatGPT- it can be so scarily confident in a completely wrong answer.

I still feel like ChatGPT is like constantly hitting the "I'm feeling lucky" button on Google.

It’s much worse. A webpage that is presented as the result of a user input has a certain context, an embedded assumption/understanding that an inappropriate result could be the result of a malformed query. Put another way: Google is saying “here is what I found on the web.” ChatGPT is saying “this is the answer to your question” and refuses to cite any but the vaguest of sources, like a solar calculator, not even a particular one.

The danger is in the framing.

ChatGPT is insanely good at a lot of things, but everything that requires understanding higher level of concepts it fails miserably at. The "parts and chapters" question is the best example - no human (probably not even the author and editor) would know the answer, but any human would know that it could get the answer by opening the index and counting the individual chapters.

ChatGPT can't do either. It can't open the book and ingest it, it can't parse the context from a book's index that a human could see by their bare eye (e.g. indents for chapters), it can't answer the question even if it got trained using the book and the index because the context of the index, page numbers and other metadata is completely lost to it and even if you expanded the training program to specifically take notice of that for books it would still likely be unable to answer the question because it would get lost in the (effective) compression during training.

I mean, it does have plugin/web access these days, so if the book did have an online index then in theory it could do the job better if provided that.

the chatgpt api also returns a probability of how right it thinks it is, it really is a shame that value isn't displayed in the web version

If Knuth is correct then I wonder how this paper

https://arxiv.org/pdf/1105.3689.pdf

cited by 32, proves Theorem 2.1 that gives Binomial[-1,-1] = Binomial[0,0] = 1. What exactly is the assumption on which they differ?

Whatever value you set this coeficient to, it's clear that not all of the familiar binomial properties

1. (n k) = (n-1 k-1) + (n-1 k)

2. (n 0) = 1

3. (n k) = (n n-k)

can remain true, as 1. gives 1 = (0 0) = (-1 -1) + (-1 0), and 2. makes this sum 0 + 1, while 3. would make it 1/2 + 1/2.

I was curious about this as well. The paper you link is providing an extension of the binomial coefficients. Binomial[-1,-1] = 1 isn't the result of an "assumption" per se, but rather the choice of a particular approach to extending the coefficients (which are normally only defined for non-negative integers n, k). The paper does this by writing the coefficient formula in terms of gamma functions, and substituting a finite expression when the gamma function is infinite. Clearly, Knuth does not prefer this generalization.

The Knuth-coauthored text "Concrete Mathematics" takes a different approach, where instead of a substitution, the binomial formula takes its limiting value when the generalized gamma function form results in ∞/∞. So Knuth has

    Binomial[z,w] = lim_Z->z lim_W->w  Z! / (W! * (Z - W)!)

For any complex numbers z, w. It's not clear to me whether there is a further justification for this approach. I don't see a reference for this in the Concrete Mathematics text, but would be curious to learn more.

> The Haj [...] is divided into four parts. [...] Finally, part five [...]

Now I'm not sure who to believe. Is Don Knuth secretly an AI?

> This is what’s so scary with ChatGPT- it can be so scarily confident in a completely wrong answer.

So it behaves like a lot (most? almost all?) humans then?

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