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That's 300 words for saying "in order for n to satisfy the requirements, it must satisfy the requirements." I tried an easier problem, extending Rolle's theore
by magnio 3y ago
That's 300 words for saying "in order for n to satisfy the requirements, it must satisfy the requirements."
I tried an easier problem, extending Rolle's theorem to the multidimensional case with Poe.com:
Prompt: Let U be an open subset of R^n with compact closure K. Suppose f: K → R is continuous on K, differentiable on U, and satisfies f(x) = 0, for all x in K \ U. Show that there exists a in U with grad f (a) = 0.
Poe: To prove that there exists a point a in U such that grad f(a) = 0, we will utilize the concept of connectedness and the Mean Value Theorem.
Since U is an open subset of R^n with compact closure K, we know that K is a closed and bounded subset of R^n. Furthermore, since K is a compact set, it is also a connected set.
Now, let's define a new function g: K → R^n such that g(x) = (f(x), f(x), ..., f(x)) for all x in K. In other words, each component of g is equal to f(x) for all x in K.
Since f is continuous on K and differentiable on U (which is an open subset containing K), it follows that g is continuous on K and differentiable on U.
Consider the function h: K → R defined as h(x) = ||g(x)||^2 = f(x)^2 + f(x)^2 + ... + f(x)^2 = nf(x)^2.
Note that h is continuous on K since g is continuous on K and ||.||^2 is a continuous function.
Now, let's consider the set S = {x in K : f(x) = 0}. Since f(x) = 0 for all x in K \ U, we have K \ U ⊆ S. Therefore, the set S is closed.
...
Stochastic parrot indeed.
- dj_mc_merlin 3y agoYeah, this is actually a really good example of its weaknesses.. it's the equivalent of a bullshit essay in math form. Completely tautological, with just a bit of the correct definitions thrown it to seem like it understands.
- Davidzheng 3y agohttps://chat.openai.com/share/f710b53c-1d87-4228-8940-f48ec69f7af0 https://chat.openai.com/share/f710b53c-1d87-4228-8940-f48ec6... GPT-4 for me solves it.
- mahogany 3y agoThat is impressive! It does look like a textbook question so I wouldn’t be surprised if a formulation almost identical to this is in the training data. But the ability to parse the “mathiness” is still impressive. Higher level math becomes pretty verbose so enters the domain of language more so than pure symbolic computation. However there is still complex reasoning under the hood. Gpt-4 in ChatGPT flounders when I ask it questions from my thesis, so I’m not sure how it will do with problems it hasn’t seen, where it needs to apply “new” reasoning. I’d love to know if in your example, GPT is reciting something it’s seem verbatim, or if it is taking multiple sources and combing them.
- eigenket 3y ago> Now, let's define a new function g: K → R^n such that g(x) = (f(x), f(x), ..., f(x)) for all x in K. In other words, each component of g is equal to f(x) for all x in K. That's hilarious