>GPT-4 with code interpreter is better at math than elite Math undergrads.
Yeah, and I just spent $20 to call BS on what you said.
GPT-4 just bullshits with confidence. I don't see how code interpeter can possibly help here.
Behold:
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Me: I drew three arrows on a piece of paper starting at the same point, pointing left, up, and diagonally (up and right). Do they represent linearly independent vectors?
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ChatGPT: In this case, you have described three different arrows pointing in unique directions:
One pointing left, which can be represented as a vector, say, [-1,0] in a two-dimensional space (using standard Cartesian coordinates).
One pointing up, which can be represented as a vector, say, [0, 1].
One pointing diagonally up and to the right, which could be represented as a vector, say, [1,1].
For these vectors to be linearly independent, no vector can be expressed as a linear combination of the others. In other words, there's no way to add or subtract scaled versions of any two vectors to get the third.
Looking at the vectors you've described:
The first vector represents movement purely in the horizontal direction.
The second vector represents movement purely in the vertical direction.
The third vector represents movement in both directions simultaneously.
Because none of these vectors can be created by scaling and adding the other two (for example, you can't add or subtract any combination of the first and second vectors to get the third one), they are linearly independent.
In mathematical terms, if we form a matrix with these vectors as columns, the determinant of this matrix would not be zero (a non-zero determinant indicates linear independence for the columns of the matrix). This is a more formal way to show their independence, but based on their directions and the explanation above, these vectors are indeed linearly independent.
You must be using a different GPT-4 than me. I recently tried to get it to reason about probability distributions arising from combining multiple probability distributions and it immediately started hallucinating.
I co-authored a published mathematics paper on knot theory[1] and wrote software that did the computations (and generated diagrams) in that paper as a Math undergrad, and I don't consider myself elite (though I did went on to get a PhD).
It seems like you have a very low bar for "elite", a very limited definition of "math", and a very peculiar one of "better at".
Absolutely not. Try asking for it to solve something not well covered in the training data (e.g. a relatively performant polyhedral homotopy continuation solver) and it completely falls apart regardless of how much hand holding you give it.
