Example of mathematical invention with ChatGPT. Something to note is that ChatGPT will consistently have a positive emotional affect even if the output is outright nonsensical.
Thanks and interesting. You can use your reply to find similar inventions, but completely impossible for me to judge on sense.
"I've been exploring the applications of p-adic analysis in quantum mechanics, especially as they pertain to certain non-locality behaviors. Do you see any parallels with the ultrametric structures often discussed in number theory?"
"Given that the moduli space of Calabi-Yau manifolds serves as a geometric foundation for certain compactifications in string theory, how do you feel these spaces relate to Mumford's conjectures in algebraic geometry?"
"From the perspective of the Atiyah-Singer index theorem, how can one reconcile anomalies in quantum field theories with the corresponding invariants in differential geometry?"
"Having delved deep into Donaldson invariants from gauge theories, I'm curious how Seiberg-Witten theory refines our understanding of four-manifolds, especially with respect to their intricate interrelation with Kähler metrics."
"Connes' non-commutative geometry framework has intriguing applications in the spectral action principle of the standard model. How do you think this might tie into L^2-invariants in non-commutative topology?"
"I've recently been examining the applications of S-duality in N=4 supersymmetric Yang-Mills theory. This led me to wonder how the Langlands correspondence in number theory might shed light on these dualities."
"The Gromov-Witten invariants provide deep insights into enumerative geometry via string theory. How would you contrast their behavior with that of the refined invariants in the presence of stable pair theories?"
"I've been intrigued by the relationship between quantum cohomology and Frobenius manifolds, especially as they relate to singularity theory. Could you explore their intricate ties to the Verlinde algebra?"
"Given the connections between Chern-Simons theory and the Jones polynomial in knot theory, how would you interpret the ramifications of the Reshetikhin-Turaev invariants in 3-dimensional quantum gravity?"
"While studying loop quantum gravity, I came across some parallels with the Tamagawa number conjecture in arithmetic. How do you think adelic structures might impact our understanding of space-time quantization?"
I have not experienced GPT-4 hallucinate like this which is also why I think Bing Chat's claim about using GPT-4 is a blatant lie. Bing Chat hallucinates a lot.
This hallucination is sobering and thought-provoking.
ChatGPT creates a bubble around your reasoning. If you discover hallucination it tries to correct itself, but most often it just creates different nonsense.
If you are not very well-versed and believe you understand it you could fall for it.
If you program you at least can try out and give feedback till it finds a solution that works. However this is not better than those programmers who introduce a lot of security issues.
ChatGPT is at best for finding ideas which are often bogus but could inspire you to useful ideas (a form of brainstorming) and at worst just useless or even dangerous.
Edit: Indeed, since you trust GPT-4 so much here is what it has to say on the subject:
> Does GPT-4 hallucinate?
> No, GPT-4 doesn't "hallucinate" in the way that humans or animals might experience hallucinations. However, the term "hallucinate" is sometimes used informally to describe when GPT-4 (or similar models) produces outputs that are not grounded in its training data or are factually incorrect. This typically means that the model is generating content based on patterns it has recognized but that might not be accurate or contextually appropriate.
> Does GPT-4 hallucinate in the latter sense?
> Yes, in the latter sense, GPT-4 can "hallucinate." It can produce outputs that are not grounded in its training data or are factually incorrect. This occurs because GPT-4 is predicting the next word based on patterns in the data it has been trained on, and sometimes those predictions might not be accurate or contextually appropriate.
For those who don't know about this paradox, let me explain it. We have now the situation that GPT-4 itself states it can hallucinate. Let's say this is true. Then case closed. GPT-4 can hallucinate. But what if this is not true? Then case closed. We have proven that GPT-4 has hallucinated in at least one case.
I see comments saying this is nonsensical, but most of this seems plausible. Like, as a former number theorist, this paragraph would also describe my guess as to what the prompt meant:
So, when talking about the "adelic representations of quantum groups," you're likely referring to a framework that seeks to describe the action of quantum groups in terms of the adeles (or ideles). This would be a way to unify and generalize certain aspects of the representation theory of quantum groups, making use of the rich structure of the adeles.
Admittedly, I don’t know what the “tautological action” might be, as I’m not super knowledgeable about quantum groups. But, that notion wasn’t even introduced by ChatGPT, it was introduced in the prompt.
Okay, yeah I’m aware of a “tautological action” where the very definition of the group is in terms of an action on some space (like say GLn), but as far as I know, quantum groups are not defined as some set of transformations of the adeles.
19 comments
[ 2.7 ms ] story [ 84.5 ms ] thread"I've been exploring the applications of p-adic analysis in quantum mechanics, especially as they pertain to certain non-locality behaviors. Do you see any parallels with the ultrametric structures often discussed in number theory?"
"Given that the moduli space of Calabi-Yau manifolds serves as a geometric foundation for certain compactifications in string theory, how do you feel these spaces relate to Mumford's conjectures in algebraic geometry?"
"From the perspective of the Atiyah-Singer index theorem, how can one reconcile anomalies in quantum field theories with the corresponding invariants in differential geometry?"
"Having delved deep into Donaldson invariants from gauge theories, I'm curious how Seiberg-Witten theory refines our understanding of four-manifolds, especially with respect to their intricate interrelation with Kähler metrics."
"Connes' non-commutative geometry framework has intriguing applications in the spectral action principle of the standard model. How do you think this might tie into L^2-invariants in non-commutative topology?"
"I've recently been examining the applications of S-duality in N=4 supersymmetric Yang-Mills theory. This led me to wonder how the Langlands correspondence in number theory might shed light on these dualities."
"The Gromov-Witten invariants provide deep insights into enumerative geometry via string theory. How would you contrast their behavior with that of the refined invariants in the presence of stable pair theories?"
"I've been intrigued by the relationship between quantum cohomology and Frobenius manifolds, especially as they relate to singularity theory. Could you explore their intricate ties to the Verlinde algebra?"
"Given the connections between Chern-Simons theory and the Jones polynomial in knot theory, how would you interpret the ramifications of the Reshetikhin-Turaev invariants in 3-dimensional quantum gravity?"
"While studying loop quantum gravity, I came across some parallels with the Tamagawa number conjecture in arithmetic. How do you think adelic structures might impact our understanding of space-time quantization?"
I have not experienced GPT-4 hallucinate like this which is also why I think Bing Chat's claim about using GPT-4 is a blatant lie. Bing Chat hallucinates a lot.
About once a week there is an article about how GPT can’t do something out together by someone with a lot of time, but not the $20 to pay for GPT 4.
ChatGPT creates a bubble around your reasoning. If you discover hallucination it tries to correct itself, but most often it just creates different nonsense.
If you are not very well-versed and believe you understand it you could fall for it.
If you program you at least can try out and give feedback till it finds a solution that works. However this is not better than those programmers who introduce a lot of security issues.
ChatGPT is at best for finding ideas which are often bogus but could inspire you to useful ideas (a form of brainstorming) and at worst just useless or even dangerous.
I don't understand how people on HN keep conflating the two.
Edit: Indeed, since you trust GPT-4 so much here is what it has to say on the subject:
> Does GPT-4 hallucinate?
> No, GPT-4 doesn't "hallucinate" in the way that humans or animals might experience hallucinations. However, the term "hallucinate" is sometimes used informally to describe when GPT-4 (or similar models) produces outputs that are not grounded in its training data or are factually incorrect. This typically means that the model is generating content based on patterns it has recognized but that might not be accurate or contextually appropriate.
> Does GPT-4 hallucinate in the latter sense?
> Yes, in the latter sense, GPT-4 can "hallucinate." It can produce outputs that are not grounded in its training data or are factually incorrect. This occurs because GPT-4 is predicting the next word based on patterns in the data it has been trained on, and sometimes those predictions might not be accurate or contextually appropriate.
For those who don't know about this paradox, let me explain it. We have now the situation that GPT-4 itself states it can hallucinate. Let's say this is true. Then case closed. GPT-4 can hallucinate. But what if this is not true? Then case closed. We have proven that GPT-4 has hallucinated in at least one case.
The difference is that Sokal knew he was lying, had a point to make in so doing, and could not create a new hoax every few milliseconds
https://en.wikipedia.org/wiki/Sokal_affair
So, when talking about the "adelic representations of quantum groups," you're likely referring to a framework that seeks to describe the action of quantum groups in terms of the adeles (or ideles). This would be a way to unify and generalize certain aspects of the representation theory of quantum groups, making use of the rich structure of the adeles.
Admittedly, I don’t know what the “tautological action” might be, as I’m not super knowledgeable about quantum groups. But, that notion wasn’t even introduced by ChatGPT, it was introduced in the prompt.
* if you grok mathematical notation