3 ms·

Funny this came up. Just today I was reading about a letter Gödel wrote to von Neumann in 1956 asking him about an idea relating to the precursors of P vs NP an

Funny this came up. Just today I was reading about a letter Gödel wrote to von Neumann in 1956 asking him about an idea relating to the precursors of P vs NP and Hutter search (https://ecommons.cornell.edu/server/api/core/bitstreams/46aef9c4-288b-457d-ab3e-bb6cb1a4b88e/content https://ecommons.cornell.edu/server/api/core/bitstreams/46ae...). Gödel was quite ahead of his time. Even though he tended to prefer theory over practical applications, it would be really interesting to have his thoughts on prediction and induction as they relate to the recent advances in machine learning.

atticora 3y ago
> it would be really interesting to have his thoughts on prediction and induction as they relate to the recent advances in machine learning. For a possibly not useless proxy, train an LLM on his oeuvre and ask it.
- teacpde 3y ago
  I can understand the downvotes, but at the same time it does makes sense, especially given the target is a logician.
- mingus88 3y ago
  You’ll get an answer with all of the linguistic flavor of Gödel’s texts and absolutely none of his insights or reasoning. LLM is a language model, not Deep Thought from the hitchhikers guide.
  - atticora 3y ago
    It works on embeddings, which are concepts. So it would at least be manipulating a similarly weighted network of concepts, though much shallower. It would be an answer from his literal domain of discourse. It would be evocative rather than authoritative.
    - piaste 3y ago
      What you describe would be literally worse than nothing. The use of those concepts would make about as much sense as the use of topological concepts in a Lacanian text. A typewriting monkey in a trenchcoat.
- riku_iki 3y ago
  > For a possibly not useless proxy, train an LLM on his oeuvre and ask it. all major LLMs are likely already trained on his works, so you can already ask: "Hi, ChatGPT, pretend to be Kurt Godel"
cushpush 3y ago
Reflecting deeply on Gödel's Completeness and Incompleteness Theorems, I'd like to contribute to the discussion. The essence of Completeness lies in the notion that a system, constructed on foundational axioms (such as facts, truths, or mathematical operations), encompasses all possible facts derivable within its framework. Interestingly, this completeness paradoxically begets contradictions. It implies the existence of 'axiomatic islands' - distinct realms of truth where a fact valid in one may contradict another in a different realm. Hence, Completeness inherently entails contradiction. On the other hand, Incompleteness, which targets a specific subset of all possible truths or derivations, can achieve absolute consistency by avoiding these contradictions. This is because it deliberately excludes certain 'axiomatic regions,' maintaining coherence within its defined scope but at the expense of being incomplete. Gödel's genius lay in unveiling the deceptive nature of mathematics when navigating these axiomatic territories. His theorems present a profound duality: the local context, where truths are confined to specific axiomatic systems, and the universal context, which can be akin to an 'ultimate truth' in spiritual terms. This ultimate truth transcends individual systems, offering a broader, more holistic understanding.
- danans 3y ago
  > This ultimate truth transcends individual systems, offering a broader, more holistic understanding. But the unknowability of that so-called "ultimate truth" implies that it's not offering anything, much less any understanding. The article says Gödel thought that escape hatch for these contradictions is death, because all the local consistency and rationality hints at some greater consistency that must be there. However, that sounds a lot like wishful thinking.
  - cushpush 3y ago
    Your insights on the inherent contradictions in rational frameworks, especially in the context of Gödel's Completeness and Incompleteness Theorems, bring us to an intriguing crossroads. Initially, Gödel’s Completeness Theorem suggests that within a certain logical system, all truths can be proven. However, his subsequent Incompleteness Theorem introduces a profound paradox: there are truths which, though existent within the system, cannot be proven by it. This contradiction between completeness and incompleteness in logic itself leads us to question: can these contradictions be reconciled to provide clarity? Here, we might consider shifting our perspective from a purely rational approach to one that embraces intentionality and clarity. The journey from understanding Gödel’s theorems to grappling with the notion of contradictions could be seen as a metaphor for our search for truth. It’s not merely about accumulating information or solving logical puzzles; it's about the intentional pursuit of clarity. This pursuit often takes us beyond the realm of intellectual reasoning into a space where understanding becomes more about intuition and less about calculation. As we delve deeper into this journey, we arrive at a crucial realization: perhaps the reconciliation of these contradictions and the understanding of 'ultimate truth' is less about logical resolution and more about experiential realization. In this light, truth is not something to be dissected in the confines of rational thought alone but to be lived and experienced. The clarity we seek may not lie in the resolution of logical paradoxes but in embracing the experiential wisdom that comes from directly engaging with these truths. Thus, while Gödel’s work brilliantly navigates the complexities of logical systems, it also inadvertently points us toward a different kind of resolution - one that is realized not through further analysis but through personal experience. In essence, the journey from completeness to incompleteness, from information to intentionality, leads us to a profound experiential understanding, an ultimate truth that is realized rather than deduced.

atticora 3y ago

> it would be really interesting to have his thoughts on prediction and induction as they relate to the recent advances in machine learning. For a possibly not useless proxy, train an LLM on his oeuvre and ask it.

teacpde 3y ago
I can understand the downvotes, but at the same time it does makes sense, especially given the target is a logician.
mingus88 3y ago
You’ll get an answer with all of the linguistic flavor of Gödel’s texts and absolutely none of his insights or reasoning. LLM is a language model, not Deep Thought from the hitchhikers guide.
- atticora 3y ago
  It works on embeddings, which are concepts. So it would at least be manipulating a similarly weighted network of concepts, though much shallower. It would be an answer from his literal domain of discourse. It would be evocative rather than authoritative.
  - piaste 3y ago
    What you describe would be literally worse than nothing. The use of those concepts would make about as much sense as the use of topological concepts in a Lacanian text. A typewriting monkey in a trenchcoat.
riku_iki 3y ago
> For a possibly not useless proxy, train an LLM on his oeuvre and ask it. all major LLMs are likely already trained on his works, so you can already ask: "Hi, ChatGPT, pretend to be Kurt Godel"

cushpush 3y ago

Reflecting deeply on Gödel's Completeness and Incompleteness Theorems, I'd like to contribute to the discussion. The essence of Completeness lies in the notion that a system, constructed on foundational axioms (such as facts, truths, or mathematical operations), encompasses all possible facts derivable within its framework. Interestingly, this completeness paradoxically begets contradictions. It implies the existence of 'axiomatic islands' - distinct realms of truth where a fact valid in one may contradict another in a different realm. Hence, Completeness inherently entails contradiction. On the other hand, Incompleteness, which targets a specific subset of all possible truths or derivations, can achieve absolute consistency by avoiding these contradictions. This is because it deliberately excludes certain 'axiomatic regions,' maintaining coherence within its defined scope but at the expense of being incomplete. Gödel's genius lay in unveiling the deceptive nature of mathematics when navigating these axiomatic territories. His theorems present a profound duality: the local context, where truths are confined to specific axiomatic systems, and the universal context, which can be akin to an 'ultimate truth' in spiritual terms. This ultimate truth transcends individual systems, offering a broader, more holistic understanding.

danans 3y ago
> This ultimate truth transcends individual systems, offering a broader, more holistic understanding. But the unknowability of that so-called "ultimate truth" implies that it's not offering anything, much less any understanding. The article says Gödel thought that escape hatch for these contradictions is death, because all the local consistency and rationality hints at some greater consistency that must be there. However, that sounds a lot like wishful thinking.
- cushpush 3y ago
  Your insights on the inherent contradictions in rational frameworks, especially in the context of Gödel's Completeness and Incompleteness Theorems, bring us to an intriguing crossroads. Initially, Gödel’s Completeness Theorem suggests that within a certain logical system, all truths can be proven. However, his subsequent Incompleteness Theorem introduces a profound paradox: there are truths which, though existent within the system, cannot be proven by it. This contradiction between completeness and incompleteness in logic itself leads us to question: can these contradictions be reconciled to provide clarity? Here, we might consider shifting our perspective from a purely rational approach to one that embraces intentionality and clarity. The journey from understanding Gödel’s theorems to grappling with the notion of contradictions could be seen as a metaphor for our search for truth. It’s not merely about accumulating information or solving logical puzzles; it's about the intentional pursuit of clarity. This pursuit often takes us beyond the realm of intellectual reasoning into a space where understanding becomes more about intuition and less about calculation. As we delve deeper into this journey, we arrive at a crucial realization: perhaps the reconciliation of these contradictions and the understanding of 'ultimate truth' is less about logical resolution and more about experiential realization. In this light, truth is not something to be dissected in the confines of rational thought alone but to be lived and experienced. The clarity we seek may not lie in the resolution of logical paradoxes but in embracing the experiential wisdom that comes from directly engaging with these truths. Thus, while Gödel’s work brilliantly navigates the complexities of logical systems, it also inadvertently points us toward a different kind of resolution - one that is realized not through further analysis but through personal experience. In essence, the journey from completeness to incompleteness, from information to intentionality, leads us to a profound experiential understanding, an ultimate truth that is realized rather than deduced.