### *Formalization and Implementation*:
While the paper lays out a theoretical framework, its practical implementation may face significant challenges. For instance, generating meaningful mathematical conjectures is far more abstract and constrained than tasks like generating text or images. The space of potential theorems is vast, and training an AI system to navigate this space intelligently would require further breakthroughs in both theory and computational techniques.

### *Compression as a Measure of Theorem Usefulness*:
The notion that a good theorem compresses provable statements is intriguing but may need more exploration in terms of practical utility. While compression aligns with Occam's Razor and Bayesian learning principles, it's not always clear whether the most "compressed" theorems are the most valuable, especially when considering the depth and complexity of many foundational theorems in mathematics.

### *Human-AI Collaboration*:
The paper lightly touches on how this AI mathematician might work alongside humans, but the real power of such a system might lie in human-AI collaboration. A mathematician AI capable of generating insightful conjectures and proofs could dramatically accelerate research, but the interaction between AI and human intuition would be key.

### *Computational and Theoretical Limits*:
There are also potential computational limits to the approach. The "compression" and "conjecture-making" frameworks proposed may be too complex to compute at scale, especially when considering the vast space of possible theorems and proofs. Developing approximation methods or heuristics that are effective in real-world applications will likely be necessary.

Here's how we can unpack this paper:

### *System 1 vs. System 2 Thinking*:
- *System 1* refers to intuitive, fast, and automatic thinking, such as recognizing patterns or generating fluent responses based on past experience. AI systems like GPT-4 excel in this area, as they are trained to predict and generate plausible content based on large datasets (e.g., text completion, language generation).
- *System 2* refers to deliberate, logical, and slow thinking, often involving reasoning, planning, and making sense of abstract ideas—such as solving a mathematical proof, engaging in formal logic, or synthesizing novel insights. The claim that AI lacks System 2 abilities suggests that while AI can mimic certain behaviors associated with intelligence, it struggles with tasks that require structured, step-by-step reasoning and deep conceptual understanding.

### "Not so much in terms of mathematical reasoning"

The claim is *partially true*, but it must be put into context:

- **Progress in AI**: AI has made **tremendous advances** in recent years, and while it may still lack sophisticated mathematical reasoning, there is significant progress in related areas like automated theorem proving (e.g., systems like Lean or Coq). Specialized systems can solve well-defined, formal mathematical problems—though these systems are not general-purpose AI and operate under specific constraints.

- **Scope of Current Models**: General-purpose models like GPT-4 weren't specifically designed for deep mathematical reasoning. Their training focuses on predicting likely sequences of tokens, not on formal logic or theorem proving. However, with enough specialized training or modularity, they could improve in these domains. We’ve already seen AI systems make progress in proving mathematical theorems with reinforcement learning and imitation learning techniques.

- **Frontiers of AI**: As AI continues to develop, future systems might incorporate elements of both System 1 and System 2 thinking by combining pattern recognition with symbolic reasoning and logical processing (e.g., systems that integrate neural networks with formal logic solvers or reasoning engines).

### Conclusion:
AI excels in tasks involving intuitive, pattern-based thinking but struggles with deliberate, goal-oriented reasoning required for deep mathematical work. However, as research evolves—especially in hybrid models that combine deep learning with symbolic reasoning and formal logic—these limitations may become less pronounced.

The future of AI may very well involve systems that are capable of the same level of mathematical reasoning (or better) as "human experts."