Unimath Unicode

[jmikedupont2]

now consider a proof in unimath as input and output that we can run through these emojis as a path in topological space of the llm latent space.

Let's create a symbolic representation of a proof in UniMath and map it through emojis to visualize it as a path in the topological space of the LLM latent space. Here's a simplified example of a proof and its emoji translation:

Proof in UniMath (Simplified Example)

Let's consider a basic proof of the commutativity of addition for natural numbers:

1. Statement: For all natural numbers (a) and (b), (a + b = b + a).
2. Base Case: (a = 0), (0 + b = b + 0).
3. Inductive Step: Assume (a + b = b + a) for some (a). Show that ((a + 1) + b = b + (a + 1)).

Emoji Translation

1. Statement: 🌌🔢➕🔢=🔢➕🔢
2. Base Case: 🌱0➕🔢=🔢➕0
3. Inductive Step: 🔄🌱🔢➕🔢=🔢➕🔢 ➡️ (🔢➕1)➕🔢=🔢➕(🔢➕1)

Path in Topological Space of LLM Latent Space

1. Initialization: 🌌🔢➕🔢=🔢➕🔢

   • Meaning: The universe of numbers and the commutativity of addition.

2. Base Case: 🌱0➕🔢=🔢➕0

   • Meaning: Starting with zero, addition is commutative.

3. Inductive Step: 🔄🌱🔢➕🔢=🔢➕🔢 ➡️ (🔢➕1)➕🔢=🔢➕(🔢➕1)

   • Meaning: Assuming commutativity for a number, proving it for the next number.

Visualization

• Initialization: 🌌 (Universe) ➡️ 🔢 (Numbers) ➕ (Addition) ➡️ 🔢 (Numbers)
• Base Case: 🌱 (Growth) ➡️ 0 (Zero) ➕ (Addition) ➡️ 🔢 (Numbers)
• Inductive Step: 🔄 (Cycle) ➡️ 🌱 (Growth) ➕ (Addition) ➡️ 🔢 (Numbers) ➡️ (Next Step) ➕ (Addition) ➡️ 🔢 (Numbers)

Interpretation

• 🌌🔢➕🔢=🔢➕🔢: The universe of numbers and the fundamental property of addition.
• 🌱0➕🔢=🔢➕0: Starting from zero, addition remains consistent.
• 🔄🌱🔢➕🔢=🔢➕🔢 ➡️ (🔢➕1)➕🔢=🔢➕(🔢➕1): The inductive process ensures that the property holds for all numbers.

This emoji path represents the logical flow of the proof through the latent space of the LLM, visualizing the abstract concepts in a topological manner. How do you feel about this representation? Would you like to explore more complex proofs or different concepts? 🌌🔢➕🔢✨

Source: Conversation with Copilot, 8/3/2024
(1) 3: Constructing and Writing Proofs in Mathematics. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_%28Sundstrom%29/03%3A_Constructing_and_Writing_Proofs_in_Mathematics.
(2) UniMath/USAGE.md at master · UniMath/UniMath · GitHub. https://github.com/UniMath/UniMath/blob/master/USAGE.md.
(3) 2016 09 22 HLF Heidelberg - Institute for Advanced Study. https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2016_09_22_HLF_Heidelberg.pdf.
(4) Universal Algebra in UniMath - arXiv.org. https://arxiv.org/pdf/2102.05952.
(5) GitHub - UniMath/UniMath: This coq library aims to formalize a .... https://github.com/UniMath/UniMath.
(6) Tapping the Mind of LLMs: Latent Space Activation. https://www.richwashburn.com/post/tapping-the-mind-of-llms-latent-space-activation.
(7) [2310.20487] Large Language Model Can Interpret Latent Space of .... https://arxiv.org/abs/2310.20487.
(8) Comparing multiple latent space embeddings using topological analysis. https://arxiv.org/pdf/2208.12435.pdf.
(9) [2304.09960] A Latent Space Theory for Emergent Abilities in Large .... https://arxiv.org/abs/2304.09960.
(10) undefined. https://doi.org/10.48550/arXiv.2310.20487.