Can vector spaces represent everything?

Vector spaces have proven essential in representing a wide array of systems and concepts, most notably in fields like quantum mechanics and machine learning. However, they aren't universally applicable for all types of data without additional structures.

Quantum Mechanics: Vector spaces, particularly Hilbert spaces, are fundamental in quantum mechanics. They allow for the representation of quantum states through superposition, where systems can exist in multiple states simultaneously with various probabilities 1 2.

Language and AI: In natural language processing, vector spaces are used to represent words, their meanings, and relationships. This method, known as word vectors, positions words in a high-dimensional space where similar words are closer together 3. However, purely vector-based approaches may lack the ability to capture more complex, contextual nuances fully, requiring symbolic reasoning or additional methodologies 4.

Data Embeddings: In machine learning, embedding discrete information into vector spaces facilitates various arithmetic operations and optimizations, making them practical for both humans and machines. However, embedding complex structures, like graphs, often requires non-Euclidean spaces for more efficient representation 5.

Neuroscience: The brain's processing of visual information demonstrates the creation of a minimalistic vector space of relevant objects for efficient memory use and prediction 6. This illustrates the capability and limitations of natural vector space representations.

In conclusion, vector spaces are extremely versatile and fundamental in many fields but are not always sufficient alone. Often, extra structures, reasoning methods, or non-Euclidean spaces are required to manage the complexity or specific nature of the data being represented.