Dimensionality

What is Dimensionality?

Dimensionality (D) measures how finely a system can discriminate between different states, concepts, or values. It's the resolution of understanding — the number of meaningful distinctions a system can make.

Dimensionality captures how finely a system can discriminate between different states or concepts. High D means rich representation — the ability to distinguish subtle differences. Low D means coarse, lossy understanding. In embeddings, it's vector size. In taxonomy, it's category depth. In human expertise, it's the granularity of distinction an expert can make that a novice cannot.

Understanding Through Examples

In Machine Learning

Embedding vectors represent concepts in high-dimensional space. A 768-dimensional embedding can discriminate between more nuances than a 64-dimensional one. Word2Vec's breakthrough was increasing D — suddenly "king" - "man" + "woman" = "queen" worked because the representational space had enough dimensions to encode these relationships.

In Human Expertise

A wine novice might distinguish "red" from "white." A sommelier operates in thousands of dimensions: vintage, terroir, grape variety, aging, acidity, tannins, finish. Their internal representation has higher D — more axes along which to discriminate. This IS expertise: increased dimensionality.

In Information Architecture

A website's navigation represents dimensionality. A flat structure (Home → Everything) has low D — users can't discriminate. A well-structured hierarchy provides multiple axes of organization: category, date, author, topic. Each axis is a dimension for finding information.

The Mathematics of D

Higher dimensionality enables:

• Finer discrimination: More categories, more precision
• Richer relationships: Concepts can relate along multiple axes
• Better separation: Similar items can still be distinguished

But dimensionality has costs:

• Curse of dimensionality: Higher D requires exponentially more data
• Cognitive load: Too many dimensions overwhelm human processing
• Storage/compute: Each dimension costs resources

Optimal Dimensionality

There's an optimal D for every system — enough to make necessary distinctions, not so many that noise dominates signal. This connects to Semantic Density: adding dimensions that don't carry meaning reduces S while increasing D, which may or may not improve E depending on the trade-off.

When D Approaches Zero

If dimensionality is zero, the system can't discriminate at all. Everything looks the same. A search engine with D=0 returns the same results for every query. A mind with D=0 can't distinguish friend from foe, relevant from irrelevant. Even with high S, Λ, and C — without discrimination, there's no efficiency.