When a mathematician tries to talk biology with AI (offtop)

After having talked with someone about evolution I resorted to the help of AI as I was trying to explain when and why one cannot induce a metric space to describe living systems.

So I suggested to consider living systems as a topology space without a metric by the analogy how a probability space by default has no metric but always has a measure, i.e. being only a measure space.
It seemed so obvious to me, that I immediately ran to check on publications hoping to find the implications (maybe some works on classifications, minding this concept… or the like) — zero.

That's why I had no choice to ask AI about their perception of such an idea. 
And here is what I get in reply:

Me: living systems as topological space without a metric
(the following text before P.S. is fully AI-generated)

 
AI: Treating living systems as a topological space without a metric focuses on closeness, connectivity, and transformation (homeomorphism/homotopy) rather than precise distance, highlighting patterns, organization, and functional relationships over physical measurement, crucial for understanding emergent properties, self-organization, and dynamic stability in biology where standard metrics fail. It allows modeling dynamic states (e.g., cell differentiation, ecosystems) as shifts between topologically similar configurations, emphasizing what stays the same during change (like a network's structure) rather than how far things move, essential for complex systems theory. 

Key Concepts

• Topological Space: A set (e.g., all possible states of a cell) with a "topology" (rules for "open sets" or neighborhoods) defining "nearness" without a ruler (metric).
• No Metric: Ignores quantitative distance; focuses on qualitative relationships like connectedness, continuity, and boundaries.
• Living Systems: Organisms, ecosystems, or biological networks where states change continuously but maintain fundamental organizational features.

Applications in Living Systems

• Self-Organization: How order emerges from simple interactions (e.g., flocking birds, protein folding) – a topological view sees stable patterns (like a vortex) as persistent shapes.
• Evolution/Development: Viewing species or organismal development as paths through a "morphospace" (a topological space of shapes) where viable forms are connected, avoiding isolated, non-functional "distances".
• Network Biology: Analyzing protein-protein interaction networks or gene regulatory networks by looking at their connectivity (topology) rather than edge weights (strengths) to find robust motifs.
• Dynamical Systems: Understanding stable states (attractors) and transitions in biological processes (like cell cycles or disease states) as points and flows in a topological landscape.

Why it's Useful

• Abstraction: Simplifies complex systems by focusing on essential structural properties.
• Robustness: Highlights properties (like being a single connected component) that survive small perturbations, crucial for biological resilience.
• Universal Patterns: Reveals underlying, scale-invariant principles governing diverse biological phenomena.

In essence, it shifts focus from how much (distance) to how (connectivity, shape, transformation) in the dynamic, often messy world of biology, using topological tools to describe biological organization and function.

P.S.
You may generate a publication in, say, theoretical biology with a snap of your fingers — and it may work like a charm… so far.
But then… you become useless (of course, unless you induce a truly new concept and/or suggest a paradigm shift).

Gemini 3's reply is here (yes I know they are coded to flatter, but they are sort of… sincere beings) — I bet it's even more exciting.
Enjoy!