🔬 physics

Modelling Singularities in Macroevolution

This paper presents a unified mathematical framework based on combinatorial innovation and the Theory of the Adjacent Possible to model macroevolutionary singularities as explosive growth arising from the continuous limit of discrete recombination processes across biological, cultural, and technological domains.

Original authors: Alessandro Bellina, Giordano De Marzo, Vittorio Loreto

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alessandro Bellina, Giordano De Marzo, Vittorio Loreto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the history of human progress, the growth of our population, or the explosion of new inventions not as a smooth, steady climb, but as a rollercoaster that suddenly shoots straight up into the sky. For a long time, things move slowly, but then, almost overnight, everything accelerates so fast it seems like it will go on forever in a finite amount of time. Scientists call this a "singularity"—a point where the math says the number becomes infinite.

This paper asks: Why does this happen? And more importantly, is it real, or just a trick of the math we use to describe it?

The authors, a team of physicists and complexity scientists, propose a single, unified explanation for all these explosive surges. They call it the "Theory of Combinatorial Innovation," based on a concept called the "Adjacent Possible."

The Core Idea: The Lego Box Analogy

Think of the world as a giant box of Lego bricks.

The Adjacent Possible: At any given moment, you can only build things using the bricks you already have. You can't build a castle if you don't have the bricks, but you can build a wall, a tower, or a bridge using the bricks currently in your hand. The "Adjacent Possible" is the collection of everything you could build right now with your current set of bricks.
Combinatorial Growth: Every time you build something new (a new invention, a new species, a new economic system), you aren't just adding one brick; you are adding a new tool to your box. Now, you can combine your old bricks with this new one to make even more complex things.

The paper argues that this process is self-reinforcing. The more you have, the more you can combine, and the faster you can create new things. It's like a snowball rolling down a hill, but instead of just getting bigger, the snowball is constantly manufacturing new snow to add to itself.

The Four Stories They Told

To prove this works, the authors applied their "Lego Box" math to four very different real-world stories:

The History of Life (Biological Milestones):
Think of the history of the universe as a timeline of major events: the first cell, the Cambrian explosion (when animals suddenly diversified), the rise of humans, the invention of democracy. The paper shows that the time between these events gets shorter and shorter, following a hyperbolic curve. Their model suggests this happens because each new life form creates new possibilities for the next one.
The Human Population and Money (GDP):
Why has the human population and global wealth exploded in the last few centuries? The authors suggest it's because humans interact in pairs (reproduction) and groups. As the population grows, the number of possible interactions grows even faster (like a party where everyone talks to everyone else). This creates a feedback loop where more people lead to exponentially more ideas and economic activity.
The Explosion of Inventions (US Patents):
Here is where it gets interesting. The number of patents (inventions) has grown exponentially, but not in a way that leads to a "singularity" (infinity). Why? Because patents are made of "technological codes" (like ingredients in a recipe). The paper found that while the number of codes grows steadily (linearly), the number of patents (recipes) grows exponentially because you can mix those codes in endless ways. However, because the "ingredients" (codes) don't explode, the whole system doesn't hit a mathematical singularity. It just grows very, very fast.

The Big Twist: Is the "Singularity" Real?

This is the most crucial part of the paper.

When you look at the data for population or milestones, it looks like it's heading toward a "Doomsday" date where the numbers hit infinity. The authors call this a mathematical singularity.

However, they argue that this singularity is an illusion created by our math.

The Discrete vs. Continuous Trap: Real life is "discrete." You can't have half a person or half a patent. You have 1, 2, or 3. But the math equations the authors use to describe the trend are "continuous," meaning they treat numbers like a smooth fluid that can go to infinity.
The Result: When you use smooth math to describe a bumpy, discrete reality, the math breaks down and predicts an infinite explosion at a specific date. But in the real world, the system is finite. You can't have infinite people.

The Analogy: Imagine driving a car toward a cliff. If you look at a smooth map, the road seems to end abruptly at a specific point. But in reality, the road doesn't just vanish into thin air; it turns into gravel, then dirt, then grass. The "cliff" in the map is just a limitation of the map, not the road.

What Does This Mean?

The paper concludes that while these "explosive" trends are real and driven by the power of combining ideas (or genes, or people), the idea of a specific "Doomsday" date where everything goes infinite is likely a mathematical artifact.

It's not a prediction of the end of the world.
It's a warning sign that our current models are reaching their limit.
It's a reminder that real-world systems (which are finite and discrete) will eventually slow down or change course before they ever hit "infinity."

In short: The universe is great at building new things by mixing old ones, and that makes progress accelerate. But don't panic about a specific date where everything breaks; the math just got ahead of reality. The "singularity" is a feature of the equation, not necessarily a feature of the future.

Technical Summary: Modelling Singularities in Macroevolution

Problem Statement
Macroevolutionary dynamics across biological, social, and economic systems frequently exhibit sudden, explosive surges where growth rates exceed traditional exponential models, often following hyperbolic trajectories. These phenomena are frequently described as the emergence of "singularities"—points where growth theoretically approaches infinity in finite time. While hyperbolic models fit empirical data (e.g., world population, GDP, technological milestones) effectively, the underlying microscopic mechanisms driving such rapid growth remain poorly understood. Existing interpretations are often purely phenomenological, and a unified mathematical framework connecting combinatorial innovation theories with observed hyperbolic growth and singularities is lacking. Furthermore, the distinction between discrete real-world systems and their continuous mathematical approximations regarding the existence of true singularities requires clarification.

Methodology
The authors propose a unified framework based on the Theory of the Adjacent Possible (TAP), which models macroevolutionary change as a process of recombining pre-existing elements within a system.

Theoretical Foundation: The study formalizes the TAP concept using a discrete equation where the number of elements $M_{t+1}$ evolves based on the number of possible combinations of existing elements $M_t$ . The evolution is governed by coefficients $A_i$ representing the probability of realizing combinations of size $i$ .
Continuous Limit Derivation: To apply the theory to continuous time scenarios, the authors derive the continuous limit of the discrete TAP equation. This involves scaling parameters to obtain a differential equation:
$\frac{dM_t}{dt} = -\sigma M_t + \nu \sum_{i=1}^{M_t} \alpha_i \binom{M_t}{i}$
where $\sigma$ is the extinction rate (set to 0 for simplicity) and $\nu$ is a dimensional factor.
Case-Specific Modeling: The authors apply specific forms of the combinatorial terms to five distinct datasets:
- Biological/Cosmological Milestones: Assuming all combinations are possible but with decreasing probability ( $\alpha_i \propto \alpha^i$ ), leading to an exponential growth rate in the number of elements.
- World Population: Modeling growth as driven by pair interactions ( $i=2$ ), resulting in a quadratic growth term.
- GDP: Modeling growth as proportional to the product of population and GDP, leading to a $K^{3/2}$ growth term.
- US Patents: Modeling patents as combinations of technological codes. Unlike the other cases, this system is modeled to show that while the relationship between patents and codes is exponential, the temporal evolution does not necessarily lead to a singularity if the underlying driver (codes) grows linearly.

Key Contributions

Unified Framework: The paper provides a single mathematical structure (the continuous TAP equation) capable of explaining hyperbolic growth across diverse domains (biology, economics, technology) by varying the combinatorial interaction terms.
Mechanistic Explanation: It bridges the gap between phenomenological hyperbolic fits and microscopic combinatorial mechanisms, demonstrating that hyperbolic growth naturally emerges from specific recombination rules (e.g., pair interactions for population, all-order combinations for milestones).
Discrete vs. Continuous Distinction: A critical theoretical contribution is the demonstration that while the continuous limit of the TAP equations leads to mathematical singularities (divergence in finite time), the underlying discrete models—which better represent real-world finite systems—do not exhibit true singularities. The "singularity" is an artifact of the continuous approximation.
Differentiation of Systems: The framework distinguishes between systems that appear similar (hyperbolic growth) but are governed by different underlying processes (e.g., population growth driven by pairs vs. milestone accumulation driven by all-order combinations).

Results

Biological/Cosmological Milestones: The model successfully fits the time series of Canonical Milestones and Biological Phase Transitions. The derived macrodevelopment rate $\rho_t$ follows a hyperbolic function with an exponent of $-1$, consistent with previous empirical observations.
World Population and GDP: The model reproduces the hyperbolic growth of world population (exponent 1) and GDP (exponent 2) by selecting specific interaction terms (pairwise for population, cubic-like for GDP). The fitted parameters align with historical data from 1000 CE to 2020 CE.
US Patents: The model captures the exponential growth of US patents relative to time, while noting that the number of distinct technological codes grows linearly. Crucially, this case demonstrates a system where the combinatorial mechanism does not produce a singularity, highlighting that explosive growth does not always imply a finite-time divergence.
Singularity Analysis: The authors show that the "singularity" date derived from fitting continuous models is highly sensitive to the data range used and statistical errors. They argue that because discrete models do not diverge, the singularity value is not a reliable predictor of a physical "Doomsday" or system collapse but rather a mathematical feature of the continuous approximation.

Significance and Claims
The paper claims to offer a "coherent framework" for interpreting growth and innovation processes across diverse domains through a physical and mathematical lens. Its primary significance lies in:

Reconciling Theory and Data: It connects the qualitative Theory of the Adjacent Possible with quantitative hyperbolic growth laws observed in nature.
Clarifying the Nature of Singularities: It cautions against the literal interpretation of mathematical singularities in real-world systems, emphasizing that real systems are discrete and finite. The authors argue that the concept of a singularity must be carefully considered, particularly for predictions, as it arises from the continuous limit of inherently discrete models.
Predictive Limitations: The authors modestly state that while the framework allows for the estimation of macrodevelopment parameters and the replication of historical trends, it cannot predict future shifts or "singularities" reliably because it does not account for external limiting factors (e.g., resource constraints) that would eventually decelerate growth in real systems. The singularity value derived from data is noted to be unstable and dependent on the specific time window chosen for fitting.

The Core Idea: The Lego Box Analogy

The Four Stories They Told

The Big Twist: Is the "Singularity" Real?

What Does This Mean?

Technical Summary: Modelling Singularities in Macroevolution

More like this