Notes on Engineering Health, March 2025: Notes on Recursion

Recursion is simultaneously a mathematical concept, a computational strategy, a biological principle, and a way of understanding complex systems.

Recursion as a fundamental principle manifests in numerous contexts, representing a powerful mechanism for generating complex structures, processes, and systems through repeated self-similar patterns. At its core, recursion demonstrates how simple rules, when applied repeatedly, can create intricate and sophisticated phenomena.


Mathematics
In mathematics, recursion is a method of defining objects, functions, sequences, or sets in terms of themselves, typically using simpler or smaller instances. It's a form of self-reference that relies on two key components:

Base Case
This is the simplest, explicitly defined instance that does not rely on self-reference. It provides a stopping point for the recursion, preventing infinite regress. Without a base case, a recursive definition is ill-defined.

Recursive Step (or Inductive Step)
This rule defines how to construct or compute a more complex instance of the object/function/sequence in terms of one or more simpler or preceding instances of the same object/function/sequence. The “simpler” aspect is crucial—it ensures that repeated application of the recursive step eventually leads back to the base case.

One of the most common applications is found in the recursive definitions of functions and sequences:

Example #1: Factorial Function (n!)
•  Base Case: 0! = 1
•  Recursive Step: n! = n * (n-1)! for n > 0
•  To find 4!, you use the recursive step: 4! = 4 * 3!. Then 3! = 3 * 2!, then 2! = 2 * 1!, then 1! = 1 * 0!. Finally, you hit the base case 0! = 1, allowing the calculation to unwind: 1! = 1, 2! = 2, 3! = 6, 4! = 24.

Example #2: Fibonacci Sequence (F_n)
•  Base Cases: F_0 = 0, F_1 = 1
•  Recursive Step: F_n = F_(n-1) + F_(n-2) for n > 1
•  Each term is defined using the two preceding terms, ultimately relying on the initial two base cases.


Computing
In computing, recursion is a fundamental  principle where a process or algorithm calls itself to solve complex problems by breaking them down into smaller, more manageable subproblems. Nearly every sophisticated computational system relies on recursive principles, including artificial intelligence algorithms, machine learning models, network routing protocols, and data compression techniques.

A concrete example is the traversing of a decision tree in an AI system. Consider an AI that plays a strategy game like chess. The AI needs to evaluate potential future moves by exploring a tree of possible game states:

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The recursive process involved here is:
1. Initial Call: The AI starts at the current game state
2. Recursive Exploration:
   •  Generates possible moves
   •  Creates child nodes for each move
   •  Calls the same evaluation method on each child
3. Base Case: Stops when maximum depth is reached or game ends
4. Aggregation: Combines results to determine the best move


Biology
In biological systems, beyond abstract mathematics or in silico computing, recursion emerges as a fundamental choreographer of life itself—a hidden algorithm that nature has been coding and replicating since the dawn of biological complexity.

For example a lung's bronchial network, a body’s vascular network, or the branching patterns of a tree are not mere aesthetic coincidences, but deep demonstrations of recursive principles at work. These structures demonstrate self-similar branching patterns where smaller components closely resemble the overall structure, creating efficient networks for gas exchange, nutrient distribution, and signal transmission.

In cellular biology, recursion manifests most eloquently in the process of protein folding. Here, amino acid chains do not simply arrange themselves in linear sequence but fold and refold, calling back to previous configurations, creating three-dimensional structures that are at once precise and probabilistic. The three-dimensional configuration of proteins emerges through iterative computational processes where amino acid sequences recursively explore conformational states, ultimately converging on stable structural configurations. The protein is in essence a living mathematical equation, solving itself through repeated iterations.

Similarly, the immune system can be seen as a recursive phenomenon through its mechanisms of antibody production, immune cell proliferation, and recursive identification and elimination of pathogens.

Genetic inheritance itself can be understood as a recursive system. DNA replication is fundamentally a recursive process—a self-referential algorithmic mechanism where genetic information serves as both the template and the instruction set for its own reproduction. The genetic code is less a static blueprint and more a dynamic, self-referential system that perpetually regenerates and adapts.

Evolutionary mechanisms, too, bear the unmistakable signature of recursive thinking. Natural selection is a recursive optimization process, where successful genetic variations are repeatedly tested, retained, and amplified. Each generation calls back to its predecessors, yet introduces subtle variations—a mathematical dance of inheritance and innovation.

Neuroscience offers perhaps the most profound illustration of recursion's biological impact. Neural networks in the brain operate through recursive computational principles, with neurons forming complex, self-referential circuits that continuously modulate and reinterpret incoming information. Consciousness itself might be understood as an emergent property of these recursive neural interactions.

The boundary between mathematical abstraction and biological reality becomes quite porous in this recursive landscape. What appears at first as a dry mathematical concept reveals itself as the very rhythm of life—a fundamental principle of emergence, complexity, and continuous self-generation.

One of the most notable scholars who has extensively explored the role of recursion in biology is Stuart Kauffman, a theoretical biologist and complex systems researcher. Kauffman has written extensively about self-organization, complexity, and recursive processes in biological systems. In his works like The Origins of Order: Self-Organization and Selection in Evolution (1993) and At Home in the Universe (1995), he developed groundbreaking ideas about how recursive principles underpin biological complexity. Specifically, Kauffman explored how recursive networks of molecular interactions can spontaneously self-organize, demonstrating that certain complex biological systems have inherent tendencies toward order that go beyond traditional Darwinian natural selection. His work suggests that recursive processes are fundamental to understanding how biological complexity emerges.


Complex Systems
In understanding complex systems, recursion is a fundamental strategy for solving problems by breaking them into smaller, more manageable pieces. In this case, recursion isn’t just a mathematical abstraction, it’s how humans naturally think. When you face a complicated task, you instinctively break it down into smaller steps. Recursion is that cognitive process formalized.

Linguists such as Noam Chomsky have argued (not without controversy) that recursive thinking is fundamental to human language, and thus potentially to consciousness itself. Our ability to nest ideas within ideas and to think abstractly might fundamentally rely on recursive cognitive mechanisms.

At a philosophical level, recursion represents a powerful way of understanding emergence—complex behaviors spontaneously arising from simple, self-referential rules.

More practically, medical decision making often employs recursive strategies. For example, the medical diagnosis process can be reduced to this common recursive pattern:
1. Initial Call: current symptom assessment
2. Recursive Exploration: differential diagnosis decision trees that branch and refine diagnostic hypotheses
3. Base Case: stop when potential diagnosis space is saturated
4. Aggregation: combine possibilities to make base treatment decision


Conclusion
As these notes point out, recursion is a profound conceptual thread that weaves through the fabric of mathematics, computing, biology, and complex systems, revealing a fundamental principle of pattern and self-reference that transcends disciplinary boundaries. Far from being a mere technical mechanism, recursion represents a deep epistemological approach to understanding complexity through elegant, self-similar structures.

Ultimately, recursion teaches us a fundamental lesson about knowledge and complexity: understanding emerges not through linear reduction, but through recognizing the interconnectedness of systems. It reminds us that complexity is not chaos, but a sophisticated interplay of simple rules generating remarkable diversity and emergence. By embracing recursive thinking, we open ourselves to a more nuanced, holistic understanding of the world. Future interdisciplinary research has the opportunity to focus on better models that can even more precisely articulate these recursive mechanisms across a range of scales from the molecular through to the systemic.

– Geoffrey W. Smith



First Five
First Five is our curated list of articles, studies, and publications for the month.

1/ The End of Science’s Peacetime
“Initially, it was hard to separate truth from rumor, but enough has happened in recent days that the worry cannot be branded as paranoia. A string of events affecting federal agencies and programs, set in motion since the presidential inauguration, has sent the nation into a frenzy of uncertainty. The consternation is justified. Although science is hardly the only or most consequential sector affected, it was an early target for a reason: Science is a battleground in the modern war against ideas.” Read more here >

2/ Will all our drugs come from China?
“It was not too long ago that China’s main contribution to the pharma industry was the raw chemical material, the active pharmaceutical ingredients (APIs), that went into finished drug products discovered, designed, and developed by Western (and Japanese) innovators. However, if you’ve been paying attention, you’ll have noticed the steady rise in Chinese companies as a source of genuinely new drugs (i.e. drug discovery). Chinese companies are now responsible for about a quarter of new trial starts—more than Europe. It is in early-stage (phase I), oncology, and cell and gene therapy where Chinese companies are particularly active.” Read more here >

3/ Machines of Loving Grace
A little late getting to this essay by Dario Amodei, the CEO and founder of Anthropic, but it is well worth the time to read his view of how AI could transform the world for the better. Read more here >
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4/ Megatrends
“The Global 50 report shares the Dubai Future Foundation’s view of the future and 50 opportunities for future growth, prosperity and well-being. Some opportunities may be in their early days of exploration, some require reflection, and some feel very far away.” Read more here >

5/ Questions, Questions, Questions
“Today’s global crises—from climate change to inequality—have demonstrated the need for a broader conceptual transformation in how to approach societal issues. Focusing on the questions can transform our understanding of today’s problems and unlock new discoveries and innovations that make a meaningful difference. Yet, how decision-makers go about asking questions remains an underexplored topic. Much of our recent work has focused on advancing a new science of questions that uses participatory approaches to define and prioritize the questions that matter most.” Read more here >



Did you Know?
In this section of our newsletter, we seek to demystify common terms and practices in our work as investors.

Capital Calls / Distributions
In the realm of venture capital, a capital call represents a request to investors for additional capital as part of their commitments to the fund. These calls are made by the fund manager when there is a need for more capital to seize investment opportunities or meet operational requirements. Investors, known as limited partners, are obligated to respond to these capital calls according to the terms of their agreements with the fund. Capital calls ensure that the fund has the financial resources necessary for its investments and operations.

On the other side, a distribution signifies the return of funds from the investments made by the fund to its investors. These returns are distributed to the investors based on their proportionate share of the profits earned by the fund. Distributions can take the form of cash, securities, or other assets and occur for various reasons, such as the successful sale of portfolio companies or the receipt of dividend payments. Distributions are a crucial part of the investment cycle, allowing investors to realize the benefits of their investments in the fund's portfolio companies.

– Haiming Chen & Dylan Henderson

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