The Reasonable Ineffectiveness of Mathematics in the Biological Sciences
Exploring why mathematics, so potent in physics, often stumbles when confronted with the messy complexity of living systems
Of Equations and Organisms: When Math Meets Life
In 1960, physicist Eugene Wigner penned a famous essay marveling at "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." He was primarily astonished by how perfectly mathematics described the physical world, from planetary orbits to subatomic particles. Yet, as the renowned mathematician I. M. Gelfand later observed, there's "only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology."9
This article explores this very paradox—why does mathematics, so potent in physics, often stumble when confronted with the messy complexity of living systems? The answer, as we're discovering, isn't that biology defies mathematics altogether, but that it demands a different kind of mathematics. What appears to be ineffectiveness is actually a reasonable response to the magnificent complexity of life itself.
Physics
Simplified systems with universal laws and identical particles
Biology
Complex systems where variation is the rule rather than the exception
The Great Divide: Understanding Mathematics' Cautious Relationship with Biology
The Nature of the Beast
Mathematics and biology have historically occupied different intellectual territories. Physics often deals with simplified systems and universal laws that apply to identical particles under consistent forces. Biology, in contrast, confronts us with complex systems where variation is the rule rather than the exception9 . Consider what makes biology uniquely challenging:
Inherent Variability
No two cells, no two organisms, and no two ecosystems are truly identical. This natural variation resists the precise, repeatable predictions that work in physics.
Multiple Scales
Biological processes operate simultaneously across scales from molecular to ecological, with each level influencing the others in ways that are difficult to disentangle.
Historical Contingency
Living systems carry the baggage of evolutionary history, constraining their forms and functions in ways that don't follow from first principles.
Early Attempts at Mathematical Biology
This isn't to say that mathematics has been completely absent from biology. Some early successes include:
Population Dynamics
Fibonacci described a growing rabbit population in the 13th century with his famous sequence, and Thomas Malthus later formulated exponential growth models in 17891 .
Genetic Foundations
Ronald Fisher developed fundamental statistical methods like analysis of variance through his work on quantitative genetics in the early 20th century1 .
Predator-Prey Models
The Lotka-Volterra equations, developed in the early 1900s, described oscillating populations of predators and their prey1 .
These approaches worked reasonably well for simplified biological scenarios but often broke down when confronted with the full complexity of real ecosystems or organisms.
A Case Study in Complexity: The Orchid and the Fungus
Recent research from Kobe University provides a perfect example of why biology resists straightforward mathematical description. Scientists discovered that orchids rely on specific wood-decaying fungi to germinate, with seedlings only growing near rotting logs and forming precise fungal partnerships that mirror those in adult orchids4 .
The Orchid-Fungus Symbiosis
A complex biological relationship that challenges simple mathematical modeling.
The Experimental Challenge
Let's examine how researchers untangled this relationship and why it would be difficult to capture in a simple mathematical model:
Methodology:
- Field Observation: Researchers documented orchid growth patterns in relation to decaying wood
- Fungal Identification: Through genetic analysis, they identified specific fungal species
- Partnership Mapping: They traced fungal relationships across orchid life stages
- Carbon Tracking: Scientists verified carbon transfer from logs to orchids via fungi
Results and Analysis
The findings revealed a multi-layered biological relationship that would challenge any simple mathematical formulation:
| Biological Component | Finding | Mathematical Challenge |
|---|---|---|
| Spatial Requirement | Seedlings only grow near deadwood | Requires spatial modeling with precise distance parameters |
| Nutritional Dependence | Orchids consume carbon from rotting logs via fungi | Involves cross-species energy transfer difficult to quantify |
| Specificity | Partnership with specific fungal species | Binary or categorical variables rather than continuous functions |
| Life Stage Variation | Fungal partnerships differ between seedlings and adults | Requires different equations for different life stages |
Experimental Results: Orchid Germination Success Rates
The data show why this biological system resists simple mathematical characterization. Instead of clean continuous functions, we find conditional relationships, specific partnerships, and context-dependent outcomes.
The Scientist's Toolkit: Research Reagent Solutions
Modern biological research employs sophisticated tools to tackle this complexity. Here are some key materials and approaches used in studies like the orchid-fungus investigation:
| Reagent/Tool | Function | Application in Orchid Study |
|---|---|---|
| Polymerase Chain Reaction (PCR) kits | Amplifies specific DNA sequences | Identified specific fungal species in the partnerships |
| Stable Isotope Labeling | Tracks element flow through systems | Confirmed carbon transfer from logs to orchids via fungi |
| Fluorescent in situ Hybridization | Visualizes specific genetic material | Mapped fungal distribution in orchid root systems |
| Axenic Culture Systems | Maintains organisms in fungus-free conditions | Established control conditions for germination experiments |
| Metagenomic Sequencing | Analyzes genetic material from environmental samples | Characterized complete fungal communities in rotting logs |
Genetic Analysis Tools
Modern biology relies heavily on genetic tools to identify species and trace relationships that are invisible to traditional observation methods.
Imaging Technologies
Advanced imaging allows researchers to visualize biological processes at multiple scales, from molecular interactions to ecosystem dynamics.
The Shifting Frontier: Mathematics Finds Its Footing in Biology
Despite these challenges, the relationship between mathematics and biology is evolving. As Leonard Adleman, the computer scientist who pioneered DNA computing, noted: "Sciences reach a point where they become mathematized," starting at the fringes but eventually reaching the field's central issues9 . For biology, that time may now be arriving due to several key developments:
New Mathematical Approaches for Biological Complexity
Rather than forcing biology into physical mathematics, researchers are developing new approaches better suited to life's complexity:
Relational Biology
Instead of focusing on concrete biological structures, this approach, pioneered by Robert Rosen, develops abstract models of organization like (M,R)-systems that represent relationships between components rather than their physical properties1 .
Stochastic Processes
Unlike deterministic physics equations, these mathematical tools account for randomness and probability, better reflecting biological variability1 .
Network Theory
This approach maps relationships between biological components—whether neurons, species, or genes—as interconnected networks that can be analyzed mathematically1 .
Complex Systems Biology
An elaboration of systems biology that uses molecular set theory, relational biology, and algebraic biology to understand supercomplex life processes1 .
The Genomics Revolution
The explosion of data from genomics has created both a need and opportunity for mathematical approaches. As one source notes, "Biology was now the study of information stored in DNA – strings of four letters: A, T, G, and C and the transformations that information undergoes in the cell. There was mathematics here!"9 This recognition of biological information as a mathematical entity has opened new avenues for quantitative analysis.
Computational Power
The increase in computing power has enabled simulations and calculations that were previously impossible, allowing researchers to model complex biological systems without relying on simplified equations that strip away important details1 .
Conclusion: Embracing Reasonable Effectiveness
The apparent "ineffectiveness" of mathematics in biology is indeed reasonable when we consider the nature of living systems. Biology operates with different constraints than physics—historical, contingent, and layered with complexity that resists reduction to simple equations.
Symbiotic Relationship
Biology poses challenging problems for mathematics, while mathematics provides fresh insights into biological complexity.
Tailored Approaches
New mathematical tools are being developed specifically for biological complexity rather than forcing biology into physical models.
Quantitative Transformation
Biology has turned into a quantitative science requiring mathematical expression tailored to life's unique melody.
What appears to be mathematics' stuttering progress in biology is actually a careful, appropriate integration of quantitative thinking into the science of life. The relationship is becoming increasingly symbiotic, with biology posing challenging new problems for mathematics, and mathematics providing fresh insights into biological complexity. In the end, this "reasonable effectiveness" may prove more fruitful than physics' seemingly magical mathematical correspondence, as it respects the rich complexity of life while gradually uncovering its underlying patterns.
As a new textbook from MIT Press aptly states, "Biology has turned into a quantitative science. The core problems in the life sciences today involve complex systems that require mathematical expression"5 —but that expression must be tailored to hear life's unique melody.