Exploring Symbolic Reasoning: A Multi-Domain Approach

Introduction

Symbolic reasoning provides a powerful lens for understanding concepts across mathematics, science, philosophy, art, and more. By using formal notation, we can distill complex ideas into precise, structured relationships, making it easier to analyze patterns, resolve contradictions, and connect abstract principles across disciplines.

This document presents a set of symbolic expressions, each paired with an accessible explanation, real-world metaphor, and practical applications. By working through these examples, readers will gain a stronger intuition for symbolic thinking, enhancing their ability to approach problems from multiple perspectives.

Logic & Mathematics

Symbolic Expression 1: Transitivity of Implication

Symbolic Expression

(A → B) ∧ (B → C) ⊢ A → C

Concept Breakdown

This expression represents the transitivity of implication, a fundamental rule in logic and mathematics. It states that:

If A implies B (A → B), and B implies C (B → C),
Then A must also imply C (A → C).

This principle is the foundation of syllogistic reasoning—allowing logical conclusions to be drawn through intermediate steps. It underlies deductive reasoning and ensures that logic flows coherently from premise to conclusion.

Long-Form Metaphor

A river is to the valley as knowledge is to the student—if the river flows from the mountains to the valley, and the valley leads to the sea, then the river must eventually reach the sea.

Potential Audience

Mathematicians, logicians, computer scientists.
Philosophers and formal reasoning scholars.
Engineers designing logical circuits, AI decision models, or formal proofs.
Lawyers and debaters who use structured arguments to reach conclusions.

Use Cases

When proving mathematical theorems: This principle allows complex proofs to be broken into smaller implications.
In AI decision-making models: Ensures that multi-step reasoning can be linked through valid transitions.
In legal and philosophical debates: Helps construct airtight logical arguments by ensuring that premises lead to conclusions.
In programming and software development: Guarantees that logical dependencies are handled in order.

Symbolic Expression 2: Union vs. Intersection Relations

Symbolic Expression

A ∪ B ⊇ A ∩ B

Concept Breakdown

This expression states that the union of two sets is always at least as large as their intersection.

The union (∪) contains all elements in either set.
The intersection (∩) contains only elements that belong to both sets.
Since the intersection is a subset of both sets, it follows that the union, which includes everything in both, must always be at least as large.

This fundamental concept is crucial in set theory, probability, and data organization, ensuring that overlapping information never exceeds the total range of elements.

Long-Form Metaphor

A family reunion is to cousins as a friendship circle is to shared interests—everyone at the reunion belongs to at least one branch of the family, but only some are both cousins and best friends.

Potential Audience

Set theorists, data scientists, logicians.
Engineers and programmers working with database queries, machine learning, or data merging.
Statisticians analyzing probabilistic outcomes.
Sociologists and market researchers studying overlapping group behaviors.

Use Cases

In database theory: SQL queries that merge datasets follow this fundamental rule.
In probability and statistics: Helps in computing joint probabilities and understanding inclusion-exclusion principles.
In systems thinking: Understanding how overlapping influences (e.g., teams, departments) contribute to collective outcomes.
In information retrieval: Optimizing search results when dealing with overlapping datasets or categorization problems.

Science & Empirical Inquiry

Symbolic Expression 1: Entropy Always Increases

Symbolic Expression

ΔS ≥ 0

Concept Breakdown

This expression states the Second Law of Thermodynamics, which asserts that in any isolated system, entropy (S) never decreases—it either stays the same (in ideal cases) or increases over time.

Entropy represents disorder or the number of ways a system can be arranged. As energy is transferred or transformed, some is inevitably lost to irreversible processes, increasing disorder in the universe.

Long-Form Metaphor

A sandcastle is to time as order is to the universe—no matter how carefully built, the waves will eventually wash it away.

Potential Audience

Physicists, chemists, and thermodynamic engineers.
Data scientists and information theorists studying entropy in data compression or loss.
Philosophers and metaphysicists contemplating irreversibility and the nature of time.

Use Cases

Understanding heat transfer and efficiency: Engineers design systems that minimize wasted energy due to entropy.
Explaining why time moves forward: The increase in entropy provides an arrow of time.
In machine learning and AI: Entropy is used in algorithms such as decision trees and information gain.

Symbolic Expression 2: Observed Reality is Shaped by Interpretation

Symbolic Expression

(Observation ∩ Reality) ⊆ Interpretation

Concept Breakdown

This expression states that what we observe is always filtered through interpretation.

Observation (perception of reality) is never perfect—it is constrained by limitations in measurement and cognition.
Reality (objective state of the universe) exists independently, but what we call “real” is shaped by how we interpret our observations.

Long-Form Metaphor

A telescope is to a galaxy as a human mind is to truth—what you see depends on how powerful your lens is, but even the clearest lens only reveals part of the whole.

Potential Audience

Scientists dealing with quantum mechanics, uncertainty, and experimental limitations.
Psychologists and neuroscientists exploring how the brain processes perception.
Philosophers questioning the nature of reality and subjective experience.

Use Cases

In physics: Measurement affects what is observed (e.g., Heisenberg uncertainty principle).
In psychology: Our biases influence what we perceive and remember.
In AI and data science: Machine learning models must be trained carefully to avoid biased interpretations of data.

Philosophy & Metaphysics

Symbolic Expression 1: Truth Exists Within Perspective

Symbolic Expression

Truth ∩ Perspective ≠ ∅

Concept Breakdown

This expression states that truth and perspective are not disjoint sets—meaning that every perspective contains at least some aspect of truth, even if it is partial or distorted.

Absolute truth may exist, but it is always accessed through a perspective.
Perspectives can overlap, meaning different viewpoints may hold shared elements of truth.

Long-Form Metaphor

A mountain is to a hiker as reality is to an observer—the side you see depends on where you stand.

Potential Audience

Philosophers and epistemologists exploring subjectivity vs. objectivity.
Scientists and researchers analyzing observer effects and experimental bias.
Social theorists discussing how different cultures interpret reality differently.

Use Cases

In debates and discussions: Acknowledging that different viewpoints may contain partial truths.
In scientific inquiry: Accounting for observer bias in measurements.
In artificial intelligence: Training AI models to recognize multiple perspectives rather than a single “true” label.

Symbolic Expression 2: Necessity vs. Contingency Contradiction

Symbolic Expression

(Necessity → Existence) ⊥ (Contingency → Non-Existence)

Concept Breakdown

This expression explores a paradox in metaphysics:

If something is necessary, it must exist (Necessity → Existence).
If something is contingent, it may not exist (Contingency → Non-Existence).
Under certain logical conditions, these statements can contradict, meaning some things must necessarily exist, while others are purely conditional.

This is deeply tied to ontological arguments for existence, modal logic, and the nature of reality.

Long-Form Metaphor

A lighthouse is to the ocean as existence is to logic—some things must stand firm, while others are shaped by the waves of contingency.

Potential Audience

Metaphysicists studying necessity, existence, and possibility.
Theologians and philosophers analyzing arguments for the existence of necessary beings.
Logicians working with modal logic and paradox resolution.

Use Cases

In logic and philosophy: Understanding how modal logic governs necessity and contingency.
In theology: Arguments surrounding necessary beings and existential certainty.
In AI and decision systems: Designing logic frameworks that differentiate between necessary and optional actions.

Systems & Engineering

Symbolic Expression 1: Structured Patterns Emerge from Chaos

Symbolic Expression

Order ∩ Chaos ≠ ∅

Concept Breakdown

This expression states that order and chaos are not entirely separate—within chaos, structured patterns can emerge. Many systems that appear chaotic at first glance contain underlying order.

In nature, chaotic systems like weather, turbulence, and ecosystems exhibit structured behavior.
In engineering, complex adaptive systems function through patterns emerging from seemingly random interactions.

Long-Form Metaphor

A whirlpool is to a river as order is to chaos—it may look unpredictable, but within its motion lies a stable pattern.

Potential Audience

Engineers designing self-regulating systems and adaptive algorithms.
Scientists studying chaos theory and nonlinear dynamics.
AI researchers exploring emergent intelligence and self-organizing models.

Use Cases

In AI and machine learning: Detecting patterns in seemingly random data.
In physics and weather modeling: Understanding turbulence and complex systems.
In cybersecurity: Recognizing order in chaotic data streams to detect anomalies.

Symbolic Expression 2: Stability and Adaptation Create Resilience

Symbolic Expression

(Stability ∧ Adaptation) ⊆ Resilience

Concept Breakdown

This expression states that resilience is not just stability (resistance to change) or adaptation (ability to change) but a combination of both.

Systems that are too rigid may break under stress.
Systems that are too fluid may lack consistency.
True resilience balances stability and adaptability to withstand external forces while remaining functional.

Long-Form Metaphor

A tree is to the wind as resilience is to stability and adaptation—too rigid, and it snaps; too weak, and it bends completely. The strongest trees sway but stand.

Potential Audience

Engineers designing robust systems and fail-safe mechanisms.
Business leaders developing organizations that survive market changes.
Biologists studying how ecosystems maintain stability while evolving.

Use Cases

In infrastructure design: Creating buildings that withstand earthquakes by flexing instead of breaking.
In AI and robotics: Designing adaptable learning models that maintain core functionality while evolving.
In business strategy: Building companies that survive disruptions through a balance of structure and change.

Creativity & Art

Symbolic Expression 1: Meaning is Assigned, Not Inherent

Symbolic Expression

Meaning ∈ Perception

Concept Breakdown

This expression states that meaning is not an intrinsic property of an object or concept—it arises from perception.

The same piece of art, music, or literature can be interpreted differently based on the viewer’s perspective, emotions, and context.
Meaning is therefore not absolute but relational, existing within the interaction between the observer and the observed.

Long-Form Metaphor

A mirror is to a face as a poem is to the reader—it reflects what is already there, but in a way you’ve never seen before.

Potential Audience

Artists, musicians, and writers exploring subjective interpretation in their work.
Literary critics, philosophers, and cultural analysts studying semiotics and perception theory.
UX/UI designers focusing on user perception and experience.

Use Cases

In literature and film analysis: Understanding how stories take on different meanings for different audiences.
In branding and marketing: Recognizing that perception drives interpretation of symbols and messages.
In AI-generated art: Training models to understand context-driven aesthetic interpretation.

Symbolic Expression 2: Creativity Thrives Under Constraints

Symbolic Expression

(Constraints → Innovation) ⊢ Art

Concept Breakdown

This expression suggests that art and creativity do not emerge in boundless freedom, but through limitations that force innovative solutions.

Constraints (material, time, rules) lead to creative breakthroughs by requiring artists to work within structured boundaries.
Many great artistic movements arose as reactions to or within specific restrictions.

Long-Form Metaphor

A haiku is to poetry as a chisel is to sculpture—by limiting excess, it forces the artist to carve something precise and profound.

Potential Audience

Writers, musicians, and visual artists using constraints as a tool for creative expansion.
Game designers, filmmakers, and composers working within technological or budgetary limits.
Software developers using minimalist programming to solve complex problems.

Use Cases

In music composition: How limited scales or constraints (e.g., the 12-bar blues) lead to creative expression.
In architecture and design: Using space limitations to drive innovative structures.
In AI-generated content: Designing models that work within constraints to enhance creative output.

Human Experience & Psychology

Symbolic Expression 1: Identity is Shaped by Perception

Symbolic Expression

Self ⊆ Perception

Concept Breakdown

This expression states that our sense of self is contained within our perception—our identity is shaped by how we perceive ourselves and how others perceive us.

Perception is both internal and external—how we see ourselves and how we believe others see us.
Since perception is fluid and evolving, so too is our identity.
Identity is not separate from perception but rather a subset of it.

Long-Form Metaphor

A reflection is to a mirror as identity is to perception—it changes depending on the angle and lighting, but there is always something real behind it.

Potential Audience

Psychologists studying self-perception and identity formation.
Neuroscientists analyzing how the brain constructs self-awareness.
Philosophers discussing the nature of selfhood and consciousness.

Use Cases

In therapy and counseling: Helping individuals understand how their perceptions shape their self-identity.
In AI and virtual reality: Designing avatars or AI models that adapt based on user perception.
In social sciences: Studying how media and social influence shape identity.

Symbolic Expression 2: Memories and Emotions Shape Interpretation

Symbolic Expression

(Emotion ∩ Memory) → Interpretation

Concept Breakdown

This expression states that our emotions and memories together influence how we interpret the world.

A neutral event may be interpreted positively or negatively depending on past experiences.
Emotion acts as a filter, reinforcing how we remember and what meaning we assign to experiences.
This explains cognitive biases, emotional responses, and why people recall the same events differently.

Long-Form Metaphor

A lens is to a camera as emotion is to memory—it colors what we see and how we remember it.

Potential Audience

Cognitive scientists and psychologists studying memory and emotional processing.
AI researchers working on sentiment analysis and natural language understanding.
Historians and sociologists examining how collective memory shapes cultures and beliefs.

Use Cases

In therapy and trauma research: Understanding how past experiences shape present emotions.
In artificial intelligence: Training AI to recognize emotional tone in human interactions.
In marketing and storytelling: Crafting messages that resonate emotionally by leveraging past associations.

Social & Cultural Thought

Symbolic Expression 1: The Self is Shaped by Society

Symbolic Expression

Individual ⊆ Collective

Concept Breakdown

This expression states that individual identity is contained within and influenced by the collective.

No individual exists in isolation; personal beliefs, behaviors, and values are shaped by culture, society, and shared experiences.
The collective defines norms, and individuals interact with or challenge these norms in shaping their own identity.

Long-Form Metaphor

A single thread is to a tapestry as an individual is to society—alone, it is just a strand, but woven together, it creates something greater than itself.

Potential Audience

Sociologists and anthropologists studying cultural influence on identity.
Psychologists analyzing social conditioning and group dynamics.
Political scientists examining how individual agency interacts with societal structures.

Use Cases

In social science research: Understanding how cultural norms influence behavior.
In education and policy-making: Creating environments that balance individual expression with societal cohesion.
In AI and social media design: Building systems that account for social influence on user behavior.

Symbolic Expression 2: Progress is Found at the Intersection of Tradition and Adaptation

Symbolic Expression

(Tradition ∩ Adaptation) → Progress

Concept Breakdown

This expression highlights that true progress does not come from abandoning tradition entirely, nor from rigidly clinging to it, but from integrating adaptation within it.

Societies evolve by balancing historical wisdom with new innovations.
Progress requires preserving valuable traditions while discarding outdated constraints.

Long-Form Metaphor

A compass is to a traveler as tradition is to progress—it provides direction, but adaptation is needed to navigate new terrain.

Potential Audience

Historians and cultural theorists studying civilizational evolution.
Business leaders and policymakers managing organizational change.
AI ethicists considering how machine learning models balance past data with future adaptation.

Use Cases

In governance and reform: Crafting policies that honor history while addressing modern needs.
In education: Teaching students to respect past knowledge while fostering innovation.
In technology development: Designing AI and systems that adapt without discarding foundational ethics.

AI & Computation

Symbolic Expression 1: Data is Not Wisdom

Symbolic Expression

Knowledge ≠ Understanding

Concept Breakdown

This expression highlights a fundamental limitation of AI and computational systems:

Knowledge refers to stored data—facts, figures, and structured information.
Understanding requires context, interpretation, and the ability to derive meaning.
An AI can hold massive amounts of knowledge but lacks intrinsic understanding unless given interpretative models.

Long-Form Metaphor

A dictionary is to language as a dataset is to intelligence—it contains all the words, but it cannot write poetry.

Potential Audience

AI researchers working on context-aware models and reasoning frameworks.
Cognitive scientists exploring human vs. machine intelligence.
Developers building knowledge-based systems that require deeper inference.

Use Cases

In AI design: Differentiating between data retrieval and conceptual reasoning.
In education: Teaching students that memorization does not equal comprehension.
In decision-making systems: Ensuring AI models incorporate contextual analysis rather than pure data processing.

Symbolic Expression 2: Intelligence Depends on Context

Symbolic Expression

(Learning ∩ Context) ⊆ Intelligence

Concept Breakdown

This expression states that true intelligence emerges at the intersection of learning and contextual awareness:

Learning involves acquiring new information or skills.
Context determines how that information is applied meaningfully.
Without context, intelligence becomes a detached system of facts without relevance or adaptability.

Long-Form Metaphor

A map is to a traveler as context is to intelligence—it tells you where things are, but only experience tells you which road to take.

Potential Audience

AI developers designing context-aware reasoning systems.
Educators refining adaptive learning methodologies.
Philosophers debating what defines intelligence beyond mere data processing.

Use Cases

In AI and machine learning: Developing models that adjust based on environmental cues rather than static logic.
In natural language processing (NLP): Creating AI that understands subtext, cultural context, and implied meaning.
In business intelligence: Ensuring decision-making systems adapt to real-world conditions rather than relying on fixed models.

Temporal Logic & Causality

Symbolic Expression 1: The Future is Constrained by the Past

Symbolic Expression

Future ∈ Past

Concept Breakdown

This expression suggests that the future is not independent—it is shaped and constrained by the past:

The present state of any system is the result of past conditions and decisions.
While the future contains possibilities, it is not entirely separate from its historical trajectory.
This principle underlies determinism, probability modeling, and predictive analytics.

Long-Form Metaphor

A planted seed is to a tree as the past is to the future—though it may grow in different directions, it cannot become something other than what it was set up to be.

Potential Audience

Physicists and philosophers debating free will vs. determinism.
Data scientists working with time-series analysis and predictive models.
Historians and sociologists examining how past events shape cultural and political futures.

Use Cases

In machine learning and forecasting: Developing AI models that use past data to predict future trends.
In scientific modeling: Understanding how initial conditions define future outcomes (e.g., chaos theory).
In legal and ethical frameworks: Examining how precedent informs future decisions.

Symbolic Expression 2: Feedback Loops Arise from Bidirectional Causality

Symbolic Expression

Concept Breakdown

(Cause → Effect) ∧ (Effect → Cause) ⊢ Feedback

This expression describes feedback loops, where cause and effect continuously influence each other:

In many systems, an effect can reinforce or modify its own cause.
This leads to self-sustaining cycles, seen in biology, economics, AI, and thermodynamics.
Some feedback loops stabilize systems (negative feedback), while others amplify change (positive feedback).

Long-Form Metaphor

A thermostat is to temperature as feedback is to causality—it senses the outcome and adjusts the cause to maintain balance.

Potential Audience

Engineers designing cybernetic and control systems.
AI researchers working on reinforcement learning and self-adaptive models.
Ecologists and economists studying supply-demand cycles and population dynamics.

Use Cases

In AI and neural networks: Implementing reinforcement learning based on past outcomes.
In environmental science: Understanding climate change feedback loops.
In business strategy: Using market feedback to refine products and services.

Emergence & Evolution

Symbolic Expression 1: Complexity is Built from Simplicity

Symbolic Expression

Complexity ⊆ Simplicity

Concept Breakdown

This expression states that all complex systems are ultimately composed of simpler elements.

What appears intricate and sophisticated is often a layered structure of simple interactions.
Complex organisms, languages, economies, and even consciousness emerge from simple, fundamental rules.
In mathematics, fractals and cellular automata demonstrate how simple rules create infinitely complex patterns.

Long-Form Metaphor

“A tree is to a forest as a neuron is to thought—individually simple, yet together forming something vastly more complex.”

Potential Audience

Scientists studying emergence in natural and artificial systems.
AI researchers analyzing complex behavior from simple algorithms.
Engineers and designers seeking minimalist approaches to complex problems.

Use Cases

In AI & computation: Understanding how neural networks build intelligence from simple activation functions.
In biology: Exploring how single-cell organisms evolved into complex life forms.
In philosophy: Examining how simple ideas evolve into complex ideologies.

Symbolic Expression 2: Evolution is Shaped by Mutation and Selection

Symbolic Expression

(Mutation ∪ Selection) ⊆ Evolution

Concept Breakdown

This expression highlights that evolution does not happen randomly—it is the product of both mutation (variation) and selection (filtering for fitness).

Mutation introduces new traits, while selection determines which traits persist.
Evolution is not purely random, nor is it purely deterministic—it is a balance of change and constraint.

Long-Form Metaphor

“A sculptor is to stone as natural selection is to life—shaping what already exists, but never creating from nothing.”

Potential Audience

Biologists studying genetic adaptation and natural selection.
AI developers applying evolutionary algorithms to optimize models.
Economists and sociologists analyzing how markets and cultures evolve over time.

Use Cases

In genetics: Explaining how random DNA mutations, combined with selective pressure, shape species over generations.
In artificial intelligence: Using evolutionary programming to optimize AI architectures.
In business and culture: Understanding how innovation and adaptation drive industry and societal progress.

Cognitive Science

Symbolic Expression 1: What We See is Shaped by What We Expect

Symbolic Expression

Perception ∈ Expectation

Concept Breakdown

This expression states that what we perceive is influenced by what we anticipate.

The brain does not passively receive information—it actively predicts and interprets stimuli based on prior experience.
Optical illusions, confirmation bias, and pattern recognition are all examples of how expectation frames our perception.
In AI, predictive models work similarly by using past data to anticipate future inputs.

Long-Form Metaphor

“A map is to a traveler as expectation is to perception—it guides interpretation, but does not always match reality.”

Potential Audience

Neuroscientists studying predictive coding in cognition.
Psychologists analyzing cognitive biases and perception errors.
AI developers working on predictive models and pattern recognition.

Use Cases

In psychology: Understanding why people see what they expect to see (e.g., priming effects).
In AI & machine learning: Enhancing predictive algorithms by mimicking human perception processes.
In design & marketing: Creating visuals or messages that align with user expectations to improve engagement.

Symbolic Expression 2: Focus Determines Awareness

Symbolic Expression

(Attention ∩ Salience) → Awareness

Concept Breakdown

This expression states that awareness is created when attention is directed toward something that stands out as important (salient).

Not all information enters conscious awareness—only what the brain deems relevant at any moment.
This explains why we miss obvious details when distracted (inattentional blindness).
In AI, attention mechanisms (like in Transformers) determine which data points are most relevant for processing.

Long-Form Metaphor

“A flashlight is to a dark room as attention is to awareness—it reveals what it focuses on, while leaving everything else unseen.”

Potential Audience

Cognitive scientists studying attention and working memory.
UX/UI designers optimizing interfaces for user focus.
AI researchers improving attention mechanisms in deep learning.

Use Cases

In neuroscience: Exploring how selective attention shapes human experience.
In AI development: Using attention layers in neural networks to improve efficiency.
In productivity science: Applying focus-enhancing techniques to boost cognitive performance.

Ethics & Decision Theory

Symbolic Expression 1: Decisions are Shaped by Context

Symbolic Expression

Choice ⊆ Context

Concept Breakdown

This expression states that all choices exist within a broader context, which constrains and influences them.

Decisions are rarely made in isolation—they are shaped by environment, culture, available knowledge, and personal history.
Free will is not absolute—it operates within the limits of available options and constraints.
In AI, decision-making models must factor in contextual dependencies for ethical and optimized outputs.

Long-Form Metaphor

“A river is to a valley as a decision is to its context—free to flow, but always shaped by the terrain it moves through.”

Potential Audience

Ethicists studying free will, determinism, and moral responsibility.
AI researchers working on context-aware decision-making models.
Economists and policymakers designing choice architecture for decision-making.

Use Cases

In behavioral psychology: Understanding how external factors influence personal choices.
In AI ethics: Ensuring that machine learning models recognize the impact of context on ethical decisions.
In economics & game theory: Designing incentive structures that guide decision-making within constraints.

Symbolic Expression 2: Power Without Responsibility is Contradictory

Symbolic Expression

(Responsibility → Power) ⊥ (Power → No Responsibility)

Concept Breakdown

This expression states that true power inherently carries responsibility, and claiming power without responsibility is logically contradictory.

Authority without accountability leads to corruption and collapse.
Ethical leadership requires balancing power with moral and social obligations.
In AI, autonomous systems must be designed with accountability mechanisms to ensure ethical deployment.

Long-Form Metaphor

“A captain is to a ship as responsibility is to power—without one, the other is directionless and doomed to drift.”

Potential Audience

Political theorists discussing ethical governance and authority.
AI ethicists working on algorithmic responsibility in decision-making.
Business leaders exploring ethical leadership and corporate responsibility.

Use Cases

In politics & law: Establishing checks and balances in governance.
In AI development: Designing explainable AI (XAI) to ensure algorithmic accountability.
In organizational ethics: Creating corporate policies that tie power to social responsibility.

Conclusion

Symbolic reasoning is not just a tool for mathematics and logic—it is a framework for structured thought that applies to nearly every domain of human understanding. Whether exploring natural laws, emergent behaviors, ethical dilemmas, or artistic creativity, symbolic expressions help us clarify relationships and reveal underlying structures.

By engaging with symbolic reasoning across different fields, we bridge disciplines, foster deeper insights, and create a more integrated understanding of complex systems. We encourage readers to explore these expressions, experiment with new symbolic formulations, and apply them in their own fields of interest.