Formal Science
At its core, formal science is the study of formal systems - abstract structures built on clearly defined rules and relationships. Unlike natural sciences that study the physical world, formal sciences study the logical structures and patterns that can exist independent of physical reality.
The fundamental building blocks of formal science are:
- Axioms: These are the basic assumptions or self-evident truths that serve as starting points. They are accepted without proof and form the foundation upon which formal systems are built. For example, in mathematics, one axiom is that if A equals B, and B equals C, then A must equal C.
- Logic: This is the systematic method of reasoning that allows us to derive valid conclusions from given premises. Formal logic provides the rules for how we can manipulate and combine statements while preserving truth.
- Abstraction: This is the process of identifying and isolating essential patterns while removing specific physical details. It allows us to study the pure form of relationships and structures.
From these foundational elements, formal science develops through:
- Deductive Reasoning: Starting with axioms and using logic, we can derive necessary conclusions. Each step follows necessarily from the previous ones, building a chain of certain knowledge. This differs from the inductive reasoning common in natural sciences, where we make general conclusions from specific observations.
- Formal Languages: These are precise symbolic systems with strict rules for combining and manipulating symbols. Mathematics, logic, and computer programming languages are examples. These languages allow us to express complex relationships unambiguously.
The power of formal science comes from its ability to discover truths that are necessarily valid in all possible worlds that satisfy the initial axioms. When we prove a mathematical theorem, we show it must be true given our starting assumptions - it’s not just a description of what we observe, but a demonstration of what must be.
This leads to the main branches of formal science:
- Mathematics: Studies numbers, quantities, shapes, and patterns using rigorous logical methods.
- Theoretical Computer Science: Examines what can be computed and how, using formal models of computation.
- Statistics: Develops formal methods for analyzing data and drawing reliable conclusions from it.
- Decision Theory: Studies formal models of rational decision-making under various conditions.
- Systems Theory: Analyzes abstract principles that apply to systems of many types.
The practical value of formal science lies in its ability to provide tools and frameworks that can be applied across many domains. When we use mathematics to model physics, statistics to analyze medical trials, or computer science to process data, we’re applying formal structures to understand and manipulate the physical world.
Understanding formal science through first principles reveals its essence as the study of necessary relationships between abstract concepts. This explains both its power - conclusions that must be true given their premises - and its limitations - the need to verify that our formal models adequately represent the physical situations we apply them to.