Subject
The formal sciences study structure in its purest form: systems of symbols, rules, and proofs that do not depend on observation. Logic, mathematics, statistics, the theory of computation, and the formal study of decisions and systems supply the tools that let reasoning in every other field be checked rather than merely trusted. Mettica connects each formal system to the intuitions it makes precise.
Structured lessons for formal sciences are being designed. The disciplines and connections below show how this subject will be organized.
Disciplines
6 disciplines. Each is listed under one subject for browsing, and many connect to others.
Formal systems of inference, proof, and truth, and the limits of what can be proved or computed.
Propositional & predicate logic · Proof theory · Model theory · Set theory …
Related: Philosophy
Structures of number, space, change, and chance, developed through proof.
Foundations · Algebra · Analysis · Geometry & topology …
Related: Natural Sciences
Learning from data under uncertainty: summarizing, inferring, and designing studies.
Descriptive & exploratory statistics · Statistical inference · Experimental design & sampling · Regression & predictive modeling …
Related: Social Sciences · Applied Sciences
What can be computed, how efficiently, and how information is represented and protected.
Automata & formal languages · Computational complexity · Algorithms & data structures · Semantics & type theory …
Related: Applied Sciences
Formal models of choice, strategy, collective decision, and optimization.
Decision theory · Game theory · Social choice & mechanism design · Operations research & optimization
Related: Social Sciences
Abstract models of feedback, control, networks, and complex dynamic behavior.
Cybernetics & control theory · Dynamical systems & complexity · Network theory
Related: Natural Sciences · Applied Sciences
Connections
These concepts pass through formal sciences and continue into other fields.
Invariance under transformation: formalized as a group, tied to conservation laws, used in structural design and artistic composition, and debated as a question about sameness.
What it means for one thing to bring about another: analyzed by philosophers, formalized in causal models, tested in experiments, and inferred from social data.
A fitting relationship among parts: heard in music, explained by acoustics, and expressed as numerical ratio.
The unbounded: made rigorous in set theory, argued over in metaphysics, and evoked in art.
Outputs that loop back to change the system producing them: control in engineering, homeostasis in biology, self-reinforcing cycles in markets and societies, and stability in dynamical systems.
Depth
Lessons in this domain will follow Mettica’s depth model, from first orientation to research-level questions.
Level 1
Orient the learner before requiring specialized vocabulary.
Level 2
Make distinctions and relationships explicit.
Level 3
Introduce the technical representation appropriate to the field.
Level 4
Expose complexity rather than conceal it.
