A formal theory is a symbolic system with consistent rules for manipulation. Mathematics is a collection of formal theories. All forms of abstract thinking end in formal theories, many of which go to mathematics.

[Unifying Principle of Scientific Theories] Optimization unifies many models in the sciences (least energy, variational method, maximum entropy, rational choice, survival of the fittest), but optimization itself does not have much meaning.

## Mathematics

Equivalent theorems can always be treated as alternative definitions, and exhibit the nature of a concept.

Conditionally equivalent statements can be viewed as conditional probabilities. Think of all entities as a space, then a conditionally equivalent statement means that, under certain condition, the set characterized by the two statements are identical.

Graphical method is a handy tool to get qualitative results when limited assumptions can be put on the functional form of the math model such that no analytical solution is attained. Graphical presentation of root of an equation is suitable for general functions that shares certain qualitative property, while limited to two univariate function crossing each other.

## Examples

Mathematics Subject Classification, 2010 provides an extensive classfication of mathematical subjects.

Questions asked by different formal theories:

• Game theory: How a game SHOULD be played (under certain assumptions);
• Dynamical system: What do all POSSIBLE trajectories of system state look like;
• Agent-based modeling: Provided a system and evolution rules, PLAY the game (Too many parameters);
• System dynamics: ^%$#$%^&*(*&^%$#$%^ (Too many parameters);
• Multiscale modeling: %^&*(&^%$#@#$%&^%\$;
• Functional analysis: How to manipulate CONCEPTS in a mathematical-analytical way;

### System

System is an abstract concept without definitive characteristics. Never use the word "system" without explicit definition.

In dynamical system, it is a set of points.

“the time dependence of a point's position in its ambient space”

In system dynamics, it is some interacting quantities through feedback loops. Stock variables are related by flow variables, forming feedback loops which are enclosed in a boundary. Anything outside the boundary is the environment.

In signals and systems, it is a set of mappings.

“any physical set of components that takes a signal, and produces a signal.”

In cybernetics, it is an object that self-corrects through feedback.

In systems theory, it is a group of interacting activities.

“A system in this frame of reference is composed of regularly interacting or interrelating groups of activities.”

In system engineer, it is computer hardware and networking.

In other places, the term is used casually, merely emphasizing something that is big and complicated. But in most cases, it’s just a set, opposed of its components.

### Form and Science

"Formal science" is an inappropriate concept. Formal theories is not science, because it cannot be falsified by reality; it can only be falsified by logic. They do not describe, explain, or predict real-world phenomena. For real-world problems, mathematical models can be seen as its isomorphism and simplification.

### Form and Concept

If you are manipulating concepts, you can formalize them into symbols and will probably find some existing math tools useful. Otherwise, you may create your own formal theory. The essence of mathematics flows at the conceptual level, not the formal level. The correct usage of formal reasoning is to aid derivation and (drastically) simplify reasoning.

Designing a symbolic system is like designing a data type: it should be just enough expressive to not be error prone.

### Form and Intuition

Mathematics makes your reasoning rigorous, but intuition makes you smart. Getting useful general results from simple models is called smart.

Mathematics provide universally correct answer, but never directly useful; intuition and generalization help get extremely useful results, even though they’re not always correct.

Mathematical proof comes after intuition. Normally, results are initially found and proved through intuition in very simple cases, and then they can be abstracted into symbols and generalized by mathematics. Proof of general results are rarely straight forward; in turn, if a proposition cannot be demonstrated intuitively in simple cases, it’s unlikely to be true in general cases.

When making interpretations of the results obtained from mathematical models, ambiguity comes in with the use of human language, such as "likely" and "inevitable". Typical practice is to redefine the vocabulary with scientific/mathematical language. But we need our good intuition to take us one step further than what is strictly stated by the models to broaden the results. The proof might be impossible or at least drastically more technical.