A formal theory is a symbolic system of objects and consistent rules for manipulation. 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.
Definition: an assignment of properties to a mathematical object.
Assumption: a requirement of properties from a mathematical object.
Proposition: a mathematical statement.
Lemma: a short theorem used in proving a larger theorem.
Theorem: a proposition proved to be true by accepted mathematical operations and arguments.
Corollary: an immediate consequence of a result (e.g. theorem) already proved; usually state a complex theorem in a language simpler to use and apply.
Remark: a short commentary following a proposition, like a footnote but worth more attention, and can be referenced.
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.
Mathematics Subject Classification, 2010 provides an extensive classfication of mathematical subjects.
Questions asked by different formal theories:
^%$#$%^&*(*&^%$#$%^
(Too many parameters);%^&*(&^%$#@#$%&^%$
;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.”
For system engineers, 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.
"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.
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.
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.
现有的数学，或者说人类现有的数学能力，把我们的眼光束缚在了线性、正态分布和决定论的狭隘范围内； 然而我们的生活却充斥着非线性、非正态分布和随机性（分别意味着叠加原理、整体预期和因果律的失效）。 这种认识上的偏差注定了我们的坎坷。
数学的本质是符号化与抽象化么？ 至少我所了解的数学是全然抛却interpretation的（然而interpretation对于认识世界，或者说科学，又是如此的重要）。
全体数学好比一个球，interpretation在外面紧紧包围着。 Interpretation就是interpretation，里面没有数学；数学也并不包括interpretation的部分。 探索真实的过程就好比从外围interpretation空间的一点出发，潜入数学内部， 游历过数学的一些部分之后重新跳出水面，到达interpretation空间的另一点。 想要了解真实的人，将已有的一些认识加以整理，变换成数学形式，利用各种已有的数学理论进行一番操作得到结果， 将数学结论重新解开成interpretation，便得到了新知。
结论的推导和结论的认识是分的很开的事情。 以阿基米德式的数学理论为例，全体知识（结论/定理）可以从极少量的定理推导出，知识呈现层级结构； 而认识知识时，各条知识呈现平行的网络结构，从部分结论可以推出另一些结论，知识间互相联结。 可以借助各种展示方式（例如图表，以韦恩图、马氏链为样板）将一些知识简洁的表达出来， 使得知识得以“压缩”，便于随时取用和深入认识。 完全可能知道结论的证明方式，却不知道如何解释这个结论。（著名的薛定谔方程）
A field in industrial mathematics typically emerge as collaboration and communication between several academic and industrial communities. Mathematical models presented in initial discussions are usually tip of the iceberg; continuing effort iteratively remodel the problem.