Axiomatic Theory of Formulas. Algebraic Theory of Formulas


This new mathematical theory developed by the book author researches the properties of mathematical formulas (aka expressions). Naturally this theory is essential for the mathematics.

Formulas are encountered in all areas of mathematics. Probably of special interest are propositional formulas in math logic. Using this Theory of Formulas may probably simplify proofs of math logic theorems.

The Theory of Formulas is a new foundation for math logic which is the foundation of mathematics. That is the Theory of Formulas is the foundation of the foundation of mathematics.

This theory researches mathematical formulas as well as any other objects which have constituent parts of different kinds (without the limit that a whole cannot be a part of its part, that is loops are not disallowed). For example, consider relations between objects in the memory of a computer or components of an electronic scheme connected with different kinds of links.

The Theory of Formulas is not limited only to finite neither only to acyclic formulas. Infinite and cyclic (e.g. a(b(a(b(…))))) formulas are also modeled by this theory. A common example of a system of infinite formulas is the set of irrational numbers. Another example of an infinite formula may be the entire mathematics in the form of some formalism (e.g. all groups of Group Theory).

This book contains a new axiomatic theory, the theory of constructs, and its specializations, including the theory of formulas:

  • Informally, a construct is anything what has indexed components (parts). E.g., the a-(b+c) can be considered as a construct which has “a” as the part with index 1, “-” as the part with index 2, and (b+c) (a subformula) as the part with index 3. 
  • Formulas is a particular case of constructs. Informally, a formula is a construct which contains symbols. E.g. a-(b+c) would be a formula which contains symbols such as a, b, c, +, -. 

To increase flexibility and power of the theory I study functions on the sets of constructs rather than the constructs themselves. So different kinds of constructs can be researched with a uniform method.

An important aspect of the theory of formulas accordingly this book is so called specialization that is adding additional subexpressions to expressions. For example if + would mean an abelian group operation, +1 (+ with added additional subformula “1”) would mean e.g. addition of whole numbers which is a specialization of abelian group operation etc. Specialization can be used instead of variable substitution.

Recall that formalist philosophy is a school in philosophy of mathematics which considers mathematical formulas as physical objects. For this new theory of formulas which includes infinite logical formulas that philosophy (“formalism”) is not applicable as infinite formulas cannot be written and are not physical objects in this sense. So this theory also changes the philosophy of mathematics.