# A Book for Mathematicians and Programmers – Axiomatic Theory of Formulas

A new book for mathematicians and programmers published: *Axiomatic Theory of Formulas* or *Algebraic Theory of Formulas*. The book is an undergraduate level but contains a new theory.

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From the preface:

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 Operator 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 schema connected with different kinds of links.

The Operator 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 operators (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.

### Draft Status

*This is a preliminary version. If you will find
any mistakes, please contact me.*

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