An arbitrary and random collection of symbols is not enough to create a consistent magical system. An issue with creating a magical system from random symbols is that there is no formal ontological framework. Consistent magical ontological frameworks can emerge from magical languages, and people can create magical languages through ordered and logical methods. A magical system is an abstraction comprised of a tuple of symbols and strings. Those strings encapsulate magical sigils, axioms, rules, and operations that relate sigils and entities. In classical, western, occult “magical systems”, there are formally articulated sets of axioms and rules. The dilemma of an informal magical system is that the strings in an informal language are unordered and do not consistently lead to inferences. The inconsistency of the structure makes the resulting magical operation unreliable. If we consider that magical entities are topologies of strings, we can create consistent magical systems. We can define a string as:

A string of length n on an alphabet l of m characters is an arrangement of n not necessarily distinct symbols from l. There are mn such distinct strings.

– –Wolfram MathWorld

Strings are words comprised of a sequence of symbols and operations. An informal system does not define rules that create the structure for the formation of magical objects, so it is difficult to build magical objects with integrity because there is no formalized notion of magical objects. When a person creates sigils by crossing out letters and rearranging lines, only the form of the symbol and not the content of the concept it references is changed. It has the same semantic meaning as a phrase not made into a sigil. A person should develop a formal ontology to make a consistent and reliable system. Having a set of valid and sound axioms creates a consistent theory for your magical system. The easiest route to go about creating a solid theory is to use equivalence classes and equivalence relations as a backbone to formulate a field of sigils.

**Sections:**

*Division and Modulo Operations**Equivalence Classes and Equivalence Relations**Building A List of Magical Classes**Magical Classes and Additive Inverses**Higher-Dimensional Magical Properties and Networks**Psychic Energy, Waves, and Magical Networks**Abelian Fields*

## Division and Modulo Operations

If given the equation 3k=9, how would it be solved to find k? The equation can be solved by dividing both sides by 3. The equation would then be in the form of 3k/3=9/3, and when that form is simplified, k=3. 3*3=9, so 9 is a multiple of 3. If given the equation 3k=10, a person would solve it with a similar method: 3k/3=10/3. When simplified by a calculator, the calculator will display 3.33… as a repeating decimal. The answer is not exact because an algorithm truncates the infinite number of digits to an inexact and imprecise answer. The inexact decimal answer can also be written as the exact answer 10=3*3+1.

The division algorithm is the method generally taught in western educational systems to calculate quotients. If the division algorithm is applied to 10/3, the closest number less than 10 that 3 can divide evenly into is 9. Since 3 can divide 9 evenly, 9 is a multiple of 3. So, you would write 3 at the top and then subtract 9 from 10. That yields a remainder of 1. If you were to write it as a mixed number, you would have 3 and 1/3. Remember how we got 3.33…? You can write this as 3 remainder 1, so 10/3=3 remainder 1. That, in turn, implies 10/3 is 3 off by one. 10/3=3 remainder 1 is equal to 10=3*3+1. Integers that divide evenly have a remainder of 0. For example, 9/3=3*3+0. What was illustrated is a modulo operation. You do not need to do this by hand every time, because that gets complicated for complex Algebraic problems. On most scientific calculators, there is a button labeled mod. If you were to enter 10mod3 in a calculator, it would evaluate to 1, which is the remainder. This concept is essential to understanding equivalence classes and their relations.