Structures, Algorithms, and Functions

Structures, Algorithms, and Functions

This time, from “Algebraic Structures” by Kei Toyama, we will discuss structure, algorithms and functions.

According to the wiki, structure is “the way in which the parts of a thing are put together to make it [1]. It is a general term for the relationships of opposition, contradiction, and dependence among the elements that make up a whole [2]. In the world of mathematics, the term “complex” refers to the arrangement and relationship of parts and elements of a complex thing. In the world of mathematics, the basic approach is to abstract the “parts that make up a thing” as much as possible and find the relationship between them.

For example, geometry is the study of space and shapes, not necessarily of numbers. For example, geometry is the study of space and shapes, not necessarily of numbers. In response to this, Toyama cites mankind’s ability to recognize patterns as the origin of mathematics. According to this book, it is this ability to recognize patterns that has enabled us to cut the complex world into understandable forms, sublimate it as a discipline, and make it into mathematics. This means, for example, that the simple concept of numbers, such as 1, 2, 3,…. For example, the simple concept of numbers, 1, 2, 3, …, is not simply a number, but an abstract concept of 2, derived from the abstraction of common forms or patterns between different things, such as two apples, two oranges, two people, and two dogs.

What is important in thinking about structure (pattern) here is the process of breaking down the object into its constituent elements, considering their mutual connections, and then carefully examining the whole. In the aforementioned example, two apples have the attributes of red, round, fruit, etc., in addition to the attributes of two apples, and similarly, two oranges and two people have various attributes, and the overlapping parts (product set) of these attributes form a common form (i.e., the structure of two). It can be interpreted as a structure of 2.

This structure is a static one without a time axis. The addition of a time axis to this static structure is called an algorithm or a function. An algorithm is a chain of operations that are connected in a certain order. For example, when performing a certain calculation, it is a series of discrete operations. A differential equation is a continuous chain of these operations. This collection of operations is called a group. When operations and functions are defined as y=f(x), x and y are the objects that receive the operations, and f() is the operation or the operation itself. In other words, x and y are noun-like ‘things’, while f() is a verb-like ‘action’.

If we look at mathematics in terms of these verbs, for example, when we consider the following operations (verbs)

a: “put on a shirt”

b: “Put on a jacket.”

c: “put on the over.”

ab” means “to put on a shirt and a jacket” and “bc” means “to put on a jacket and an overcoat. In the normal mathematical world, ab=ba, but in this world, ab≠ba because the operation “put on a shirt and put on a jacket” has a different result than the operation “put on a jacket and put on a shirt. In addition, (ab)c is “put on a shirt, put on a jacket, and put on an overcoat,” and a(bc) is “put on a shirt, put on a jacket, and put on an overcoat,” and the final result is the same, which means that (ab)c=a(bc), which is the binding rule in the mathematical world.

In this way, the world of mathematical symbols that we normally use will be different in a world with different presupposed axioms and theorems. Formal logic and mathematical logic, as described in “Formal Language and Mathematical Logic,” are the study of how to deal with this abstracted world of symbols, and are an approach to the meaning of things. In addition, set theory as described in “Outline of Set Theory and Reference Books” and logic as described in “Creating Logic Part 1: Beginning Logic Reading Memo” etc. are also foundations for considering these topics.