It is said that discussing the familiar in unfamiliar terms opens up new avenues for thought and creativity. That is my goal here. By approaching games, especially RPG’s, the same way one would a theorem or scientific principle, my hope is to inspire others to rethink of their perception and conception of games themselves. It might prompt people to develop in ways they never thought of before and to explore new mechanics they discovered by codifying what just came intuitively for them.
I’ll start with something very general, but also very vital: formal systems. Formal systems have three major components (and I am borrowing heavily from Douglas Hofstadter for these terms):
Math is the most well known formal system. Geometry, for example, consists of Euclid’s famous axioms (with non-Euclidean Geometry altering one of them), a system for deriving proofs based on those axioms, and the theorems that follow from those axioms.
Games are also a formal system. We tend not to think about them as such, preferring to view them as crafts or works of art. And there is indeed a strong element of art to games, from the ideas present and the execution of their delivery. But in starkest terms, games operate on the same principles as math, language, and other systems, albeit at a less abstract level than fundamental elements. By realizing this, it can be possible to define and formalize a number of otherwise elusive terms we use in gaming, such as immersion and elegance.
In a game, the axioms are the materials required to play. The decision procedures are the rules of the game, outlining how the axioms can be utilized. The theorems are then the valid moves that can take place within the game. For example, in chess, the axioms are the board, the pieces, and the two players. The decision procedures are the way the pieces move, removal of pieces, win conditions, etc. The theorems are the set of valid moves a player can make on their turn, such as castling or getting out of check.
A trait of formal systems is the occurrence of isomorphism. Essentially, when the mind is confronted with a set of highly abstract concepts, such as the symbolic language of the 1900’s logicians, we will assign meaning to the patterns. Often these meanings will be drawn from personal experience and knowledge and used as a way to grant additional meaning to the patterns we see. For example, when school children are taught addition with the plus sign, they are prompted to think of the physical act of combining two separate piles of things into one pile. This creates an isomorphism: when you see the abstract symbol, +, think of two piles being combined.
This has the benefit of allowing us to understand something very abstract and foreign, by couching it in the familiar. However, the problem with isomorphism is that we will want to assume the system behaves like the thing we associate it with. This can cause issues in math, where not everything behaves as in the real world. For example, if you add the set of all the composite numbers and the set of all the prime numbers together, your isomorphism idea of two piles tells you that the resulting set is larger. This is false. Since there are an infinite number of elements in the two sets, the resulting set is also infinite - and the same size. (This isomorphism confounded mathematicians and philosophers for thousands of years until Georg Cantor.)
While isomorphisms might be undesirable for formal systems - Bertrand Russell certainly thought so - games distinguish themselves in that they deliberately seek to induce isomorphisms!
Think of Dungeons and Dragons. Players roll dice and assemble numbers on a piece of paper, deriving statistics and numbers. What is written down is highly abstract, the result of navigating a long series of complex rules. However, in the end, the game asks the player not to see the numbers on the paper - but a person, a character. A sentient being whose behavior they control and whose capabilities are defined by the numbers, but whose actions are their choice.
There are games that lack this appeal to isomorphism, of course. Many of the card games, such as Poker, or board games, such as Chess and Checkers, or even many sports, such as baseball and soccer. These seek to stand on their own merits, rather than insert an appeal to our imaginations. This does not make them any less of a game, it just makes their intended hook different.
The games of key interest here are those that do rely on isomorphism. These games want players to take their decision rules and the resulting theorems and translate them into other terms. When Tephra asks a player to roll strike using their d12, it is asking the player to imagine that the use of the axioms (the d12, themselves) and the theorem (the valid move) they created with the decision rule (how tiers of strike work) not as abstract numerical constructs but as a character swinging a melee weapon at another, ready to deal damage and defend themselves. This is a complex arrangement and understanding how it works is a major step in game design itself.
Next, let’s explore the concept of isomorphism further in relation to the typographical decision rules a game chooses to have and see how that leads us to a concept of immersion.
I’ll start with something very general, but also very vital: formal systems. Formal systems have three major components (and I am borrowing heavily from Douglas Hofstadter for these terms):
- Axioms
- Typographic decision procedures
- Theorems
Math is the most well known formal system. Geometry, for example, consists of Euclid’s famous axioms (with non-Euclidean Geometry altering one of them), a system for deriving proofs based on those axioms, and the theorems that follow from those axioms.
Games are also a formal system. We tend not to think about them as such, preferring to view them as crafts or works of art. And there is indeed a strong element of art to games, from the ideas present and the execution of their delivery. But in starkest terms, games operate on the same principles as math, language, and other systems, albeit at a less abstract level than fundamental elements. By realizing this, it can be possible to define and formalize a number of otherwise elusive terms we use in gaming, such as immersion and elegance.
In a game, the axioms are the materials required to play. The decision procedures are the rules of the game, outlining how the axioms can be utilized. The theorems are then the valid moves that can take place within the game. For example, in chess, the axioms are the board, the pieces, and the two players. The decision procedures are the way the pieces move, removal of pieces, win conditions, etc. The theorems are the set of valid moves a player can make on their turn, such as castling or getting out of check.
A trait of formal systems is the occurrence of isomorphism. Essentially, when the mind is confronted with a set of highly abstract concepts, such as the symbolic language of the 1900’s logicians, we will assign meaning to the patterns. Often these meanings will be drawn from personal experience and knowledge and used as a way to grant additional meaning to the patterns we see. For example, when school children are taught addition with the plus sign, they are prompted to think of the physical act of combining two separate piles of things into one pile. This creates an isomorphism: when you see the abstract symbol, +, think of two piles being combined.
This has the benefit of allowing us to understand something very abstract and foreign, by couching it in the familiar. However, the problem with isomorphism is that we will want to assume the system behaves like the thing we associate it with. This can cause issues in math, where not everything behaves as in the real world. For example, if you add the set of all the composite numbers and the set of all the prime numbers together, your isomorphism idea of two piles tells you that the resulting set is larger. This is false. Since there are an infinite number of elements in the two sets, the resulting set is also infinite - and the same size. (This isomorphism confounded mathematicians and philosophers for thousands of years until Georg Cantor.)
While isomorphisms might be undesirable for formal systems - Bertrand Russell certainly thought so - games distinguish themselves in that they deliberately seek to induce isomorphisms!
Think of Dungeons and Dragons. Players roll dice and assemble numbers on a piece of paper, deriving statistics and numbers. What is written down is highly abstract, the result of navigating a long series of complex rules. However, in the end, the game asks the player not to see the numbers on the paper - but a person, a character. A sentient being whose behavior they control and whose capabilities are defined by the numbers, but whose actions are their choice.
There are games that lack this appeal to isomorphism, of course. Many of the card games, such as Poker, or board games, such as Chess and Checkers, or even many sports, such as baseball and soccer. These seek to stand on their own merits, rather than insert an appeal to our imaginations. This does not make them any less of a game, it just makes their intended hook different.
The games of key interest here are those that do rely on isomorphism. These games want players to take their decision rules and the resulting theorems and translate them into other terms. When Tephra asks a player to roll strike using their d12, it is asking the player to imagine that the use of the axioms (the d12, themselves) and the theorem (the valid move) they created with the decision rule (how tiers of strike work) not as abstract numerical constructs but as a character swinging a melee weapon at another, ready to deal damage and defend themselves. This is a complex arrangement and understanding how it works is a major step in game design itself.
Next, let’s explore the concept of isomorphism further in relation to the typographical decision rules a game chooses to have and see how that leads us to a concept of immersion.
I am not sure this idea of all infinites are the same in size is accepted by all mathematicians. When I was in advanced calculus we compared the set of whole number to the set of rational numbers. A one to one correspondence could be made between all the whole numbers and all the rational numbers between 0 and 1, thus the infinite set of rational numbers is larger than the infinite set of whole numbers. I still find this persuasive
ReplyDeleteYou're touching on the concept of counting. All real numbers between 0 and 1 is infinite - and cannot be counted. Whole numbers are also infinite, but they can be counted. So they're both the same size (infinite), but the degree to which they can be sorted differs. That's what Cantor realized.
DeleteI understand what you're getting at, though.