Mathematics is a system for building abstract statements that can be mapped to reality. The axioms of a mathematical (or other axiomatic) model define the conditions that a system (such as a pair of apples in the real universe) must satisfy in order for the abstract model to be applicable as well as providing a schema for mapping the abstract model to the concrete system.
There are other kinds of abstractions we could meaningfully talk about and they need not be defined as precisely as an axiomatic model like mathematics. An abstract model could be defined as a relationship between abstract ideas that can be mapped to a concrete system by pinning down each of its constituent abstractions to a concrete member of the system.
An abstract model may be predictive, meaning it has an if-then structure: if some relation between abstract members holds then the model predicts that some other relation will also hold. Such a predictive model may be true or false for any given concrete system that it is applied to. The standard we expect of a mathematical model is that it is valid (true for all concrete systems that it can be applied to), yet an abstract model need not meet so high a standard for it to be useful. We can imagine much fuzzier abstract models that are true only some of the time but can be useful by providing general-purpose rules that allow us to infer information about the actual state of a concrete system that matches the criteria of the model. If we know the probability of an abstract predictive model being correct we can use it wherever it is applicable to inform the construction of causal models.
If we consider causal models to operate in the realm of first order logic where we can quantify over and describe relationships between the basic units of cause and effect in our universe, an abstract model lives in the realm of higher order logic and can describe the relationships between causal relations and lower order abstract models.
An abstract model need not be predictive to be useful. It may be defined to be applicable only where the entire relation it describes holds. In this case it simply acts as a reusable symbol that is useful for representing a model of a concrete system more compactly as in the way a function in a computer program factors out reusable logic, or a word in human language factors out a reusable abstract idea.
Justice and Mercy are both fuzzy abstract models. To the extent that people agree on their definitions they are meaningful for communicating a particular relationship between pinned-down abstractions. For example, Justice may be defined (simplistically) as describing a relationship between human deeds and subsequent events such that deeds labelled ‘bad’ result in punishment and deeds labelled ‘good’ result in reward. The particular deed and subsequent event as well as the definitions of good, bad, punishment and reward are all component abstractions of the abstract model called Justice which must be pinned down in a concrete system in order for the concept of Justice to be applied in that system.
Justice may also be used as a predictive model if you formulate it as a prediction from a good/bad deed to a future reward/punishment event (or vice versa) and it would be useful for constructing a causal model of any particular concrete system to the extent that this predicted relationship matches the actual underlying nature of that system.
Note: none of this is based on any formal study of logic outside of this Epistemology sequence so some of the terminology in this post was invented by me just now.
Mathematics is a system for building abstract statements that can be mapped to reality. The axioms of a mathematical (or other axiomatic) model define the conditions that a system (such as a pair of apples in the real universe) must satisfy in order for the abstract model to be applicable as well as providing a schema for mapping the abstract model to the concrete system.
There are other kinds of abstractions we could meaningfully talk about and they need not be defined as precisely as an axiomatic model like mathematics. An abstract model could be defined as a relationship between abstract ideas that can be mapped to a concrete system by pinning down each of its constituent abstractions to a concrete member of the system.
An abstract model may be predictive, meaning it has an if-then structure: if some relation between abstract members holds then the model predicts that some other relation will also hold. Such a predictive model may be true or false for any given concrete system that it is applied to. The standard we expect of a mathematical model is that it is valid (true for all concrete systems that it can be applied to), yet an abstract model need not meet so high a standard for it to be useful. We can imagine much fuzzier abstract models that are true only some of the time but can be useful by providing general-purpose rules that allow us to infer information about the actual state of a concrete system that matches the criteria of the model. If we know the probability of an abstract predictive model being correct we can use it wherever it is applicable to inform the construction of causal models. If we consider causal models to operate in the realm of first order logic where we can quantify over and describe relationships between the basic units of cause and effect in our universe, an abstract model lives in the realm of higher order logic and can describe the relationships between causal relations and lower order abstract models.
An abstract model need not be predictive to be useful. It may be defined to be applicable only where the entire relation it describes holds. In this case it simply acts as a reusable symbol that is useful for representing a model of a concrete system more compactly as in the way a function in a computer program factors out reusable logic, or a word in human language factors out a reusable abstract idea.
Justice and Mercy are both fuzzy abstract models. To the extent that people agree on their definitions they are meaningful for communicating a particular relationship between pinned-down abstractions. For example, Justice may be defined (simplistically) as describing a relationship between human deeds and subsequent events such that deeds labelled ‘bad’ result in punishment and deeds labelled ‘good’ result in reward. The particular deed and subsequent event as well as the definitions of good, bad, punishment and reward are all component abstractions of the abstract model called Justice which must be pinned down in a concrete system in order for the concept of Justice to be applied in that system.
Justice may also be used as a predictive model if you formulate it as a prediction from a good/bad deed to a future reward/punishment event (or vice versa) and it would be useful for constructing a causal model of any particular concrete system to the extent that this predicted relationship matches the actual underlying nature of that system.
Note: none of this is based on any formal study of logic outside of this Epistemology sequence so some of the terminology in this post was invented by me just now.