It’s interesting how we use the word “model” to mean two different, perhaps even opposite things. In this post we have “models” describing “reality”, and in logic we have “theories” describing “models”.
For some reason it felt like a big insight to me to realize that computers aren’t identical to any particular piece of math, but rather are a model of that math, which can also be studied with other math. Any given piece of computer-related math might ignore some properties of computers that another formalism would bring to the forefront.
A weak theory could have both reality and mathematically simple structures as its models, from the point of view of an informal metatheory that allows talking about reality (and motivates the theory). A familiar structure (as a “model”) can be used both to study a complicated formal system (natural numbers for PA with its other nonstandard models), and a vaguely defined reality (classical mechanics for the real world with its black holes and quantum mechanics).
It’s interesting how we use the word “model” to mean two different, perhaps even opposite things. In this post we have “models” describing “reality”, and in logic we have “theories” describing “models”.
For some reason it felt like a big insight to me to realize that computers aren’t identical to any particular piece of math, but rather are a model of that math, which can also be studied with other math. Any given piece of computer-related math might ignore some properties of computers that another formalism would bring to the forefront.
A weak theory could have both reality and mathematically simple structures as its models, from the point of view of an informal metatheory that allows talking about reality (and motivates the theory). A familiar structure (as a “model”) can be used both to study a complicated formal system (natural numbers for PA with its other nonstandard models), and a vaguely defined reality (classical mechanics for the real world with its black holes and quantum mechanics).