Language and Math Model Reality
A phrase is one or more words.
A term is a distinct concept, represented by a phrase.
A phrase may be used to refer to more than one term. Which term is intended in a particular case must be inferred.
A standard definition is a description of the terms a phrase conventionally refers to.
A local definition is a description of the term a phrase refers to within a well-defined context. E.g., legal contracts and technical standards use local definitions that only apply within themselves.
Usually there are alternate descriptions which could be used to define a term.
To aid memory, local definitions are often assigned to phrases with related standard definitions. E.g., my local definition of the words “local definition” and “standard definition” are related to definitions more broadly and to the software programmer’s use of locally defined entities as well as standard entities provided and used by a larger group.
Such documents often stylize the words that refer to these locally defined terms. E.g., they bold or italicize them. This practice prevents ambiguity if words may refer to either local and standard definitions, but I think it also muddles the documents. Thus, I only italicize locally defined terms sparingly after their first use. When I want to refer to words, as opposed to terms, I place the words in quotes.
Standard definitions change, usually slowly.
Local definitions, if provided as a precaution due to this change, may even be identical to standard definition used at a point in time.
Several standard definitions may be attached to a term simultaneously, by different groups of people. The groups can be separated geographically. E.g., American English differs subtly from British English. They can also be separated in time. E.g., a younger generation using a term differently than their parents.
While a standard definition can be thought of as a local definition whose context is a particular group of people, I will not use the term in this way.
An individual may be aware of multiple standard definitions, and may use them appropriately within context.
A local definition can not be wrong. It can only be useless.
A standard definition can be wrong if it is not what is conventionally meant by a group when they use a term.
Authors don’t always make their local definitions explicit.
Sometimes authors want their local definition to become the standard definition.
Since standard definitions change, reading older documents can be difficult. E.g., Shakespeare’s plays are difficult to read because they use old unknown words and because they use words whose meaning has changed.
To interpret documents like their original readers we must share their standard definitions. This can be a legal concern. E.g., when interpreting the original intent of the U.S. Constitution. It can also be a religious concern. E.g., when interpreting the original intent of the words of inspired religious figures.
Language and mathematics let us describe things.
Any mathematical proposition could be restated with words.
A model is a simplified description of reality that is useful for some purpose.
Most models are incomplete in the sense that there exist verifiable questions that could be asked of the thing described that can’t be answered by the model.
Most models are also approximate; there exist verifiable questions that could be asked of the thing described that would be answered incorrectly by the model.
There is a tradeoff between a model’s utility and its complexity.
Two models are equivalent if they allow one to answer the same set of questions about the thing and they give the same answers.
A model is more detailed than another if it can answer everything the other model can answer, as well as additional questions.
There can be many models of the same thing, each answering questions about a different facet of the thing.
There can conceivably be complete models. A complete model is sufficient to answer any verifiable question about the thing.
There may be non-verifiable metaphysical questions about a thing. Such questions aren’t necessarily invalid. E.g., why are the laws of physics the way they are?
What do we mean by a thing? Even the demarcation of the thing is a model, and often an approximate one. A simple solution to this is to say the entire universe is one thing—one big quantum wave function. I believe this is one interpretation of quantum physics.
How would we know if a model is complete? Typically, you don’t.
The laws of physics aim to be a complete model of how the universe’s state changes. It is not a complete model of the universe, which would require knowledge of its initial state too.
There isn’t enough space in the universe to create a complete model of itself.
We don’t know if the laws of physics completely describe how the universe’s state changes. The laws could be different in other parts of the universe, at different times, or even in different local or global states. E.g., the gravitational constant could shift over time. If matter is arranged a certain way, say into a brain-like structure, the laws of physics could change within the brain. More fantastical examples are also possible; it could be the case that if a golden box with a particular shape were created then the laws of physics would change throughout the universe.
A complete model of a thing is not the thing.
Terms are Models
Terms are models.
The statement, “there are two yellow pillows on my couch,” is a model.
Most of our terms are only good enough for their standard ways it is used in our language. Most standard uses of language don’t require particularly complex models.
Specialists tend to need more granular models. Thus, they produce local definitions within their fields to supplement the library of standard terms. E.g., lawyers, doctors, engineers, and philosophers have their own lingo.
Children learn their first terms using examples. Adults often learn new terms using other terms they are already familiar with, but not always. E.g., your friend may hold up a new fruit at the grocery and tell you it’s name.
Many abstract terms must be learned using existing terms. E.g., could democracy be taught using sensory examples?
Since terms are defined with other terms, their definitions tend to be circular.
A term can be more or less understood. This can break problems with circularity.