Metaphors are essential and important, as Lakoff and Johnson argued so convincingly in "Metaphors We Live By". Many of our discussions here are philosophical, and the main tool of Philosophy is language. The earliest types of Philosophy takes the form of a dialog or discourse (Plato or Aristotles). Since then, Philosophy has used logic and linguistic examination. Using linguistic tools, one can find metaphors and analogies or categories and classifications. Metaphors belongs to the best things we can find in Philosophy.
Metaphors lead to meaning and understanding. Without metaphors, it is impossible to achieve an understanding of abstract items. They give meaning to abstract things. Understanding, meaning and semantics require a connection or relationship between two worlds. Meaning arises from a *mapping* between two worlds..
- from connections between physical ojects and abstract symbols
- from metaphors and analogies, which map abstract domains to concrete domains
- from dictionaries which map one language to another
Equations and metaphors are similar, they relate two different terms, things, and sides. Metaphors let us express one thing in terms of another. In this sense, metaphors are the "calculus of the mind".
- A function is at it's core a mapping from one thing to another.
- An equation means to find a mapping from one thing to another, too.
- A law (of nature) is at it's core a mapping from one world to another.
According to this definition, one would say a theorem exists because there are many worlds which can be mapped to each other in a meaningful way.