This is the first of a series of blogs about representations in mathematics in a very broad sense.

Every kind of representation associates one kind of object with another kind of object, with the association limited to certain aspects of the objects. The way this association is limited is not always or even usually made explicit. There are many examples of different sorts of representations on the abstractmath website in the understanding math chapter, particularly in the articles on models and representations, and on images and metaphors. I intend to reorganize this material because my understanding of the situation has changed over the past year, so I will say some things here in g&g and hope for an informative reaction.

This posting is a summary of the various kinds of representation I want to talk about. The links above have more detail about many of them.

A representation can be physical, mental or mathematical, and what it represents can be a physical process or a mathematical object or other concepts.

Examples

• The printed graph of a function or an icosahedron made out of plastic are physical representations of math objects.

What you picture in your mind when you think about the graph of a particular function is a mental representation of a math object.

• Your visualization of a particle going faster or slower on a path may be a mental representation of both a physical process and a function of time that models the movement of the particle.
• A matrix representation of a group, or a string of digits in base 10 notation, are mathematical representations of a mathematical objects.
• The function describing the movement of the physical particle just mentioned is a mathematical model of a physical process.

Terminology
Words used for special types of representations are models, images, and metaphors.

Metaphors

Metaphors are one of the Big New Things in cognitive science and the word has had its meaning extended so much from the grammatical meaning that it may be referred to as a conceptual metaphor.

• When you say the function f(x) = x^2 “goes to infinity when x gets large” you are using a metaphor.
• When you think of the set of real numbers as an infinitely long line you are using a conceptual metaphor.

Stay tuned…

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