Vectors: The Danger of A Bad Definition

Posted 2 months ago | Originally written on 16 Feb 2024
"A bad definition is worse than useless––it is misleading and the cause of endless confusion."

The standard definition of a vector is: "a way to describe a quantity that has both magnitude and direction". Even ChatGPT4 states that: "A vector is a mathematical entity that possesses both magnitude and direction. It is often represented graphically by an arrow, where the length of the arrow indicates the vector's magnitude, and the direction in which the arrow points specifies its direction. In the context of mathematics and physics, vectors are used to describe quantities such as displacement, velocity, force, and acceleration, which have both magnitude and a specific direction in space."

While that is fine as far it goes, it seems to me that this interpretation suffers from deviating from the fundamental property of vectors. Inasmuch as vectors, in their various forms, define entities by conveying both magnitude and direction, the core property of a vector is that it is a collection of two or more values that describe a single quantity; the notions of magnitude and direction are emergent from the essence of the property of being multi-valued i.e., we have to further define how to compute magnitude and direction, but fundamentally, any collection of values to describe an entity is a vector. This only makes sense when we consider the contrapositive: can one define a vector with a single value? Of course not! It is precisely because we have multiple values that we can speak of a vector at all.

By starting with the idea that some quantities are only meaningfully described this way can we then motivate the idea of a vector space (the collection of all such possible multi-valued points) from which we can then refer to magnitudes and directions. In other words, directions are only meaningful in spaces and spaces are only meaningful in multi-valued quantities. However, (and this is critical!) the two concepts are not bound to one another: while all spaces will naturally consist of vectors, it doesn't follow that all vectors exist on spaces. Furthermore, it is often the case that the space in which the vectors inhabit tells us nothing about the vectors present in that space i.e., the properties of the space and those of the vectors on the space are independent. For instance, the flow of a fluid may be the result of the structure of the space but need not be. By contrast, in engineering waveguides, the goal is to influence the behaviour of the electromagnetic field (a vector field with each point in space defined by a vector) by the structure of the space (the physical dimensions and material properties of the waveguide).

Fortunately, Wikipedia does get it right and further motivates the distinction between the notion of a space and the content of a space: "In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces." By stating "...or to elements of some vector spaces", the Wikipedia definition grants that every point in some arbitrary space of any dimension may be associated with a vector of a totally different dimension e.g., a 1-D space (points along some line) may each point associated with an 10-D vector.

However, the bottom line is that we should drop the idea that vectors are for quantities that have both magnitude and direction; rather, they are for quantities which are only meaningfully described using multiple values. When these multi-valued quantities occupy some space then we can talk of direction as well as define magnitudes.

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OK, Paul. That's all well and good. But how would you then define a force using this idea?

I'm thinking...