A magnitude is a value that represents a size, frequency, or importance. Typically, a magnitude is taken to be a number that is either positive or zero.
A magnitude space is the mathematical space of all possible magnitudes. Intuitively, magnitudes are “positive numbers with 0 and infinity”.
Often times, magnitudes are presented as "like the real numbers, but with the negatives lobbed off". This isn't wrong, but magnitude space can stand on its own with emphasis on different properties. Magnitudes are fundamental to most scientific modeling which includes: ratios, measurements, frequency, probability, and size.
Point at Infinity
The magnitude space contains a point at infinity which represents the greatest possible magnitude. It also makes magnitude space topologically compact because without it a sequence of magnitudes could grow larger and larger without reaching a limit point. With the inclusion of the point at infinity, all sequences of magnitudes have a limit point.
Unlike other mathematical spaces, there is no ambiguity about adding a point at infinity. In the real numbers, for example, there are 2 valid ways of adding points at infinity to achieve compactification. The first way is to include a positive infinite and a negative infinity. The second way - called the projective completion - is to include a single point at infinity which connects both the positive and negative sides.
In contrast, the point at infinity in magnitude space can only be constructed one way, and it is clearly a point whose magnitude is greater than any other magnitude.
Construction with Ratios
The natural numbers (0, 1, 2, 3, etc) are a fundamental mathematical space, and magnitudes can be constructed using ratios of these numbers.
When the numerator of this ratio is 0, the magnitude is 0. When the denominator is 0, the magnitude is the point at infinity. The only ratio that isn’t allowed is 0/0 which is an indeterminate form.
See episode 94 of The Local Maximum for a discussion of infinite ratios.
All magnitudes can be approximated with ratios of natural numbers, but some might be irrational numbers. Therefore, the standard magnitude space must be constructed by completing the set of positive ratios using Dedekind completion.
Beyond the Reals
In some cases one might want to consider an infinitesimal magnitude which is non-zero but less than any real number. This is useful in a case where two objects have a magnitude of zero compared to a third, but their ratio still matters.
For example, let’s suppose that I want to consider the measurements of a unit square (1x1), a line of length 3 and a line of length 5. Let’s call these objects A, B, and C respectively.
The area of object A is 1, but the areas of objects B and C are 0 (they have no width and therefore no area). However, we can make up an infinitesimal number q that represents the length of a unit line segment. Therefore, the magnitudes of B and C are 3q and 5q respectively. This allows us to consider their ratio even if the magnitudes are both 0 compared to A.
This situation can also be found in the relative occurrence of events. We can say that event A is infinitely times more likely to happen than B or C. However, we still might want to consider the ratio of B to C with A suppressed. Therefore, we can model the frequency of B and C with infinitesimals rather than with 0.
Another way to look at this is to consider the relative magnitude of two different objects, which is the ratio of their magnitudes. Even if two objects have magnitude 0 relative to some third object, we might still be able to consider their magnitude relative to each other without getting the indeterminate form 0/0.
Finally, the richest universe of magnitudes which takes this idea to its logical conclusion is from the construction of surreal numbers which is out of the scope of this article.
Properties
In some cases, we might want to consider an “exotic” magnitude space which breaks some of these rules. The following properties are useful benchmarks in understanding a non-standard magnitude space. When left unspecified, a magnitude space is understood to just mean positive numbers (bounded by 0 and infinity)
Total Order: There is an ordering of magnitudes where we can say that one magnitude is less than another (a < b) or greater than another (b > a). This is a total order, meaning that any two magnitudes are either equal, or one is greater and the other one is lesser.
Sum: Any two magnitudes can be added together (x + y) to form another magnitude. This operation is commutative and associative. It has an identity of 0 (meaning that x + 0 = x for all x), and it has an absorbing element of infinity (let’s call it inf) where x + inf = inf.
Sums and Order: The sum of two magnitudes is always greater than or equal to its parts. Also, there is a limited form of subtraction where x + __ = y is solvable if x is less than or equal to y.
Product with Natural Numbers: With the sum as defined, any magnitude x can be multiplied by a natural number n to produce (0 + x + x + x +...) n times. Furthermore, any two magnitudes can also be multiplied together, with multiplication distributing over addition.
Inverse: Magnitudes have multiplicative inverses (1/x). Taking the inverse of two magnitudes will reverse the ordering (x < y => 1/y < 1/x). Any magnitude multiplied by its inverse is 1, and 1 is the only magnitude that is its own inverse.
Note on Inverses and 1: It only makes sense to have inverses and some notion of “1” if the magnitude represents a ratio. For example, if the magnitude represents a length that we are recording in meters, the value of “1” is arbitrary and an inverse would also be arbitrary.
Boundedness: For any magnitude x, 0 < x < inf