On the Ubiquity of Real Numbers.
Given that the reals are a unique set of objects defined by certain properties, anything that can be modeled by them is neccessarily exhibiting those defining properties.
Real numbers are the unique complete ordered field. Often times, it is advantageous for us to adopt some model of a system that utilizes this unique mathematical object.
If some physical system can in some way be modeled by the real numbers, then we'll refer to said system as a physical instance of the reals. And, these instances neccessarily exhibit the same properties as the reals, meaning they have some sense of order and completeness, as well as some operations between elements of the instance that behave nicely.
Furthermore, any theorem or mathematical machinery applicable to the reals should also apply to true physical instances of them, namely the tools of calculus and other constructions on these basic elements. So do these instances really exhibit all the properties we would expect, and if not, where do they fail?
Below, we'll take a high-level view of these properties, and then look at some instances of the reals.
Defining Properties of the Reals
It's hard to talk about the reals without mentioning the operations between them. In fact, they're essentially useless without these operations (except for ordering perhaps). The reals can be defined as the unique complete ordered abelian group, where the group action is simple addition. Then with multiplication (and it's discrete relation to addition), we get a field for free.
Of course, the reals may also be constructed as a mathematical object. This is often done in the form of Dedekind cuts of rational numbers. For the topic at hand, however, it's most advatageous to consider an axiomatization of the reals; as this facilitates our discussion of properties.
Axioms of the Reals
Field AxiomsOrder Axioms
- Addition - closed, associative, commutative, identity, inverse, distributive
- Multiplication - closed, associative, commutative, identity, inverse (exclude 0), distributive
Completeness
- All numbers satisfy \(a\leq b\) or \(a\geq b\).
- Any subset of the reals has a least upper bound and greatest lower bound that exists and is contained within the reals. (This is the axiomatic approach to Dedekind cuts)
See this overview for a more formal statement
Here, the field axioms just guarantee that addition and multiplication act in the nice ways we expect: I can add numbers in any order I like, multiply distributively over addition, etc.
The order axioms just make explicit that these objects have a definite, global order. That is, any set of elements are mutually comparable with a definite answer. We may then order all these elements according to their mutual order.
The requirement of completeness is then, perhaps, the least intuitive. A good heuristic is to think of it this way: if I keep dividing a real number in half, I should always get a real number back, and, in fact, I should be able to do this an infinite number of times (this is the conceptual baggage) and still have a real number in the end. This is a rough example of a Cauchy sequence, and the completeness condition guarantees that all Cauchy sequences (those that converge) yield points within the space in which they're formed.
Proofs of Uniqueness
The reals are, in fact, the unique (that is, only) mathematical object satisfying the above properties. This is guaranteed by a proof of uniqueness.
This is rather nice though; as it means we may then represent any instance of the reals in whatever other representation we may choose. Consider this: the concept of Arabic numerals (1, 2, 3, ....) actually are an instance of the reals themselves. They're essentially a specific set of objects that just so happen to satisfy all the properties of the reals. And, because of this, we can model every other instance of the reals with numbers alone.
As an example; I can also represent real numbers as lines of certain lengths. I can choose some length, call it one, and then linearly scale the length of all other lines relative that one-long line. So for example, instead of an arabic numeral for the concept of three, I would simply have a line of length three times that of the line of length one. I can also define addition: lining up the lines and adjoining them; and multiplication: stretching lines by a certain factor.
So, lines of certain length are an instance of the reals; and I, consequentley, could model all other instance of the reals as lines. But, this would likely be unwieldy, and human eyes are better at distinguishing characters as opposed to discriminating lengths. Hence, we more often choose to represent real numbers as characters, i.e. Arabic numerals.
Physical Instances of the Reals
Let's now consider some scenarios where the reals are often used in real life.
A classic example is measuring time for some duration or computing volumes of liquids: in both cases, I can add elements and multiply or divide the quantities to my leisure. I can also always say for certain whether one period of time was longer than another, or one reseviour has more liquid than another.
Perhaps the most impactful implementation of a model of the reals is monetary value. Via monetary value, objects have an implicit ordering in terms of worth. And I can add up currency, and divide (split bills) and multiply it (per hour wage). From a certain perspective, the cash that an object is worth is a material substantiation (or representation like Arabic numerals) of the real number value of the object.
The reals also have obvious applications in probability and speculative economics, as well as a myriad of other subjects.
Inherited Problems
Perhaps the most complex, and certainly the hardest to define, characteristic of the reals is that of completeness. In fact, one of the greatest challenges in axiomatic constructions for the numbers we know and love is the jump from algebraic numbers to the complete set of the reals. This jump requires, in some constructions, the use of Dedekind cuts. Alternatively, in the form of an axiom, we can state this property in terms of Cauchy sequences; that is, for every Cauchy sequence that is a subset of the reals, the limits of such sequences are also included in the reals.
With this complexity, we also get a good bit of what one may call continuity baggage. Uncomputable numbers raise a particular problem for our physical instance of the reals: if we can never compute certain numbers (such as infinitesimals), then we surely can never measure such a number in our physical apparatus. This seems to imply we have some more structure in our mathematical model than may be warranted. In other words, it seems we can't have any real instances of the reals.
If we consider again the idea of some volume of water, we know for certain (now) that there's some minimum unit of water. Thus, we can't infinitely divide it and we certainly don't satisfy the completeness property of the reals. Even something as abstract as economics has a lowest unit, namely for USD, the penny. And other, perhaps moral, problems may be found with the apparent linear ordering of the worth of things, inherited from the monetary value of goods. For instance, a family heirloom may appear worthless if only because it's monetary value is near nothing.
This completeness isn't as easily cast off as we might imagine, however. Completeness is a requirement for the applicability of calculus, a tool oft used in physical instances of the reals. So, then what ever could replace such an indispensible tool? I'm not sure.
Furthermore, the algebraic numbers famously lack some useful and common numbers, such as pi and e (see transcendental numbers).
These considerations of the deficieny of the reals are further explored in Alex Lamb's article.
