Though high school students make routine use of real numbers, it is in fact a strange set of numbers, that I personally struggle to understand rigorously. In particular, I think it’s a tall order to construct a physically intuitive model of the real numbers, despite their rather suggestive label, “real”, forcing mathematicians to at least entertain the idea that they in fact correspond to some aspect of our physical reality.

The work of Georg Cantor shows that if the real numbers are in fact physically real, then we are living in a truly strange place, where systems could contain themselves, be arbitrarily divisible, yet nonetheless stay the same size, and these are just a few of the bizarre properties of infinite sets that he showed must exist, if we posit the existence of infinity in the first instance, and accept some primordial, and therefore, difficult to escape notions of mappings between sets.

But there is a property of the reals that is not shared by the integers, which I’ve used to define a set of numbers I call the inarticulable numbers. These are numbers that, by virtue of the cardinality of the set of real numbers, simply cannot be expressed in any finite statement of symbols. Non-computable numbers are a common topic among computer scientists, which again must exist simply by virtue of the cardinality of the real numbers. In short, there are only countably many programs, and because there are uncountably many reals, there simply aren’t enough programs to calculate all the reals. This means that some reals simply cannot be calculated on a UTM. But we can nonetheless specify and define rigorously some non-computable numbers, such as the Chaitin numbers, which despite being non-computable, can nonetheless be defined and identified by a finite equation.

In contrast, what I call inarticulable numbers are a set of numbers that cannot be specified by any finite equation. Because every program can be expressed as a sequence of mathematical operations, it follows that if a number is inarticulable, then there cannot be any program that generates it, since that would be tantamount to an equation that specifies the number. As a result, if a number is inarticulable, it is therefore non-computable. Inarticulable numbers must exist, since any human language for doing mathematics will consistent of finite statements. This means that even if our alphabet is infinite, which, e.g., the integers are, the statements that express the math to be done must be finite, in that we can write only a finite number of etchings on a page before we die. It’s really that simple.

As a result, it must be the case that there is a subset of the real numbers comprised of numbers that are not only non-computable, but also incapable of definite expression in a written human language. Because there are only countably many finite expressions over any given finite or countable alphabet, it follows that the complement of the set of inarticulable numbers (i.e., the numbers that can be expressed in a finite statement) is countable, which implies that the set of inarticulable numbers is uncountable. In plain English, basically all of the real numbers defy definite expression in a written human language.

This is a shocking conclusion, in my opinion, because it shows that it’s not just computation that is incapable of expressing most real numbers – it’s written human language itself. The most advanced means by which human beings can describe the external world is through the language of mathematics. And it turns out, that even our greatest and most expressive means of description is incapable of describing basically all real numbers. This means that if real numbers are in fact real, then we can’t describe most of what’s around us, implying that the portion of information about the external world described by our written descriptions of its machinations is effectively zero.

All of this is completely true, in the sense that this follows from the application of logic to the basic premises of mathematics, so you can either take issue with logic or very primordial assumptions about the operations of mathematics, or you can accept what I’ve said above, which I believe to be true, because despite being very conscious of my limitations as a human being, I nonetheless believe that believing in logic and mathematics is the most productive way to exist.

That said, I spend a lot of time thinking about the nature of computation that goes beyond the power of a Turing Machine, and if we are going to make use of most of the real numbers as a physical matter, then we’ll have to learn how to do exactly that – which is to build a machine that is categorically superior to a UTM. This means developing a language that goes beyond the limits of written language, since as I’ve shown above, most of the reals defy definite expression in a written language. It turns out, human beings have spent a significant amount of their time developing a different language that might make the cut, and this is the language of art, where symbolism in particular, could serve as a pointer to an otherwise inarticulable quantity. That is, even though I can’t write down an equation for a given inarticulable quantity, I can still assign it a label, and if both you and I have observed the same inarticulable quantity, we can both agree to call it “blue“.