Richard Dedekind formalized the real numbers in 1872. But Dedekind did not take the step into infinite complexity. “...we must endeavor completely to define irrational numbers by means of the rational numbers alone.”12 From here, Dedekind then steps backward into infinity.
> I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed.13
The idea of a Dedekind cut is to define the spaces between rational numbers as irrational. Real numbers are a boundary between two sets of rational numbers, one set all being smaller, the other set all being larger. It is as though we could identify an irrational such as π by bounding it from above and from below with successively closer rationals. Just like the Greeks did 2500 years earlier in approximating the value of π by successively nesting geometric shapes.
Wittgenstein found the now accepted definition of the real numbers to be nonsense. The Dedekind description does not provide an algorithm with which to construct or to enumerate the reals. Numbers are operations, they are rules as well as objects. We can make closer and closer approximations of an irrational such as √2; the notation for the irrational number itself is the algorithm to make these approximations. However, √2 does not exist as an object because for Wittgenstein no number is an object:
> In mathematics everything is algorithm and nothing is meaning; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.14
From the perspective of computer science, Wittgenstein’s philosophy of mathematics is appealing. It is unnecessary, and perhaps unreasonable, for a finite system to be able to represent any arbitrary irrational number. Mathematical forms that are encoded by a finite binary sequence are constructible, indeed algorithmic. Each James form, for example, defines a process for generating a particular number. The form of an irrational number, for both boundary and conventional notations, is a computational specification, and computational specifications cannot call upon infinite processes.
To Wittgenstein, there is no reason or criterion “for the irrational numbers being complete”.15 Wittgenstein thus bans our third category of irrational numbers, the lawless irrationals that have no identifiable structure in their (infinite) sequence of decimals. James forms cannot express lawless irrationals, by definition no notation can. Some lawless irrational numbers are chaotic. We cannot know in advance which digit in a lawless number comes next since, being lawless, there is no mathematical process available to predict the structure. These numbers are not random, they are indeed deterministic. They are just immune to mathematical abstraction. There are lawless numbers that cannot be known or represented by any process, again not necessarily random since statistical randomness can be evaluated. We are fast approaching a central question about real numbers: are numbers that cannot be identified by any mathematical technique still legitimate as numbers?
Only a very special subgroup of real numbers can be specified, only those for which there is an algorithmic process that generates them to any specified degree of accuracy. The continuum of real numbers is not only imaginary, it is irretrievable by imagination. We are but a small step away from philosopher Mary Leng’s modern perspective that mathematics is a well-constructed fiction.
> Mathematical hypotheses, on my view, are best thought of not as truths by convention (for they do not have the status of truths), but rather, as conventionally adopted useful fictions.16
Abstract mathematics has evolved into a somewhat self-indulgent dance. Like writing it is a communion between our thoughts inside and our symbolic squiggles outside. Brian Rotman:
> In other words, mathematics is a rigorous inscriptional fantasy: the insistence on writing determining what can be legitimately imagined, and the ongoing process of imagining controlling what mathematicians can meaningfully and usefully write down.17
Drawing icons and diagrams and containment boundaries places a different type of rigorous constraint upon mathematical structure, that of physical realizability. James forms thus identify the constructible real numbers.