Notes on Russel's Introduction to Mathematical Philosophy

In the Introduction to Mathematical Philosophy, Bertrand Russell describes a philosophical ethos in relation to mathematics, that stands in contrast with the standard practice of mathematicians. Rather than progressing from a set of established principles into complex operations -- from the arithmetics of natural numbers to the algebra of real numbers, and from that to the differentiation and integration calculus -- Russel describes the philosophical ethos as the research of the essence of mathematics, and the logical operations by which we can deduce these foundational principles, such as numbers, sequences, sets and functions.

As mathematics undergoes a process of arithmetisation, in a divergence from formal logic, it follows that, in order to take on the investigation of its principles, it is necessary to define first the fundamental concept of number. As Russell describes it, though the number is an ubiquitous concept in the hundreds of years of the history of mathematics, a clear and correct definition was only formalised and published in recent years, by Frege (Foundations of Arithmetics, 1884). Frege's definition is different to that of Peano, who defines numbers in terms of sequences or progressions, and also subtly different to the philosophers who defined numbers simply in terms of plurality -- one apple, two apples, three apples.
The problems in defining numbers in terms of progressions or plurality are:

• Progressions, while they provide a correct knowledge about the relationships of numbers -- according to the function f(x) that determines the successor of x -- they do not provide any knowledge about the number itself and what it represents; if we change the reference to the first element of the sequence, the reference to all elements in the sequence is lost.
• Plurality, on the other hand, while it does provide knowledge of a single number in terms of the quantity of objects, it tells us nothing about the number as a number -- and, more importantly, about what numbers are; in other words, it provides a sense of what "three" is, but not of what "number" is.

A precise definition of what a number is requires that we abstract a single quantity into the general concept of all quantities, therefore putting forward the concept of groups -- or, more precisely, classes. Just as we can define a class a with three apples, and a class b with three bananas, we can define a class of all classes with three elements (i.e. all classes similar to a class with three elements, in terms of quantity of elements). Similarly, we can describe a class of all classes with thirty elements, or a class of all classes with three hundred elements.

The number of a class, then, will be the class of all classes similar to it. A number is any entity that is the number of a class.

The definition of numbers in terms of classes is interesting to the student of computer sciences, because of the importance that term has gained since the advent of object-oriented programming. Though I want to tread carefully and avoid the sin of anachronism (we are talking about a text written in 1919, when modern computation by electronic machines was a distant dream), it is worthy of note how Russell describes classes as "logical fictions" -- not real things in and of themselves, but rather a sort of imaginary line traced around objects, grouping them together in terms of some arbitrary common description (generative function; e.g. the class of men, the class of cars).

Am I then accusable of being too hasty and unconsidered in seeing a similarity between this definition of classes with the concept of abstract data types, which is so cherished in the field of object-oriented programming? Of course classes in OOP languages are much more complex structures -- with attributes, behaviours, contracts, interfaces -- than the simple abstractions of classes as "logical fictions" described by Russell; but I do see how we could interpret the latter as the former's source and inspiration.

We could, for example, describe the type specification of a class constructor as its intentional definition, and the instances of a class as its extensional definition. As Russell writes, classes can be defined in two ways: extensionally, in terms of its members, or intentionally, in terms of an abstract description; so, a class A could be defined as:

• A = {Jack, Jim}
• A = "men in this room now"

As for methods, they describe the relations between objects of this class with other objects in the world.

This approximation was brought to my mind because of a tweet someone posted some other day, about the weirdness of writing numbers as class objects in OOP languages, which leads to constructs such as 10.is_pair(). As we see in Russell's writing, the idea of describing numbers as abstract classes capable of having properties is not an invention of object-oriented programming.