Husserl’s Conceptions of Formal Mathematics
Edmund Husserl’s conception of mathematics was a unique blend of Platonist and formalist ideas. He believed that mathematics had reached a mixed state combining Platonic and formal elements and that both were important for the pursuit of the sciences, as well as for each other. However, he seemed to believe that only the Platonic aspects had significance for his science of phenomenology. Because of the significance of the distinction between these two types of mathematics, I will always use one of the adjectives “material” or “formal” when discussing any branch of mathematics, unless I specifically mean to include both.
First, I must specify what I mean by each of these terms. By material mathematics, I will mean mathematics as it had traditionally been done before the conceptions of imaginary numbers and non-Euclidean geometry. Thus, any branch of material mathematics seeks to describe how some class of existing things actually behaves. So material geometry seeks to describe how objects lie in space, material number theory seeks to describe how the actual natural numbers are related, and material logic seeks to describe how concepts actually relate to one another. Some of these areas (like material geometry) seek to deal with the physical world, while others (like material logic) deal with abstract objects, so I avoid using the word “Platonic”, which suggests only the latter. By formal mathematics, I will mean mathematics done as is typical in the 20th century, purely axiomatically, without regard to what sorts of objects it might actually describe. Thus, for formal geometry it is irrelevant whether the objects described are physical objects in actual space, or n-tuples of real numbers in Cartesian space. Some areas of mathematics, such as algebra and perhaps topology, seem to exist only in formal and not material guises. This is probably a result of their more recent development.
A significant development brought about by the change from material to formal mathematics is the move from conceptions of “truth” to “correctness”. That is, where material mathematics sees every statement in the field as being “true” or “false”, formal mathematics sees them as being either consequences or negations of consequences of the axioms.1 (Or, as Gödel’s incompleteness theorem showed, neither). This change came about because without a specific class of objects to describe, it is unclear whether a statement can meaningfully be said to be true. In addition, the inaccessible nature of many mathematical objects (or at least, the unreliability of our perceptions of them) makes it difficult to claim certainty, as mathematics likes to do. The circumstance of following from the axioms seems much more of a reliable claim to be able to make.2
In any case, both material and formal mathematics deal with classes of objects rather than particular individuals, and are thus considered “eidetic sciences” by Husserl. As a...