Dominique Pradelle. Être et genèse des idéalités. Un ciel sans éternité

In Intuition et idéalités: Phénoménologie des objets mathématiques (2020), Dominique Pradelle questioned the nature of mathematical knowledge–the status of mathematical objects as well as the kind of evidence they are objects of (see my review 1) . His new book continues this investigation by challenging what appears to be a paradox: mathematical objects are not real but ideal, which means that their being somehow depends on their “constitution” (by several acts of abstraction, idealization or formalization) and even on their “genesis” (in the sense that some constitutive acts became possible only within some theoretical situation, which has some history), and yet they appear as “being there” and “having always been there”, independently of those constitutive acts and of this genesis.

That idealities are “there” and “timelessly there” is a claim that not only goes back to Plato; it has also been defended by authors who simultaneously insisted on the fact that idealities first and foremost are contents of thought (and of mental acts). Thomas Aquinas provided this view with a theological ground: even though finite beings think only of some contents and know truths about them only at some moments of time, these contents and these truths stay in God’s infinite understanding without any temporal restriction. Bernard Bolzano’s (and later Gottlob Frege’s) semantic objectivism will replicate this view without invoking God, relying only on intersubjectivity: while an idea or presentation (Vorstellung) is a mental act, temporally situated in some subject’s mental life, its content, which Bolzano names “idea in itself”, is neither subjective nor temporal since it can be the content of any subject’s thought at any time; and propositions made of such “ideas in themselves” stay the same for whoever thinks about them or states them, and even stay the same–and keep their truth value–if no one ever thinks about them. In his own fight against psychologism, but also against anthropologism or historicism, Edmund Husserl will similarly claim that ideal contents of mental acts should be distinguished from the mental acts in which they appear: while the latter actually happen at some moment in time in some subjective mind, the former are ideal and available for any subject at any time, and so are the propositions involving these contents, which means that neither their meaning nor their truth value are subject-relative or even species-relative or time-relative.