In a first section called “Metaphysical preamble”, the author presents three pictures about mathematics, which he calls: fictionalist, standard Platonism, plenitudinous Platonism. The paper focuses mainly on the third option.
Plenitudinous Platonism is taken as the view that whenever one has a consistent theory of pure mathematics, then there are mathematical objects that satisfy the theory under a perfectly standard satisfaction relation. In a second section he states the main concern of the paper: the objectivity issue. The issue is formulated, putting aside the fictionalist option, as: which of our mathematical sentences have determinate truth value? He rephrases the question as the title of the paper: “Which undecidable mathematical sentences have determinate truth value?” The issue is dealt with in two sections with the titles: “Putnam’s models” and “Reality and the concepts of finiteness and natural numbers; extreme antiobjectivism”. The titles of the sections suggest their contents. A good set of notes enriches the interesting arguments of the paper.
For the entire collection see [Zbl 0901.00023].